Active Flow Control: Papers contributed to the Conference Active Flow Control 2006, Berlin, Germany, September 27 to 29, 2006 (Notes on Numerical Fluid Mechanics and Multidisciplinary Design)

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95

Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM)

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Active Flow Control Papers contributed to the Conference “Active Flow Control 2006”, Berlin, Germany, September 27 to 29, 2006 Rudibert King (Editor)

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Library of Congress Control Number: 2007922725 ISBN-10 3-540-71438-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-71438-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 Printed in Germany

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and SPS Cover-Design: deblik, Berlin Printed on acid-free paper SPIN:12037247 89/3141/SPS 543210

NNFM Editor Addresses

Prof. Dr. Ernst Heinrich Hirschel (General Editor) Herzog-Heinrich-Weg 6 D-85604 Zorneding Germany E-mail: [email protected] Prof. Dr. Wolfgang Schröder (Designated General Editor) RWTH Aachen Lehrstuhl für Strömungslehre und Aerodynamisches Institut Wüllnerstr. zw. 5 u. 7 52062 Aachen Germany Email: ofﬁ[email protected] Prof. Dr. Kozo Fujii Space Transportation Research Division The Institute of Space and Astronautical Science 3-1-1, Yoshinodai, Sagamihara Kanagawa, 229-8510 Japan E-mail: fujii@ﬂab.eng.isas.jaxa.jp Dr. Werner Haase Höhenkirchener Str. 19d D-85662 Hohenbrunn Germany E-mail: [email protected] Prof. Dr. Bram van Leer Department of Aerospace Engineering The University of Michigan Ann Arbor, MI 48109-2140 USA E-mail: [email protected] Prof. Dr. Michael A. Leschziner Imperial College of Science Technology and Medicine Aeronautics Department Prince Consort Road London SW7 2BY U. K. E-mail: [email protected]

Prof. Dr. Maurizio Pandolﬁ Politecnico di Torino Dipartimento di Ingegneria Aeronautica e Spaziale Corso Duca degli Abruzzi, 24 I-10129 Torino Italy E-mail: pandolﬁ@polito.it Prof. Dr. Jacques Periaux Dassault Aviation 78, Quai Marcel Dassault F-92552 St. Cloud Cedex France E-mail: [email protected] Prof. Dr. Arthur Rizzi Department of Aeronautics KTH Royal Institute of Technology Teknikringen 8 S-10044 Stockholm Sweden E-mail: [email protected] Dr. Bernard Roux L3M – IMT La Jetée Technopole de Chateau-Gombert F-13451 Marseille Cedex 20 France E-mail: [email protected] Prof. Dr. Yurii I. Shokin Siberian Branch of the Russian Academy of Sciences Institute of Computational Technologies Ac. Lavrentyeva Ave. 6 630090 Novosibirsk Russia E-mail: [email protected]

Preface

The dramatically increasing requirements of mobility through road-, rail- and airborne transport systems in the future necessitate non-evolutionary improvements of transportation systems. Without severe implications concerning the environment or restrictions concerning the performance, these requirements will only be met by a concerted action of many disciplines. It is believed that with ACTIVE FLOW CONTROL a key technology exists to supply an important block in the mosaic to be laid in the pursuit of best and sustainable solutions. Manipulation of ﬂuid ﬂows is highly advantageous in many cases. Aerodynamic or ﬂuid ﬂows around or inside bodies impose drag, lift and moments on the body, remove or supply energy by convection. Flow-induced noise may be produced by the interaction of a body with the surrounding air. Moreover, the interaction with the body changes the state of the ﬂow drastically. A neatly aligned laminar ﬂow around a wing of an aircraft giving enough lift, can become highly irregular and separated from the surface, with the result of a loss of lift. For cooling of engines of transport and other systems highly irregular turbulent ﬂuid ﬂows across the components are needed to guarantee a large heat transfer. In future engines of airplanes complying for example with the EU Vision 2020 an increased heat transfer, on the other hand, has to be avoided by all means in some parts of the engine. Turbine stages may be exposed here to extremely hot gases, needed for high eﬃciency, which would destroy the blades. In this application, more laminar ﬂow regimes would be advantageous yielding a poorer heat transfer. The irregular ﬂow in a combustor e.g., of an aero engine, determines to a great extent the thermodynamic eﬃciency of and the reactions occurring inside the system. Complying with environmental constraints concerning the production of exhaust gases like NOx will necessitate a low mixture ratio of fuel to air. However, this together with the requirements of light engines will result in thermo-acoustic instabilities in burners. Noise emission will only be one problem related with this situation. More severe consequences will result from the massive mechanical burden the burner has to sustain. Improving the eﬃciency of airplanes by ﬂow control will signiﬁcantly reduce the production of greenhouse gases and save money for the airlines and for the customers. It is not just the increased lift by ﬂow control, which will allow smaller, lighter engines which consume less fuel. It is of course as well the reduced ﬂow-induced drag exerted by the diﬀerent components of an aircraft e.g., engines nacelles, fuselage, landing gear and so forth, which will allow for smaller engines. When steeper climbing and landing paths are possible thanks to higher lift obtained by ﬂow control, the noise exposure on ground would be reduced quantitatively. Moreover, it is the improved design of the engines: If an engine can be run with fewer compressor stages, its size and weight will be

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reduced. Additionally, the drag induced by this component might be lowered. More ﬂowrelated challenges for air-transport systems could easily be listed. From this selection of examples it can be concluded that manipulation of ﬂuid ﬂows is of paramount importance. Flows can be manipulated either by passive or by active means. Passive methods comprise the shaping of the bodies, the inclusion of devices such as spoilers, riblets or vortex generators. It goes without saying that manipulation of ﬂows with these passive means has a very long, well established tradition in ﬂuid mechanics. In contrast to passive means, active concepts of ﬂow control are still in their infancy. Much basic and applied research will be needed to fully exploit the beneﬁts associated with it. The most obvious advantage of active over passive manipulation is the possibility to adapt the kind and size of manipulation in a desired, if possible optimal way to the actual operating conditions of a process. Active methods are needed when passive ones reach their limits to extend regions of operation, when passive methods have positive and negative eﬀects depending on the operating conditions, or when due to active methods conventional design constraints can be relaxed. Most importantly, active methods of ﬂow control can result in a nongradual, drastic improvement of a system, as desired at the beginning of this section. Active ﬂow control can be done by blowing, suction, acoustic actuation, and in some cases by magneto-hydrodynamic forces. It can be further subdivided in steady and in unsteady actuation. The simplest form is steady actuation. Fluid is blown into or sucked out of a ﬂow over the contour of a body. It can be energized so that it can withstand an adverse pressure gradient which otherwise would lead to an undesired separation of the ﬂow from the surface. A desired degree of mixing between diﬀerent ﬂow regimes can be inﬂuenced as well, just to name one more example. Unsteady ﬂow control is known to oﬀer the same or even larger control authority as steady control, but with much smaller control inputs. The art of active, unsteady ﬂow control is to play around, to exploit instabilities present in the ﬂow system. Due to their very nature as an unstable process, only a rather small amount of energy i.e., periodic mass ﬂow is necessary to trigger the instability. The frequency and the size of actuation have to be matched exactly, to meet the demands of the control and to synchronize it with the operating point dependent character of the ﬂow. Moreover, as the onset of instability itself is not the control goal, it has to be implemented such that the lift of a wing is increased, the drag reduced, the mixing in a burner improved, etc. Most of the work done so far in the area of active ﬂow control is devoted to open-loop control. For open-loop control concepts extensive real and/or numerical experiments have to be performed to determine for example, the size, frequency or shape of the control signal necessary for every operating condition. This information is stored in the controller and retrieved afterwards when a speciﬁc situation is met. Various experiments in wind tunnels and for a real aircraft have shown the power of this dynamic active ﬂow control. However, when the well- deﬁned conditions of a wind tunnel are left and abundant disturbances act

Preface

IX

on the wing, car, or other ﬂow systems, this open-loop control is bound to fail. It will fail also when operating conditions are chosen for which no information is available in the open-loop controller. Hence, to exploit the enormous potentials of active ﬂow control, a control loop has to be established. By measuring the eﬀect of the control with appropriate sensors, any disturbance or deterioration due to changing operating conditions is observed and a closed-loop controller can react accordingly. Besides a much better disturbance rejection and robustness with respect to process uncertainties, closed-loop control often increase the achievable goals, such as maximal lift. From a control engineering point of view, ﬂow control systems are counted among the most diﬃcult ones due to their nonlinearity and inﬁnite parameter dimension. A modeling based on physics leads to the Navier-Stokes together with the governing mass conservation equations for which huge computer resources are needed to ﬁnd numerical solutions. This complexity necessitates complementary approaches of controller synthesis, such as robust, adaptive, or nonlinear methods based on low-dimensional models if these controllers have to be brought into a real system. These challenges and perspectives in active ﬂow control laid the ground at Technische Universit¨ at Berlin (TUB) almost 10 years ago to build up a Collaborative Research Center (SFB) 557 on CONTROL OF COMPLEX TURBULENT SHEAR FLOWS. It is ﬁnanced by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) with signiﬁcant contributions by TUB, Deutsches Zentrum f¨ ur Luft und Raumfahrt e.V. (DLR, German Aerospace Center), ZuseInstitut Berlin (ZIB) and Freie Universit¨at Berlin. This SFB 557 initiated the Conference ACTIVE FLOW CONTROL 2006, held September 27-29, 2006, at the TUB, Germany. The goal was to attract leading experts in the area of active ﬂow control from around the world discussing the state-of-the-art of the aforementioned ﬂow control aspects. Plenary and invited papers were dedicated to all important ﬁelds of active ﬂow control, such as sensing, actuators, aerodynamics, acoustics, turbo-machinery, numerics or control. Preparing this book which was published shortly after the conference was challenging. First, due to size limitations the INTERNATIONAL PROGRAM COMMITTEE and the REVIEWERS had the diﬃcult task to select a subset of the excellent papers presented at the conference to be included in this book1 . Reviewers were: A. Banaszuk, East Hardford; P. Bonnet, Poitiers; H. Choi, Seoul; H. Fernholz, Berlin; M. Gad-el-Hak, Richmond; K. Kunisch, Graz; V. Mehrmann, Berlin; M. M¨ oser, Berlin; W. Neise, Berlin; W. Nitsche, Berlin; B.R. Noack, Berlin; C.O. Paschereit, Berlin; W. Schr¨oder, Aachen; A. Seifert, Tel Aviv; S. Siegel, Colorado Springs; G. Tadmor, Boston; F. Thiele, Berlin. Their help is highly acknowledged. Second, deciding on the sequence of chapters was not straightforward. Due to the interdisciplinary character of ﬂow control many

1

More papers can be found in the conference CD.

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Preface

contributions comprise results from diﬀerent areas concerning methods and/or applications. Neither an ordering according to sensors, actuators, controllers, nor to the various areas of applications, such as airfoils, bluﬀ bodies, cavities etc., was possible without arbitrarily grouping some papers which cover diﬀerent aspects at the same time. As a compromise, chapter headings were chosen according to the session headings of the conference. The international conference on ACTIVE FLOW CONTROL 2006 was used at the same time to highlight recent achievements obtained in the Collaborative Research Center in Berlin. The SFB 557 concentrates in its third period in the years 2004-2007 on four ﬂow conﬁgurations, namely (1) a high-lift wing, (2) turbo-machines, (3) a generic car model, and (4) a burner. This concentration on very few, but theoretically and technically demanding problems has led to an increased cooperation and has improved drastically the insights into the beneﬁts of synergy and mutual understanding. By this very simple instrument, mathematicians, theoreticians and experimenters from ﬂuid and aerodynamics, control engineers, computational ﬂuid dynamicists, acousticians, etc. are working together on the very same technical problems. Some of the results obtained are included in this volume as well. A main focus of the SFB 557 for all four ﬂow conﬁgurations is dedicated to closed-loop ﬂow control. As a result, the SFB 557 oﬀers today a vast experience in experimental closed-loop ﬂow control, not only for the above mentioned ﬂow conﬁgurations but for other conﬁgurations, too, such as the backward-facing step, diﬀusers, etc. Again some related work is described in this volume as well. The ORGANIZING COMMITTEE is indebted to the DFG. Without their substantial ﬁnancial support the making of this book would not have been possible. Furthermore, the editor is grateful to Mrs. Steﬃ Stehr for the signiﬁcant help in the organization of the conference and the compilation of this volume.

Berlin, September 2006

Rudibert King

Table of Contents

The Taming of the Shrew: Why Is It so Difficult to Control Turbulence? Mohamed Gad-el-Hak …..……………………………………………………........

1

Part I: Actuators Electromagnetic Control of Separated Flows Using Periodic Excitation with Different Wave Forms Christian Cierpka, Tom Weier, Gunter Gerbeth …………………………………

27

Pulsed Plasma Actuators for Active Flow Control at MAV Reynolds Numbers B. Göksel, D. Greenblatt, I. Rechenberg, Y. Kastantin, C.N. Nayeri, C.O. Paschereit …………………………………………………………………...

42

Experimental and Numerical Investigations of Boundary-Layer Influence Using Plasma-Actuators S. Grundmann, S. Klumpp, C. Tropea …………………………………………… 56 Designing Actuators for Active Separation Control Experiments on High-Lift Configurations* Ralf Petz, Wolfgang Nitsche ……………………………………………………... 69 Closed-Loop Active Flow Control Systems: Actuators A. Seifert ………………………………………………………………………….

85

Part II: State Estimation and Feature Extraction State Estimation of Transient Flow Fields Using Double Proper Orthogonal Decomposition (DPOD) Stefan Siegel, Kelly Cohen, Jürgen Seidel, Thomas Mclaughlin ……………...... 105 A Unified Feature Extraction Architecture* Tino Weinkauf, Jan Sahner, Holger Theisel, Hans-Christian Hege, Hans-Peter Seidel …………………………………………………….…………

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Part III: Air Foils Control of Wing Vortices I. Gursul, E. Vardaki, P. Margaris, Z. Wang …………………………………… 137 Towards Active Control of Leading Edge Stall by Means of Pneumatic Actuators C.J. Kähler, P. Scholz, J. Ortmanns, R. Radespiel …………………………….... 152 Computational Investigation of Separation Control for High-Lift Airfoil Flows* Markus Schatz, Bert Günther, Frank Thiele …………………………………… 173 Steady and Oscillatory Flow Control Tests for Tilt Rotor Aircraft M. Schmalzel, P. Varghese, I. Wygnanski ………………………………………

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Part IV: Cavities Reduced-Order Model-Based Feedback Control of Subsonic Cavity Flows – An Experimental Approach M. Samimy, M. Debiasi, E. Caraballo, A. Serrani, X. Yuan, J. Little, J.H. Myatt ………………………………………………………………………. 211 Supersonic Cavity Response to Open-Loop Forcing David R. Williams, Daniel Cornelius, Clarence W. Rowley ……………………

230

Part V: Bluff Bodies Active Drag Control for a Generic Car Model* A. Brunn, E. Wassen, D. Sperber, W. Nitsche, F. Thiele ………………………..

247

Continuous Mode Interpolation for Control-Oriented Models of Fluid Flow* Marek MorzyĔski, Witold Stankiewicz, Bernd R. Noack, Rudibert King, Frank Thiele, Gilead Tadmor ………………………….………………………………………. 260 Part VI: Turbomachines and Combustors Active Management of Entrainment and Streamwise Vortices in an Incompressible Jet D. Greenblatt, Y. Singh, Y. Kastantin, C.N. Nayeri, C.O. Paschereit …………..

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Active Control to Improve the Aerodynamic Performance and Reduce the Tip Clearance Noise of Axial Turbomachines with Steady Air Injection into the Tip Clearance Gap L. Neuhaus, W. Neise …………………………………………………………... 293

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Part VII: Optimal Flow Control and Numerical Studies Drag Minimization of the Cylinder Wake by Trust-Region Proper Orthogonal Decomposition Michel Bergmann, Laurent Cordier, Jean-Pierre Brancher ……………………. 309 Flow Control on the Basis of a FEATFLOW-MATLAB Coupling* Lars Henning, Dmitri Kuzmin, Volker Mehrmann, Michael Schmidt, Andriy Sokolov, Stefan Turek ……………………………………..…………….

325

On the Choice of the Cost Functional for Optimal Vortex Reduction for Instationary Flows Karl Kunisch, Boris Vexler ……………………………………………………..

339

Flow Control with Regularized State Constraints* J.C. de los Reyes, F. Tröltzsch ………………………………………………......

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Part VIII: Closed-Loop Flow Control Feedback Control Applied to the Bluff Body Wake* Lars Henning, Mark Pastoor, Rudibert King, Bernd R. Noack, Gilead Tadmor …………………………………………………….…………….. 369 Active Blade Tone Control in Axial Turbomachines by Flow Induced Secondary Sources in the Blade Tip Regime* O. Lemke, R. Becker, G. Feuerbach, W. Neise, R. King, M. Möser …………….. 391 Phase-Shift Control of Combustion Instability Using (Combined) Secondary Fuel Injection and Acoustic Forcing* Jonas P. Moeck, Mirko R. Bothien, Daniel Guyot, Christian Oliver Paschereit …………………………………………………….. 408 Vortex Models for Feedback Stabilization of Wake Flows Bartosz Protas …………………………………………………………………… 422 Keyword Index …………………………………………………………………

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* These papers are results of the Collaborative Research Center 557 “Control of complex turbulent shear flows” (SFB 557, TU Berlin) and are printed according to the regulations of and funded by the DFG (German Research Foundation).

The Taming of the Shrew: Why Is It so Difﬁcult to Control Turbulence? Mohamed Gad-el-Hak Virginia Commonwealth University, Richmond, VA 23284-3015, USA [email protected] http://www.people.vcu.edu/∼gadelhak

Summary In the present chapter I shall emphasize the frontiers of the ﬁeld of ﬂow control, pondering mostly the control of turbulent ﬂows. I shall review the important advances in the ﬁeld that took place during the past few years and are anticipated to dominate progress in the future. By comparison with laminar ﬂow control or separation prevention, the control of turbulent ﬂow remains a very challenging problem. Flow instabilities magnify quickly near critical ﬂow regimes, and therefore delaying transition or separation are relatively easier tasks. In contrast, classical control strategies are often ineffective for fully turbulent ﬂows. Newer ideas for turbulent ﬂow control to achieve, for example, skin-friction drag reduction focus on the direct onslaught on coherent structures. Spurred by the recent developments in chaos control, microfabrication and soft computing tools, reactive control of turbulent ﬂows, where sensors detect oncoming coherent structures and actuators attempt to favorably modulate those quasi-periodic events, is now in the realm of the possible for future practical devices. In this chapter, I shall provide estimates for the number, size, frequency and energy consumption of the sensor/actuator arrays needed to control the turbulent boundary layer on a full-scale aircraft or submarine.

1 Introduction 1.1 The Taming of the Shrew Considering the extreme complexity of the turbulence problem in general and the unattainability of ﬁrst-principles analytical solutions in particular, it is not surprising that controlling a turbulent ﬂow remains a challenging task, mired in empiricism and unfulﬁlled promises and aspirations. Brute force suppression, or taming, of turbulence via active, energy-consuming control strategies is always possible, but the penalty for doing so often exceeds any potential beneﬁts. The artiﬁce is to achieve a desired effect with minimum energy expenditure. This is of course easier said than done. Indeed, suppressing turbulence is as arduous as the taming of the shrew. R. King (Ed.): Active Flow Control, NNFM 95, pp. 1–24, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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1.2 Control of Turbulence Numerous methods of ﬂow control have already been successfully implemented in practical engineering devices. Delaying laminar-to-turbulence transition to reasonable Reynolds numbers and preventing separation can readily be accomplished using a myriad of passive and predetermined active control strategies. Such classical techniques have been reviewed by, among others, Bushnell (1983; 1994), Wilkinson et al. (1988), Bushnell and McGinley (1989), Gad-el-Hak (1989), Bushnell and Hefner (1990), Fiedler and Fernholz (1990), Gad-el-Hak and Bushnell (1991), Barnwell and Hussaini (1992), Viswanath (1995), and Joslin et al. (1996). Yet, very few of the classical strategies are effective in controlling free-shear or wall-bounded turbulent ﬂows. Serious limitations exist for some familiar control techniques when applied to certain turbulent ﬂow situations. For example, in attempting to reduce the skin-friction drag of a body having a turbulent boundary layer using global suction, the penalty associated with the control device often exceeds the saving derived from its use. What is needed is a way to reduce this penalty to achieve a more effective control. Flow control is most effective when applied near the transition or separation points; in other words, near the critical ﬂow regimes where ﬂow instabilities magnify quickly. Therefore, delaying/advancing laminar-to-turbulence transition and preventing/ provoking separation are relatively easier tasks to accomplish. To reduce the skinfriction drag in a non-separating turbulent boundary layer, where the mean ﬂow is quite stable, is a more challenging problem. Yet, even a modest reduction in the ﬂuid resistance to the motion of, for example, the worldwide commercial airplane ﬂeet is translated into fuel savings estimated to be in the billions of dollars. Newer ideas for turbulent ﬂow control focus on the direct onslaught on coherent structures. Spurred by the recent developments in chaos control, microfabrication and soft computing tools, reactive control of turbulent ﬂows is now in the realm of the possible for future practical devices. The primary objective of the present chapter is to advance possible scenarios by which viable control strategies of turbulent ﬂows could be realized. As will be argued in the following presentation, future systems for control of turbulent ﬂows in general and turbulent boundary layers in particular could greatly beneﬁt from the merging of the science of chaos control, the technology of microfabrication, and the newest computational tools collectively termed soft computing. Control of chaotic, nonlinear dynamical systems has been demonstrated theoretically as well as experimentally, even for multidegree-of-freedom systems. Microfabrication is an emerging technology which has the potential for producing inexpensive, programmable sensor/actuator chips that have dimensions of the order of a few microns. Soft computing tools include neural networks, fuzzy logic and genetic algorithms and are now more advanced as well as more widely used as compared to just few years ago. These tools could be very useful in constructing effective adaptive controllers. Such futuristic systems are envisaged as consisting of a large number of intelligent, interactive, microfabricated wall sensors and actuators arranged in a checkerboard pattern and targeted toward speciﬁc organized structures that occur quasi-randomly within a turbulent ﬂow. Sensors detect oncoming coherent structures, and adaptive controllers process the sensors information and provide control signals to the actuators which in

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turn attempt to favorably modulate the quasi-periodic events. Finite number of wall sensors perceive only partial information about the entire ﬂow ﬁeld above. However, a low-dimensional dynamical model of the near-wall region used in a Kalman ﬁlter can make the most of the partial information from the sensors. Conceptually all of that is not too difﬁcult, but in practice the complexity of such a control system is daunting and much research and development work still remain. 1.3 Outline In the present chapter, I shall review the important developments in the ﬁeld of ﬂow control that took place during the past few years and suggest avenues for future research. The emphasis will be on reactive ﬂow control for future vehicles and other industrial devices. The present chapter is organized into four sections. Reactive ﬂow control and the selective suction concept are described in Section 2. The number, size, frequency and energy consumption of the sensor/actuator units required to tame the turbulence on a full-scale air or water vehicle are estimated in that same section. Section 3 considers the emerging area of chaos control, particularly as it relates to reactive control strategies. Finally, brief concluding remarks are given in Section 4.

2 Reactive Control 2.1 Introductory Remarks Targeted control implies sensing and reacting to a particular quasi-periodic structure in the boundary layer. The wall seems to be the logical place for such reactive control, because of the relative ease of placing something in there, the sensitivity of the ﬂow in general to surface perturbations, and the proximity and therefore accessibility to the dynamically all important near-wall coherent events. According to Wilkinson (1990), there are very few actual experiments that use embedded wall sensors to initiate a surface actuator response (Alshamani et al., 1982; Wilkinson and Balasubramanian, 1985; Nosenchuck and Lynch, 1985; Breuer et al., 1989). This decade-old assessment is fast changing, however, with the introduction of microfabrication technology that has the potential for producing small, inexpensive, programmable sensor/actuator chips. Witness the more recent reactive control attempts by Kwong and Dowling (1993), Reynolds (1993), Jacobs et al. (1993), Jacobson and Reynolds (1993a; 1993b; 1994; 1995; 1998), Fan et al. (1993), James et al. (1994), and Keefe (1996). Fan et al. and Jacobson and Reynolds even consider the use of self-learning neural networks for increased computational speeds and efﬁciency. Recent reviews of reactive ﬂow control include those by Gad-el-Hak (1994; 1996), Lumley (1996), McMichael (1996), Mehregany et al. (1996), and Ho and Tai (1996). Numerous methods of ﬂow control have already been successfully implemented in practical engineering devices. Yet, limitations exist for some familiar control techniques when applied to speciﬁc situations. For example, in attempting to reduce the drag or enhance the lift of a body having a turbulent boundary layer using global suction, global heating/cooling or global application of electromagnetic body forces, the actuator’s energy expenditure often exceeds the saving derived from the predetermined active

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control strategy. What is needed is a way to reduce this penalty to achieve a more efﬁcient control. Reactive control geared speciﬁcally toward manipulating the coherent structures in turbulent shear ﬂows, though considerably more complicated than passive control or even predetermined active control, has the potential to do just that. As will be argued in this and the following sections, future systems for control of turbulent ﬂows in general and turbulent boundary layers in particular could greatly beneﬁt from the merging of the science of chaos control, the technology of microfabrication, and the newest computational tools collectively termed soft computing. Such systems are envisaged as consisting of a large number of intelligent, communicative wall sensors and actuators arranged in a checkerboard pattern and targeted toward controlling certain quasi-periodic, dynamically signiﬁcant coherent structures present in the near-wall region. 2.2 Targeted Control Successful techniques to reduce the skin friction in a turbulent ﬂow, such as polymers, particles or riblets, appear to act indirectly through local interaction with discrete turbulent structures, particularly small-scale eddies, within the ﬂow. Common characteristics of all these methods are increased losses in the near-wall region, thickening of the buffer layer and lowered production of Reynolds shear stress (Bandyopadhyay, 1986). Methods that act directly on the mean ﬂow, such as suction or lowering of near-wall viscosity, also lead to inhibition of Reynolds stress. However, skin friction is increased when any of these velocity-proﬁle modiﬁers is applied globally. Could these seemingly inefﬁcient techniques, e.g. global suction, be used more sparingly and be optimized to reduce their associated penalty? It appears that the more successful drag-reducing methods, e.g. polymers, act selectively on particular scales of motion and are thought to be associated with stabilization of the secondary instabilities. It is also clear that energy is wasted when suction or heating/cooling is used to suppress the turbulence throughout the boundary layer when the main interest is to affect a near-wall phenomenon. One ponders, what would become of wall turbulence if speciﬁc coherent structures are to be targeted, by the operator through a reactive control scheme, for modiﬁcation? The myriad of organized structures present in all shear ﬂows are instantaneously identiﬁable, quasi-periodic motions (Cantwell, 1981; Robinson, 1991). Bursting events in wall-bounded ﬂows, for example, are both intermittent and random in space as well as time. The random aspects of these events reduce the effectiveness of a predetermined active control strategy. If such structures are nonintrusively detected and altered, on the other hand, net performance gain might be achieved. It seems clear, however, that temporal phasing as well as spatial selectivity would be required to achieve proper control targeted toward random events. A nonreactive version of the above idea is the selective suction technique which combines suction to achieve an asymptotic turbulent boundary layer and longitudinal riblets to ﬁx the location of low-speed streaks. Although far from indicating net drag reduction, the available results are encouraging and further optimization is needed. When implemented via an array of reactive control loops, the selective suction method is potentially capable of skin-friction reduction that approaches 60%.

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The genesis of the selective suction concept can be found in the papers by Gadel-Hak and Blackwelder (1987; 1989) and the patent by Blackwelder and Gad-el-Hak (1990). These researchers suggest that one possible means of optimizing the suction rate is to be able to identify where a low-speed streak is presently located and apply a small amount of suction under it. Assuming that the production of turbulence kinetic energy is due to the instability of an inﬂectional U (y) velocity proﬁle, one needs to remove only enough ﬂuid so that the inﬂectional nature of the proﬁle is alleviated. An alternative technique that could conceivably reduce the Reynolds stress is to inject ﬂuid selectively under the high-speed regions. The immediate effect of normal injection would be to decrease the viscous shear at the wall resulting in less drag. In addition, the velocity proﬁles in the spanwise direction, U (z), would have a smaller shear, ∂U/∂z, because the suction/injection would create a more uniform ﬂow. Since Swearingen and Blackwelder (1984) and Blackwelder and Swearingen (1990) have found that inﬂectional U (z) proﬁles occur as often as inﬂection points are observed in U (y) proﬁles, suction under the low-speed streaks and/or injection under the high-speed regions would decrease this shear and hence the resulting instability. The combination of selective suction and injection is sketched in Figure 1. In Figure 1a, the vortices are idealized by a periodic distribution in the spanwise direction. The instantaneous velocity proﬁles without transpiration at constant y and z locations are shown by the dashed lines in Figures 1b and 1c, respectively. Clearly, the U (yo , z) proﬁle is inﬂectional, having two inﬂection points per wavelength. At z1 and z3 , an inﬂectional U (y) proﬁle is also evident. The same proﬁles with suction at z1 and z3 and injection at z2 are shown by the solid lines. In all cases, the shear associated with the inﬂection points would have been reduced. Since the inﬂectional proﬁles are all inviscidly unstable with growth rates proportional to the shear, the resulting instabilities would be weakened by the suction/injection process. The feasibility of the selective suction as a drag-reducing concept has been demonstrated by Gad-el-Hak and Blackwelder (1989) and is indicated in Figure 2. Low-speed streaks were artiﬁcially generated in a laminar boundary layer using three spanwise suction holes as per the method proposed by Gad-el-Hak and Hussain (1986), and a hot-ﬁlm probe was used to record the near-wall signature of the streaks. An open, feedforward control loop with a phase lag was used to activate a predetermined suction from a longitudinal slot located in between the spanwise holes and the downstream hot-ﬁlm probe. An equivalent suction coefﬁcient of Cq = 0.0006 was sufﬁcient to eliminate the artiﬁcial events and prevent bursting. This rate is ﬁve times smaller than the asymptotic suction coefﬁcient for a corresponding turbulent boundary layer. If this result is sustained in a naturally developing turbulent boundary layer, a skin-friction reduction of close to 60% would be attained. Gad-el-Hak and Blackwelder (1989) propose to combine suction with non-planar surface modiﬁcations. Minute longitudinal roughness elements if properly spaced in the spanwise direction greatly reduce the spatial randomness of the low-speed streaks (Johansen and Smith, 1986). By withdrawing the streaks forming near the peaks of the roughness elements, less suction should be required to achieve an asymptotic boundary layer. Experiments by Wilkinson and Lazos (1987) and Wilkinson (1988) combine suction/blowing with thin-element riblets. Although no net drag reduction is yet attained in these experiments, their results indicate some

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Fig. 1. Effects of suction/injection on velocity proﬁles. Broken lines: reference proﬁles. Solid lines: proﬁles with transpiration applied. (a) Streamwise vortices in the y-z plane, suction/injection applied at z1 , z2 and z3 . (b) Resulting spanwise velocity distribution at y = yo . (c) Velocity proﬁles normal to the surface.

advantage of combining suction with riblets as proposed by Gad-el-Hak and Blackwelder (1987; 1989). The recent numerical experiments of Choi et al. (1994) also validate the concept of targeting suction/injection to speciﬁc near-wall events in a turbulent channel ﬂow. Based on complete interior ﬂow information and using the rather simple, heuristic control law proposed earlier by Gad-el-Hak and Blackwelder (1987), Choi et al.’s direct numerical simulations indicate a 20% net drag reduction accompanied by signiﬁcant suppression of the near-wall structures and the Reynolds stress throughout the entire

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Fig. 2. Effects of suction from a streamwise slot on ﬁve artiﬁcially induced burstlike events in a laminar boundary layer (from Gad-el-Hak and Blackwelder, 1989). (a) Cq = 0.0. (b) Cq = 0.0006.

wall-bounded ﬂow. When only wall information was used, a drag reduction of 6% was observed; a rather disappointing result considering that sensing and actuation took place at every grid point along the computational wall. In a practical implementation of this technique, even fewer wall sensors would perhaps be available, measuring only a small subset of the accessible information and thus requiring even more sophisticated control algorithms to achieve the same degree of success. Low-dimensional models of the nearwall ﬂow (Section 3) and soft computing tools can help in constructing more effective control algorithms. Time sequences of the numerical ﬂow ﬁeld of Choi et al. (1994) indicate the presence of two distinct drag-reducing mechanisms when selective suction/injection is used. First, deterring the sweep motion, without modifying the primary streamwise vortices above the wall, and consequently moving the high-shear regions from the surface to the interior of the channel, thus directly reducing the skin friction. Secondly, changing the evolution of the wall vorticity layer by stabilizing and preventing lifting of the near-wall spanwise vorticity, thus suppressing a potential source of new streamwise vortices above the surface and interrupting a very important regeneration mechanism of turbulence. Three modern developments have relevance to the issue at hand. Firstly, the recently demonstrated ability to revert a chaotic system to a periodic one may provide

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optimal nonlinear control strategies for further reduction in the amount of suction (or the energy expenditure of any other active wall-modulation technique) needed to attain a given degree of ﬂow stabilization. This is important since net drag reduction achieved in a turbulent boundary layer increases as the suction coefﬁcient decreases. Secondly, to selectively remove the randomly occurring low-speed streaks, for example, would ultimately require reactive control. In that case, an event is targeted, sensed and subsequently modulated. Microfabrication technology provides opportunities for practical implementation of the required large array of inexpensive, programmable sensor/actuator chips. Thirdly, newly introduced soft computing tools include neural networks, fuzzy logic and genetic algorithms and are now more advanced as well as more widely used as compared to just few years ago. These tools could be very useful in constructing effective adaptive controllers. 2.3 Reactive Feedback Control A control device can be passive, requiring no auxiliary power, or active, requiring energy expenditure. Active control is further divided into predetermined or reactive. Predetermined control includes the application of steady or unsteady energy input without regard to the particular state of the ﬂow. The control loop in this case is open, and no sensors are required. Because no sensed information are being fed forward, this open control loop is not a feedforward one. Reactive control is a special class of active control where the control input is continuously adjusted based on measurements of some kind. The control loop in this case can either be an open, feedforward one or a closed, feedback loop. The distinction between feedforward and feedback is particularly important when dealing with the control of ﬂow structures which convect over stationary sensors and actuators. In feedforward control, the measured variable and the controlled variable differ. For example, the pressure or velocity can be sensed at an upstream location, and the resulting signal is used together with an appropriate control law to trigger an actuator which in turn inﬂuences the velocity at a downstream position. Feedback control, on the other hand, necessitates that the controlled variable be measured, fed back and compared with a reference input. Moin and Bewley (1994) categorize reactive feedback control strategies by examining the extent to which they are based on the governing ﬂow equations. Four categories are discerned: adaptive, physical model-based, dynamical systems-based, and optimal control. Note that except for adaptive control, the other three categories of reactive feedback control can also be used in the feedforward mode or the combined feedforwardfeedback mode. Also, in a convective environment such as that for a boundary layer, a controller would perhaps combine feedforward and feedback information and may include elements from each of the four classiﬁcations. Each of the four categories is brieﬂy described below. Adaptive schemes attempt to develop models and controllers via some learning algorithm without regard to the details of the ﬂow physics. System identiﬁcation is performed independently of the ﬂow dynamics or the Navier–Stokes equations which govern this dynamics. An adaptive controller tries to optimize a speciﬁed performance index by providing a control signal to an actuator. In order to update its parameters,

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the controller thus requires feedback information relating to the effects of its control. The most recent innovation in adaptive ﬂow control schemes involves the use of neural networks which relate the sensor outputs to the actuator inputs through functions with variable coefﬁcients and nonlinear, sigmoid saturation functions. The coefﬁcients are updated using the so-called back-propagation algorithm, and complex control laws can be represented with a sufﬁcient number of terms. Hand tuning is required, however, to achieve good convergence properties. The nonlinear adaptive technique has been used with different degrees of success by Fan et al. (1993) and Jacobson and Reynolds (1993b; 1995; 1998) to control, respectively, the transition process and the bursting events in turbulent boundary layers. Heuristic physical arguments can instead be used to establish effective control laws. That approach obviously will work only in situations in which the dominant physics are well understood. An example of this strategy is the active cancellation scheme, used by Gad-el-Hak and Blackwelder (1989) in a physical experiment and by Choi et al. (1994) in a numerical experiment, to reduce the drag by mitigating the effect of nearwall vortices. As mentioned earlier, the idea is to oppose the near-wall motion of the ﬂuid, caused by the streamwise vortices, with an opposing wall control, thus lifting the high-shear region away from the surface and interrupting the turbulence regeneration mechanism. Nonlinear dynamical systems theory allows turbulence to be decomposed into a small number of representative modes whose dynamics are examined to determine the best control law. The task is to stabilize the attractors of a low-dimensional approximation of a turbulent chaotic system. The best known strategy is the OGY method which, when applied to simpler, small-number of degrees of freedom systems, achieves stabilization with minute expenditure of energy. This and other chaos control strategies, especially as applied to the more complex turbulent ﬂows, will be revisited in Section 3.2. Finally, optimal control theory applied directly to the Navier–Stokes equations can, in principle, be used to minimize a cost function in the space of the control. This strategy provides perhaps the most rigorous theoretical framework for ﬂow control. As compared to other reactive control strategies, optimal control applied to the full Navier– Stokes equations is also the most computer-time intensive. In this method, feedback control laws are derived systematically for the most efﬁcient distribution of control effort to achieve a desired goal. Abergel and Temam (1990) developed such optimal control theory for suppressing turbulence in a numerically simulated, two-dimensional Navier–Stokes ﬂow, but their method requires an impractical full ﬂow-ﬁeld information. Choi et al. (1993) developed a more practical, wall-information-only, sub-optimal control strategy which they applied to the one-dimensional stochastic Burgers equation. Later application of the sub-optimal control theory to a numerically simulated turbulent channel ﬂow has been reported by Moin and Bewley (1994) and Bewley et al. (1997; 1998). The recent book edited by Sritharan (1998) provides eight articles that focus on the mathematical aspects of optimal control of viscous ﬂows. 2.4 Required Characteristics The randomness of the bursting events necessitates temporal phasing as well as spatial selectivity to effect selective control. Practical applications of methods targeted at

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controlling a particular turbulent structure to achieve a prescribed goal would therefore require implementing a large number of surface sensors/actuators together with appropriate control algorithms. That strategy for controlling wall-bounded turbulent ﬂows has been advocated by, among others and in chronological order, Gad-el-Hak and Blackwelder (1987; 1989), Lumley (1991; 1996), Choi et al. (1992), Reynolds (1993), Jacobson and Reynolds (1993b; 1995), Moin and Bewley (1994), Gad-el-Hak (1994; 1996; 1998), McMichael (1996), Mehregany et al. (1996), Blackwelder (1998), Delville et al. (1998), and Perrier (1998). It is clear that the spatial and temporal resolutions for any probe to be used to resolve high-Reynolds-number turbulent ﬂows are extremely tight. For example, both the Kolmogorov scale and the viscous length-scale change from few microns at the typical ﬁeld Reynolds number—based on the momentum thickness—of 106 , to a couple of hundred microns at the typical laboratory Reynolds number of 103 . MEMS sensors for pressure, velocity, temperature and shear stress are at least one order of magnitude smaller than conventional sensors (Ho and Tai, 1996; 1998; L¨ofdahl et al., 1996; L¨ofdahl and Gad-elHak, 1999). Their small size improves both the spatial and temporal resolutions of the measurements, typically few microns and few microseconds, respectively. For example, a micro-hot-wire (called hot-point) has very small thermal inertia and the diaphragm of a micro-pressure-transducer has correspondingly fast dynamic response. Moreover, the microsensors’ extreme miniaturization and low energy consumption make them ideal for monitoring the ﬂow state without appreciably affecting it. Lastly, literally hundreds of microsensors can be fabricated on the same silicon chip at a reasonable cost, making them well suited for distributed measurements and control. The UCLA/Caltech team (see, for example, Ho and Tai, 1996; 1998, and references therein) has been very effective in developing many MEMS-based sensors and actuators for turbulence diagnosis and control. The handbook edited by Gad-el-Hak (2006) offers a comprehensive coverage of the broad ﬁeld of microelectromechanical systems. It is instructive to estimate some representative characteristics of the required array of sensors/actuators. Consider a typical commercial aircraft cruising at a speed of U∞ = 300 m/s and at an altitude of 10 km. The density and kinematic viscosity of air and the unit Reynolds number in this case are, respectively, ρ = 0.4 kg/m3 , ν = 3 × 10−5 m2 /s, and Re = 107 /m. Assume further that the portion of fuselage to be controlled has a turbulent boundary layer characteristics which are identical to those for a zero-pressure-gradient ﬂat plate at a distance of 1 m from the leading edge. In this case, the skin-friction coefﬁcient1 and the friction velocity are, respectively, Cf = 0.003 and uτ = 11.62 m/s. At this location, one viscous wall unit is only ν/uτ = 2.6 microns. In order for the surface array of sensors/actuators to be hydraulically smooth, it should not protrude beyond the viscous sublayer, or 5ν/uτ = 13 µm. Wall-speed streaks are the most visible, reliable and detectable indicators of the preburst turbulence production process. The detection criterion is simply low velocity near the wall, and the actuator response should be to accelerate (or to remove) the lowspeed region before it breaks down. Local wall motion, tangential injection, suction, 1

Note that the skin friction decreases as the distance from the leading increases. It is also strongly affected by such things as the externally imposed pressure gradient. Therefore, the estimates provided in here are for illustration purposes only.

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heating or electromagnetic body force, all triggered on sensed wall-pressure or wallshear stress, could be used to cause local acceleration of near-wall ﬂuid. The recent numerical experiments of Berkooz et al. (1993) indicate that effective control of bursting pair of rolls may be achieved by using the equivalent of two wall-mounted shear sensors. If the goal is to stabilize or to eliminate all low-speed streaks in the boundary layer, a reasonable estimate for the spanwise and streamwise distances between individual elements of a checkerboard array is, respectively, 100 and 1000 wall units,2 or 260 µm and 2600 µm, for our particular example. A reasonable size for each element is probably one-tenth of the spanwise separation, or 26 µm. A (1 m × 1 m) portion of the surface would have to be covered with about n = 1.5 million elements. This is a colossal number, but the density of sensors/actuators could be considerably reduced if we moderate our goal of targeting every single bursting event (and also if less conservative assumptions are used). It is well known that not every low-speed streak leads to a burst. On the average, a particular sensor would detect an incipient bursting event every wall-unit interval of P + = P u2τ /ν = 250, or P = 56 µs. The corresponding dimensionless and dimensional frequencies are f + = 0.004 and f = 18 kHz, respectively. At different distances from the leading edge and in the presence of nonzero-pressure gradient, the sensors/actuators array would have different characteristics, but the corresponding numbers would still be in the same ballpark as estimated in here. As a second example, consider an underwater vehicle moving at a speed of U∞ = 10 m/s. Despite the relatively low speed, the unit Reynolds number is still the same as estimated above for the air case, Re = 107 /m, due to the much lower kinematic viscosity of water. At one meter from the leading edge of an imaginary ﬂat plate towed in water at the same speed, the friction velocity is only uτ = 0.39 m/s, but the wall unit is still the same as in the aircraft example, ν/uτ = 2.6 µm. The density of required sensors/actuators array is the same as computed for the aircraft example, n = 1.5 × 106 elements/m2 . The anticipated average frequency of sensing a bursting event is, however, much lower at f = 600 Hz . Similar calculations have been recently made by Gad-el-Hak (1993; 1994; 1998), Reynolds (1993) and Wadsworth et al. (1993). Their results agree closely with the estimates made here for typical ﬁeld requirements. In either the airplane or the submarine case, the actuator’s response need not be too large. According to Gad-el-Hak (2000), wall displacement on the order of 10 wall units (26 µm in both examples), suction coefﬁcient of about 0.0006, or surface cooling/heating on the order of 40◦ C/2◦ C (in the ﬁrst/second example, respectively) should be sufﬁcient to stabilize the turbulent ﬂow. As computed in the two examples above, both the required size for a sensor/actuator element and the average frequency at which an element would be activated are within the presently known capabilities of microfabrication technology. The number of 2

These are equal to, respectively, the average spanwise wavelength between two adjacent streaks and the average streamwise extent for a typical low-speed region. One can argue that those estimates are too conservative: once a region is relaminarized, it would perhaps stay as such for quite a while as the ﬂow convects downstream. The next row of sensors/actuators may therefore be relegated to a downstream location well beyond 1000 wall units. Relatively simple physical or numerical experiments could settle this issue.

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elements needed per unit area is, however, alarmingly large. The unit cost of manufacturing a programmable sensor/actuator element would have to come down dramatically, perhaps matching the unit cost of a conventional transistor,3 before the idea advocated in here would become practical. An additional consideration to the size, amplitude and frequency response is the energy consumed by each sensor/actuator element. Total energy consumption by the entire control system obviously has to be low enough to achieve net savings. Consider the following calculations for the aircraft example. One meter from the leading edge, the skin-friction drag to be reduced is approximately 54 N/m2 . Engine power needed to overcome this retarding force per unit area is 16 kW/m2 , or 104 µW/sensor. If a 60% drag-reduction is achieved,4 this energy consumption is reduced to 4320 µW/sensor. This number will increase by the amount of energy consumption of a sensor/actuator unit, but hopefully not back to the uncontrolled levels. The voltage across a sensor is typically in the range of V = 0.1–1 V, and its resistance in the range of R = 0.1–1 MΩ. This means a power consumption by a typical sensor in the range of P = V 2 /R = 0.1–10 µW, well below the anticipated power savings due to reduced drag. For a single actuator in the form of a spring-loaded diaphragm with a spring constant of k = 100 N/m and oscillating up and down at the bursting frequency of f = 18 kHz with an amplitude of y = 26 microns, the power consumption is P = (1/2) k y 2 f = 600 µW/actuator. If suction is used instead, Cq = 0.0006, and assuming a pressure difference of ∆p = 104 N/m2 across the suction holes/slots, the corresponding power consumption for a single actuator is P = Cq U∞ ∆p/n = 1200 µW/actuator. It is clear then that when the power penalty for the sensor/actuator is added to the lower-level drag, a net saving is still achievable. The corresponding actuator power penalties for the submarine example are even smaller (P = 20 µW/actuator for the wall motion actuator, and P = 40 µW/actuator for the suction actuator), and larger savings are therefore possible.

3 Chaos Control 3.1 Nonlinear Dynamical Systems Theory In the theory of dynamical systems, the so-called butterﬂy effect denotes sensitive dependence of nonlinear differential equations on initial conditions, with phase-space solutions initially very close together separating exponentially. The solution of nonlinear dynamical systems of three or more degrees of freedom may be in the form of a strange attractor whose intrinsic structure contains a well-deﬁned mechanism to produce a chaotic behavior without requiring random forcing. Chaotic behavior is complex, aperiodic and, though deterministic, appears to be random. A question arises naturally: just as small disturbances can radically grow within a deterministic system to yield rich, unpredictable behavior, can minute adjustments to 3

4

The transistor was invented in 1947. In the mid 1960s, a single transistor sold for around $70. In 1997, Intel’s Pentium II processor (microchip) contained 7.5 × 106 transistors and cost around $500, that is less than $0.00007 per transistor! A not-too-farfetched goal according to the selective suction results discussed earlier.

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a system parameter be used to reverse the process and control, i.e. regularize, the behavior of a chaotic system? Recently, that question was answered in the afﬁrmative theoretically as well as experimentally, at least for system orbits which reside on lowdimensional strange attractors (see the review by Lindner and Ditto, 1995). Before describing such strategies for controlling chaotic systems, we ﬁrst summarize the recent attempts to construct a low-dimensional dynamical systems representation of turbulent boundary layers. Such construction is a necessary ﬁrst step to be able to use chaos control strategies for turbulent ﬂows. Additionally, as argued by Lumley (1996), a lowdimensional dynamical model of the near-wall region used in a Kalman ﬁlter (Banks, 1986; Petersen and Savkin, 1999) can make the most of the partial information assembled from a ﬁnite number of wall sensors. Such ﬁlter minimizes in a least square sense the errors caused by incomplete information, and thus globally optimizes the performance of the control system. Boundary layer turbulence is described by a set of nonlinear partial differential equations and is characterized by an inﬁnite number of degrees of freedom. This makes it rather difﬁcult to model the turbulence using a dynamical systems approximation. The notion that a complex, inﬁnite-dimensional ﬂow can be decomposed into several low-dimensional subunits is, however, a natural consequence of the realization that quasi-periodic coherent structures dominate the dynamics of seemingly random turbulent shear ﬂows. This implies that low-dimensional, localized dynamics can exist in formally inﬁnite-dimensional extended systems—such as open turbulent ﬂows. Reducing the ﬂow physics to ﬁnite-dimensional dynamical systems enables a study of its behavior through an examination of the ﬁxed points and the topology of their stable and unstable manifolds. From the dynamical systems theory viewpoint, the meandering of low-speed streaks is interpreted as hovering of the ﬂow state near an unstable ﬁxed point in the low-dimensional state space. An intermittent event that produces high wall stress—a burst—is interpreted as a jump along a heteroclinic cycle to different unstable ﬁxed point that occurs when the state has wandered too far from the ﬁrst unstable ﬁxed point. Delaying this jump by holding the system near the ﬁrst ﬁxed point should lead to lower momentum transport in the wall region and, therefore, to lower skin-friction drag. Reactive control means sensing the current local state and through appropriate manipulation keeping the state close to a given unstable ﬁxed point, thereby preventing further production of turbulence. Reducing the bursting frequency by say 50%, may lead to a comparable reduction in skin-friction drag. For a jet, relaminarization may lead to a quiet ﬂow and very signiﬁcant noise reduction. In one signiﬁcant attempt the proper orthogonal, or Karhunen-Lo`eve, decomposition method has been used to extract a low-dimensional dynamical system from experimental data of the wall region (Aubry et al. 1988; Aubry, 1990). Aubry et al. (1988) expanded the instantaneous velocity ﬁeld of a turbulent boundary layer using experimentally determined eigenfunctions which are in the form of streamwise rolls. They expanded the Navier–Stokes equations using these optimally chosen, divergence-free, orthogonal functions, applied a Galerkin projection, and then truncated the inﬁnitedimensional representation to obtain a ten-dimensional set of ordinary differential equations. These equations represent the dynamical behavior of the rolls, and are shown to exhibit a chaotic regime as well as an intermittency due to a burst-like phenomenon.

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However, Aubry et al.’s ten-mode dynamical system displays a regular intermittency, in contrast both to that in actual turbulence as well as to the chaotic intermittency encountered by Pomeau and Manneville (1980) in which event durations are distributed stochastically. Nevertheless, the major conclusion of Aubry et al.’s study is that the bursts appear to be produced autonomously by the wall region even without turbulence, but are triggered by turbulent pressure signals from the outer layer. More recently, Berkooz et al. (1991) generalized the class of wall-layer models developed by Aubry et al. (1988) to permit uncoupled evolution of streamwise and cross-stream disturbances. Berkooz et al.’s results suggest that the intermittent events observed in Aubry et al.’s representation do not arise solely because of the effective closure assumption incorporated, but are rather rooted deeper in the dynamical phenomena of the wall region. The book by Holmes et al. (1996) details the Cornell research group attempts at describing turbulence as a low-dimensional dynamical system. In addition to the reductionist viewpoint exempliﬁed by the work of Aubry et al. (1988) and Berkooz et al. (1991), attempts have been made to determine directly the dimension of the attractors underlying speciﬁc turbulent ﬂows. Again, the central issue here is whether or not turbulent solutions to the inﬁnite-dimensional Navier–Stokes equations can be asymptotically described by a ﬁnite number of degrees of freedom. Grappin and L´eorat (1991) computed the Lyapunov exponents and the attractor dimensions of two- and three-dimensional periodic turbulent ﬂows without shear. They found that the number of degrees of freedom contained in the large scales establishes an upper bound for the dimension of the attractor. Deane and Sirovich (1991) and Sirovich and Deane (1991) numerically determined the number of dimensions needed to specify chaotic Rayleigh-B´enard convection over a moderate range numbers, Ra. of Rayleigh They suggested that the intrinsic attractor dimension is O Ra2/3 . The corresponding dimension in wall-bounded ﬂows appears to be dauntingly high. Keefe et al. (1992) determined the dimension of the attractor underlying turbulent Poiseuille ﬂows with spatially periodic boundary conditions. Using a coarse-grained numerical simulation, they computed a lower bound on the Lyapunov dimension of the attractor to be approximately 352 at a pressure-gradient Reynolds number of 3200. Keefe et al. (1992) argue that the attractor dimension in fully-resolved turbulence is unlikely to be much larger than 780. This suggests that periodic turbulent shear ﬂows are deterministic chaos and that a strange attractor does underlie solutions to the Navier– Stokes equations. Temporal unpredictability in the turbulent Poiseuille ﬂow is thus due to the exponential spreading property of such attractors. Although ﬁnite, the computed dimension invalidates the notion that the global turbulence can be attributed to the interaction of a few degrees of freedom. Moreover, in a physical channel or boundary layer, the ﬂow is not periodic and is open. The attractor dimension in such case is not known but is believed to be even higher than the estimate provided by Keefe et al. for the periodic (quasi-closed) ﬂow. In contrast to closed, absolutely unstable ﬂows, such as Taylor-Couette systems, where the number of degrees of freedom can be small, local measurements in open, convectively unstable ﬂows, such as boundary layers, do not express the global dynamics, and the attractor dimension in that case may inevitably be too large to be determined experimentally. According to the estimate provided by Keefe et al. (1992), the colossal

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data required (about 10D , where D is the attractor dimension) for measuring the dimension simply exceeds current computer capabilities. Turbulence near transition or near a wall is an exception to that bleak picture. In those special cases, a relatively small number of modes are excited and the resulting simple turbulence can therefore be described by a dynamical system of a reasonable number of degrees of freedom. 3.2 Chaos Control There is another question of greater relevance here. Given a dynamical system in the chaotic regime, is it possible to stabilize its behavior through some kind of active control? While other alternatives have been devised (e.g., Fowler, 1989; H¨ubler and L¨uscher, 1989; Huberman, 1990; Huberman and Lumer, 1990), the recent method proposed by workers at the University of Maryland (Ott et al., 1990a; 1990b; Shinbrot et al., 1990; 1992a; 1992b; 1992c; 1998; Romeiras et al., 1992) promises to be a significant breakthrough. Comprehensive reviews and bibliographies of the emerging ﬁeld of chaos control can be found in the articles by Shinbrot et al. (1993), Shinbrot (1993; 1995; 1998), and Lindner and Ditto (1995). Ott et al. (1990a) demonstrated, through numerical experiments with the Henon map, that it is possible to stabilize a chaotic motion about any pre-chosen, unstable orbit through the use of relatively small perturbations. The procedure consists of applying minute time-dependent perturbations to one of the system parameters to control the chaotic system around one of its many unstable periodic orbits. In this context, targeting refers to the process whereby an arbitrary initial condition on a chaotic attractor is steered toward a prescribed point (target) on this attractor. The goal is to reach the target as quickly as possible using a sequence of small perturbations (Kostelich et al., 1993a). The success of the Ott-Grebogi-Yorke’s (OGY) strategy for controlling chaos hinges on the fact that beneath the apparent unpredictability of a chaotic system lies an intricate but highly ordered structure. Left to its own recourse, such a system continually shifts from one periodic pattern to another, creating the appearance of randomness. An appropriately controlled system, on the other hand, is locked into one particular type of repeating motion. With such reactive control the dynamical system becomes one with a stable behavior. The OGY-method can be simply illustrated by the schematic in Figure 3. The state of the system is represented as the intersection of a stable manifold and an unstable one. The control is applied intermittently whenever the system departs from the stable manifold by a prescribed tolerance, otherwise the control is shut off. The control attempts to put the system back onto the stable manifold so that the state converges toward the desired trajectory. Unmodeled dynamics cause noise in the system and a tendency for the state to wander off in the unstable direction. The intermittent control prevents that and the desired trajectory is achieved. This efﬁcient control is not unlike trying to balance a ball in the center of a horse saddle (Moin and Bewley, 1994). There is one stable direction (front/back) and one unstable direction (left/right). The restless horse is the unmodeled dynamics, intermittently causing the ball to move in the wrong direction. The OGY-control needs only be applied, in the most direct manner possible, whenever the ball wanders off in the left/right direction.

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Desired trajectory

Unstable manifold

Stable manifold

Fig. 3. The OGY method for controlling chaos

The OGY-method has been successfully applied in a relatively simple experiment by Ditto et al. (1990) and Ditto and Pecora (1993) at the Naval Surface Warfare Center, in which reverse chaos was obtained in a parametrically driven, gravitationally buckled, amorphous magnetoelastic ribbon. Garﬁnkel et al. (1992) applied the same control strategy to stabilize drug-induced cardiac arrhythmias in sections of a rabbit ventricle. Other extensions, improvements and applications of the OGY-strategy include higher-dimensional targeting (Auerbach et al., 1992; Kostelich et al., 1993b), controlling chaotic scattering in Hamiltonian (i.e., nondissipative, area conservative) systems (Lai et al., 1993a; 1993b), synchronization of identical chaotic systems that govern communication, neural or biological processes (Lai and Grebogi, 1993), use of chaos to transmit information (Hayes et al., 1994a; 1994b), control of transient chaos (Lai et al., 1994), and taming spatio-temporal chaos using a sparse array of controllers (Chen et al., 1993; Qin et al., 1994; Auerbach, 1994). In a more complex system, such as a turbulent boundary layer, there exist numerous interdependent modes and many stable as well as unstable manifolds (directions). The ﬂow can then be modeled as coherent structures plus a parameterized turbulent background. The proper orthogonal decomposition (POD) is used to model the coherent part because POD guarantees the minimum number of degrees of freedom for a given model accuracy. Factors that make turbulence control a challenging task are the potentially quite large perturbations caused by the unmodeled dynamics of the ﬂow, the non-stationary nature of the desired dynamics, and the complexity of the saddle shape describing the dynamics of the different modes. Nevertheless, the OGY-control strategy has several advantages that are of special interest in the control of turbulence: (1) the mathematical model for the dynamical system need not be known; (2) only small changes in the control parameter are required; and (3) noise can be tolerated (with appropriate penalty). Recently, Keefe (1993a; 1993b) made a useful comparison between two nonlinear control strategies as applied to ﬂuid problems. Ott-Grebogi-Yorke’s feedback method

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described above and the model-based control strategy originated by H¨ubler (see, for example, H¨ubler and L¨uscher, 1989; L¨uscher and H¨ubler, 1989), the H-method. Both novel control methods are essentially generalizations of the classical perturbation cancellation technique: apply a prescribed forcing to subtract the undesired dynamics and impose the desired one. The OGY-strategy exploits the sensitivity of chaotic systems to stabilize existing periodic orbits and steady states. Some feedback is needed to steer the trajectories toward the chosen ﬁxed point, but the required control signal is minuscule. In contrast, H¨ubler’s scheme does not explicitly make use of the system sensitivity. It produces general control response (periodic or aperiodic) and needs little or no feedback, but its control inputs are generally large. The OGY-strategy exploits the nonlinearity of a dynamical system; indeed the presence of a strange attractor and the extreme sensitivity of the dynamical system to initial conditions are essential to the success of the method. In contrast, the H-method works equally for both linear and nonlinear systems. Keefe (1993a) ﬁrst examined numerically the two schemes as applied to fullydeveloped and transitional solutions of the Ginzburg-Landau equation, an evolution equation that governs the initially weakly nonlinear stages of transition in several ﬂows and that possesses both transitional and fully-chaotic solutions. The Ginzburg-Landau equation has solutions that display either absolute or convective instabilities, and is thus a reasonable model for both closed and open ﬂows. Keefe’s main conclusion is that control of nonlinear systems is best obtained by making maximum use possible of the underlying natural dynamics. If the goal dynamics is an unstable nonlinear solution of the equation and the ﬂow is nearby at the instant control is applied, both methods perform reliably and at low-energy cost in reaching and maintaining this goal. Predictably, the performance of both control strategies degrades due to noise and the spatially discrete nature of realistic forcing. Subsequently, Keefe (1993b) extended the numerical experiment in an attempt to reduce the drag in a channel ﬂow with spatially periodic boundary conditions. The OGY-method reduces the skin friction to 60–80% of the uncontrolled value at a mass-ﬂux Reynolds number of 4408. The H-method fails to achieve any drag reduction when starting from a fully-turbulent initial condition but shows potential for suppressing or retarding laminar-to-turbulence transition. Keefe (1993a) suggests that the H-strategy might be more appropriate for boundary layer control, while the OGYmethod might best be used for channel ﬂows. It is also relevant to note here the work of Bau and his colleagues at the University of Pennsylvania (Singer et al., 1991; Wang et al., 1992), who devised a feedback control to stabilize (relaminarize) the naturally occurring chaotic oscillations of a toroidal thermal convection loop heated from below and cooled from above. Based on a simple mathematical model for the thermosyphon, Bau and his colleagues constructed a reactive control system that was used to alter signiﬁcantly the ﬂow characteristics inside the convection loop. Their linear control strategy, perhaps a special version of the OGY’s chaos control method, consists simply of sensing the deviation of ﬂuid temperatures from desired values at a number of locations inside the thermosyphon loop and then altering the wall heating either to suppress or to enhance such deviations. Wang et al. (1992) also suggested extending their theoretical and experimental method to more complex situations such as those involving B´enard convection (Tang and Bau, 1993a; 1993b). Hu and Bau (1994) used a similar feedback control strategy to demonstrate that

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the critical Reynolds number for the loss of stability of planar Poiseuille ﬂow can be signiﬁcantly increased or decreased. Other attempts to use low-dimensional dynamical systems representation for ﬂow control include the work of Berkooz et al. (1993), Corke et al. (1994), and Coller et al. (1994a; 1994b). Berkooz et al. (1993) applied techniques of modern control theory to estimate the phase-space location of dynamical models of the wall-layer coherent structures, and used these estimates to control the model dynamics. Since discrete wallsensors provide incomplete knowledge of phase-space location, Berkooz et al. maintain that a nonlinear observer, which incorporates past information and the equations of motion into the estimation procedure, is required. Using an extended Kalman ﬁlter, they achieved effective control of a bursting pair of rolls with the equivalent of two wallmounted shear sensors. Corke et al. (1994) used a low-dimensional dynamical system based on the proper orthogonal decomposition to guide control experiments for an axisymmetric jet. By sensing the downstream velocity and actuating an array of miniature speakers located at the lip of the jet, their feedback control succeeded in converting the near-ﬁeld instabilities from spatial-convective to temporal-global. Coller et al. (1994a; 1994b) developed a feedback control strategy for strongly nonlinear dynamical systems, such as turbulent ﬂows, subject to small random perturbations that kick the system intermittently from one saddle point to another along heteroclinic cycles. In essence, their approach is to use local, weakly nonlinear feedback control to keep a solution near a saddle point as long as possible, but then to let the natural, global nonlinear dynamics run its course when bursting (in a low-dimensional model) does occur. Though conceptually related to the OGY-strategy, Coller et al.’s method does not actually stabilize the state but merely holds the system near the desired point longer than it would otherwise stay. Shinbrot and Ottino (1993a; 1993b) offer yet another strategy presumably most suited for controlling coherent structures in area-preserving turbulent ﬂows. Their geometric method exploits the premise that the dynamical mechanisms which produce the organized structures can be remarkably simple. By repeated stretching and folding of “horseshoes” which are present in chaotic systems, Shinbrot and Ottino have demonstrated numerically as well as experimentally the ability to create, destroy and manipulate coherent structures in chaotic ﬂuid systems. The key idea to create such structures is to intentionally place folds of horseshoes near low-order periodic points. In a dissipative dynamical system, volumes contract in state space and the co-location of a fold with a periodic point leads to an isolated region that contracts asymptotically to a point. Provided that the folding is done properly, it counteracts stretching. Shinbrot and Ottino (1993a) applied the technique to three prototypical problems: a one-dimensional chaotic map; a two-dimensional one; and a chaotically advected ﬂuid. Shinbrot (1995; 1998) and Shinbrot et al. (1998) provide recent reviews of the stretching/folding as well as other chaos control strategies.

4 Conclusions In the present chapter, I have emphasized the frontiers of the ﬁeld of ﬂow control, reviewing the important advances that took place during the past few years and providing

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a blueprint for future progress. In two words, the future of ﬂow control is in taming turbulence by targeting its coherent structures: reactive control. Recent developments in chaos control, microfabrication and soft computing tools are making it more feasible to perform reactive control of turbulent ﬂows to achieve drag reduction, lift enhancement, mixing augmentation and noise suppression. Field applications, however, have to await further progress in those three modern areas. Other less complex control schemes, passive as well as active, are more market ready and are also witnessing resurgence of interest. The outlook for reactive control is quite optimistic. Soft computing tools and nonlinear dynamical systems theory are developing at fast pace. MEMS technology is improving even faster. The ability of Texas Instruments to produce an array of one million individually addressable mirrors for around 0.01 cent per actuator is a foreteller of the spectacular advances anticipated in the near future. Existing automotive applications of MEMS have already proven the ability of such devices to withstand the harsh environment under the hood. For the ﬁrst time, targeted control of turbulent ﬂows is now in the realm of the possible for future practical devices. What is needed now is a focused, well-funded research and development program to make it all come together for ﬁeld application of reactive ﬂow control systems. In parting, it may be worth recalling that a mere 10% reduction in the total drag of an aircraft translates into a saving of $3 billion in annual fuel cost (at 2005 prices) for the commercial ﬂeet of aircraft in the United States alone. Contrast this beneﬁt to the annual worldwide expenditure of perhaps a few million dollars for all basic research in the broad ﬁeld of ﬂow control. Taming turbulence, though arduous, will pay for itself in gold. Reactive control as difﬁcult as it seems, is neither impossible nor a pie in the sky. Beside, lofty goals require strenuous efforts. Easy solutions to difﬁcult problems are likely to be wrong as Henry Louis Mencken once lamented.

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Shinbrot, T., and Ottino, J.M. (1993a) “Geometric Method to Create Coherent Structures in Chaotic Flows,” Phys. Rev. Lett. 71, pp. 843–846. Shinbrot, T., and Ottino, J.M. (1993b) “Using Horseshoes to Create Coherent Structures in Chaotic Fluid Flows,” Bul. Am. Phys. Soc. 38, p. 2194. Shinbrot, T., Bresler, L., and Ottino, J.M. (1998) “Manipulation of Isolated Structures in Experimental Chaotic Fluid Flows,” Exp. Thermal & Fluid Sci. 16, pp. 76–83. Shinbrot, T., Ditto, W., Grebogi, C., Ott, E., Spano, M., and Yorke, J.A. (1992a) “Using the Sensitive Dependence of Chaos (the “Butterﬂy Effect”) to Direct Trajectories in an Experimental Chaotic System,” Phys. Rev. Lett. 68, pp. 2863–2866. Shinbrot, T., Grebogi, C., Ott, E., and Yorke, J.A. (1992b) “Using Chaos to Target Stationary States of Flows,” Phys. Lett. A 169, pp. 349–354. Shinbrot, T., Grebogi, C., Ott, E., and Yorke, J.A. (1993) “Using Small Perturbations to Control Chaos,” Nature 363, pp. 411–417. Shinbrot, T., Ott, E., Grebogi, C., and Yorke, J.A. (1990) “Using Chaos to Direct Trajectories to Targets,” Phys. Rev. Lett. 65, pp. 3215–3218. Shinbrot, T., Ott, E., Grebogi, C., and Yorke, J.A. (1992c) “Using Chaos to Direct Orbits to Targets in Systems Describable by a One-Dimensional Map,” Phys. Rev. A 45, pp. 4165– 4168. Singer, J., Wang, Y.-Z., and Bau, H.H. (1991) “Controlling a Chaotic System,” Phys. Rev. Lett. 66, pp. 1123–1125. Sirovich, L., and Deane, A.E. (1991) “A Computational Study of Rayleigh–B´enard Convection. Part 2. Dimension Considerations,” J. Fluid Mech. 222, pp. 251–265. Sritharan, S.S., editor (1998) Optimal Control of Viscous Flow, SIAM, Philadelphia, Pennsylvania. Swearingen, J.D., and Blackwelder, R.F. (1984) “Instantaneous Streamwise Velocity Gradients in the Wall Region,” Bul. Am. Phys. Soc. 29, p. 1528. Tang, J., and Bau, H.H. (1993a) “Stabilization of the No-Motion State in Rayleigh–B´enard Convection through the Use of Feedback Control,” Phys. Rev. Lett. 70, pp. 1795–1798. Tang, J., and Bau, H.H. (1993b) “Feedback Control Stabilization of the No-Motion State of a Fluid Conﬁned in a Horizontal Porous Layer Heated from Below,” J. Fluid Mech. 257, pp. 485–505. Viswanath, P.R. (1995) “Flow Management Techniques for Base and Afterbody Drag Reduction,” Prog. Aero. Sci. 32, pp. 79–129. Wadsworth, D.C., Muntz, E.P., Blackwelder, R.F., and Shiﬂett, G.R. (1993) “Transient Energy Release Pressure Driven Microactuators for Control of Wall-Bounded Turbulent Flows,” AIAA Paper No. 93-3271, AIAA, Washington, D.C. Wang, Y., Singer, J., and Bau, H.H. (1992) “Controlling Chaos in a Thermal Convection Loop,” J. Fluid Mech. 237, pp. 479–498. Wilkinson, S.P. (1988) “Direct Drag Measurements on Thin-Element Riblets with Suction and Blowing,” AIAA Paper No. 88-3670-CP, Washington, D.C. Wilkinson, S.P. (1990) “Interactive Wall Turbulence Control,” in Viscous Drag Reduction in Boundary Layers, eds. D.M. Bushnell and J.N. Hefner, pp. 479–509, AIAA, Washington, D.C. Wilkinson, S.P., and Balasubramanian, R. (1985) “Turbulent Burst Control through Phase-Locked Surface Depressions,” AIAA Paper No. 85-0536, New York. Wilkinson, S.P., and Lazos, B.S. (1987) “Direct Drag and Hot-Wire Measurements on ThinElement Riblet Arrays,” in Turbulence Management and Relaminarization, eds. H.W. Liepmann and R. Narasimha, pp. 121–131, Springer-Verlag, New York. Wilkinson, S.P., Anders, J.B., Lazos, B.S., and Bushnell, D.M. (1988) “Turbulent Drag Reduction Research at NASA Langley: Progress and Plans,” Int. J. Heat and Fluid Flow 9, pp. 266–277.

Electromagnetic Control of Separated Flows Using Periodic Excitation with Different Wave Forms Christian Cierpka, Tom Weier, and Gunter Gerbeth Forschungszentrum Rossendorf, P.O. Box 510119, 01314 Dresden, Germany [email protected] http://www.fz-rossendorf.de/pls/rois/Cms?pNid=226

Summary Time periodic Lorentz forces have been used to inﬂuence the separated ﬂow on an inclined ﬂat plate in deep stall at a Reynolds number of 104 . The inﬂuence of the control parameters effective momentum coefﬁcient and excitation frequency as well as excitation wave form is discussed based on phase averaged PIV measurements. As expected, control authority depends strongly on momentum input and excitation frequency, but effects of the excitation wave form can be shown as well.

1 Introduction Owing to its technological importance, ﬂow separation and its control is a persistent topic of ﬂuid dynamic research. A comprehensive and recent review can be found in [1]. While steady blowing is a tool investigated for almost eight decades, separation control by periodic addition of momentum has been a subject of intense research only since the early 1990s. Its most striking feature is that a control goal, e.g. a speciﬁc lift increase, can typically be attained by orders of magnitude smaller momentum input compared to steady actuation [2]. Control authority depends mainly on time averaged momentum input and excitation frequency. Greenblatt and Wygnanski [3] and more recently Seifert et al. [4] reviewed the state of the art of “active ﬂow control”, a term now commonly used for periodic excitation. If the ﬂuid is electrically conducting, like seawater or ionized air, momentum addition, usually accomplished by imposing mass ﬂuxes, can be achieved as well by electromagnetic, i.e. Lorentz forces. The Lorentz force density F appears as a body force term on the right hand side of the Navier–Stokes–Equation for incompressible ﬂow p F ∂u + (u · ∇)u = − + ν∇2 u + , ∂t ρ ρ

(1)

where u denotes the velocity, p the pressure, and t the time, respectively. ρ is the density and ν the kinematic viscosity of the ﬂuid. As can be seen from (1), the electromagnetic force density acts as a momentum source for the ﬂow. The Lorentz force density itself is the vector product of a current density j and a magnetic induction B F = j × B. R. King (Ed.): Active Flow Control, NNFM 95, pp. 27–41, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

(2)

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Ohm’s law j = σ(E + u × B)

(3)

for moving conductors describes the current density. E denotes an electric ﬁeld, and σ the electrical conductivity of the ﬂuid. In liquid metal Magnetohydrodynamics (MHD), the Lorentz force density and the ﬂow are usually strongly coupled, since the ﬂow induces currents via the u × B term in (3), these currents generate Lorentz forces (2), and the Lorentz forces change the ﬂow (1). The reason for this strong coupling is the very high electrical conductivity of liquid metals, typically σ = O(106 ) S/m. In the case of seawater or other electrolytes, σ is small (∼ 10 S/m). Therefore, the induced currents are very low for moderate applied magnetic ﬁelds (B0 ∼ 1 T). Accordingly, the Lorentz forces due to these currents are negligible and too weak to act on the ﬂow. In order to generate forces large enough to inﬂuence the ﬂow, an additional electric ﬁeld has to be applied. The ratio of the applied E0 to the electric ﬁeld induced by the free stream velocity U∞ in the presence of the applied magnetic ﬁeld B0 is commonly termed load factor [5] φ=

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(4)

For seawater ﬂow control with moderate magnetic ﬁelds, it follows from the above φ 1.

(5)

This implies on one hand that the force density distribution can be calculated independently of the ﬂow ﬁeld. On the other hand, a large load factor means a small efﬁciency of momentum generation since the ratio of mechanical (∼ jU∞ B0 ) to electrical (∼ jE0 ) power is the reciprocal of φ. Strictly speaking, the ﬂow acts not only on the electric ﬁeld distribution, but deforms the magnetic ﬁeld as well. However, even in the case of most liquid metal MHD problems – apart from dynamo experiments – it is well justiﬁed to ignore the induced magnetic ﬁelds when determining the generated Lorentz force. Despite its low energetical efﬁciency, the Lorentz force has several appealing features qualifying it as an interesting actuator for basic research: momentum is directly generated in the ﬂuid without associated mass ﬂux, the frequency response of the actuation is practically unlimited, no moving parts are involved. First investigations of the electromagnetic control of electrolyte ﬂows date back to the 1950’s. Already in 1954, Crausse and Cachon [6] gave experimental evidence of successful separation postponement as well as separation provocation on a half cylinder. Similar experiments were performed later by Lielausis [7]. After a few investigations on laminar ﬂow control [8,9] and related topics [10,11], activities in that ﬁeld declined with the beginning 1970’s. A renewed interest in electromagnetic ﬂow control arose in the 1990’s. The majority of publications concentrated on the topic of turbulent skin friction reduction, e.g. [12,13,14,15,16]. Compared to skin friction reduction, the use of Lorentz forces to control ﬂow separation received less attention. A circular cylinder equipped with electrodes as well as permanent magnets generating a wall

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29

parallel force in streamwise direction was used in the experiments and numerical calculations of Weier et al. [17]. Similar conﬁgurations were investigated later by Kim and Lee [18], Posdziech and Grundmann [19], and Chen and Aubry [20]. Results on separation prevention on hydrofoils using steady Lorentz forces have been reported by Weier et al. [21]. Weier and Gerbeth [22] examined time periodic Lorentz forces to control the separated ﬂow around a NACA 0015 at chord length Reynolds numbers 5.2 × 104 < Re < 1.5×105. Essential features like charakteristic efﬁcient excitation frequencies, effective momentum coefﬁcients and resulting lift gain compare well to that found with alternative methods for periodic addition of momentum. Of special interest for the present paper is the inﬂuence of different excitation wave forms on the resulting lift and drag of the hydrofoil. Force balance measurements from [22] demonstrate clearly that under otherwise identical conditions different excitation wave forms can change the attainable lift by up to 70%. A further ﬁnding of interest by Weier and Gerbeth [22] is the scaling of the lift increase with the peak instead of the effective momentum coefﬁcient under certain conditions. These results motivated the present paper which aims by help of particle image velocimetry (PIV) measurements to further investigate excitation wave form effects. This topic is of potential interest as well for active ﬂow control by means other than electromagnetic forces, since using similar nondimensionalized excitation frequencies and momentum coefﬁcients, different excitation methods achieve comparable results [3,22]. Up to now, the inﬂuence of the excitation wave form has received comparably little attention. Bouras et al. [23] used an elaborate device to apply pulsations of sinusoidal, triangular and square wave forms to the leading edge slot of a lambda wing. However, the authors found no signiﬁcant effect of the pulsation wave form. Piezoelectric actuators are often used in active ﬂow control experiments. Their utility is yet limited to a narrow frequency band around their resonance frequency. To partly overcome this limitation, Wiltse and Glezer [24] applied amplitude modulated driving signals. This technique has later been used by a number of researchers to control separated ﬂows (e.g. [25,26,27,28]). Margalit et al. [25] report a distinct effect of the modulation wave form. Traditionally, the rms value of the excitation is used in active ﬂow control applications to characterize momentum input [3]. However, supporting the above mentioned lift gain scaling with the peak momentum coefﬁcient [22], alternative approaches are discussed as well. In reference to Chang et al. [29], Wu et al. [30] used the peak excitation velocity in the deﬁnition of their momentum coefﬁcient. Recently, Kiedaisch et al. [31] proposed the ratio of peak excitation velocity to local ﬂow velocity as a dominant parameter.

2

Experimental Setup and Parameters

Gailitis and Lielausis [8] and Rice [32] proposed the arrangement of ﬂush mounted electrodes and permanent magnets shown in Fig. 1 to generate a wall parallel Lorentz force. Magnets and electrodes are of the same width a, a condition maximizing the attainable integral force density [33]. Apart from end effects, both electric as well as magnetic

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ﬁelds have only components in wall normal (y) and spanwise (z) direction. From the vector product (2) follows that the Lorentz force possesses a streamwise (x) component Fx only. The force density distribution shows non–uniformities in z–direction in the range of 0 ≤ y a [34]. Averaged over z, the mean force density decreases exponentially with increasing wall distance Fx =

π π j0 M0 e− a y . 8

(6)

M0 denotes the magnetization of the permanent magnets and j0 the applied current density, respectively. The magnetic induction at the surface of the magnetic poles can be calculated from the geometry of the magnets and their magnetization. For the experiments described here, the electrode/magnet–array sketched in Fig. 1 has been mounted to the leading edge at the suction side of an inclined ﬂat plate as sketched in Fig. 2. The body of the plate consists of polyvinyl chloride (PVC). It has a circular leading edge, a span width of 140 mm, a chord length of 130 mm, and a thickness of 10 mm. Shape and material were chosen as to allow for ease of manufacturing and durability in the electrolyte solution (0.25 M NaOH). Both, magnets as well as electrodes, have a quadratic surface with an edge length of a = 5 mm. At the surface of the magnets, a mean magnetic induction of B0 = 0.35 T was determined. A high power ampliﬁer FM 1295 from FM Elektronik Berlin has been used to feed the electrodes. It was driven by a frequency generator 33220A from Agilent. The applied Lorentz force strength is given as an effective momentum coefﬁcient T 1 aB 1 l 0 cµ = · · · j(t)2 dt (7) 2 2 ρU∞ c T 0

relating the rms momentum added by the Lorentz force to that in the free stream. In (7) j(t) denotes the time–depended current density, T its period of oscillation, and l the length of the actuator, respectively. For the plate investigated here l equals a. As common practice [3], we use percentage terms for cµ for convenience. A reduced frequency

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fe c U∞

(8)

F+ =

characterizes the time dependency of the momentum input. fe = 1/T denotes the excitation frequency. The chord length c was chosen as a characteristic length in (8) since in our case it is in a good approximation identical to the distance from the excitation location to the trailing edge. The PIV measurements reported in the following were done in a small electrolyte channel. This channel is driven by a centrifugal pump. A settling chamber with a free surface and equipped with a ﬁlter pad, two honeycombs and a set of four screens results nevertheless in a relatively small turbulence level in the test section. The latter, featuring a free surface as well, is 1 m long and has a 0.2×0.2 m cross section. For more details we refer to [35]. The channel has been operated with a mean velocity of U∞ = 8 cm s−1 resulting in a chord length Reynolds number of Re = 1.04 × 104 for the plate. It has been mounted between rectangular endplates made from PMMA extending from the bottom of the test section to the free surface in vertical and from 3 cm in front of the leading edge to 3 cm behind the trailing edge in horizontal direction. The angle of attack of the plate was kept constant at α = 16◦ throughout the measurements. The PIV setup consists of a Spectra Physics continuous wave Ar+ –Laser type 2020–5 as light source and a Photron Fastcam 1024PCI 100K for recording the images. A light sheet, formed by two cylindrical lenses, was placed at mid–span of the plate extending in the direction of the ﬂow (x) and normal to the test section bottom wall (y). The ﬂow was seeded with polyamide particles of 25 µm mean diameter. In the plane of the light sheet, x and y velocity components have been calculated from the images using PIVview–2C 2.3 from PivTec. The camera was operated at 60 Hz frame rate, while the single image exposure time was set to 2 ms by the camera shutter. For each conﬁguration a total of 6400 single images of 1024×512 pixel2 have been recorded synchronized to the excitation signal. Each image was correlated with its successor using multigrid interrogation with a ﬁnal window size of 32×32 pixel2 and 50% overlap, image deformation and sub pixel shifting.

3

Results

3.1 Inﬂuence of the Excitation Frequency As noted above, a crucial role for the characteristics of the ﬂow plays the excitation frequency. Although different observations exist, e.g. [36], it is widely recognized that excitation with a dimensionless frequency F + = O(1) has the greatest effect in re– establishing the lift of a stalled airfoil, at least for low to moderate momentum input [3]. In Fig. 3 the mean values for the streamwise velocity for sinusoidal excitation at F + = 0.5, 1, 2 and 4 are shown. As an indicator for the size of the separated region, areas with streamwise back ﬂow, depicted by the dashed lines, can be used. As clearly visible, F + = 1 results in the smallest separation region, whereas for frequencies F + > 1 and F + < 1 its area increases signiﬁcantly. This effect can possibly be explained by the different vortex structures resulting from forcing at different frequencies. On the left hand side in Fig. 4

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the phased averaged vorticity distribution for F + = 0.5 and 4 is given. Phase averaging of the synchronized PIV data was performed based on the excitation frequency. One period has been split into 20 equally wide bins. In the ﬁrst half period, the Lorentz force points downstream, while it changes sign in the second half period. For the lower excitation frequency F + = 0.5, no discernible vortex structures can be found in Fig. 3. Instead, a region of higher vorticity indicates the shear layer, which originates at the leading edge and, instead of developing straight in downstream direction, is bended by the upstream pointing Lorentz force.

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For the excitation with higher frequency several distinct regions of high vorticity appear within the shear layer above the front part of the plate. As the plot of the λ2 criteria, see [37] for deﬁnition and discussion, indicates, these regions are small and compact vortices, which loose their identity rapidly during downstream advection. Phase averaged vorticity and λ2 plots for forcing at the “optimum” frequency F + = 1 are part of Fig. 9 and Fig. 10, respectively. During most of the forcing period, two pronounced vortices are present on the suction side. It may be inferred from the averaged velocity contours in Fig. 3 and the phase averaged measurements discussed above that the momentum exchange between the separated region and the outer ﬂow is most intensive for forcing at F + = 1. For the lower excitation frequency F + = 0.5, no discernible vortices are formed, while at the higher excitation frequency of F + = 4 small and compact vortices originate in the actuator region. However, due to the high excitation frequency, these structures are small. During their downstream advection, they neither grow markedly, nor do they merge. On the contrary, they dissipate quickly and are not any more detectable above the downstream half of the plate in the λ2 plot of Fig. 3. Consequently, they are not able to transfer much momentum from the free stream to the separated region. 3.2 Inﬂuence of the Excitation Amplitude In Fig. 5 the phase averaged ﬂow ﬁelds for sinusoidal excitation at F + = 1 are shown for cµ = 1.3% and cµ = 5.2%. For the larger momentum coefﬁcient, signiﬁcant effects on the ﬂow ﬁeld can be observed. Due to the strong acceleration in the actuator region during the phase with downstream Lorentz force, the ﬂuid follows the contour of the plate beginning at the leading edge. The vortices produced are comparably strong, the ﬂow inbetween them is attached to the plate. In case of the lower momentum coefﬁcient, the structures forming at the leading edge do not penetrate the separated region down to the surface of the plate. Their effect is limited to the shear layer region. Fig. 6 displays vorticity and λ2 contours for F + = 1 at the two momentum coefﬁcients discussed above. As can be seen from the vorticity plot for cµ = 1.3%, the initial shear layer extends relatively far downstream. It is rolled up in the phase with upstream Lorentz force to an elongated vortex structure which dissipates quickly. For the higher momentum coefﬁcient of cµ = 5.2%, the initial shear layer is considerably shorter, its roll up happens faster and more close to the leading edge. The vortex structures contain more concentrated vorticity and therefore have a longer lifetime than these generated with lower momentum coefﬁcients. Presumably, the strong and compact vortex structures remain discernible vortices in the wake of the plate. Due to their strength, they will cause an intensive momentum exchange between freestream and separated region. Therefore, and due to the partial reattachement in the phase with downstream Lorentz force, the mean ﬂow reattaches completely to the plate for cµ = 5.2% (not shown). 3.3 Inﬂuence of the Excitation Wave Form Fig. 7 shows contours of the mean ﬂow component in streamwise direction u for the baseline ﬂow and excitation with F + = 1 and cµ = 2.6% and different wave forms. At the low Reynolds number of Re = 1.04 · 104 a ﬂat plate will experience leading edge

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stall already at inclination angles of about α = 5◦ [38]. Consequently, the plate is in deep stall at α = 16◦ and the large separation zone on the suction side shown in Fig. 7 results. This recirculation region extends downstream far behind the trailing edge with the maximum of backﬂow situated in the aft position of the plate and rearward of the trailing edge. While in all cases with excitation, separation is obviously still present, the structure of the separated region is markedly modiﬁed. First, its size is strongly reduced, and second, the region with maximum backﬂow is shifted towards the leading edge. The thin area of attached ﬂow directly at the plate is due to the electrolysis bubbles generated at the electrodes, which are moved upwards by buoyancy. They carry along with them the wall near ﬂuid. This happens, since the plate is mounted top down in the channel, while it has been ﬂipped over for the graphs. Different excitation wave forms under otherwise identical conditions result in separated regions of different extension and shape. Excitation with a rectangular wave form generates the smallest separation bubble. Stronger backﬂow is localized in a relatively small region downstream of the leading edge. For excitation with a sinusoidal wave form, the separation bubble grows in downstream direction. As well, the region with stronger backﬂow is lengthened. Finally, excitation with a triangular wave form shows the largest separation bubble and consequently the greatest extend of the region with stronger backﬂow. The latter extends now almost to the trailing edge. Fig. 8 displays power spectra of the v–component of the velocity versus frequency normalized by the excitation frequency of 0.6Hz. The position of the points A, B, C, and D is denoted in Fig. 7. To compute the power spectra, time series of the velocity signal have been extracted from the PIV measurements at the respective points. The initial spectrum of the baseline ﬂow directly behind the leading edge is very noisy. Subsequently, it develops a broadband peak with a maximum at 2.2 Hz in point B. Further downstream, larger vortex structures corresponding to lower frequencies dominate the ﬂow, consequently the peak frequency shifts to lower values of 1.2 Hz in point C and 1 Hz in point D. The

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spectrum in point A under excitation with a rectangular wave form shows a sharp peak at the excitation frequency and lower but not less sharp peaks at their odd harmonics. These odd harmonics correspond to the Fourier coefﬁcients of the rectangular signal. Their peak height decreases rapidly with increasing distance from the point of excitation. Only the 3rd and 5th harmonic are still discernible at point B, while in points C and D the excitation frequency alone dominates the power spectrum. Excitation with a pure sine wave results in a pronounced peak at the excitation frequency at all points, the same holds true for excitation with a triangular wave form. There, higher harmonics as in the case of the square wave could be expected, but the Fourier coefﬁcients are much smaller in case of the triangle wave and therefore probably not resolved by the measurements. Investigating the excitation by Lorentz forces includes a wide range of frequencies and momentum coefﬁcient as well as the freedom of the excitation wave form. For the present study sinusoidal, triangular and rectangular excitation was investigated. Keeping the integral of the applied Lorentz force, deﬁned √ with the rms of the current density (7), = 2 jrec for excitation by sinusoidal and to constant, the peak value changes to j sin √ jtri = 3 jrec for excitation by triangular wave forms. In Fig. 9 the phase averaged vorticity distributions and in Fig. 10 the λ2 -criteria for the different wave forms are shown.

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Fig. 9. Phased averaged values of the vorticity ωz for F + = 1, cµ = 2.6% for excitation with rectangular (left), sinusoidal (middle), and triangular wave forms (right)

For the excitation with the rectangular wave form, the ﬂow at the leading edge is accelerated constantly in streamwise direction during the ﬁrst half period. The shear layer has already rolled into a vortex like structure and due to the change from an upstream force to a downstream one, a small vortex separates, moves downstream and merges with the previously shed vortex. The shear layer starts again to grow and is lifted from the surface of the plate during the second half period. During number 13, 14, and 15 of the 20 intervals used for phase averaging, another merging of small vortices near the leading edge takes place, which is not shown here due to space limitations. The square wave excitation produces large structures of relatively high vorticity. Their size and energy remains roughly constant while they are convected downstream towards the trailing edge. Due to the jump from a constant positive to a constant negative Lorentz force the formation of compact and long living vortex structures is favoured. The vortex formations resulting from triangular and sinusoidal excitation wave forms are relatively similar to each other, but different from the vortices produced by square wave excitation. Especially the roll up of the shear layer into larger vortex structures

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occurs differently. Due to the much more gentle time dependence of the force, the shear layer is gradually wind up into a large vortex during the period with upstream pointing Lorentz force. This large vortex leaves the region near the leading edge, if the Lorentz force changes sign and moves the structure downstream. Only one large vortex is shed per period. In contrast to square wave excitation, no vortex amalgamation takes place for sine and triangle waves. However, a remarkable difference between sine and triangle wave excitation remains: the vortex structures forming under triangular excitation contain more vorticity and have a longer lifetime than that produced by sine wave forcing.

4 Conclusions Time periodic Lorentz forces have proven to be a viable tool to control separated ﬂows. The Lorentz force actuator provides a great ﬂexibility to choose excitation frequencies,

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amplitudes as well as wave forms. As expected, control authority of the excitation frequency and amplitude is most striking. However, the excitation wave form as well has a large inﬂuence on the mean ﬂow and on the phase averaged structures. Even if the low energetical efﬁciency of the Lorentz force actuator will hamper its use in industrial applications, it is a valuable tool for basic research.

Acknowledgements Financial support from Deutsche Forschungsgemeinschaft (DFG) in frame of the Collaborative Research Centre (SFB) 609 is gratefully acknowledged.

References [1] Gad-el Hak, M.: Flow control: passive, active, and reactive ﬂow management. Cambridge University Press (2000) [2] Wygnanski, I.: Boundary layer and ﬂow control by periodic addition of momentum. AIAA–paper 97–2117 (1997) [3] Greenblatt, D., Wygnanski, I.: The control of ﬂow separation by periodic excitation. Prog. Aero. Sci. 36 (2000) 487–545 [4] Seifert, A., Greenblatt, D., Wygnanski, I.: Active separation control: an overview of Reynolds and Mach number effects. Aerosp. Sci. Techn. 8 (2004) 569–582 [5] Sutton, G., Sherman, A.: Engineering Magnetohydrodynamics. McGraw Hill, New York (1965) ´ Cachon, P.: Actions e´ lectromagn´etiques sur les liquides en mouvement, [6] Crausse, E., notamment dans la couche limite d’ obstacles immerg´es. Comptes rendus hebdomadaires des s´eances de l’ Acad´emie des Sciences 238 (1954) 2488–2490 [7] Lielausis, O.: Effect of electromagnetic forces on the ﬂow of liquid metals and electrolytes. PhD thesis, Academy of Sciences of the Latvian SSR, Institute of Physics, Riga (1961) in Russian. [8] Gailitis, A., Lielausis, O.: On a possibility to reduce the hydrodynamic resistance of a plate in an electrolyte. Appl. Magnetohydrodynamics, Rep. Phys. Inst. 12 (1961) 143–146 in Russian. [9] Tsinober, A.B., Shtern, A.G.: On the possibility to increase the stability of the ﬂow in the boundary layer by means of crossed electric and magnetic ﬁelds. Magnitnaya Gidrodinamica 3 (1967) 152–154 (in Russian). [10] Meyer, R.: Magnetohydrodynamic method and apparatus. US Patent 3,360,220 (1967) [11] Shtern, A.: Feasibility of modifying the boundary layer by crossed electric and magnetic ﬁelds. Magnitnaya Gidrodinamika 6 (1970) 124–128 [12] Nosenchuck, D., Brown, G., Culver, H., Eng, T., Huang, I.: Spatial and temporal characteristics of boundary layers controlled with the lorentz force. In: 12th Australian Fluid Mechanics Conference, Sydney (1995) [13] Henoch, C., Stace, J.: Experimental investigation of a salt water turbulent boundary layer modiﬁed by an applied streamwise magnetohydrodynamic body force. Phys. Fluids 7 (1995) 1371–1383 [14] Crawford, C.H., Karniadakis, G.E.: Reynolds stress analysis of EMHD–controlled wall turbulence. Part I. Streamwise forcing. Phys. Fluids 9 (1997) 788–806 [15] Berger, T.W., Kim, J., Lee, C., Lim, J.: Turbulent boundary layer control utilizing the lorentz force. Phys. Fluids 12 (2000) 631–649

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[16] Du, Y., Symeonidis, V., Karniadakis, G.: Drag reduction in wall–bounded turbulence via a transverse travelling wave. J. Fluid Mech. 457 (2002) 1–34 [17] Weier, T., Gerbeth, G., Mutschke, G., Platacis, E., Lielausis, O.: Experiments on cylinder wake stabilization in an electrolyte solution by means of electromagnetic forces localized on the cylinder surface. Experimental Thermal and Fluid Science 16 (1998) 84–91 [18] Kim, S., Lee, C.: Investigation of the ﬂow around a circular cylinder under the inﬂuence of an electromagnetic force. Exp. Fluids 28 (2000) 252–260 [19] Posdziech, O., Grundmann, R.: Electromagnetic control of seawater ﬂow around circular cylinders. European Journal of Mechanics–B/Fluids 20 (2001) 255–274 [20] Chen, Z., Aubry, N.: Active control of cylinder wake. Communications in Nonlinear Science and Numerical Simulation 10 (2005) 205–216 [21] Weier, T., Gerbeth, G., Mutschke, G., Lielausis, O., Lammers, G.: Control of ﬂow separation using electromagnetic forces. Flow, Turbulence and Combustion 71 (2003) 5–17 [22] Weier, T., Gerbeth, G.: Control of separated ﬂows by time periodic Lorentz forces. Eur. J. Mech. B/Fluids 23 (2004) 835–849 [23] Bouras, C., Nagib, H., Durst, F., Heim, U.: Lift and drag control on a lambda wing using leading-edge slot pulsation of various wave forms. Bulletin of the American Physical Society 45 (2000) 30 [24] Wiltse, J., Glezer, A.: Manipulation of free shear ﬂows using piezoelectric actuators. J. Fluid Mech. 249 (1993) 261–285 [25] Margalit, S., Greenblatt, D., Seifert, A., Wygnanski, I.: Active ﬂow control of a delta wing at high incidence using segmented piezoelectric actuators. In: 1st Flow Control Conference, St. Louis, MO (2002) AIAA–paper 2002–3270. [26] Pack, L.G., Schefﬂer, N.W., Yao, C.S.: Active control of separation from the slat shoulder of a supercritical airfoil. In: 1st Flow Control Conference, St. Louis, MO (2002) AIAA–paper 2002–3156. [27] Washburn, A., Amitay, M.: Active ﬂow control on the stingray UAV: Physical mechanism. In: 42nd Aerospace Sciences Meeting & Exhibit, Reno, NV (2004) AIAA–paper 2004–0745. [28] Pack Melton, L.G., Yao, C.S., Seifert, A.: Application of excitation from multiple locations on a simpliﬁed high–lift system. In: 2nd Flow Control Conference, Portland, OR (2004) AIAA–paper 2004–2324. [29] Chang, R., Hsiao, F.B., Shyu, R.N.: Forcing level effects of internal acoustic excitation on the improvement of airfoil performance. J. Aircraft 29 (1992) 823–829 [30] Wu, J.Z., Lu, X.Y., Denny, A., Fan, M., Wu, J.M.: Post–stall ﬂow control on an airfoil by local unsteady forcing. J. Fluid Mech. 371 (1998) 21–58 [31] Kiedaisch, J., Demanett, B., Nagib, H.: Active ﬂow control of large separation: A new look at scaling parameters. In: 58th Annual Meeting of the APS Division of Fluid Dynamics. (2005) [32] Rice, W.: Propulsion system. US Patent 2,997,013 (1961) [33] Grienberg, E.: On determination of properties of some potential ﬁelds. Applied Magnetohydrodynamics. Reports of the Physics Institute 12 (1961) 147–154 (in Russian). [34] Weier, T., Fey, U., Gerbeth, G., Mutschke, G., Lielausis, O., Platacis, E.: Boundary layer control by means of wall parallel Lorentz forces. Magnetohydrodynamics 37 (2001) 177–186 [35] Weier, T., Gerbeth, G., Fey, U., Mutschke, G., Posdziech, O., Platacis, E., Lielausis, O.: Some results on electromagnetic control of ﬂow around bodies. In: Int. Symp. on Seawater Drag Reduction. (1998) 229–235

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[36] Amitay, M., Glezer, A.: Role of actuation frequency in controlled ﬂow reattachement over a stalled airfoil. AIAA J. 40 (2002) 209–216 [37] Jeong, J., Hussain, F.: On the identiﬁcation of a vortex. J. Fluid Mech. 285 (1995) 69–94 [38] Schmitz, F.: Aerodynamik des Flugmodells. Tragﬂ¨ugelmessungen I. C.J.E. Volckmann Nachf. E. Wette, Berlin–Charlottenburg (1942)

Pulsed Plasma Actuators for Active Flow Control at MAV Reynolds Numbers B. Göksel1, D. Greenblatt2, I. Rechenberg1, Y. Kastantin2, C.N. Nayeri2, and C.O. Paschereit2 1

Technical University Berlin, Institute of Process Engineering Ackerstr. 71-76, Secr. ACK1, D-13355 Berlin, Germany [email protected] 2 Technical University Berlin, Hermann-Föttinger-Institute of Fluid Mechanics MüllerBreslau-Strasse 8, D-10623 Berlin, Germany

Summary An experimental investigation of separation control using steady and pulsed plasma actuators was carried out on an Eppler E338 airfoil at typical micro air vehicle Reynolds numbers (20,000Re140,000). Pulsing was achieved by modulating the high frequency plasma excitation voltage. The actuators were calibrated directly using a laser doppler anemometer, with and without free-stream velocity, and this allowed the quantification of both steady and unsteady momentum introduced into the flow. At conventional low Reynolds numbers (Re>100,000) asymmetric single phase plasma actuators can have a detrimental effect on airfoil performance due to the introduction of low momentum fluid into the boundary layer. The effect of modulation, particularly at frequencies corresponding to F+≈1, became more effective with decreasing Reynolds number resulting in significant improvements in CL,max. This was attributed to the increasing momentum coefficient, which increased as a consequence of the decreasing free-stream velocities. Particularly low duty cycles of 3% were sufficient for effective separation control, corresponding to power inputs on the order of 5 milliwatts per centimeter.

1 Introduction Achieving sustained flight of micro air vehicles (MAVs) bring significant challenges due to their small dimensions and low flight speeds. This combination results in very low flight Reynolds numbers (Re<200,000), where conventional low-Reynoldsnumber airfoils perform poorly, or even generate no useful lift. Some of the best performing airfoils in this Re range are cambered flat plates and airfoils with a thickness to chord ratio (t/c) of approximately 5% [1], [2]. MAV are usually designed with surveillance, sensing or detection in mind. Hence, a typical MAV mission should include a “high speed dash” (V~65km/h, 18m/s) to or from a desired location with significant head or tail winds, and low-speed loiter (V~30km/h, 8.3m/s) while maneuvering, descending and climbing [3]. Mueller defines two MAV sizes, which we can call “large” (b=15cm, M=90g) and “small” (b=8cm, M=30g) [1]. R. King (Ed.): Active Flow Control, NNFM 95, pp. 42–55, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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The generation of useful lift at Re<50,000 is particularly challenging because passive tripping of the boundary layer is virtually impossible [5]. Consequently, unconventional approaches have been pursued, such as ornithopters that are inspired by bird and insect flight. Active control methods are also pursued. For example, Greenblatt & Wygnanski investigated perturbing an airfoil leading-edge boundary layer via twodimensional periodic excitation at Re=50,000 and 30,000 [8]. Near-sinusoidal pertubations at F+≈1 resulted in the restoration of conventional low-Reynolds-number lift and aerodynamic efficiency, while excitation-induced lift oscillations were small and hysteresis associated with stall was eliminated. However, with decreasing Re larger periodic perturbations (expresses as ¢Cµ ² ) were required to generate useful lift. A similarity between the timescales associated with excitation and those characterizing dynamic stall in small flying creatures provided some insight into these observations. They observed that typical MAV dimensions are suited to actuation by means of micro-electromechanical systems (MEMS)-based devices. It was also noted that the effectiveness and efficiency of actuators required to supply the prescribed excitation will ultimately determine the success and limitations of the method.

2 Motivation for the Present Study To illustrate the challenges facing development of these vehicles, let us define the wing aspect ratio: AR = b / c where c is the standard mean chord and assume that for typical MAVs: 1 ≤ AR ≤ 2 . Furthermore, we define a characteristic Reynolds number Re = Vc /ν and lift coefficient:

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Now, using definitions of “small” and “large” MAVs defined above, we generate Vstall versus CL curves corresponding to AR=1 and 2, Figure 1. Also shown are the target loiter speed and corresponding Reynolds numbers. It is evident that the smaller vehicle requires a larger CL , max with simultaneously lower Reynolds number at the loiter target. Furthermore, wings with AR>1 are required to produce significantly larger CL , max at lower Reynolds number. Conventional low Reynolds number UAVs, where typically Re>200,000, achieve loiter targets by deploying flaps. This is not considered practical for MAVs loitering at Re<50,000, where passive tripping of the boundary layer in order to generate useful lift is not possible. Figure 2 shows airfoil section CL , max for conventional low Reynolds number airfoils and reflects the well-known performance deterioration with reducing Re. Thus the problem of attaining low loiter speeds is compounded because performance

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degradation due to lower Reynolds number conflicts with higher CL , max requirements. It is emphasized that loiter is a mission critical flight regime, where the MAV performs its primary function such as surveillance or sensing. Plasma-based actuators have recently demonstrated application to separation control [9], [10], [11], [13], [14], [15]. The first separation flow control on airfoils at typical MAV Reynolds numbers (13,000
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actuation using high voltage (10–20 kV) charged corona discharge wires in 1999 [9], [10]. Göksel demonstrated significant improvement to an Eppler E338 airfoil performance [e.g. CL , max , (l / d ) max ], particularly for 10,000
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The potential application of plasma actuators to the MAV is clearly evident by comparing Figures 1 and 2. Here it is seen that the requirement for high CL in the loiter regime can be met by plasma actuation. However, the power requirement was relatively high (~8.5W) corresponding to CW , max ≤ 137 for the range of Reynolds numbers considered. Moreover, calibration of the corona wires by measuring downstream mean velocity profiles, indicated that Cµ,max>10%, thereby indicating that at low Reynolds numbers circulation control is possible [17]. Several comparisons of separation control by periodic excitation versus steady blowing have indicated that similar performance benefits (e.g. ∆CL ) can be achieved where ¢Cµ ² is up to two orders of magnitude smaller than Cµ . Using plasma actuators in a pulsed mode, Corke et al. have shown that steady forcing produced negligible changes to CL , max while unsteady forcing at F+ =1 resulted in ∆C L, max ≈ 0.2 [11]. Performance improvements using pulsed actuation were demonstrated on a delta wing using piezo-electric actuators by Margalit et al. [16]. The present investigation was undertaken to examine the possibility of controlling separation using plasma actuators in a pulsed mode at typical MAV Reynolds numbers. A pulsed plasma jet, generated using the single phase actuation technique near the leading edge of the airfoil (x/c=1%) was utilized for this purpose [15]. Data were compared with both 2D and 3D boundary layer tripping. The momentum added to a flow by means of pulsed actuation introduces both time-mean and unsteady components of momentum. To quantify this, a separated experiment was conducted to calibrate the actuators and hence estimate both steady and unsteady components of momentum ( Cµ , ¢Cµ ² ).

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3 The Experiments 3.1 Actuator Calibration Setup

Calibration of the actuator was conducted in a closed-loop wind tunnel with a 2m long test section of 400 x 280mm in a quiescent environment ( U ∞ = 0 ) and at freestream velocities corresponding to the Reynolds numbers tested here. All boundary layers were laminar at the test location. A splitter plate with an elliptical leading edge was installed in the tunnel. The plasma actuator was placed 0.57m downstream of the leading-edge and consisted of two thin metal electrodes separated by a thin dielectric layer, Figure 3 [11], [12], [13]. Sufficiently high voltages (at low radio frequencies in the kHz-range) supplied to the actuator causes the air to weakly ionize at the edges of the upper electrodes. These are regions of high electric field potential. In this asymmetric configuration, the plasma is only generated at one edge, Figure 3. The plasma moves to regions of increasing electric field gradients and induces a 2-D wall jet in the flow direction along the surface, thereby adding momentum to the boundary layer [6].

Fig. 3. Schematic of the plasma actuator used for the present experiments

Performing LDV profile measurements, at 3mm, 12mm and 25 mm downstream of the actuator, the steady momentum in the jet was quantified by ∞

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where u is the time-mean velocity profile without plasma actuation. For the purpose of pulsed (or unsteady) actuation, the wave modulation method was employed where the kHz carrier wave is modulated by a square-wave that correspond to low frequencies appropriate for separation control [11], [14], [15], [16]. This introduces mean ( u J ) and unsteady ( u~J and ~ vJ ) velocity components and thus the jet momentum is made up of time-mean and oscillatory component quantified by ∞ ∞ J tot = J + ¢ J ² = ³ ρ (u J2 − u 2 )dy + ³ ρ (u~J2 + v~J2 ) dy , 0

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as Cµ , tot = Cµ + ¢Cµ ² and also expressed as ( C µ , ¢C µ ² ). For all data acquired here, the actuator was excited with a signal of intermittent bursts of 4.0 kHz that were modulated in the range of 2.5 to 100 Hz. The duty cycle was varied from 1% to 100% at constant voltage. 3.2 Airfoil Setup and Testing

Experiments were performed on an Eppler E338 airfoil (c=17.8cm, b=50cm) mounted between circular endplates downstream of the exit of a 600mm and a 1200mm diameter low speed open jet wind tunnel. Lift and drag were measured using a two component balance. This airfoil was previously used for flow control experiments with high voltage (10–20kV) charged corona discharge wires, and a full description of the setup can be found in [9], [13], [15]. The plasma actuator consisted of two thin metal electrodes separated by a dielectric layer which formed part of the airfoil surface, Figure 3 [11], [12], [13]. Airfoil performance was also assessed by tripping the boundary layer using a three-dimensional (3D) turbulator of height 200 microns and a twodimensional (2D) step of height 100 microns, at x/c=1%.

4 Discussion of Results 4.1 Actuator Calibration

LDV for data for u and u~J at 3mm downstream of the actuator are shown for U ∞ = 0.83m/s and U ∞ = 5.79m/s in Figures 4a, b and 5a, b respectively. For all data acquired v~J2 << u~J2 and could consequently be ignored without materially changing the results of equation (7). With no actuation (plasma off), a laminar Blasius boundary layer forms on the plate. At the lower velocity, jet actuation at all duty cycles considered here produces a significant steady and unsteady near wall momentum. In general, 16

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Fig. 5. Normalized mean velocity and turbulence intensity at an intermediate free-stream velocity 4

4

100% duty cycle, 24W/m

100% duty cycle, 24W/m

50% duty cycle, 13W/m

3

50% duty cycle, 13W/m

3

10% duty cycle, 2.5W/m

y (mm)

y (mm)

10% duty cycle, 2.5W/m

5% duty cycle, 1.3W/m

1% duty cycle, 0.6W/m

5% duty cycle, 1.3W/m

1% duty cycle, 0.6W/m

2

2

1

1

0

0 0

1

2

u (m/s)

3

4

0

0.5

1

u' (m/s)

1.5

Fig. 6. Actuator calibration at 3mm downstream for different duty cycles at U ∞ = 0

larger duty cycles produce larger near-wall mean-flow jets. On the other hand, driving the actuator in burst mode produces larger oscillatory components of momentum, Table 1. Driving the actuator at 100% duty cycle produces a momentum deficit from approximately 2-3mm from the wall. This is believed to be a consequence of the vortical flow associated with the wall jet, Figure 4a. Also, a mild momentum surplus is generated for all actuator duty cycles in the outer part of the boundary layer. Note that no distinction has been drawn here between purely periodic perturbations and turbulent fluctuations, consequently u~J is representative of the overall unsteadiness u’. As the free-stream velocity increases the relative momentum added to the flow decreases significantly. At U ∞ = 5.79m/s corresponding to Re=70,000, both steady and unsteady components of momentum are negligible, Figures 5a and 5b. Based on these

Pulsed Plasma Actuators for Active Flow Control at MAV Reynolds Numbers

49

Table 1. Steady and unsteady actuator calibrations at various free stream velocities

Duty Cycle (%) 100 50 10 100 50 10 100 50 10

Cµ (%)

¢Cµ ² (%)

8.31 5.41 1.76 1.05 0.36 0.018 0.74 0.38 0.02

0.25 0.93 0.54 0.025 0.054 0.018 0.009 0.014 0.008

U∞ (m/s) 0.83 0.83 0.83 2.50 2.50 2.50 4.15 4.15 4.15

data it is not expected that the plasma actuators will have a significant separation control effect for Re>70,000. Figure 6 shows actuator calibration data for U ∞ = 0 . In this case the duty cycle was gradually increased from 1% to 100%. It was noted that a duty cycle threshold between 2% and 4% is reached where there is a significant increase in near-wall unsteady momentum. Peak unsteady momentum is reached at a duty cycle of approximately 10%. Further increases in duty cycle result in decreases to both steady and unsteady near wall momentum. At 100% duty cycle a near-steady wall jet is formed with relatively large mean near wall momentum. 4.2 Airfoil Performance Data

The airfoil data is presented below in terms of deceasing Reynolds number, starting at typical low Re~140,000 (conventional low Re, Figure 7) and reducing to ~20,000 (approximate lower MAV limit). Maximum errors associated with CL and CD were ±0.02, and this unfortunately precluded the recording of meaningful CD measurements at lower Reynolds numbers. We note that plasma control at 100% duty cycle has a detrimental effect and reduces CL , max . This is because a relatively slow speed steady jet is being generated by the plasma actuator that is much less than the free-stream velocity with C µ ≈0.1% (see section 2). Hence, the low momentum fluid introduced near the wall, in fact, promotes separation. This may appear counterintuitive, but a similar effect was noted when using conventional steady slot blowing with U J / U ∞ < 1 [4]. All other duty cycles considered (≤50%, corresponding to F+=1) have a net positive post-stall effect with relatively low ¢C µ ² < 0.1% . Changes to post-stall lift and small changes to CL , max at conventional low Reynolds numbers have been observed by others (e.g. [11], [15]). Interestingly, data is marginally superior when the duty cycle is reduced from 50% to 10%. This might have been expected when considering the data in Figure 6b, which shows that the 10% duty cycle actuation produced greater

50

B. Göksel et al. 1,5

Re = 140,000

1

CL

0,5

Baseline

100% duty cycle

50% duty cycle

10% duty cycle 0 0

5

10

15

20

25

Į (°)

Fig. 7. Example of the effect of plasma actuation at F+=1 on airfoil performance at conventional low Reynolds numbers

unsteady near-wall momentum. Moreover, this result is even more significant when we account for the fact that duty cycle percentage correlates linearly with power consumption. With Reynolds number reduced to 80,000, the near wall jet velocity is comparable to that in the near wall boundary layer and the detrimental effect on CL , max disappears (not shown). At high post-stall angles, when the airfoil is fully stalled, the jet has a positive effect on CL (not shown). At Re=65,000 the baseline clean airfoil performed poorly, but its performance improved with the addition of either 2D or 3D tripping, Figure 8a. 2D tripping was slightly superior, but the airfoil still suffered from significant hysteresis., In contrast, pulsed control at F+=1 and 3% duty cycle virtually eliminating hysteresis and produced a slight increase in CL , max . At Re=50,000, also shown here for F+=1.0, the effect of plasma actuation can be far more clearly observed, Figure 9a,b. As mentioned above [5], and shown in Figure 9a, it is virtually impossible to effectively promote transition passively at these Reynolds numbers, although the 2D trip was more effective than the 3D trip. This is reflected in the poor performance of the airfoil with CL , max < 0.8 . For the purposes of presenting an unbiased evaluation, all plasma actuation data presented in the remainder of this paper were compared with that of the 2D trip. In this instance, the 100% duty cycle actuation has a net positive effect on CL , max and this is because it generates a steady wall jet corresponding to C µ =0.74%, Table 1. Successive reductions in duty cycle clearly result in improvements in performance, both with respect to the CL - α linearity as well as CL , max . Note, in addition, that

Pulsed Plasma Actuators for Active Flow Control at MAV Reynolds Numbers

51

CL , max is larger than that at the higher Reynolds numbers. It is assumed that this is due to the larger C µ values which increase as a consequence of the reducing freestream velocity. This runs counter to the typical baseline trends and has clear potential for reducing loiter speed discussed in the introduction. 1.5

1,5

Re = 65,000

Re = 65,000

1

1

CL

CL

0.5

0,5

Baseline: clean airfoil

Baseline

Baseline: 2D trip

3% duty cycle

Baseline: 3D trip 0 -10

-5

0

0

5

10

Į (°)

15

20

25

-10

-5

0

5

10

15

20

25

Į (°)

Fig. 8. Example of the effect of plasma actuation at F+=1 on airfoil performance at Reynolds numbers Re=65,000

Traditional steady separation is usually characterized by a proportionality between a performance indicator (e.g. Cl,max) and Cµ [4], but this is not always the case when control is periodic [7]. For the data present in Figure 5, the conventional arguments of additional unsteady near wall momentum can be applied for duty cycles between 100% and 10% as discussed above. However, performance continues to improve as the duty cycle is reduced from 10% to 3%, Figure 9, despite the decreasing near wall momentum, Figure 6. This is a perplexing phenomenon, but has practical ramifications when it is considered that power supplied to the actuators is proportional to duty cycle. Further reductions in Reynolds number to 35,000 and 20,500 showed ever greater effects on control. For example, in the latter case (Re=20,500) which is very near the low end of the MAV Reynolds number range, significant effect were observed and hence additional data were acquired in an attempt to optimize control. Employing a 5% duty cycle and placing the airfoil at a post stall angle of attack ( α = 18°) a frequency scan was performed for the range 0.25 ≤ F + ≤ 10.4 , Figure 10. The optimum is seen to be at F + ≈ 1 and this is consistent with conventional low Reynolds number data [7]. Corke et al. observed that, using plasma actuators, the minimum voltage required to attach a post-stall separated flow was at F + ≈ 1 [11].

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1.5

1,5

Re = 50,000 Re = 50,000

Baseline: clean airfoil Baseline: 2D trip 1

Baseline: 3D trip

1

CL

CL

0.5

0,5 Baseline 100% duty cycle 50% duty cycle 10% duty cycle 3% duty cycle

0 -10

-5

0

0

5

10

Į (°)

15

20

25

-10

-5

0

5

10

15

20

25

Į (°)

Fig. 9. Example of the effect of plasma actuation at F+=1 on airfoil performance at Reynolds numbers Re=50,000 0,9

0,8

0,7

∆CL 0,6

0,5 0

2

4

6

8

10

12

F+

Fig. 10. Effect of reduced frequency on post-stall (α=18° airfoil lift at a low MAV Reynolds number; Re=20,500). Cµ=0.05% and duty cycle = 3%.

Similar effects have also been observed on delta wings using zero mass-flux jets [16]. Further attempts at optimisation considered variation of the duty cycle. It was observed that the optimum lies somewhere between 3% and 8%. Figure 11. Interestingly, this is the range where the maximum oscillatory momentum is added to the flow. Finally, the effect of input voltage on the CL versus α curves was investigated. It was determined that for V>8kVpp (corresponding to 0.5W/m), the effect on the airfoil

Pulsed Plasma Actuators for Active Flow Control at MAV Reynolds Numbers

53

0,9

0,8

0,7

∆CL 0,6

0,5 0

10

20

30

40

50

60

Duty Cyle (%)

Fig. 11. Effect of duty cycle on post-stall (α=18° airfoil lift at a low MAV Reynolds number; Re=20,500)

performance is clearly significant and CL , max is larger than at the higher Reynolds numbers, Figure 12. Note that here the optimum F+ and duty cycles have been used. Data was generated for increasing α (filled symbols) and decreasing α (open symbols). Note that below 10kVpp the CL versus α curve is non linear, but this non linear feature does not show any significant hysteresis trend repeats for decreasing α , Figures 12 and 13. 1,5 Re = 20,500

1

CL 0,5

0 Baseline F+=1, Cw=1.0

-0,5 -10

0

10

Į (°)

20

30

Fig. 12. Effect of plasma actuation on airfoil performance at a low MAV Reynolds number illustrating non-linear behavior at low CW. Cµ=0.04% and duty cycle = 3%.

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1,5 Re = 20,500

1

CL 0,5

0 Baseline F+=1, Cw=1.7

-0,5 -10

0

10

Į (°)

20

30

Fig. 13. Effect of plasma actuation on airfoil performance at a low MAV Reynolds number illustrating the minimum CW required for linear behavior. Cµ=0.05% and duty cycle = 3%.

5 Conclusion The present investigation considered separation control using steady and pulsed plasma actuation on an airfoil at typical MAV Reynolds numbers. Pulsing was achieved by modulating the high frequency plasma excitation voltage. The actuators were calibrated directly and variations of the duty cycle showed large differences between the steady and unsteady components of momentum addition. Calibration of the actuators provided a basic explanation of the observed airfoil performance. For example, steady, relatively low momentum steady actuation was detrimental at Re>100,000, while beneficial at Re=50,000 due to the four-fold increase relative momentum addition. Modulating the actuators at frequencies corresponding to F+≈1, resulted in improvements to CL,max, which increased with reductions in Re. At the low end of the MAV Reynolds number range (Re=20,500) modulation increased CL,max by more than a factor of 2. In addition, hysteresis associated with the baseline airfoil was eliminated. Of particular interest from an applications perspective was that performance, measured here by CL,max, was shown to increase with decreasing duty cycle, and hence power input. In fact, duty cycles of around 3% were sufficient for effective separation control, corresponding to power inputs on the order of 500 milliwatts per unit length.

References [1] T.J. Mueller. “Aerodynamic Measurements at Low Reynolds Numbers for Fixed Wing Micro-Air Vehicles”. Presented at the RTO AVT/VKI Special Course on Development and Operation of UAVs for Military and Civil Applications, VKI, Belgium, September 13-17, 1999.

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[2] T.J. Mueller (ed.). "Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications". In Progress in Aeronautics and Astronautics, Volume 195, 2001, pp. 115-142. [3] S.J. Morris. “Design and Flight Test Results for Micro-Signed Fixed-Wing and VTOL Aircraft”. 1st International Conference on Emerging Technologies for Micro Air Vehicles, Georgia Institute of Technology, Atlanta Georgia, February 1997, pp. 117-131. [4] J.S. Attinello. “Design and Engineering Features of Flap Blowing Installations”. In G.V. Lachmann. “Boundary Layer and Flow Control. Its Principles and Application”, Volume 1, 1961, Pergamon Press, New York, pp. 463-515. [5] B.H. Carmichael. “Low Reynolds Number Airfoil Survey”. NASA Contractor Report 165803, Volume I, November 1981. [6] J.R. Roth, D. Sherman and S. Wilkinson. "Boundary Layer Flow Control with One Atmosphere Uniform Glow Discharge Surface Plasma". AIAA 1998-0328, 1998. [7] D. Greenblatt and I. Wygnanski. “Control of separation by periodic excitation” In Progress in Aerospace Sciences, Volume 37, Issue 7, 2000, pp. 487-545. [8] D. Greenblatt and I. Wygnanski. “Use of Periodic Excitation to Enhance Airfoil Performance at Low Reynolds Numbers”. In AIAA Journal of Aircraft, Volume 38, Issue 1, 2001, pp. 190-192. [9] B. Göksel. “Improvement of Aerodynamic Efficiency and Safety of Micro Aerial Vehicles by Separation Flow Control in Weakly Ionized Air”. (in German) In Proceedings of the German Aerospace Congress, Leipzig, DGLR Paper JT-2000-203, 2000, pp. 13171331. [10] B. Göksel and I. Rechenberg. “Active Separation Flow Control Experiments in Weakly Ionized Air”. In Andersson H. I. and Krogstad P.-Å. (eds.) Advances in Turbulence X, Proceedings of the 10th Euromech European Turbulence Conference, CIMNE, Barcelona, 2004. [11] T.C. Corke, C. He and M.P. Patel. “Plasma flaps and slats: An application of weakly ionized plasma actuators”. AAIA Paper 2004-2127, 2nd AIAA Flow Control Conference, Portland, Oregon, 2004. [12] C.L. Enloe, T.E. McLaughlin, R.D. Van Dyken, K.D. Kachner, E. J. Jumper and T.C. Corke. “Mechanism and Responses of a Single Dielectric Barrier Plasma Actuator: Plasma Morphology”. In AIAA Journal, Volume 42, Issue 3, 2004, pp. 589-594. [13] B. Göksel and I. Rechenberg. “Active Flow Control by Surface Smooth Plasma Actuators”. In Rath H. J., Holze C., Heinemann H.-J., Henke, R. and Hönlinger H. (eds.) “New Results in Numerical and Experimental Fluid Mechanics V”, Contributions to the 14th STAB/DGLR Symposium Bremen, Germany 2004, NNFM Vol. 92, Springer, 2006 (in press). [14] B. Göksel and I. Rechenberg “Experiments to Plasma Assisted Flow Control on Flying Wing Models”. Submitted for Proceedings of CEAS/KATnet Conference on Key Aerodynamic Technologies To Meet the Challenges of the European 2020 Vision, Bremen, 2005. [15] B. Göksel, D. Greenblatt, I. Rechenberg and C. O. Paschereit “Plasma Actuators for Active Flow Control”. DGLR Paper 2005-210, to be published in the Proceedings of the German Aerospace Congress 2006, Friedrichshafen, Germany, 2005. [16] S. Margalit, D. Greenblatt, A. Seifert and I. Wygnanski “Delta wing stall and roll control using segmented piezoelectric fluidic actuators”. In AIAA Journal of Aircraft, Volume 42, Issue 3, 2005, pp. 698-709. [17] Ph. Poisson-Quinton and L. Lepage “Survey of French research on the control of boundary layer and circulation”, in Lachmann, G. V., “Boundary layer and Flow Control. Its Principles and Application”, Volume 1, Pergamon Press, New York, 1961, pp. 21-73.

Experimental and Numerical Investigations of Boundary-Layer Influence Using Plasma-Actuators S. Grundmann, S. Klumpp, and C. Tropea Institute of Fluid Mechanics and Aerodynamics, Technische Universität Darmstadt Flughafenstrasse 19, D-64347 Griesheim

Summary This is a fundamental study about the influence of plasma-actuators on boundarylayer flows, including both experimental and numerical investigations. The first set of experiments is conducted in quiescent air and these results are used to calibrate a numerical model which simulates the plasma-actuator in an existing RANS (Reynolds Averaged Navier-Stokes) code. The second set of experiments involves a flat-plate boundary layer at various free-stream velocities, where the actuator adds momentum to the boundary layer. The previously calibrated numerical model is used to simulate the influence of the actuator on the boundary layer. The agreement between simulation and experiment is very good and the simulations with the new model be considered a reliable predictive tool.

Nomenclature f E

[N/m³] Body force [V/m] Electric field strength

eC V0

[C] [V]

E0

[V/m]

k1, k2 [-]

ȡC

[1/m³]

Electric field strength at design point Charge number density

a0

[m]

ȡC0

[1/m³]

b0

[m]

İ0

[F/m]

Charge number density at design point Permittivity of free space

d

[M]

Elementary charge Applied voltage at design point Electric field gradients Normal plasma breakdown length Tangential plasma breakdown length Gap between electrodes

1 The Plasma Actuator Plasma actuators consist of two electrodes separated by insulation covering the lower electrode completely, while the upper one remains exposed to the flow. They are driven by a high frequency AC-voltage of several thousand volts and several thousand R. King (Ed.): Active Flow Control, NNFM 95, pp. 56–68, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

Experimental and Numerical Investigations of Boundary-Layer Influence

57

Hertz. The actuator generates a body force tangential to the surface which provides a variety of possibilities to manipulate or to create flows. In quiescent fluid, this force creates a wall jet with a velocity of several meters per second. With a mean flow, the actuator can be used to impart momentum into boundary layers and to alter their velocity profile, their turbulence distribution or to promote transition from a laminar to turbulent. The amplitude of the force can be altered by varying the driving voltage. The physical working principle has been described by Font [3]. During the first half-cycle, the upper electrode is negative and the high voltage causes it to emit electrons which ionize the neutral air molecules on their way to the lower electrode of opposite charge. A weakly ionized plasma is generated. The electrons gather on the insulation film of the lower electrode, while the predominantly positive ions are accelerated towards the negative upper electrode. Colliding with neutral air molecules, they transfer their momentum to the air and cause a body force affecting the air. This body force is directed opposite to the observed wall jet. These electrons gather on the insulation film, like on a barrier and build up a secondary potential, which limits the intensity of the electric field. This feature gives the plasma its name: Dielectric Barrier Discharge Plasma. After the change of the polarity of the driving voltage, the number of electrons is one order higher than that present at the beginning of the first half-cycle. Therefore the number of ions created by the moving electrons is much higher during the second half-cycle than that during the first. The ions now move towards the lower electrode: due to their larger number they cause a body force about ten times larger than that in the first half-cycle. This time the force is in the same direction as the observed flow. The electrons can reach the upper exposed electrode and are neutralized. That is the reason why the next cycle starts again with a small number of electrons. After one entire cycle a net force remains, causing a wall jet. The actuators can be implemented as a single actuator or as an array of actuators. The actuators in an array can be driven with voltages of the same phase or with a phase shift between them, which causes an electrostatic wave to move along the surface. This technique results in higher wall-jet velocities than with actuators in phase [8]. Several experiments show the successful application of plasma actuators as transition strips [10], as active separation control due to transition of the flow above the suction side of airfoils [2, 4], and as wake control of bluff bodies like cylinders [9]. The investigations conducted at the Institute of Fluid Mechanics and Aerodynamics of the Technische Universität Darmstadt concentrate on the fundamentals of the ability of the plasma actuator to influence a flow. The goal is to determine the possibilities of manipulating boundary layers, for example promoting transition, sustaining laminar conditions or possibly even re-laminarization of a turbulent boundary layer through strong streamwise acceleration. To achieve this, an open circuit wind tunnel has been equipped with a test section that allows measurements on a flat plate with and without a pressure gradient. Single actuators and actuator-arrays can be placed at different positions on the plate.

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2 The Numerical Model and Its Calibration To reduce the number of experiments, to design the experimental setup and to provide additional insight into the functioning of the actuators, numerical simulations accompany the experiments. To reproduce the effect of a plasma- actuator numerically the induced body force must be modelled. 2.1 The Numerical Model In general, two different effects of the plasma are taken into account. On the one hand, the accelerated plasma imparts momentum on the flow field, which can be expressed by a body force. On the other hand, the plasma leads to a local rise in temperature, which causes a reduction of viscosity. Measurements by Jacob et al. [5] show that the latter effect is negligible. So it seems reasonable to model the actuator as a body force, which can easily be implemented in a RANS code. Preliminary simulations using unsteady RANS confirmed that it is unnecessary to resolve the time dependent behaviour of the plasma operating at the driving voltage frequency of several thousand Hertz, because the flow will not react significantly to disturbances at these frequencies. To model the actuator’s body force, Jayaraman et al. [6] assume that the plasma only exists in a triangular region above the actuator, while no charges exist outside this region. The exact spatial charge density, the electric field strength and temporal dependency have not yet been explored, therefore several assumptions have to be made. The governing equations are the equation for the force acting on the charges (1) and the Poisson equation (2). The charge distribution inside the triangular domain is determined by using equation (2). G G f = E ȡC eC (1)

G G ȡ e ∇⋅E = C C İ0

(2)

Jayaraman et al. [6] assume a linear decrease of the field strength with its maximum at the point of the shortest distance between the electrodes. It can be expressed by

E = E 0 − k 1x − k 2 y

(3)

Additionally a linear growth of the triangular domain with increasing voltage is assumed. This leads to an expression for the total force acting on the flow field, which becomes negative at higher voltages. This is not reasonable and results from the fact that the linear decreasing electric field strength becomes negative at a certain distance from its maximum. To avoid this unphysical behaviour, a new distribution of the electric field has been introduced, in which the field strength approaches zero and does not become negative. The following equation is used. E = E 0 e − m1x e − m 2 y

(4)

Experimental and Numerical Investigations of Boundary-Layer Influence

59

The gradients of the electric field should be of the same value like in the model of Jayaraman et al. [6]. This leads to the expressions

m1 =

a 0 ȡC0eC d a 0 + b 0 İ 0 V0

and m 2 =

b 0 ȡ C0 e C d a 0 + b 0 İ 0 V0

(5)

The field strength still has its maximum at the same point as the linear model (Eq. 3), but has an asymptotic approach to zero and therefore never becomes negative. This distribution leads to the following expression of the body force. G ȡ V2 G f = C 0 e − 2 ( m x + m y ) ne C (6) V0 d 1

2

This force occurs only inside the triangular domain of influence and is implemented in the momentum equations of the Navier-Stokes equations. The size of the triangular domain of influence is set to be the same length as the lower electrode at the maximum of the operating voltage. Figure 1 shows the total force induced by the actuator, calculated with the model of Jayaraman et al. [6] and calculated with the modified model used in this work. 10

4

x 10

Force in x-Direction e-Function E-Field Distrib. Linear E-Field Distrib.

Force [N/m]

3

2

1

0

-1 0

0.5

1

1.5

Voltage [V]

2

2.5 7

x 10

Fig. 1. Total body force for different models

2.2 PIV Measurements for Calibration

Since the numerical model contains several assumptions, it is necessary to calibrate three coefficients. To create an adequate data base for calibrating the developed model, PIV measurements of the flow induced by the plasma-actuator in otherwise quiescent flow have been performed. 2.2.1 Experimental Setup A container of dimensions 1.2m by 0.75m by 0.6m has been lined with black fabric to prevent reflections from disturbing the measurements. The plasma-actuator was placed on a flat plate in the centre of the box. The plate was elevated about 0.1m above the floor of the box, where the seeding particles resided in the container. The measurement domain starts 50mm upstream and ends 250mm downstream of the

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S. Grundmann, S. Klumpp, and C. Tropea

plasma-actuator and the velocity field is resolved with 159x112 vectors. The results presented have been averaged over 400 single image pairs. The coordinate origin has been place on the wall at the right corner of the upper electrode of the plasmaactuator. The generator for the driving voltage of the actuator has a mass of 0.7kg and its power supply of 2kg. The size of the generator is 280mm x 110mm x 40mm. It consists of a resonant circuit and a cascade of transformers controlled by computer through a multifunction IO card. The software allows the operating frequency of the actuator and the duty cycle in the pulsating mode to be controlled. The operating voltage of the actuator is manually adjusted through the supply voltage of the generator. Two studies were conducted: The influence of the operating voltage and the influence of a variation of the duty cycle in pulsation mode. Due to the finite size of the container it was necessary to allow the generated mean flow in the box to dissipate during breaks between the measurements. Due to reflections of the laser sheet on the flat plate and the illumination of the plasma-actuator it was not possible to resolve the flow immediately adjacent to the surface. The closest measurable points at the wall start at a height of y=2.1mm. The consequence is that the very thin wall jet created by the actuator could not be detected in the range 0<x<50mm. 2.2.2 Actuator-Induced Flow Figure 2 shows the mean U and V velocities of the wall-jet created by a plasmaactuator driven by a voltage of ue=10kVpp at a frequency of fm=5kHz. The left diagram of the figure 2 clearly shows the wall-jet that grows in thickness with increasing x-coordinate. In the range 0<x<50mm the wall-jet and the entrained air are apparently thinner than 2.1mm. In the right diagram the mean V velocity has been plotted. A distinct region of flow directed downwards at x=0mm is apparent. This demonstrates the sink-like effect of the actuator on the flow. It draws the surrounding air to the right edge of the upper electrode and diverts it away tangential to the wall. Downstream the jet diverges due to deceleration and mixes with the surrounding air, yielding positive V velocities.

Fig. 2. Measured (PIV) mean U and V velocities for ue=10kVpp and fm=5kHz

Experimental and Numerical Investigations of Boundary-Layer Influence

61

2.2.3 Variation of the Operating Voltage The influence of the operating voltage was studied for a frequency of fm=6kHz. The voltage was varied in steps of 1000 Volts, beginning with ue=5kVpp up to ue=10kVpp. The length and the speed of the wall-jet rise with increasing voltage. At the lowest driving voltage a weak jet is generated, which immediately turns up and back towards the area of downward suction. A vortex results which remains fixed above the actuator. Increasing the voltage causes the vortex to move downstream and to expand in the x-direction, until it breaks down and no direct feedback between the wall-jet and the downward suction area develops. Another parameter study reveals the influence of lowering the duty cycle. The influence of reducing the duty cycle is very similar to reducing the voltage. These investigations involving unsteady operation of the plasma-actuator are not discussed further in the context of the present study; however they will also require an unsteady numerical simulation. 2.2.4 Calibration of the Numerical Model The numerical simulations have been carried out on a 2D grid of 5500 cells with the SST-turbulence model and the low Reynolds treatment that the solver Fluent provides for boundary layers. The inlet boundary conditions were set to laminar with a turbulent viscosity ratio of one. The product of the momentum flow rate and the mass flow rate is constant for a wall-jet. Therefore it is justified to calibrate the model at a single downstream position of the jet. To adjust the charge density ρc, within the triangular region of influence of height a0(V=V0) and length b0(V=V0), the maximum horizontal velocities at x=100mm were examined and the model coefficients were tuned until the simulations showed the same magnitude for all measured voltages. All following simulations were conducted with these coefficents. The correlation between the operating voltage and the maximum velocity, is given by U§V(a-ebV)½. The correlation of Jayaraman [6] is of the type U§(aV4-bV5)½ whereas Corke et al. [1] found what they call the 7/2-power-law, which is U§V7/2. As figure 3 shows, all the correlations fit quite well with the experimental results. The entrainment of surrounding air by the wall jet is highly sensitive to changes of the viscosity of the fluid. Especially at low velocities, like in these experiments, the choice of turbulence model and wall treatment also has a strong influence on the numerical viscosity of the fluid. Figures 4 to 6 show the velocity profiles at the downstream positions x=60mm, x=100mm and x=160mm, showing rather good agreement between experiment and simulations. In figure 7 the measured and simulated momentum flow rate is shown as a function of downstream distance. The y-resolution of the PIV measurements (ymin=2.1mm) is too coarse to resolve the wall-jet completely below x=100mm, and this is responsible for the larger variations between experiment and simulations in this region. The sudden rise of the momentum transport at x=0 could also not be measured for the same reason.

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Maximum Velocity at x=100mm

Velocityprofile with V=8kV at x=60mm

0.7

0.1

Simulation Experiment

0.6

Experiment Simulation

0.0002*V7/2

0.08

0.06

0.4

y [m]

um ax [m/s]

0.5

0.3

0.04

0.2 0.02

0.1 0 6

7

8

9

0 -0.2

10

0

0.2

0.4

0.6

u [m/s]

V [kV]

Fig. 3. Maximum velocity

Fig. 4. Experimental and simulated velocity profiles at x=60mm

Velocityprofile with V=8kV at x=100mm

Velocityprofile with V=8kV at x=160mm

0.1

0.1 Experiment Simulation

Experiment Simulation 0.08

0.06

0.06 y [m]

y [m]

0.08

0.04

0.04

0.02

0.02

0 -0.1

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Fig. 5. Experimental and simulated velocity profiles at x=100mm -3

1.4

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Momentum Flow Rate V=8kV

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Fig. 7. Momentum flow rate for V=8kVpp

6

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Fig. 8. Momentum flow rates as a function of driving voltage

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In figure 8 the experimental and simulated momentum flow rate at x=100mm is shown as a function of the high voltage. Calibration experiments were only performed up to a value of 8kVpp; the value shown for 12kVpp is simulation only. Downstream of x=100 the experimental and numerical results in figures 7 and 8 agree well enough to conclude that the model is adequately calibrated to proceed with the simulations of the subsequent experiments.

3 Accelerating the Boundary Layer of a Flat Plate The following experiments have been conducted on a flat plate. The wind tunnel has a test section of 0.45m width and 0.45m height. The plate is placed in the middle of the test section and the leading edge has an elliptic profile to prevent the flow from separating. The plasma-actuator is placed 150mm downstream of the leading edge, oriented in the spanwise direction. To maintain a smooth surface the actuator is fitted in a notch. Experiments have been carried out for a free-stream velocity between 6m/s and 12m/s, for which the boundary layer is laminar at the actuator position. The resulting Reynolds number based on the displacement thickness at the position of the actuator varies between 1200 and 1700. 3.1 Experimental Investigation

In all of the following experiments the actuator was driven at a voltage of 12kVpp and a frequency of 6 kHz. The applied power to reach this voltage with an actuator of 450mm length was 35W, corresponding to 78W per meter of actuator length, including all the electrical losses in the high-voltage generator and the losses due to thermal heating of the actuator and its insulation film. The distinction between the power that the actuator transfers to the air, the power absorbed by the plasma and the power consumption of the high-voltage generator will be made in future studies. The velocity profiles are measured at the positions 30mm, 40mm, 60mm and 80mm downstream of the actuator. The measurements were performed with two constant temperature hot-wire probes (CTA). Both signals are linearised by an analog amplifier and are passed through a low-pass filter to avoid aliasing effects when digitized. Both probes are mounted on a single traversing system with a vertical separation of 40mm. The lower probe is traversed throughout the boundary layer, while the upper one measures the free-stream velocity, which is necessary to compensate for velocity fluctuations due to weather influences on the wind tunnel at low speeds. The electric field of the actuator disturbs the CTA measurements leading to an apparent turbulence, while the mean value is not influenced. The magnitude of this offset of the measured variance of the velocity fluctuations is independent of the local velocity; hence it is treated as system noise and it can be removed in the signal processing step. Figure 9 shows an example velocity profile for a boundary layer 60mm downstream of the actuator with and without the actuator turned on. This is a flat

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plate boundary layer with a very weak adverse pressure gradient. The acceleration of the lower parts of the boundary layer is obvious. The plots of the velocity differences between the affected and the unaffected boundary layer in figure 10 show that the maximum acceleration occurs below one millimetre height. The velocity difference decreases rapidly after x=60mm and the velocity maximum moves slightly upwards. The difference of the momentum flow rate between the affected and the unaffected boundary layer is plotted in figure 11 for all measured velocities at the positions x=30mm, x=40mm, x=60mm and x=80mm downstream of the actuator. The added momentum rises with increasing mean flow velocity at the first two positions. The reason for this observation cannot yet be explained and further measurements are pending. A first explanation is to assume that the momentum transfer from the very fast ions to the neutral air molecules happens more effectively when the air has a higher velocity. The momentum differences rise also with increasing distance from the actuator for the first three positions. A consequence of this observation is that momentum transfer does not only happen in the triangular region around the actuator but also several millimetres downstream of the actuator. Laser Doppler measurements are planned for the future to investigate these observations further.

Boundarylayer [Umean=8m/s, Pos=60mm] 4

Velocity Difference [Umean=8m/s] 4

off 12kV, 6kHz, 100%

40mm 60mm 80mm

3.5

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U/U0

Fig. 9. Velocity profile with and with actuator

0

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Up -U [m/s]

Fig. 10. Velocity difference with and without actuator, Umean=8m/s

The momentum difference decreases after x=60mm due to turbulent mixing and dissipation in the boundary layer. At velocities of 10m/s and 12m/s the momentum difference disappears almost completely at x=80mm. The RMS-measurements show that the amount of turbulent kinetic energy in the boundary layers at x=80mm is twice as large as at x=60mm. The point of transition moves upstream at higher free stream velocities and has more influence on the boundary layer development near the actuator than at lower velocities.

Experimental and Numerical Investigations of Boundary-Layer Influence

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Momentum Flow Rate Difference [12kVpp] 0.04 0.035

2

I [kgm/s ]

0.03 0.025 0.02 0.015 6m/s 7m/s 8m/s 10m/s 12m/s

0.01 0.005 0 20

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x [mm]

Fig. 11. Difference in momentum flow rate with and without actuator

3.2 Numerical Investigation

The numerical investigation of the second experimental setup was conducted on a 2Dgrid of 156000 cells. The turbulence intensity of the inlet data was set to T i=0.3%, which is the turbulence intensity of the wind tunnel. The SST-model with the LowReynolds treatment in the commercial code Fluent was used. The grid is a simple rectangle with a no-slip wall on the lower boundary, a constant velocity inlet at the left side, a symmetry plane at the upper side and a constant pressure outlet on the right side. 4

Plasma off (Sim.) 12 kV (Sim.) Plasma off (Exp.) 12kV (Exp.)

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Fig. 12. Boundary layer profiles for U0=6m/s

Figure 12 shows the development of the measured and simulated boundary layer at the positions x=40mm, x=60mm and x=80mm. The profiles are plotted for the flow with the actuator turned on and off. The comparison of the measured and simulated profiles shows good agreement. Figure 13 shows the momentum flow rate differences for all simulations corresponding to the measurements presented in figure 11. Two major differences

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between experiment and simulation can be identified. One is the steadily decreasing momentum difference with increasing x-coordinate and the other is its slower decrease for the simulations. The first agrees with the observation in the first part of the study. The momentum difference does not rise during the first few millimetres downstream of the actuator as in the experiment. The reason for the increase of the momentum difference in the experiment has to be analysed in future work. The second is the faster decrease of momentum difference for all velocities in the experiment and can be explained by the occurrence of transition in the experiment that does not occur in the simulation. The increasing turbulence intensity in the experiment dissipates the added momentum faster than in the simulation. In summary, the value of added momentum in the experiment and in the simulations is nearly identical and the correlation between both is acceptable. Boundary Layer Thickness (U0=6m/s)

Momentum Flow Difference 0.04

12 plasma off 12 kV

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Fig. 14. Simulated boundary-layer thickness

One of the main long-term goals of these investigations of plasma-actuators is to sustain a laminar boundary layer in an adverse pressure gradient. The first step in this direction has been achieved, as figure 14 demonstrates. It shows the boundary layer thickness of the simulated boundary layer with and without actuator for Umean=6m/s. The boundary layer thickness grows from zero to 2mm in height at x=0mm, where the actuator is placed. It continues growing without the actuator, but with the actuator working, its thickness is reduced by 60%. The point of transition is seen to be much later, at approximately x=600mm. These numerical results can only be considered qualitatively valid, especially concerning the exact position of transition because no validation data is currently available. In any case, the growth of the boundary layer can be delayed and the condition of the boundary layer immediately following the actuator corresponds to a position 100mm upstream. As a result, the point of transition is delayed by approximately 90mm. Figure 15 shows the simulated local skin friction coefficient. The numerical solution of the unaffected flow follows the Blasius solution until transition, after which it follows the accepted solution of a turbulent boundary layer. The actuator

Experimental and Numerical Investigations of Boundary-Layer Influence

67

Local Skin Friction Coefficient (U 0=6m/s) 0.02 Plasma off 12 kV c f ,lam 0.015

cf

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Fig. 15. Simulated local skin friction coefficient

causes a peak of skin friction at the position of the actuator. Behind that peak the cfvalue does not completely align with the laminar values but the transition is delayed compared to the unaffected flow. The net benefit in reducing the skin friction by delaying transition is partially compensated by the peak produced by the acceleration of the boundary layer. Future work will concentrate on an optimisation of the number and position of the actuators and the driving voltage to achieve a maxim net skin friction reduction.

4 Conclusions An existing numerical model to simulate the body force created by a plasma-actuator has been improved and calibrated using experimental data obtained from velocity measurements around an actuator placed in quiescent flow. The comparison of the simulations with measurements is very satisfactory and the capability of the method to predict the effect of an actuator is considered sufficient to be used as a predictive tool. In a flat-plate boundary layer the plasma actuator is shown to significantly reduce boundary-layer thickness, leading also to delayed transition.

References [1] Corke, T.C., Matlis, E. (2000) Phased plasma arrays for unsteady flow control. Fluids 2000 Conference and Exhibit, Denver, CO, June 19-22, 2000, AIAA-2000-2323 [2] Corke, T.C.,He, C. (2004) Plasma Flaps and Plasma Slats: An Application of WeaklyIonized Plasma Actuators. 2nd AIAA Flow Control Conference, Portland, Oregon, June 28-1, 2004, AIAA-2004-2127 [3] Font, G.I., Morgan, W. (2005) Plasma Discharges in Atmospheric Pressure Oxygen for Boundary Layer Separation Control, 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, Ontario, June 6-9, 2005, AIAA-2005-4632 [4] Grundmann, S., Tropea,C. (2005) Gepulste Plasma Actuatoren zur aktiven Grenzschichtbeeinflussung. 12. STAB Workshop 2005, DLR Göttingen

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[5] Jacob, J., Rivir, R., Campbell, C., Estevedoreal, J. (2004) Boundary Layer Flow Control Using AC Discharge Plasma Actuators. 2nd Flow Control Conference, Portland, OR, AIAA-2004-2128 [6] Jayaraman, B., Shyy, W. (2003) Flow Control and Thermal Management Using Dielectric Glow Discharge Concepts. 33rd AIAA Fluid Dynamics Conference and Exhibit, Orlando, Florida, June 23-26, 2003, AIAA-2003-3712 [7] Orlov,D., Corke, T. (2005) Numerical Simulation of Aerodynamic Plasma Actuator Effects. AIAA 43rd Aerospace Sciences Meeting, 10-13 January 2005, AIAA 2005-1083 [8] Roth, J.R. (2003) Aerodynamic Flow Acceleration using Paraelectric and Peristaltic Electrohydrodynamic (EHD) Effects of a One Atmosphere Uniform Glow Discharge Plasma, Physics of Plasmas, Vol. 10, No. 5 [9] Touchard, G., Artana, G., Sosa, R., Moreau, E. (2003) Control of the near-wake flow around a circular cylinder with electrohydrodynamic actuators. Experiments in Fluids 35, 580-588 [10] Velkoff, H.R., Ketcham, J. (1968) Effect of an Electrostatic Field on Boundary-Layer Transition. AIAA Journal 6 (7), pp. 1381-1383.

Designing Actuators for Active Separation Control Experiments on High-Lift Conﬁgurations Ralf Petz and Wolfgang Nitsche Berlin University of Technology, Marchstr.12, Skr. F2, 10625 Berlin, Germany [email protected] http://www.aero.tu-berlin.de

Summary Designing actuators for experimental investigations that deal with the active control of separation by periodic excitation is of immense importance. The conclusions drawn from such experiments heavily rely on the actuator performance. Actuation frequency and amplitude as well as actuator location and jet direction play an important part that is still not fully understood in some aspects. Once the experiments are successfully performed the everlasting question of power consumption versus aerodynamic beneﬁt arises. In order to estimate an overall ﬁgure of merit not only the aerodynamic beneﬁts have to be taken into account but also actuator weight and volume as well as initial and maintenance costs and system complexity. Although many experiments are performed in this ﬁeld of research scaling actuator performance from wind tunnel experiments to full-size applications is only possible for those who use zero-net-mass-ﬂux actuators which require only electrical input. Once compressed air in combination with valves is used to excite the ﬂow it is difﬁcult to estimate the actuator performance due to longer ducting system and different tube diameters with additional pressure losses. The paper gives an overview of three different actuator designs that aim at enhancing the aerodynamic performance of high-lift conﬁgurations by suppressing ﬂow separation on a single slotted ﬂap. All experiments were performed in the Collaborative Research Centre 557 Control of complex shear ﬂows set up at the Berlin University of Technology.

1 Introduction Active ﬂow control plays an ever-growing part in aerodynamic research [1]. As the conventional aerodynamic designs are pushed to their limits, active ﬂow control seems to be one possibility to overcome certain aerodynamic limitations, e.g. ﬂow separation. The Collaborative Research Centre at TU-Berlin deals with the control of complex turbulent shear ﬂows and includes a couple of experimental and numerical projects that are related to active separation control by periodic excitation [2,3,4]. Prerequisite for experiments with active ﬂow control is the design and production of a suitable excitation apparatus. Depending on the size of the wind tunnel model, actuator dimensioning may be crucial for the whole project. R. King (Ed.): Active Flow Control, NNFM 95, pp. 69–84, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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Periodic excitation in the sense of alternating suction and blowing is used to delay boundary layer separation in diffusers [5], turbomachinery blades [6] or on wings/highlift conﬁgurations [7,8,9]. It has also been tested successfully in order to manage the wake structure behind ﬂapped wings [10]] or for the purpose of jet vectoring [11,12]. Active ﬂow control by means of alternating suction and blowing has been investigated successfully at high Reynolds number [13] and high Mach numbers [14]. Slot location [15] and jet direction [16,17] have to be considered as they have an impact on frequency and amplitude. Despite the multitude of ﬂow control experiments conducted, it is still challenging to set up a complex ﬂow control experiments. One reason for this is the difﬁculty to experimentally isolate a single excitation parameter and document its impact on the ﬂow because excitation parameters often heavily interact. A further problem emerges when the power consumption and weight of the complete excitation system is compared to the aerodynamic beneﬁts. In laboratory environments this is hardly ever realised because the actuator weight does not scale with the model weight. Most investigations are conducted using small or medium-size wind tunnel models, which have different actuator requirements than full-size applications. Although a lot of data is presented in journals and conference proceedings scaling an excitation apparatus to a full-size application on the basis of wind tunnel data only, is almost impossible because there are still too many unknown factors. However, it has been proven by a lot of experiments that periodic excitation in terms of alternating suction/blowing or pulsed jets is a very effective tool to delay boundary layer separation and it is more efﬁcient than steady blowing under certain conditions up to two orders of magnitude [18]. A full-size demonstration has also been performed very successfully on the tilt-rotor aircraft in order to minimize the download alleviation by periodic excitation [19]. Besides aerodynamic experiments, a lot of investigations about ﬂow control actuators are carried out mostly regarding zero-net-mass-ﬂux actuators [20,21,22] with a few addressing the scaling of actuators [23]. The purpose of this paper is to describe some aspects of the actuator design process for three different experiments that are all related to active separation control in order to enhance the aerodynamic performance of high-lift conﬁgurations. The ﬁrst two wind tunnel models have exactly the same two-dimensional generic set-up and differ only in size whereas the third conﬁguration is a more realistic model with a fuselage and a ﬁnite swept back wing. The excitation systems are demonstrated as well as their restrictions, constraints and part of their performance. Exemplary results in terms of lift and drag enhancement are presented for all three cases [24,25]. A short note on actuators that were used for closed-loop control is added because they require special treatment [26].

2 Actuator Design Process To design an actuator for separation control is a very complex and important task. Flow control experiments heavily rely on the precise and faultless work of the excitation system. If these systems are unable to perform within certain design parameters, e.g. frequency and amplitude, they do not have an impact on the ﬂow at all. If experiments do not work as expected it is hard to tell whether the actuator itself is the cause or the ﬂow condition. This is one reason why the actuation has to be monitored constantly

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throughout the experiment. However, the actuator itself frequently limits the possibilities to investigate more realistic ﬂow conditions, e.g. higher free-stream velocities, because it fails to produce sufﬁciently high excitation amplitudes or frequencies, for instance. As a consequence, it is very important to have an excitation mechanism at hand that is reliable, robust, has a adequate frequency range and produces adequate amplitudes. In order to give some numbers for excitation frequency and magnitude the following example for a typical passenger aircraft is provided. Frequency and amplitude are estimated for the airplane in landing conﬁguration with periodic excitation applied to the trailing edge ﬂap only. The following equations are used: F+ = f - frequency in Hz,

f ·l u∞

l - ﬂap chord, cµ = 2 ·

h · c

f requency

(1)

u∞ -free stream velocity uRMS u∞

2 amplitude

(2)

h - slot width, c - airfoil chord, uRMS - RMS of the excitation jet velocity, u∞ - free stream velocity. A typical wing span of 60m is assumed with an aspect ratio of about 10 giving an average chord length of 6 meters. The ﬂap chord length is estimated with 25% of the main chord length giving a ﬂap chord of 1.5 meters. This is an estimation as inbord and outbord ﬂap chords differ due to sweep an taper. The average landing speed of this type of aicraft is 140kt (u∞ = 75m). The left hand diagram of Figure 1 shows the calculated excitation frequency in Hertz based on equation 1 for varying ﬂap chord length and different nondimensional frequencies F + . Assuming a recued frequency of F + = 1 results in an excitation frequency of 40Hz to 60Hz, F + = 3 yields frequencies around 150hz. The results for estimating the RMS values (based on equation 2) of the excitation jet velocity are displayed in the right hand side plot of ﬁgure 1 for different slot widths. A slot width of 2mm and a cµ of 0.02% would require an RMS jet velocity of about 40m/s. Therefore, a full-size application requires an actuator that has a realtive low frequency but produces high jet velocities. In contrast, wind tunnel investigations require small actuators with higher frequencies due to the smaller model size. It should be a general goal in actuator development, for the periodic oscillations to be generated as close to the excitation slot as possible in order to minimize pressure losses caused by long ducts, pressure lines or guiding vanes and to minimize the volume necessary for actuator housing. It is even more desirable to have an actuator that relies on electrical energy only and requires no additional plumbing for pressure lines. These are some of the reasons why such efforts are made to design zero-net-mass-ﬂux actuators of a very small size that can be placed inside a very small wing. The disadvantage is the extensive and often time-consuming research required to establish a system that is reliable, has high excitation velocities, a wide frequency range, does not overheat or gets damaged easily. In respect of experiments featuring closed-loop control the actuators should be able to change frequency and amplitude very fast in order to minimize the dead time for the controller. In order to meet all those requirements,

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Fig. 1. left: Estimation of excitation frequency for a typical passenger aircraft with trailing edge excitation for various ﬂap chord lengths and nondimensional forcing frequencies. right: Estimation of excitation jet velocity (RMS) for different oscillatory blowing momentum coefﬁcient cµ and slot widths.

e.g. small size, high frequency, high amplitude, fast response, reliable, robust, easy to calibrate etc. extensive research has to be carried out including totally different ﬁelds of research. Although very sophisticated actuators do exist, most research teams do not have this type of actuator at their disposal. A lot of investigations are conducted in a laboratory environment using fairly small to medium-size wind tunnel models. But the smaller the model gets the smaller the excitation system has to be, which makes it even more difﬁcult to design and produce. Most models are of the generic type because it makes actuator installation easier. Once wings or wing sections are investigated most actuators are especially designed to ﬁt inside the wing and are often not scalable to full-size applications. This paper describes three different excitation systems that were specially designed to ﬁt in the respective high-lift model. The investigation is aimed at enhancing the aerodynamic performance of simple and later of complex high-lift conﬁgurations by suppressing the ﬂow separation on the single slotted ﬂap high-lift system, as shown in ﬁgure 2.

Fig. 2. Snapshots of PIV data for an unexcited (left hand side) and excited ﬂow (right hand side), grayed areas representing vorticity strength (water tunnel, Rec = 0.16 · 105 )

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Very early tests on high-lift ﬂows were carried out in a small water tunnel (using LDV and PIV) with a specially adopted excitation system for liquids. Once is was clear that the excitation enhances the aerodynamic performance, e.g. lift, more sophisticated wind tunnel models were constructed, which are described in the next few chapters. One special characteristic of high-lift models with slotted ﬂaps is the difﬁculty to install the excitation system inside a small ﬂap because of insufﬁcient space and limited access.

3 Small High-Lift Model Early wind tunnel experiments were conducted using a small model that consists of a main wing (chord length=180mm) and a single slotted ﬂap (chord length= 72mm). The span of the model amounts to 400mm, giving an aspect ratio of 2.2 for the main wing and 5.5 for the ﬂap. These small dimensions were chosen because of wind tunnel constraints and because it was assumed at the time that a smaller model could be handled more easily, was cheaper and that the excitation system was easier to design. In retrospect, it can be said that a small model is merely cheaper but makes things even more complicated in terms of the excitation apparatus because of size constraints and the need for higher frequencies.

Fig. 3. Set-up of the small generic high-lift model (Rec = 1.6 · 105 ) [27]

Figure 3 shows the high-lift conﬁguration with the main wing and the single slotted trailing edge ﬂap. Angle of attack α and ﬂap deﬂection δf describe the geometric angles while ﬂap gap and ﬂap overlap give the position of the ﬂap relative to the main wing. The excitation is located on the upper surface of the ﬂap near the leading edge. At this position a slot (width 0.3mm) is made in the surface of the model to allow the excitation to emerge perpendicular to the surface. Unsteady excitation, i.e. alternating suction and blowing, was used in order to enhance shear layer mixing, thus moving high-energy ﬂuid from the shear layer to the wall. The strengthened boundary layer is then able to sustain the severe adverse pressure without separating. This allows higher lift coefﬁcients in the post-stall region as well as higher ﬂap deﬂections with single slotted ﬂap systems. To develop an excitation system for a model that small was very complicated and probably took up 60-70% of the overall time allotted to construct the complete set-up. The ﬂap has a maximum thickness of 11mm and consists of machined aluminium. An attempt was made to integrate a specially designed excitation system into the ﬂap, but lack of space did not allow for a reliable system. After a series of setbacks, a system

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consisting of two sideways mounted speakers was chosen. Figure 4 shows the model and the two speakers mounted to the ﬂap. The speakers generate pressure oscillations with a desired frequency and amplitude, which are directed into an internal chamber inside the ﬂap which is hollow in that region (right picture in ﬁg. 4). A special set of guiding vanes distribute the air along a span of 380mm which were optimzed to work for the blowing and the suction phase.

Fig. 4. left: Excitation system of the small high-lift model using a speaker system right: View of the inside of the hollow ﬂap structure [27]

Actuator Performance: Speakers produce sound effectively but not much pressure, which is needed in order to achieve a high jet velocity [28]. Due to this special set-up it was very difﬁcult to get the velocity distribution as constant along the span as possible without reducing the excitation amplitude too much. The distribution depends strongly on the excitation frequency and less on amplitude. Figure 5 shows the maximum jet velocity along the actuator span for three different frequencies. The velocity is measured using a movable single hot-wire that is set as close to the slot as possible without destroying it. As was to be expected, the velocity at both ends - close to the speakers - is higher than at mid-span (200Hz) which is even worse for higher frequencies (400Hz). For very low frequencies the maximum velocity occurs at midspan and not at the ends. The peaks at both ends result from support struts inside the ﬂap, which stabilize the hollow structure and the slot width. The distribution (200Hz) was achieved only after extensive tests of different guiding vane spacings inside the ﬂap. The right hand side plot shows a hot-wire signal close to the slot. The hot-wire in this case was set 0.2mm above the slot which explains the velocity differences of the blowing and the suction phase. In order to calibrate the excitation system, a fast pressure transducer was placed inside the cavity, which measured the pressure ﬂuctuations generated by the speakers. The velocity ﬂuctuation at discrete points along the span were then measured for all combinations of frequencies and amplitudes and plotted against the cavity pressure ﬂuctuation (ﬁg. 6). As can bee seen in the plot, the velocity ﬂuctuations are proportional to the pressure ﬂuctuations [13] showing lower gradients for higher frequencies. Higher velocities are not possible due to limitations of the speaker system. This time-consuming procedure is essential in order to calculate a nondimensional blowing coefﬁcient and

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Fig. 5. left: Velocity distribution along the actuator span for three different frequencies [27], right: Hot-wire signal about 0.2mm above the slot at midspan

Fig. 6. Calibration of the excitation apparatus [27]

to ensure that for every combination of frequency and amplitude the correct amount of momentum is used to excite the ﬂow. The cited momentum coefﬁcient is the averaged value of different spanwise positions. One lesson learned was that an actuator has to be very reliable in operation, which led to larger and more sophisticated speakers than originally intended. Nevertheless, during operation both speakers had to be monitored very closely (voltage and power) and were turned off automatically when an overload occurred. Once a speaker failed the system had to be taken out of the tunnel and recalibrated. Results: Extensive measurements with and without periodic forcing were conducted using a six-component wind-tunnel balance. The results prove that the onset of separation is delayed and lift in the post-stall region is increased as well as the maximum ﬂap deﬂection angle. The small dimensions of the model facilitate ﬂow ﬁeld measurements with a movable single hot-wire. Two results are plotted in ﬁgure 7 for a ﬁxed conﬁguration of α = 7◦ , δf = 36◦ . In the unexcited case (left-hand side diagram) the massive separation is clearly visible resulting in an almost horizontal jet ﬂow coming through the ﬂap gap. By comparison, the diagram on the right shows the excited case with periodic forcing at the leading edge of the ﬂap. Alternating suction and blowing reattaches the jet ﬂow almost completely once excitation frequency and amplitude are tuned to effective values.

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Fig. 7. Flow velocities around the trailing edge ﬂap for unexcited and excited case α = 7◦ , δf = 36◦ , (Rec = 0.15 · 106 , F + = 0.5, cµ = 0.045%, hot-wire measurement taken from ref. [27])

4 Larger High-Lift Model In the second phase of the project a completely new model was built with the aim to test much higher and more relevant Reynolds numbers. The geometric constraints, e.g. proﬁle shape, gap, overlap, remained the same while the proﬁle chord was extended to 500mm and the ﬂap chord to 200mm. The total span was raised to 1550m in order to improve the two-dimensional ﬂow quality resulting in an aspect ratio of 3.1 for the main wing and 7.75 for the ﬂap (ﬁg. 8). This new design with an almost tripled ﬂap chord and a span that is almost four times as large necessitated a completely new actuator design. High Reynolds numbers up to a million also called for higher excitation amplitudes. The experience gained with the speaker actuator, velocity distribution, reliability and calibration led us to prefer a system consisting of fast switching solenoid valves and compressed air. At ﬁrst glance, this may appear to be a hurdle because a system using compressed air needs a lot of additional plumbing but after a thorough comparison of several actuator designs this was the best choice at the time. One major difference between the new valve approach and the speaker system is the time-dependent velocity signal of the excitation. The speaker system produces an alternating suction/blowing signal whereas the compressed air-solenoid valve approach generates a pulsed blowing signal without any suction phase, but with signiﬁcantly improved spanwise uniformity and options for intended 3D excitation modes. This type of excitation mechanism enables steady blowing, pulsed blowing, steady suction, pulsed suction and with additional plumbing alternating blowing and suction. In order to supply 1.5m of excitation slot with compressed air the system had to be divided into smaller segments (eleven in this case) to guarantee a constant velocity distribution along the complete span. A pressure transducer is connected to each actuator cavity in order to measure the pressure ﬂuctuations for monitoring reasons and calibration purposes. The larger ﬂap made it possible to install the valves (thickness 10mm) inside the ﬂap and minimize pressure oscillation losses through long tubes. Due to the plumbing, the volume of one actuator segment (span x height x length) is 114mm

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Fig. 8. Schematic view of the excitation apparatus of the larger high-lift model showing the eleven solenoid valves inside the ﬂap (Rechord = 106 )

x 15mm x 110mm leaving enough space inside the ﬂap for additional measurement equipment. This assembly has two major draw-backs, one being the additional plumbing required to supply each of the eleven actuator segments with the same amount of air, the second being the relatively low frequency of the solenoid valves compared to the speaker system. They allow Strouhal numbers up to a value of F + = 2 at free stream velocities of 20m/s. The system is capable of producing a steady blowing momentum coefﬁcient cµ,std. = 0.4% wheras the max. oscillatory momentum coefﬁcient is somewhat lower at cµ = 0.2%. The advantage is a very reliable (no maintenance needed), very versatile excitation mechanism with high jet velocities. The weight of the excitation system components -including all solenoid valves, pressure regulator valves, tubes, pressure transducers etc.- adds up to around 2kg. As the solenoid valves are originally manufactured for industrial needs the have a service life of over 2 · 108 cycles at very low cost. Actuator Performance: Figure 9 gives an indication of the velocity distribution along one actuator span of 114mm. The minimum and maximum velocities do not differ by more than ±10% from the mean value (left diagram). This result is a trade-off between using as little plumbing as possible while maintaining a constant velocity distribution. In this case, a minimum of four pressure connections and one solenoid valve are necessary. Eleven of these actuator segments were then placed alongside each other to form the complete excitation apparatus with a 2D slot (segment width=114mm). The distance between the segements is less then 0.5mm. The spanwise segmentation coupled with individual control of each valve gives a very versatile excitation system that makes it possible to achieve three-dimensional excitation modes (varying excitation amplitude and phase along the span) and to gain rolling moment control by simply using actuators on parts of the ﬂap. One major issue of this type of system is its calibration, which is a time-consuming procedure. A single hot-wire is moved from segment to segment very close to the slot measuring the unsteady velocity. This is carried out for different frequencies and amplitudes. Hot-wire signals for three different excitation frequencies are shown in ﬁgure 9 (right diagram). The low frequency signal has the steepest rising and falling edges if compared to the higher frequencies (100Hz and 160Hz). The repetition accuracy of the

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Fig. 9. left: Velocity distribution along one actuator span of 115mm (pressure supply 0.5 bar gauge, frequency 110Hz, slot width 0.3mm) right: Hot-wire signals close to the slot for three different excitation frequencies

solenoid valves is better than 0.2ms allowing frequencies up to 200Hz. This is just one exemplary result of the actuator test programme performed. The result is heavily inﬂuenced by pressure line length as is the signal quality of the pulses generated. A fast pressure transducer was attached to each actuator segment for calibration and health monitoring while installed in the wind tunnel. One major advantage of the solenoid valves is the outstanding reliability, which was very important as wind tunnel time is always limited. Although there is access to the ﬂaps interior on the lower side, an actuator failure can hardly be repaired while the model is mounted in the tunnel. As closed-loop separation control is successfully performed using this model, some changes to the actuator had to be made in order to allow a very fast and easy change of excitation parameters. Closed-loop control, where the ﬂow condition is measured by sensors and feedback to a digital signal processor (DSP), which subsequently changes the excitation parameters according to a control theory, demands fast adjustment of excitation parameters. Closed-loop performance should therefore be taken into consideration, when designing actuators [29]. Results: The model is placed on a six-component balance, which ensures simultaneous measurements of all forces and moments acting on the model. These integral values provide instantaneous information on the impact of different excitation parameters. However, the balance system is not capable of measuring lift ﬂuctuations as it allows data acquisitions with a few Hertz only. The lift and drag data shown is averaged over one second at each angle of attack. As the angles of attack and ﬂap deﬂection are automatically adjustable, angle sweeps are possible within a very short time. Figure 10 shows some exemplary results of the lift and drag enhancement that was achieved with the pulsating jet. At low angles of attack (α = −3◦ to α = 1◦ ) the ﬂow on the ﬂap is only partially separated and a complete separation occurs at α = 2◦ . By effectively delaying the onset of separation the lift is increased by as much as 12% while the drag is at the same time decreased by about 11%. By improving lift and drag the resulting lift-to-drag ratio is improved by 20-25%. By having a spanwise segmented actuator with individually addressable valves the ﬂow can be forced to reattach only to certain parts of the ﬂap, i.e. some actuator segments are systematically shut off while others are turned on. The right-hand side picture of ﬁgure 10 indicates the control of the rolling

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Fig. 10. left: Lift and drag polars for the unexcited and excited case (Rec = 0.55·106 , F + = 0.9, cµ = 0.06%) right: Control of the rolling motion is possible only with a segmented actuator design and individually addressable valves [26]

motion by actuating only on the right side of the ﬂap. Wool tufts on the ﬂap visualize the ﬂow and clearly mark attached and separated ﬂow regions.

5 Half-Model with Swept Back Wing The third model in this series is aerodynamically more complex than the previous two test cases. The model consists of a fuselage including a peniche in order to minimize the inﬂuence of the wind tunnel boundary layer and a swept back wing with a constant chord of 450mm (clean conﬁguration) and a ﬁnite wing span of 1120mm. The wing is equipped with an expendable leading edge slat (slat deﬂection δs = 26◦ ) and a single slotted fowler ﬂap (ﬁg. 11). Some minor changes have been made to the model in order to deﬂect the ﬂap automatically. Due to the use of modern airfoil shapes, compared to the NACA airfoils used in the generic set-up, the integration of actuators inside the ﬂap is more difﬁcult. The model has originally never been intended for the use of ﬂow control techniques. Neither main or ﬂap have a hollow structure as the large twodimensional model has. The wing is made completely out of aluminium and therefore does not allow the actuators to be placed inside the main wing. As a result, the ﬂap was rebuilt out of composites in order to include at least part of the actuator mechanism inside the ﬂap close to the excitation slot. The ﬂap itself has a maximum thickness of about 10mm leaving enough space for an actuator of about 7 to 8mm in height. There have been numerous tests of different actuator working principles (voice coil, piezo, magnetic) of no more than 7mm in height, but non gave the indication that it could be realized in a very limited time frame and without exceeding the budget. Instead, the highly versatile actuator system of the large two-dimensional high lift model was basically transferred to the half-model. The small ﬂap does not allow the installation of the valves (thickness 10mm) inside the ﬂap, so a trade-off has to be made between actuator efﬁciency and system complexity. The actuator design process resulted in a very similar approach again using a system containing compressed air and fast switching solenoid valves as shown in ﬁgure 11. Eleven valves are placed inside the fuselage and are connected to eleven specially designed spanwise oriented pressure chambers inside the ﬂap that end in a very narrow slot. Each valve is individually

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Fig. 11. Schematic drawing of the half-model with a swept back wing and the actuator assembly, of which one half is placed inside the fuselage and the other half inside the ﬂap

addressable which, in addition to two-dimensional excitation modes, also facilitates three-dimensional modes and the control of rolling moments. Actuator Performance: An effort was made to minimize the plumbing necessary to supply each actuator segment with compressed air. The large generic high-lift needed four connections for an actuator span of 115mm while the new design needs only one to cover 90 mm of actuator span. The velocity distribution for two different actuator frequencies along one of these actuator segments is plotted on the left-hand side of ﬁgure 12. Evidently, the results are better than in the previous cases and show an almost constant velocity proﬁle. The drawback of this type of assembly, where the valves are placed a long way from the excitation slot, is that the long pressure lines generate a signiﬁcant pressure loss and therefore reduce the excitation amplitude and deteriorate the signal quality. The default length for all results presented herein is 1000mm because of the wing span of 1140mm. All valves have to be connected to the pressure chambers with exactly the same tube length to keep them all in phase. Different tube lengths have been tested in order to document the dramatic inﬂuence on the mean jet velocity. Figure 12 (diagram on the

Fig. 12. left: Mean jet velocity along an actuator span (pressure supply = 2 bar gauge, frequency f=50 Hz and f=100 Hz, slot width 0.3mm) right: Inﬂuence of compressed air line length (pressure supply p= 3.5bar gauge, tube diameter 4mm)

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Fig. 13. Signal quality of the time-dependent jet velocity with a tube length of 150mm, 1000mm, 2000mm, pressure supply p = 2 bar gauge, frequency f = 50 Hz

right) demonstrates this effect by showing the velocity distributions along one actuator span for three different tube lengths. It is obvious that the longer the tubes, the worse the time-dependent signal quality of the unsteady jet velocity gets. Figure 13 shows the results of the default conﬁguration with 150mm, 1000mm and 2000m long tubes, a pressure supply of 2 bar gauge and a frequency of 50 Hz. The pulses are clearly visible but the long tube deteriorates the signal quality. The shape of the pressure chamber inside the ﬂap has a substantial inﬂuence on the signal quality and was in this case optimised to achieve a constant velocity distribution along the span. Results: Some results of the experimental investigation are presented here in order to document the impact of pulsed blowing, particularly on the lift behaviour. The lefthand side diagram of ﬁgure 14 shows lift polars obtained with a ﬁxed ﬂap deﬂection of 47◦ and extended slat (δs = 26◦ ). The lift is measured ﬁrst without ﬂow control and afterwards with all actuators turned on at the desired frequency and amplitude. Due to the large ﬂap deﬂection of 47◦ the ﬂow on the ﬂap is completely separated at all angles of attack. Pulsed blowing is able to attach the ﬂow and increases lift by about 10-12% at all angles of attack below α = 19◦ . Maximum lift is increased by about 5% which is less than in the two-dimensional case. Although the investigations are still under way, it turns out, that the wing tip vortex caused by the ﬁnit wings span interacts heavily with the ﬂap ﬂow. Since the actuator is segmented in spanwise direction local control of the ﬂow is possible and thus the control of the rolling moment as presented in the two dimensional test case. The picture on the right-hand side of ﬁgure 14 gives a good indication of the asymmetric ﬂow conditions on the ﬂap while only the inner six actuators are active and control the ﬂow. The outer segments are inactive which results in a separated ﬂow [30].

6 Conclusion This paper describes some of the aspects of actuator design using three examples from a high-lift separation control project within the framework of the Collaborative Research Centre Control of complex turbulent shear ﬂows set up at the Berlin University of Technology. It is not the purpose of this paper to provide a manual for designing

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Fig. 14. Lift versus angle of attack and wool tuft visualisation (Rec = 0.35 · 106 , slat deﬂection δs = 26◦ , F + = 0.78, cµ = 0.065%)

excitation mechanisms but rather to point out some of the problems related especially to the models currently used. As mentioned in the introduction, the excitation systems presented are very speciﬁc and adjusted to the test case because of the particular problems related to small models. Scaling of such results especially when using compressed air and ducting system in terms of actuator efﬁciency or supply pressure needed seems highly doubtful. Using zero-net-mass-ﬂux actuators makes the scaling to full-size applications much more easier. However, this does not effect in any way the aerodynamic enhancements that are possible with unsteady excitation. All experiments presented herein clearly show the dramatic inﬂuence of periodic excitation on lift and drag. The aerodynamic performance of single slotted high-lift conﬁgurations can be dramatically improved up to a lift-to-drag ratio enhancement of 25%. Further optimization and adaptation of actuator parameters will result in even better performance gains. Dividing the actuator in spanwise segments enables control of the rolling motion [31,32]. The ongoing research in the ﬁeld of active ﬂow control and especially on active control of separation is and will be very important for future aerodynamic developments. Despite the remarkable results that have been achieved in the past 10-15 years it will need more time for this new technique to be applied to a civil aircraft. Today, many laboratory experiments with very different constraints, e.g. geometry, Reynolds number, boundary layer or excitation apparatus, are conducted that are only comparable to a limited extent. There is a lack of large-scale if not full-size wind tunnel experiments with actuators designed for practical applications, where system weight, system complexity, actuator efﬁciency as well as high Reynolds number tests may be performed. This data would be very helpful to verify small-scale generic tests in the laboratory. A ﬁrst full-size demonstration has already be performed on the XV-15 tilt-rotor aircraft as described in detail in [33]. Nevertheless, high-quality and carefully performed smallscale tests have to be carried out. If conducting active ﬂow control experiments many important parameters related to the excitation system have to be considered, such as excitation frequency, amplitude, duty cycle, jet direction, excitation modes (3D or 2D)

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[34,32], excitation location (validated CFD if available). Some of the questions that come up in the presented investigations still have to be investigated, such as: 1. Alternating suction/blowing or pulsed blowing: How does the suction phase contribute to the reattachment process? 2. Best excitation jet direction: Tangential, perpendicular, or something in between? 3. Excitation slot width: Does the slot width have an impact on the reattachment process other than inﬂuencing the jet velocity?

References [1] Gad-el-Hak, M.: Flow control: The future. Journal of Aircraft Vol. 38 (2001) [2] Brunn, A., Nitsche, W.: Active control of the separated shear layer in an axisymmetric diffuser ﬂow by means of periodic forcing. Notes on Numerical Fluid Mech. and Multidisciplinary Design Vol. 87 (2004) pp. 335–342 [3] Schatz, M., Thiele, F., Petz, R., Nitsche, W.: Separation control by periodic excitation and its application to a high lift conﬁguration. AIAA Paper 04-2507 (2004) [4] Wassen, E., Thiele, F.: Separation control in diffuser ﬂow using periodic excitation. IN: New Results in Numerical and Experimental Fluid Mechanics IV, Notes on Numerical Fluid Mechanics and Multidisciplinary Design Vol. 87 (2004) pp. 327–334 [5] Feakins, S.H., MacMartin, D.G., Murray, R.M.: Dynamic separation control in a low-speed asymmetric diffuser with varying downstream boandary condition. AIAA Paper 03-4161 (2003) [6] Culley, D.E., Bright, M.M., Prahst, P.S., Strazisar, A.J.: Active ﬂow separation control of a stator vane using surface injection in a multistage compressor experiment. NASA TM212356 (2003) [7] Nishri, A., Wygnanski, I.: Effects of periodic excitation on turbulent ﬂow separation from a ﬂap. AIAA Journal 36 (1998) pp. 547–556 [8] Melton, L.P., Yao, C.S., Seifert, A.: Application of excitation from multiple locations on a simpliﬁed high-lift system. AIAA Paper 04-2324 (2004) [9] Melton, L.P., Yao, C.S., Seifert, A.: Active control of separation from the ﬂap of a supercritical airfoil. AIAA Journal Vol. 44 (2006) pp. 34–41 [10] Greenblatt, D.: Management of vortices trailing ﬂapped wings via separation control. AIAA Paper 05-0061 (2005) [11] Smith, B.L., Glezer, A.: Jet vectoring using synthetic jets. Journal of Fluid Mechanics 458 (2002) pp. 1–34 [12] Rapoport, D., Fono, I., Cohen, K., Seifert, A.: Closed-loop vectoring control of a turbulent jet using periodic excitation. Journal of Propulsion and Power Vol.19 (2003) pp. 646–654 [13] Seifert, A., Pack, L.G.: Oscillatory control of separation at high reynolds numbers. AIAA Journal 37 (1999) pp. 1062–1071 [14] Seifert, A., Greenblat, D., Wygnanski, I.: Active separation control: an overview of reynolds nad mach numbers effects. Aerospace Science and Technology Vol. 8 (2004) pp. 569–582 [15] Seifert, A., Pack, L.G.: Compressibility and excitation location effects on high reynolds numbers active separation control. Journal of Aircraft 40 (2003) pp. 110–126 [16] Yehoshua, T., Seifert, A.: Boundary condition effects on oscillatory momentum generators. AIAA Paper 03-3710 (2003) [17] Greenblatt, D., Wygnanski, I.: Effect of leading-edge curvature on airfoil separation control. Journal of Aircraft Vol.40 (2003) pp. 473–481

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[18] Seifert, A., Darabi, A., Wygnanski, I.: Delay of airfoil stall by periodic excitation. Journal of Aircraft 33 (1996) pp. 691–698 [19] Kjellgren, P., Anderberg, N., Wygnanski, I.: Download alleviation by periodic excitation on a typical tilt-rotor conﬁguration and experiment. AIAA Paper 00-2697 (2000) [20] McCormick, D.C., Lozyniak, S.A., MacMartin, D.G., Lorber, P.: Compact, high-power boundary layer separation control actuation development. ASME 2001-18279 (2001) [21] Gallas, Q., Holman, R., Nishida, T., Carroll, B., Sheplak, M., Cattafesta, L.: Lumped element modeling of piezoelectric-driven synthetic jet actuators. AIAA Journal Vol. 41 (2003) pp. 240–247 [22] Gilarranz, J.L., Traub, L.W., Rediniotis, O.K.: A new class of synthetic jet actuators - part i: Design, fabrication and bench top characterization. Journal of Fluids Engineering Vol. 127 (2005) pp. 367–376 [23] Lachowicz, J., Yao, C.S., Wlezien, R.W.: Scaling of an oscillatory ﬂow-control actuator. AIAA Paper 98-0330 (1998) [24] Tinapp, F., Nitsche, W.: On active control of high-lift ﬂow. In Rodi, W., Laurence, D., eds.: Engineering and Turbulence Modelling and Experiments 4. Elsevier Science Ltd. (1999) 619–626 [25] Petz, R., Nitsche, W.: Active separation control on a high-lift conﬁguration by a periodically pulsating jet. ICAS 2004-118 (2004) [26] Petz, R., Nitsche, W., Becker, R., King, R.: Lift, drag and moment control on a high-lift conﬁguration by means of active ﬂow control. In: CEAS/KATnet Conference on Key Aerodynamic Technologies. Number Paper no. 53, 20-22 June, Bremen, Germany, European Union, Deutsche Gesellschaft fr Luft- und Raumfahrt (2005) [27] Tinapp, F.: Aktive Kontrolle der Str¨omungsabl¨osung an einer Hochauftriebs-Konﬁguration. PhD thesis, Berlin University of Technology (2001) [28] Williams, D.R., Acharya, M., Bernhardt, J., Yang, P.: The mechanism of ﬂow control on a cylinder with the unsteady bleed technique. AIAA Paper 91-0039 (1991) [29] Becker, R., King, R., Petz, R., Nitsche, W.: Adaptive closed-loop separation control on a high-lift conﬁguration using extremum seeking. AIAA Paper 06-3493 (2006) [30] Petz, R., Nitsche, W.: Active control of ﬂow separation on a swept constant chord halfmodel in high-lift conﬁguration. AIAA Paper 06-3505 (2006) [31] Seifert, A., Bachar, T., Wygnanski, I., Kariv, A., Cohen, H., Yoeli, R.: Application of active separation control to a small uav. Journal of Aircraft Vol. 36 (1999) pp. 474–477 [32] Timor, I., Ben-Hamou, E., Guy, Y., Seifert, A.: Maneuvering aspects and 3d effects of active airfoil ﬂow control. AIAA Paper 04-2614 (2004) [33] Nagib, H., Kiedaisch, J., Wygnanski, I., Stalker, A., Wood, T., McVeigh, M.: First-in-ﬂight full-scale application of active ﬂow control: The xv-15 tiltrotor download reduction. In: Enhancement of NATO Military Flight Vehicle Performance by Management of Interacting Boundary Layer Transition and Separation. Paper no. 29, Prague, Czech Republic, Research and Technology Organization AVT-111 Specialists’ Meeting (2004) [34] Seifert, A., Eliahu, S., Greenblatt, D., Wygnanski, I.: Use of piezoelectric actuator for airfoil separation control. AIAA Journal Vol. 36 (1998) pp. 1535–1537

Closed-Loop Active Flow Control Systems: Actuators A. Seifert* School of Mechanical Engineering, Faculty of Engineering Tel-Aviv University, Tel-Aviv 69778, Israel [email protected]

Summary Closed-loop active flow control (CLAFC), the capability to estimate, efficiently alter and maintain a flow state, relies on the control authority of available actuators as a primary enabling technology. The requirements from the actuation systems are outlined and a critical review of available actuation technology is offered. Since the relevance of a given actuator depends on the application, separation control is considered over a wide range of operational conditions. Unsteady zero-net-mass-flux (ZNMF) periodic excitation was proven to be significantly more effective than steady blowing and simpler to apply than steady suction for the control of boundary layer separation. Furthermore, it can utilize flow instability as efficiency magnifier. When generated by Piezo-fluidic actuators, it has a bandwidth that is suitable for a wide range of feedback control applications. However, the current state-of-the-art ZNMF actuators lack, for certain applications, sufficient control authority. Therefore, effective methods for coupling the excitation to the most unstable modes of the flow should preferably be sought after and utilized. A new robust and simple actuator concept that combines steady suction and pulsed blowing is presented. It can generate wide band-width near sonic oscillations. Its performance was modeled and validated in several scales. The valve design allows highly efficient operation, not nullifying the favorable effects of future CLAFC schemes. Three-dimensional (3D) excitation modes should be explored, as the flow naturally becomes 3D even if the baseline flow and the excitation are nominally twodimensional (2D). To be for industrial applications, overall system efficiency should always be assessed, not only the improvement in aerodynamic performance. Three performance based criteria for comparing different actuation concepts are presented and discussed. The first criterion evaluates the actuator based on its force or linear momentum generation capability as it operates in still fluid, while considering its weight, volume and power consumption. The second criterion is simply the actuator peak velocity Mach number relative to the free-stream Mach number, where it is rare to find any benefit from actuator with Mach ratio smaller than 0.1 and/or momentum coefficient smaller than 0.01%. The third, application dependent, criterion is the Aerodynamic figure of merit, an energy efficiency criterion, based on the improvement of the controlled performance (e.g., lift to drag ratio) of a certain *

Senior lecturer of Mech. Eng., member EUROMECH.

R. King (Ed.): Active Flow Control, NNFM 95, pp. 85–102, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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application, when the power consumption (and also the weight) of the actuation system are taken into account.

1 Background and Motivation Active flow control (AFC) is a fast growing multidisciplinary science and technology thrust aimed at altering a natural flow state or development path into a more desired state (or path). Flow control was simultaneously introduced with the boundary layer concept by Prandtl [1] at the turn of the 20th century. In the period leading to and during World-War-Two, as well as in the Cold War era, flow control was extensively studied and applied mainly to military fluid related systems. A comprehensive historical review and analysis was provided by Lachmann [2] and more recently by Gad-el-Hak [3, 4]. All known flow control efforts preceding the pioneering work of Schubauer and Skramstad [5] used steady-state tools and mechanisms for flow management. Though the fluid mechanics aspect can be robust, steady flow control methods proven to be of inherently marginal power efficiency, and therefore limited the implementation of the resulting systems. Unsteady flow control using periodic excitation and utilizing flow instability phenomena (such as the control of flow separation [6]) has the potential of overcoming the efficiency barrier. The example shown in Fig. 1 demonstrates that separation control using periodic excitation at a reduced frequency of the same order, but higher than the natural vortex shedding frequency, can save 90 to 99% of the momentum required to obtain similar gains in performance, as compared to steady blowing. The latter utilizes the well known Coanda [7] effect to reattach the massively separated flow. The feasibility of increasing the efficiency and simplifying fluid related systems (e.g. for high-lift) is very appealing if one considers that 1% saving in world consumption of commercial jet fuel is worth more than $1 millions a day of direct operating costs, while the environmental effects are difficult to quantify. The progress in miniaturization, actuators, sensors, simulation techniques and system integration enables the utilization of wide bandwidth unsteady flow control methods in a closed-loop AFC (CLAFC) system. However promising the technology might look, significant barriers exist between the capabilities available to the technologist and the successful field application. Comprehensive experiments are required to close the gap between our theoretical understanding and our computational capabilities and real-world problems. Theory of AFC is of limited scope due to the inherently non-linear nature of the underlying physical processes, but should still provide physical insight, directions for computational studies and design guidelines. Identifying the above limitations, this manuscript does not attempt to review the AFC nor the CLAFC state-of-the-art (e.g. see [8]). Rather, an attempt is made to critically review and identify future directions in one key issue, i.e., actuators, and discuss it in a rather comprehensive manner. Naturally and conveniently, the review will focus on activities conducted at Tel-Aviv University and its collaborators, and therefore is not a comprehensive literature review.

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10

Fig. 1. Recovery of effective lift at post-stall condition on a NACA 0015 airfoil using steady blowing or periodic excitation with different magnitudes (Cµ) applied at the leading edge [6]. Empty squares; steady, filled diamonds; oscillatory.

2 Review of Actuation Methods, Benefits and Limitations 2.1 Actuation System Properties and Comparison Criteria A prerequisite for comparing actuation systems is identifying clear operational criteria. Even the output to be compared is difficult and somewhat vague to identify. The fluidic output can be a local or a global velocity fluctuation, unsteady momentum injection and mass removal, vorticity flux and other flow properties. The independent parameters could be size, weight, energy consumption, cost, maintenance, robustness, life span, band-width, compatibility and complexity. Additional parameters to compare are sensitivity to secondary parameters as temperature, humidity, dust as well as environmental impacts such as heat, noise, electro-magnetic emissions and light radiation. A possible measure of an actuator output could be its thrust, measured while the actuator is operating in still fluid (air is presently considered), divided by its weight times its energy consumption. An estimate of the overall energy efficiency (considering the signal generation and amplification system) should preferably be provided along with the estimated additional weight associated with the signal generation and amplification system, based on current technology. If thrust is not measured, it could be estimated by the ejected fluid impulse. A candidate actuator dimensionless robustness and efficiency performance parameter, termed Overall Figure of Merit might be:

OFM ≡

Fa2U p Wa P

(1)

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In the above formulae,

Wa is the (preferably) total weight of the actuation system or

of the actuator alone, P is the total actuation system power consumption or just the power delivered to the actuator, U p is the peak or typical actuator output velocity and

Fa is the thrust generated by the actuator when operating in still fluid. The

actuator thrust might be replaced by: Fa

≈ CρAaU p2 , where Aa is a typical cross

section from which excitation is affected (e.g., cross section of a slot). Scaling arguments should be provided to guide the evaluation of the constant C. Its value should be between 0.5 to 0.25 accounting for the blowing stage only and for the shape of the time dependent blowing velocity profile. When searching for a suitable actuator for a given application, a candidate applicability criterion is simply the actuator peak velocity Mach number relative to the free-stream Mach number, where it is rare to find performance benefits from actuator with Mach ratio smaller than 0.1 and/or oscillatory momentum coefficient smaller than 0.01%. In addition, those who applied the actuator for the delay of boundary layer separation may find it appropriate to use the first Aerodynamic Figure of Merit (AFM1) defined as [9]:

U∞L AFM 1 ≡

(U ∞ D + P)

(L D )

(2 )

baseline

Where, L is the lift, D is the drag, the subscript baseline refers to uncontrolled-natural conditions and P is the power provided to the actuators. It should be indicated if form or total drag was measured and how and also if total system power consumption or only actuator power is reported. A second Aerodynamic Figure of Merit (AFM2) might be defined as well, where the weight of the actuator ( Wa ) is taken off the lift of the controlled case:

U ∞ (L − Wa ) AFM 2 ≡

(L D )

(U ∞ D + P)

(3)

baseline

This approach is not always justified because in certain applications the actuators replace or at least reduce the weight of an existing system, e.g., simplified high-lift system [27], so Wa should be the net weight change which could be optimally negative. Additionally, an economy actuator efficiency parameter could be defined. A possible definition can be the inverse of the product of the fractional savings in direct operation costs for the entire system life multiplied by the fractional change in system cost. This however is application dependent and certainly beyond the scope of this paper.

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2.2 External Actuation Systems Early studies of active separation control utilized external actuation systems for the production of periodic excitation [25-27, 29 and many more]. This approach was justified at the time, and is still justified today in limited scope flow-physics related studies, because progress had to be made on the route to real-time, real-world AFC application before the massive investment in flight worthy actuator development could be justified. The limited budget (as always), time and space constrains forced the pioneering AFC researcher to use whatever was available to generate oscillatory flow excitation. Two of the prominent methods, e.g., speakers and oscillatory blowing actuators are reviewed below. Speaker Systems Speakers were the first devices used to generate acoustic excitation [42, 45] in wind tunnel test sections. These studies were initially aimed at either acoustic receptivity, transition promotion and later for separation control, due to the tight connection between transition, separation and reattachment of low Reynolds numbers boundary layers. The study of Zaman et al, (1987 [42]), clearly demonstrated the limitations of such approaches for the performance of boundary layer separation control experiments. The proof that standing waves in the test section were actually responsible for the resonating nature of the flow response to the acoustic excitation emanating from speakers installed on the wind tunnel walls and the lack of capability to generate such a phenomenon in flight ruled out external speakers as viable actuation system. An additional use for speakers (voice coil devices or more generally electromagnetic actuators or even shakers) was to use the large surface membranes, coupled with finite displacement to change the volume of a cavity. This cavity could be placed outside the airfoil and the wind tunnel (as in [43,44,48,49]), where the pressure waves are routed through a conduit or flexible tube into an additional cavity inside the airfoil. This cavity, in turn, is communicating the pressure oscillations through a slot or array of holes to the controlled boundary layer. This approach could create significant amplitude and phase variations across the span of the airfoil for frequencies that are higher than quarter the acoustic wave length normalized by the airfoil span according to f max ≤

a , where a is the speed of sound and b is the 4b

airfoil span. If the available internal volume is large enough, if a half-span model is used or a diffuser is tested, the voice coils can be inserted into the geometry to be controlled or interact with the cavity in a more span-uniform manner [16,38,39,40,41]. Generally, voice-coil electro-magnetic actuators are too large, heavy, expensive and energy consuming to be considered for fielded flow control systems. The loading on the membranes cause increased fatigue and frequent failure. Nevertheless, the first successful full-scale flight demonstration of AFC was performed using such actuation system [38].

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Oscillatory Blowing Actuators Oscillatory blowing actuators are a class of devices with a common operational principle, i.e., alternatively opening and closing the connection between a pressure source(s) and a cavity leading to a slot or opening and closing connections between a pressurized and a partially vacuumed chambers and a cavity. This class of devices was successfully applied during the first high Reynolds numbers demonstration of AFC [27]. The experimental set-up at the Langley 0.3m cryogenic tunnel was capable of zero-mass-flux excitation due to the pressurized tunnel. Figure 2 shows the set-up. By adjusting the inlet pressure and exit resistance, it was possible to achieve zeromass flux. The amplitude and phase variations across the 13" wide test section at a frequency of 350 Hz were negligible, even though the excitation was introduced into a span-uniform cavity in the NACA0015 airfoil from one side only. An additional innovative use of oscillatory blowing valve was its application to a small UAV [12]. Figure 3 below shows the experimental set-up used for both large scale wind tunnel and flight tests of the same vehicle. An axial fan ingested flow into a duct leading into a "tube and barrel" assembly. The barrel had two openings. One at the front, to allow the air to enter and one on the circumference, rotating and getting into and out of alignment with two stationary openings, each leading to a cavity in one of the wings. This way, the flow entering the valve was switched between the right and left wings at a rate of 50-70Hz. These cavities were open to the external flow through the slots at the flap shoulder, as seen in Fig. 3.

Fig. 2. A sketch of the experimental set-up at the 0.3m TCT with the oscillatory blowing valve on one side of the airfoil, while the other side of the airfoil cavity is vented to the atmosphere [27]

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Fig. 3. The use of oscillatory blowing valve to switch the flow sucked into the fan inlet between the right and left wings of a small AFC UAV [Seifert et al, 99, Ref. 12]

Yet another class of oscillatory blowing actuators, such as commercial pneumatic valves, adopted to AFC applications or Piston actuators, similar to internal combustion engines that are driven by electric motors [36]. Both concepts might be suitable for lab applications, but are too heavy, space consuming and low efficiency to be considered for flight worthy systems. 2.3 Internal Actuation Systems Surface Mounted Piezo “Benders” Surface mounted, mechanical actuators rely on the interaction with the incoming boundary layer shear to generate the excitation. An experiment utilizing ten (10) segmented Piezo benders placed across the span of a 2D airfoil, (Fig. 4a) was performed and reported by Seifert et al (1998 [9]). The amplitude and phase of the benders was individually controlled, and a cross section and top view of the installation on a low Reynolds airfoil developed by Israeli Aircraft Industries is shown in Figure 4b. Several points are worth mentioning. The fluidic output of the mechanical actuator, interacting with an incoming turbulent boundary layer (TBL) is limited by the available reservoir (i.e., the mean shear). The actuator's free-end velocity is small compared to the fluidic velocities required for flow control (e.g., at least 0.1 U ∞ ). It was demonstrates [9] that the peak RMS velocity fluctuations close to the wall are about 0.2 U ∞ when the actuator’s tip amplitude was about 3mm, spanning the entire high shear region of the TBL. This level of excitation is marginal at best. Additional complications limit the utilization of surface mounted actuators in

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general, and Piezo benders in particular. However, an interesting feature that was utilized in the course of that investigation was the operation of the actuators out of phase (i.e., π radians from each other). This mode of operation was found to consume only 25% of the power required to operate all actuators “in phase”, presumably due to compression of the cavity air under the actuators (Fig. 4b) that assists the upwards motion of every second actuator as its neighbors are moving down. The resulting liftdrag ratio [9] indicated that the 2D mode (i.e., uniform phase and amplitude across the entire span) is slightly superior to the three-dimensional (3D, i.e., alternating phase in the current case) mode in terms of aerodynamic performance. However, the fact that the generation of the 3D mode consumed only 25% of the power, led to an earlier achievement of larger than unity “Figure of Merit”, (AFM1, Eq. 2; [12]). The meaning of AFM1>1 indicates that it is more energy efficient to introduce power to the actuators, in order to improve L/D, rather than to the powerplant, to overcome drag. Besides the important applicability aspect, the fundamental nature of the interaction between the multiple modes of excitation resulting from the complex 3D actuator motion indicates the importance and relevance of studying 3D instability modes to enhance AFC effectiveness and efficiency. Margalit et al. [10] also cite AFM1>1 data using cavity installed internal Piezo fluidic actuators applied to control the flow over a delta wing at high incidence. It is highly desirable that the research associated with actuator development and utilization will include comprehensive measurement and reporting of the energetic and environmental impact of the actuator under consideration to supplement fluidic output characterization.

Fig. 4. The application of 10 surface mounted Piezo "benders" for separation control on the IAI-Pr8 airfoil (Seifert et al, 1998 [9]). Bottom right figure shows AFM1>1

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Piezo fluidic actuators Cavity based fluidic actuators are a class of devices capable of producing oscillatory flow excitation [35]. These devices contain three elements; pressure fluctuations generating mechanism a cavity and a slot, communicating the cavity with the flow to be controlled. Though many concepts have been suggested for generating the cavity pressure oscillations (e.g., a loudspeaker, a piston, a diaphragm, oscillating materials, pulsed heat or mass injection and removal), most of these concepts are impractical, due to large size, weight, compatibility issues and energy consumption. In the current application, the cavity pressure fluctuations are generated by an oscillating Piezoelectric elements that form at least one of the cavity walls and are driven by an electric AC signal. Consequently, velocity oscillations result at the actuator’s exit slot. The Helmholtz resonance mechanism can be utilized to generate larger velocities than one could obtain only due to volumetric cavity changes, but over a narrow frequency range. Higher magnitude velocity fluctuations could be achieved at the mechanical resonance frequency of the actuators, though at a narrower frequency range. The operation at the mechanical resonance is undesirable also from damage tolerance aspect and uniformity of multiple actuator application point of views. The problem at hand (“system”) contains two major components leading to the following groups of parameters: The actuator geometry and the actuator operating conditions. The geometry of the cavity is of lesser importance for the current discussion, but the neck geometry is of significant importance [34], determining the shape of the slot exit velocity profile. This is because, the slot vorticity flux during the blowing stage is crucial to the evolution of the external flow field. An application dependent optimum should be found between peak ejection velocity, maximum momentum and the largest vorticity flux out of the actuator's slot.

Fig. 5. Schematic structure of a generic oscillatory vorticity generator (OVG) configuration with leading parameters [32,33]

The main geometrical slot-exit boundary conditions and operational parameters of the 2D generic OVG configuration are shown in Fig. 5. Note that 3D effects are not considered here and that the exterior lines are treated as semi-infinite boundaries. As

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mentioned before, the actuator contains three main components – oscillating element(s), a cavity and a slot. Here, the cavity and the oscillating components are excluded from the dimensional considerations, since the current emphasis is on the relationship between the periodic pressure oscillation and the velocity oscillation at the slot and especially the external flow, and not on the mechanism that creates the pressure oscillations inside the cavity. Given that the flow is periodic, and for a pure sine wave excitation, we can write the slot exit velocity, Vs(r, t), as

~

Vs(r, t) = Vs(r)·Sin(2πft) = Up· V (r)·Sin(2πft)

(4)

Where: Up is the peak magnitude of the velocity profile at the slot (in both space and

~

time) and V (r) is a normalized velocity vector distribution, r is a location vector and t is time. For order of magnitude analysis, the velocity profile can be treated as a "tophat", while for slots and operational conditions dominated by viscosity, it could be considered parabolic. Therefore, the slot velocity, for a given geometry, is a function of the excitation frequency, f, the kinematic viscosity and time. If we apply Buckingham's Pi theorem on the parameters indicated in Fig.5 we obtain: (5) F (h, b1, H1, α1, β1, b2, H2, α2, β2, f, Up, ρ, µ) = 0 Since we have nine dimensional parameters, a group of six independent dimensionless parameters can be formed (angles are already dimensionless). A possible set is: (6)

§b b H H F¨ 1 , 2 , 1 , 2 , ¨h h h h ©

· fh 2 U P h , , α 1 , α 2 , β1 , β 2 ¸ = 0 ¸ v ν ¹

The above dimensionless parameters can be divided into two groups: The boundary conditions group:

§ b1 b2 H 1 H 2 · , , α 1 , α 2 , β1 , β 2 ¸ ¨ , , ©h h h h ¹

§ and the excitation parameters group: ¨ ¨ ©

fh 2 U P h ·¸ , ν v ¸¹

(7)

(8)

The excitation parameters group includes, the excitation Reynolds number, based on the peak velocity at the slot and its width, in the form: Re ≡ U p h ν . The Stokes number, defined as:

S tk ≡

fh 2

ν

, provides a measure of the slot unsteady boundary

layers penetration into the core of the theoretically “top hat” velocity distribution. When the actuator configuration is fixed, this number is simply related to the square

Closed-Loop Active Flow Control Systems: Actuators 2

root of the excitation frequency. The ratio S tk form:

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Re , is the Strouhal number in the

S tr ≡ fh U p , which can be treated as a dimensionless excitation frequency-

amplitude parameter. Since the slot “neck” length (b1=b2 in the current configuration) used in common devices is significantly smaller than the unsteady wavelength, this parameter is important only outside the slot. The Strouhal number could also be viewed as a ratio of length scales, the excitation wavelength and the ejected fluid typical convection distance, which would become important while considering the resulting external flow field. For S tk approaching zero, the unsteady effects are negligible, and for sufficiently low Re numbers, the flow at the actuator's neck would resemble a plane Poisille flow. As S tk increases, the unsteady effects become important, and the velocity profile at the actuator's neck would become more like a top-hat profile (with an overshoot and a phase lag at the unsteady boundary layers forming at the neck). The velocity slug Reynolds number is probably the least representative Reynolds number of the problem at hand; however it is used for lack of a better, conveniently measured Reynolds number. The boundary conditions group also plays a significant role in the resulting flow field, influencing, for instance, vortex pair circulation and asymmetry. The actuator's "lips" width to slot width ratio, Hi/h, would most likely play a significant role in the suction process, influencing the flow field at the vicinity of the slot, together with the angles setup of the actuator. It is well known that suction and blowing are different processes in viscous fluid flow, where flow separation could take place. The same fluid mass leaving the actuator has to be entrained back into its cavity during the suction stage. Since sink flow is less affected than source (jet) flow by the details of the boundary conditions at the slot, the difference between the suction and blowing stages of the excitation cycle is expected to be strongly dependent on the boundary condition [32,33]. Piezo fluidic actuators have been successfully developed and applied during the last decade. Flight demonstration has not been performed yet, to the best knowledge of the author. It is expected that with proper design, careful fabrication and effective implementation, Piezo fluidic actuators will become a valuable tool for the AFC practitioners. Current trends demonstrate peak exit velocities reaching M=0.3, deemed sufficient for many AFC applications. To date, the Piezo fluidic actuators were the only ones to demonstrate AFM>1, still in laboratory experiments. There are many AFC applications where Piezo fluidic actuators are not compatible or suitable, or lack sufficient control authority. This also holds where there is no internal volume to use or the system is already in existence. In these cases surface mounted actuators is the way to go. In compressible flow and especially in supersonic flight speeds it is not expected that peak excitation Mach number with Mpeak=0.3 would suffice. This calls for further development of more robust actuation technology. 2.4 Additional Actuator Concepts Many other types of actuators are currently at different levels of development and application stages. A partial list and brief discussion follows. This is by no means a comprehensive review of the state-of-the-art or current trends.

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Combustion jet actuators are based on subsonic wave combustion process, and operate at a time scale much greater then the time scale of the wave expansion within the device. Their operation is similar to a small pulsejet engines or pulsed detonation engines [18]. They consist of an internal combustion chamber, an inlet valve that periodically introduces mixed fuel and Oxygen into the combustion chamber, an igniter to initiate the combustion process and exit slot to communicate the hot combustion gases to the external flow. Combustion jets actuators are capable of providing high impulse at a relatively wide frequency range (about 100Hz demonstrated to date) and therefore they can provide good solution to boundary layer separation control at high speeds and large scales, where such low frequencies might be adequate [20]. The main disadvantage of such devices is the need to use high temperature, exotic materials, the internal combustion process that consumes fuel, requires a pre-mixing process and a mixing device, complexity, safety, installation, weight and compatibility concerns. The Hartmann tube actuator is a device that uses shock wave oscillations to produce high amplitude sound waves. The phenomenon was discovered By Julius Hartmann in 1919 while using a Pitot probe in a compression region of a jet. The use of high intensity sound waves for flow control application was the focus of many studies, and shown to provide a limited control authority at low-speeds [21]. The combination of high-frequency-high-intensity pressure waves, may be suitable for high-speed flowcontrol- applications where marginal control authority may be sufficient. However, the receptivity of a separating turbulent boundary layer to sound waves is rather low. These devices are usually large and heavy. The installation issues of this class of actuators in an aerodynamic surface have not been assessed yet. The fundamental operational principle of plasma actuators is imposing kinetic energy on a limited region inside the boundary layer using a body-force. Methods of plasma generation include DC, AC, RF, microwave, arc, corona and spark electric discharge. Labergue et al (2004, [22]) explained that, for the case of a DC corona discharge established between two electrodes flush mounted along a non-conducting surface in air at atmospheric pressure, ions are produced at the anode and electrons at the cathode (see Fig. 6a). In their drift motion between the two electrodes under electrostatic Coulomb forces, these ions exchange momentum with neutral particles and induce an airflow termed “ionic wind”. The common plasma actuators operate in a similar manner. The above described process creates steady or pulsed near-wall velocity perturbations, which can be used to modify the near-wall flow. The amplitude of the plasma actuators depends on the supplied current and voltage and the fluid properties. Temperature has a strong effect on the actuator performance. Plasma actuators were proven to be efficient in high-speed cold jet [23], where the receptivity to the excitation is almost singular, and in boundary layer control at low-speeds [24]. One of the shortcomings of plasma actuators is the need to generate high voltage; accompanied with energy losses and excessive weight. Another disadvantage of plasma actuators is the sensitivity to the controlled fluid properties (e.g., humidity can cause dramatic changes in the efficiency and lead to damage), emitted light and electromagnetic radiation.

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(a)

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(b)

Fig. 6. (a)Sketch of the Plasma actuator operation principle. [31]. (b) Sketch of the Spark jet actuator. [30].

The Spark jet actuator [30] is an actuator concept, similar to the combustion jet actuator, only that the entire energy deposition into the actuators cavity is due to an electric spark generated between two electrodes situated inside a small cavity. This type of actuator is under study for quite some time. The energy deposited by the spark into the cavity increases its temperature and pressure, causing the heated compressed gas to exit the cavity through a hole. This process have been demonstrated to be capable of generating supersonic exit speeds and been numerically simulated as well. However, it suffers from similar disadvantages as the combustion based actuator, namely heating, exotic materials, rapid degradation and compatibility issues. It has not been demonstrated in repeated operation at any applicable rate and has not been applied yet for flow control applications. Suction and oscillatory blowing actuator A new actuator for active flow control (AFC) applications is under development at the Tel-Aviv University, Meadow Aerodynamics laboratory. The new fluidic device combines steady suction and oscillatory blowing, both proven to be very effective for AFC (US Patent 2006-0048829-A1). The New fluidic device is a combination of a bi-stable fluidic amplifier and an ejector (Fig. 7). The bi-stable device is based on the principle of wall attachment (“Coanda effect”) to one of the two walls of a diverging diffuser. When a fluid jet is flowing in the proximity of a wall, a low-pressure region is formed between the jet and the wall. The low pressure draws the jet towards the wall, the jet will deflect and eventually become attached to the wall. In the case of two close and symmetric walls, such as in the configuration shown in Figure 7, the jet will randomly re-attach to one wall. If a pressure pulse is introduced at one of the control ports (control R or control L in Fig. 7) the jet will detach from that wall and reattach to the opposite wall, due to the two marginally stable conditions it can be in. The bi-stable fluidic amplifier can self-oscillate by connecting the two control ports by a tube. In this configuration the oscillation frequency is related to the tube length, volume, and resistance, the speed of the sound wave in the tube and the flow rate through the actuator’s main path. A detailed model of these aspects is provided in an accompanying paper [28]. The frequency of the valve, when operating in selfoscillating mode, can be adjusted by the length, diameter and resistance of the tube connecting the two control ports (Fig, 7).

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For active flow control use, and especially during the actuator design and development stages, there is a need to control the flow rate (which would dictate the magnitude of the external excitation) and the oscillation frequency. Therefore, there is a need to produce controlled frequency oscillatory pulses in the control ports. This can be done in many ways; piezoelectric actuators, spinning valve, solenoid valves and other devices and methods. The ejector is a simple fluidic device based on Bernoulli’s principle. When a jet stream expands into larger cylindrical cavity, it creates a low pressure region around it. If the cavity behind the jet is open to the free atmosphere or to a higher pressure environment (such as the surface of an airfoil) the pressure gradient will cause the external air to be sucked into the cavity around the internal jet. The purpose of the ejector in the current device is to increase the flow rate. However, if the fluid would be extracted from an aerodynamic surface it would create suction flow across the aerodynamic surface through slots or holes. The proposed fluidic actuator is intended to function over a wide range of frequencies and flow rates. In the self-oscillating mode, there are no moving or active elements while the frequency and amplitudes are coupled, which is desired from an applicability point of view. Furthermore, the distribution of suction and oscillatory-blowing over the surface to be controlled offer many intriguing combinations for effective AFC.

Fig. 7. Schematic of the suction and oscillatory blowing actuator

A detailed model and performance characteristics are presented in an accompanying paper (Arwatz et al, 2006, [28]). It was demonstrated that the new actuator is capable of near sonic output velocities and frequencies of order 1kHz. It is insensitive to exit load variations and demonstrated extremely efficient operation. A small scale (~2cm) device is being tested and its application for separation control is due next.

3 Actuation Systems Performance Comparison A detailed comparison of different actuators and actuation schemes, based on the three criteria suggested above was attempted. However, very little data is available in the open literature that is required for the calculations of the performance criteria suggested above. Only through private communication with the IIT group were we

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able to obtain sufficient information for comparison of TAU developed actuators with the actuators used for the flight tests of the XV-15 (Nagib et al, [38]). Further information that is provided below is based on conservative evaluations by the author and can be adjusted in the future when more information becomes available. The Overall Figure of Merit (OFM) compares the actuator performance as it is operated in still air, taking its fluidic output, its weight and power consumption into account. For the actuators used during the XV-15 fight tests and at the flight test conditions, the OFM=0.014 with Up=80.4m/s. Comparing this number to those of the TAU applied Piezo fluidic actuators results in OFM=0.052 (Up=20m/s, Margalit et al, [10]) and OFM=0.107 (Up=40m/s, Timor et al, [11]). For the compact actuator used by Yehoshua and Seifert (Up=60m/s, [32,33]) the OFM=0.295. However, the latest actuator was tested in free air and was not restricted by installation considerations. Note also the different peak slot velocities in the above comparisons. An estimation of the OFM for plasma actuators such as those used in [24,31], results in OFM<1.0x10-4. This was evaluated for Up=3.5m/s, a region of influence extending 1mm above the electrode, a total actuator weight of about 4gm/meter and steady-state operation. The Aerodynamic Figure of Merit (AFM1 above) was available only in two TAU publications, where values above unity were reported [10,12].

4 Summary, Conclusions and Recommendations A critical review of past, present and future actuation technology and approaches have been conducted. Three criteria for selecting and comparing actuators have been proposed, mainly focusing on boundary layer separation control. It is now becoming evident that AFC experiments should include energetic characterization of the actuators and actuation system as a whole. Considerations such as weight, volume, cost, and compatibility should be addressed. From the actuation concepts on which sufficient information is available, it is clear that Piezo-fluidic technology is significantly superior to electro-magnetic and two-three orders of magnitude more efficient than plasma actuators. Piezo-fluidic actuators were the only to report greater than unity aerodynamic figure of merit to date, still in laboratory experiments.

Acknowledgment The author would like to thank the entire staff of the TAU Meadow Aerolab, especially Ilan Fono and Shlomo Paster for the major contributions during the years. Personal communications with H. Nagib, J. Kiedaisch and B. Göksel are acknowledged.

References 1. Prandtl, L., “Motion of Fluids with Very Little Viscosity," English Translation of ”Uber Flussigkeitsbewegug bei sehr kleiner Reibung", Third International Congress of Mathematicians at Heidelberg, 1904, from Vier Abhandlungen zur Hydro-dynamik und Aerodynamik", pp. 1-8, Gottingen, 1927, NACA TM-452, March 1928.

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2. Lachman, G. V. (Ed.) Boundary Layer and Flow Control. Vol. 1 and 2, Pergamon Press. 1961 3. Gad-el-Hak, M., Pollard, A., Bonnet, J. P. (Eds.) Flow Control: Fundamentals and Practices, Springer-Verlag, New York, 1998, 527 pp. 4. Gad-el-Hak, M. Flow Control: Passive, Active, and Reactive Flow Management, Cambridge University Press, Cambridge, 2000, 421 pp. 5. Schubauer G.B. and Skramstad H.K., Laminar boundary layer oscillations and stability of laminar flow. Journal of the Aeronautical Sciences, Vol. 14, No. 2, 1947, pp. 69-78. 6. Seifert, A., Darabi, A. and Wygnanski, I., 1996, “Delay of Airfoil Stall by Periodic Excitation”, J. of Aircraft. Vol. 33, No. 4, pp. 691-699. 7. Newman, B,G, “The deflection of plane jets by adjacent boundaries – Coanda effect”, in Lachmann, Boundary layer and Flow Control, V. 1, Pergamon Press, 1961 8. Collis, S.S., Joslin, R.D, Seifert, A. and Theofilis, V., “Issues in active flow control: theory, simulation and experiment”, Prog. Aero Sci., V40, N4-5, May-July 2004 (previously AIAA paper 2002-3277). 9. Seifert, A., Eliahu, S., Greenblatt, D. and Wygnanski, I., 1998, “Use of Piezoelectric Actuators for Airfoil Separation Control (TN)”, AIAA J., Vol. 36, No. 8, pp. 1535-1537. 10. Margalit, S., Greenblatt, D., Seifert A. and Wygnanski, I., “Delta Wing Stall and Roll Control using Segmented Piezoelectric Fluidic Actuators”, (previously AIAA paper 20023270), AIAA J. of Aircraft, May-June 2004. 11. Timor, I., Ben-Hamou, E., Guy, Y. and Seifert, A., “Maneuvering Aspects and 3D Effects of Active Airfoil Flow Control”, AIAA paper 2004-2614, June, 2004. Combustion, Turbulence and Control, to be published. 12. Seifert, A., Bachar, T., Wygnanski, I., Kariv, A., Cohen, H. and Yoeli, R., “Application of Active Separation Control to a Small UAV”, J. Aircraft, Vol. 36, No. 2, March-April 1999, pp. 474-477. 13. Yom-Tov, Y., Dabush, E., Belson, R. and Seifert, A., ”Roll control of an MAV using active flow control, Paper presented at the 43rd ISR Aero conf., Feb. 2003 14. Lee, G.-B., Shih, C., Tai, Y.-C., Tsao, T., Liu C., Huang, A., and Ho, C.-M., “Robust Vortex Control of a Delta Wing Using Distributed MEMS Actuators,” Journal of Aircraft, Vol. 37, No. 4, pp. 697-706, July-Aug 2000. 15. A. Naim, A. Seifert and I. Wygnanski, “Active Control of Cylinder Flow With and Without a Splitter Plate Using Piezoelectric Actuators”, AIAA Paper 2002-3070, June 2002. 16. Seifert, A. and Pack, L.G., “Active Control of Separated Flow on a Wall-mounted “Hump” at High Reynolds Numbers”, AIAA J., V. 40, No. 7, July, 2002, pp. 1363-1372. (Part of AIAA paper 99-3430). 17. Huerre, P. and Monkewitz, P. “Local and Global Instabilities in Spatial Developing Flows,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 473-537. 18. Kailasanath, K., “Review of Propulsion Applications of Detonation Waves”, AIAA Journal, Vol. 38, No. 9, 2000, pp. 1698–1708. 19. Cutler, A.D., Beck, B.T., Wilkes, J.A., Drummond, P.J., Alderfer, W.D., Paul, M. and Danehy, M.P.,” Development of a Pulsed Combustion Actuator for High-Speed Flow Control”, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, AIAA 2005-1084, January 10-13, 2005, pp. 16. 20. Ahuja, K.K. and Burrin, R .H., “Control of Flow Separation by Sound”, AIAA Paper-84 2298, Oct. 1984.

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21. Kastner, J. and Samimy, M., "Development and Characterization of Hartmann Tube Fluidic Actuators for High-Speed Flow Control”, AIAA Journal, Vol. 40, No. 10, 2002, pp. 1926-1934. 22. Labergue, A., leger, L., Moreau, E. and Touchard, G., ”Effect of a Plasma Actuator on an Airflow along an Inclined Wall - P.I.V. and Wall Pressure Measurements”, Journal of Electrostatics, 10th international Conference on electrostatics 2005 June 15-17 Espool/Helsinki Finland. 23. Samimy, M., Adamovich, L., Webb, B., Kastner, J., Hileman, J., Keshav, S. and Palm, P., “Development and characterization of plasma actuators for high-speed jet control”, Experiments in Fluids, Vol.37, 2004, pp. 577–588. 24. Post, M.L. and Corke, T.C., ”Separation control on high angle of attack airfoil using plasma actuators”, AIAA Journal, Vol. 42, No. 11, 2004, pp. 2177-2182. 25. Seifert, A., Bachar, T., Koss, D., Shepshelovits, M. and Wygnanski, I., 1993, "Oscillatory Blowing, a Tool to Delay Boundary Layer Separation", AIAA J. Vol. 31, No. 11, pp. 20522060. 26. Seifert, A., Darabi, A. and Wygnanski, I., 1996, “Delay of Airfoil Stall by Periodic Excitation”, J. of Aircraft. Vol. 33, No. 4, pp. 691-699. 27. Seifert, A. and Pack, L.G., “Oscillatory Control of Separation at High Reynolds Numbers”, AIAA J. Vol. 37, No. 9, Sep. 1999, pp. 1062-1071. 28. Gilad Arwatz, Ilan Fono and Avi Seifert, "Suction and Oscillatory Blowing Actuator", Paper presented at the IUTAM MEMS and AFC meeting, London, September, 2006. 29. M. Hites, H. Nagib, T. Bachar and I. Wygnanski, "Enhanced Performance of Airfoils at Moderate Mach Numbers Using Zero-Mass Flux Pulsed Blowing," AIAA Paper 20010734, 2001. 30. B. Cybyk, K. Grossman and J. Wilkerson, "Single-Pulse Performance of the SparkJet Flow Control Actuator", AIAA paper 2005-0401, 2005. 31. Göksel, B., Rechenberg, I. (2004) Active Separation Flow Control Experiments in Weakly Ionized Air. In Andersson H. I. and Krogstad P.-Å. (eds.) Advances in Turbulence X, Proceedings of the 10th Euromech European Turbulence Conference, CIMNE, Barcelona, 2004. 32. Yehoshua, T. and Seifert, A. “Active boundary layer tripping using oscillatory vorticity generator”, Aerospace Science and Technology, 10 (3): 175-180 APR 2006. 33. Yehoshua, T. and Seifert, A. “Boundary Condition Effects on the Evolution of a Train of Vortex Pairs in Still Air”, Aeronautical J., 110 (1109): 397-417 JUL 2006. 34. Lee, C Y. and Goldstein D B., “Two-Dimensional Synthetic Jet Simulation”, AIAA Journal, V.40 ,N.3, March 2002. 35. Glezer, A. and Amitay, M., “Synthetic Jets”, Annu. Rev. Fluid Mech., Vol. 34, 2002, pp. 503-529. 36. Gilarranz, J. L., Traub, L. W. and Rediniotis, O.K., "A New Class of Synthetic Jet Actuators—Part I: Design, Fabrication and Bench Top Characterization", Journal of Fluids Engineering, March 2005, Vol. 127 p 367.-376. 37. M. Schatz, F. Thiele, R. Petz, W. Nitsche, "Separation Control by Periodic excitation and its Application to a High Lift Configuration, AIAA Paper 2004-2507, 2004. 38. Nagib, H.M., Kiedaisch, J.W., Wygnanski, I.J., Stalker, A.D., Wood, T. and McVeigh, M.A., "First-In-Flight Full-Scale Application of Active Flow Control: The XV-15 Tiltrotor Download Reduction", RTO-MP-AVT-111, 2005, see also: http://fdrc.iit.edu/research/ docs/MAFC_XV_15_Briefing_Final.pdf

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39. Greenblatt, D., Paschal, K. B., Yao, C.-S., Harris, J., Schaeffler, N. W. & Wasbarn, A. E., 2004, "A separation control CFD validation test case part 1", baseline & steady suction. AIAA paper 2004-2220. 40. Halfon, E, Nishri, B., Seifert, A. and Wygnanski, I., “Effects of Elevated Free-stream Turbulence on active control of a Separation Bubble”, J. Fluids Eng, V. 126, pp. 10151024, Nov. 2004. 41. Lutz and Wygnanski, "Controlling the Flow around a Swept Back Circular Cylinder Using Periodic Excitation", APS/DFD meeting, Nov. 2005. http://meetings.aps.org/Meeting/DFD05/Event/36530. 42. Zaman, KBMQ, Bar-Sever A, Mangalam, S.M., (1987). "Effect of acoustic excitation on the flow over a low Re airfoil". Journal of Fluid Mech., 182, pp. 127-148. 43. Chang, R.C., Hsiao, F.B. and Shyu, R.N. (1992). "Forcing level effects of internal acoustic excitation on the improvement of airfoil performance". AIAA Journal 29(5), 823-829. 44. Hsiao, F.B., Liu, C.F. and Shyu, J.Y. (1990). "Control of wall separated flow by internal acoustic excitation", AIAA Journal 28(8), 1440-1446. 45. Collins F.G., Zelenevitz, J., (1975). "Influence of sound upon separated flow over wings". AIAA Journal, 13(3), 408-410. 46. Ahuja, K.K., Whipkey, R.R., Jones, G.S. (1983). "Control of turbulent boundary layer flow by sound", AIAA Paper 83-0726. 47. Ahuja, K.K., Burrin, R.H., (1984). "Control of flow separation by sound", AIAA Paper 842298. 48. K. Zaman and D. Culley, "A Study of Stall Control over an Airfoil Using 'Synthetic Jets'", AIAA paper 2006-98. 49. Shepshelovich, M., Koss, D., Wygnanski, I. And Seifert, A., "An experimental evaluation of a low-Reynolds number high-lift airfoil with vanishingly small pitching moment", AIAA paper 1989-538, Jan 1989. 50. Cutler, A.D., Beck, B.T., Wilkes, J.A., Drummond, P.J., Alderfer, W.D., Paul, M. and Danehy, M.P.,” Development of a Pulsed Combustion Actuator for High-Speed Flow Control”, AIAA paper 2005-1084, 2005. 51. Kastner, J. and Samimy, M., "Development and Characterization of Hartmann Tube Fluidic Actuators for High-Speed Flow Control”, AIAA Journal, Vol. 40, No. 10, 2002, pp. 1926-1934.

State Estimation of Transient Flow Fields Using Double Proper Orthogonal Decomposition (DPOD) Stefan Siegel, Kelly Cohen, Jürgen Seidel, and Thomas McLaughlin US Air Force Academy, Colorado Springs, Colorado 81001, USA

Abstract For successful feedback flow control, an accurate estimation of the flow state is necessary. Proper Orthogonal Decomposition (POD) has been used to achieve this. However, if the POD modes are derived from a set of snapshots obtained from one flow condition only, the resulting modes will become less and less valid for a flow field that is for example altered by the effect of feedback flow control. In the past, a shift mode has been added to account for the change in the mean flow. Here, we present a new scheme that allows for the derivation of shift modes for all of the original POD modes. This DPOD mode set thus may span a range of flow conditions that are different in forcing, Reynolds number or other parameters affecting the modes. Artificial Neural Network Estimation (ANNE) allows for real time monitoring of the time coefficients associated with these DPOD modes.

1 Introduction The usefulness of POD in estimating and analyzing complex flow fields like wake flows is well established in literature. Two features of POD prove very advantageous when applied to flow field data: First, the optimality in terms of capturing most of the energy of the flow with the least possible number of modes. This allows reducing large data sets obtained from computational fluid dynamics or particle image velocimetry drastically, while still preserving the most important features of the flow. And second, when applied to time periodic flow fields, the temporal coefficients represent the physical vortex shedding phenomena with good accuracy, and can therefore be used for feedback flow control (Siegel et al. 2003). However, a POD model is strictly speaking only valid for the flow situation from which it is derived. Therefore, any modification of the flow field by means of feedback flow control may bring about a reduction and in some cases absolute nullification of the validity of the model. In order to address this problem, modified POD models have been proposed both by Noack et al. (2003) and Siegel et al. (2003). Both modifications improved the POD model by allowing adjustments of the mean flow using a shift or mean flow mode in addition to the POD derived modes themselves. These ad hoc modifications do capture the change in the mean flow reasonably well, but fail to address the spatial changes seen in the fluctuating modes. Therefore, a POD procedure that can adapt the fluctuating modes as well may be advantageous since it would maintain the validity of the model through transient flow situations. This has been realized by other research R. King (Ed.): Active Flow Control, NNFM 95, pp. 105–118, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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groups as well, and different attempts have been made to interpolate sets of similar POD modes. For example, Luchtenburg et al. (2006) use the shift mode information in order to select a proper set of fluctuating modes from a larger data basis. The advantage of this approach is that the number of POD modes used for flow estimation at any point in time is the same as what was used as the initial POD basis. However, it is impossible to derive a low dimensional model from the associated time coefficients of these modes without further analysis and simplifying assumptions about the spatial changes in these modes. The reason is that the dynamic behavior of the slower mode changing phenomena is not represented by the time coefficients that may be derived for this type of decomposition. In this work, we seek to develop a low dimensional POD basis that spans transient data sets with the least possible modes, and includes the entire dynamic behavior of the flow in the resulting time coefficients. The first step we take in developing such a procedure is to explore the behavior of the POD method as the snapshot ensemble is reduced in size. This has been done by Siegel et al. (2005) using a short time POD procedure (SPOD) that leads to physically correct spatial modes. “Short” in this context refers to the length of the snapshot ensemble compared to a cycle of the fundamental frequency in the flow field. Siegel et al. (2005) explored the SPOD method by applying it to various transient data sets. The goal of this research was to find a better set of POD modes in order to obtain an accurate global estimate of the flow field. It was demonstrated that SPOD delivers accurate spatial POD modes for snapshot ensembles as short as one shedding cycle. This capability can be used to analyze transient flow situations in great detail by deriving POD bases for each individual shedding cycle. However, this results in as many POD bases as there were shedding cycles in the original data set, which makes the resulting assembly of bases relatively large and in general unusable for both real time estimation and low dimensional model building. The SPOD mode basis is therefore very similar to the “tunable model“ described by Luchtenburg et al. (2006). In this effort, we build on the success of the SPOD method and develop means to reduce the many SPOD bases in an effective manner. This process we refer to as Double Proper Orthogonal Decomposition or DPOD, since the POD decomposition is applied twice in order to arrive at the DPOD basis. We demonstrate the accuracy of the resulting basis for the estimation of transient flow fields, and discuss features that make it particularly suitable for flow field estimation and low dimensional model building, as well as feedback flow control. While we demonstrate the DPOD procedure on a single transient data set, the method can be applied to an arbitrarily large number of SPOD bases that may include actuation effects, Reynolds number changes and other parameter variations. The use of both linear and nonlinear projection methods to perform sensor based flow state estimation is demonstrated. For low-dimensional control schemes to be implemented, a real-time estimation of the modes present in the flow is necessary, since it is not possible to measure them directly especially in real-time. After the time histories of the temporal coefficients of the POD model are determined using the DPOD procedure, the estimation of the lowdimensional states is needed for closing the loop. Sensor measurements may take the form of wake velocity measurements, as in this paper, or for an application be based on surface-mounted pressure measurements and/or shear stress sensors. For practical applications it is desirable to reduce the information required for estimation to the

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minimum. In this research effort, an 8 sensor configuration is employed together with a non-linear dynamic estimator which is based on a Artificial Neural Networks design. The remainder of the paper is structured as follows: Section II describes the numerical simulation of the Navier Stokes equations, using a computational fluid dynamics (CFD) solver. This is followed by a short description of the DPOD procedure in section III. Sections IV and V provide an extended discussion of the results for the SPOD and DPOD decompositions, respectively. Section VI demonstrates how the time coefficients of the DPOD model can be estimated from sensor readings using linear and nonlinear methods. Section VII provides conclusions of the current research. An outlook for future work is presented in section VIII.

2 D Shaped Cylinder Simulation For the scope of this work, a D-Shaped two-dimensional bluff body geometry was chosen as a generic wake flow developing a von Karman vortex street. The geometry is a semi ellipse with an aspect ratio of 35:2. This geometry was chosen to provide room for actuator implementation in future wind tunnel studies. The reference length for Reynolds and Strouhal numbers is the base height, H. The Reynolds number based on H was 300. The wake can be controlled by two blowing and suction slots which are located at the rear corners of the body and are angled at 30 degrees to the free stream direction. A structured, body fitted “D” grid with about 200,000 points is used for the CFD simulation (see Figure 1). This grid is stretched to cluster the grid points tightly in the shear layers of the near wake, as well as around the blowing and suction slots. Uniform flow boundary conditions at the far field are imposed using Riemann invariants. To ensure computational efficiency, the Mach number is set to M=0.1. The time step is ∆t=0.005d/U, where U is the free stream velocity. A previous study using this grid showed good comparison of the resulting Strouhal number with experimental results (Siegel et al. 2005b). For the purpose of POD model development, the computational results are truncated in space to the bounding box shown in Figure 1. The bounding box extends

Fig. 1. Body Geometry, CFD gridpoints and POD bounding box

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from 1H upstream of the base, to 9H downstream. In the vertical direction, the flow is truncated to ±2H. This region of interest encloses the vortex formation region entirely; outside of it, the flow is mostly steady. The bounding box contains about 40 000 grid points. During the simulation, the flow is impulsively started from rest at time t=0s. As a result of the wake instability, the limit cycle oscillation known as the von Karman vortex street develops. This is shown in Figure 2 where it can be seen that the flow requires about 20 shedding cycles to develop a limit cycle oscillation as witnessed by the lift force. 0.06

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The goal in modeling this flow field is both in the accurate representation of the flow energy, as well as in deriving spatial modes that capture the physical behavior with one or a pair of modes being attributable to a certain aspect of the physical flow behavior, like mean flow, von Karman vortex shedding modes, and higher order vortex shedding modes. This separation of the flow into individual modes representing physical phenomena is a key requirement for the development of feedback controllers that control these particular flow features.

3 Double POD Decomposition The POD decomposition for a 2D scalar spatial field u evolving over J time steps as described by Holmes, Lumley and Berkooz (1996) is shown in equation (1). J

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data set, it is typically possible to truncate the POD model at a relatively low number of modes while retaining most of the energy of the original flow field. This can be done by either inspecting the energy distribution in the modes, or by inspection of the spatial modes that will not show any discernible structure beyond a certain mode number. If POD is performed on the flow field without subtracting the mean flow, the first mode will typically be the mean flow followed by large scale fluctuating modes, in the case of the cylinder wake the two modes representing the von Karman vortex street. Since in almost all cases the modes obtained by the decomposition described above have a physical interpretation, we will refer to them as the physical modes. The decomposition works particularly well for flow fields with large, time periodic features like the periodic vortex shedding in wake flows. Siegel et al. (2005) have shown that for time periodic flows modes identical to those obtained from snapshot ensembles containing large number of shedding cycles can be obtained using snapshot ensembles of small integer number of cycles, down to a minimum of one shedding cycle. This allows for decomposition of a time evolving flow field into individual events the size of exactly one cycle of the frequency of interest: J

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described by Noack and Siegel, respectively. Thus the modes with indices k>1 can be referred to as first, second and higher order “shift” modes that allow the POD mode ensemble to adjust for changes in the spatial modes. We will refer to all of these additional modes obtained by the DPOD decomposition as shift modes, since they modify a given physical mode to match a new flow state due to either a formation length or recirculation zone length change. This may be due to effects of forcing, a different Reynolds number, feedback or open loop control or similar events. Thus, in the truncated DPOD mode ensemble for each physical mode one or more shift modes may be retained based on inspection of energy content or spatial structure of the mode. We will now demonstrate how this DPOD procedure can be used to create mode ensembles that cover the entire unforced transient startup of the D-shaped cylinder wake. This mode ensemble will thus cover not just the limit cycle, but also the steady flow state before the onset of the vortex shedding. The latter is of particular importance since vortex shedding suppression control targets the steady flow state as the control goal.

4 SPOD Modes of Transient Startup Figure 2 indicates the boundaries of the bins used to derive SPOD modes. 21 SPOD mode sets were derived, the individual bins are numbered in Figure 2. As may be expected, the POD modes derived from the bins closer to the start of the simulation are quite different than those derived from snapshot sets from the end of the simulation. Figure 3 compares the second POD mode from bins 4 and 18. The second POD mode is the first mode of a mode pair representing the von Karman vortex street, and contains about 45% of the fluctuating energy. The peak amplitude of this mode can be found about 6d downstream of the base for the mode derived from bin 4, and about 2 base heights downstream of the base for the mode derived from bin 18. This indicates that the vortices are forming further downstream at the beginning of the simulation, and move closer to the base as the limit cycle develops. SPOD Mode 2 Bin 18 2

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5 DPOD Model of the D-Shaped Cylinder The DPOD procedure introduced in a previous section achieves just that: Derivation of shift type modes for all physical modes that are deemed of importance in a transient flow field. Figure 4 shows a DPOD mode set for the transient simulation introduced in Figure 2, retaining 5 physical modes and 3 shift modes. Modes (1,n) represent the mean flow, Modes (2,n) and (3,n) the von Karman vortex street, while Modes (4,n) and (5,n) represent a higher order vortex shedding mode with twice the spatial frequency of the von Karman vortex street. Inspecting the energy contents of these modes presented in Figure 5, it can be seen that most energy is contained in the mean flow, followed by the von Karman vortex street modes which appear as pairs of approximately equal energy. This behavior is identical to a conventional POD procedure performed on a time periodic flow field. The energy content of the entire mode basis is now a two dimensional energy plane, with steep energy dropoff both towards higher order physical and higher order shift modes. While the energy content dropoff of the shift modes of the mean flow mode (Modes (1,2) and (1,3)) is fairly steep, it can be seen that the dropoff for the shift modes of the von Karman modes (Modes (2,2), (2,3) and (3,2), (3,3)) is far less steep, demonstrating the importance of including these modes in a low order model. Since the dropoff in energy is not uniform for all shift modes, one might retain different number of shift modes for each physical mode based on energy considerations. The spatial distribution of the shift modes demonstrates how they manage to adjust the distribution of each physical mode to the changes in the vortex shedding pattern seen during the transient startup simultation: The first shift mode of the mean flow mode, for example, will extend the length of the recirculation zone which is what can be observed during the development of the vortex shedding limit cycle. This behavior is almost identical the artificially created shift modes suggested by Noack et al. (2003) or the so called mean flow mode introduced by Siegel (2003). A consequence of the much longer recirculation zone at the startup of the limit cycle is also that the vortices are forming further downstream of the body, which can be seen by inspecting the SPOD Modes shown in Figure 3. This effect is modeled mostly by the first shift mode of the two von Karman vortex shedding modes, i.e. Modes (2,2) and (3,2). These have a similar appearance than the von Karman vortex shedding modes themselves, but with their maximum mode amplitude shifted further downstream. The same effect can be seen for the higher order vortex shedding modes, where the maximum of modal activity of the shift modes (Modes (4,2) and (5,2)) is also further downstream than for the modes themselves (Modes (4,1) and (5,1)). Since all of the spatial modes have been normalized to unity in magnitude, the temporal coefficients provide insight into the energy content as time evolves during the simulation. Moreover, the energy exchange between modes and their shift modes is also captured in these time coefficients. Thus it is of interest to inspect the time coefficients in order to understand the dynamics of energy exchange between the modes during the transient simulation. For example, the time coefficients of the two mean flow mode shift modes, Mode (1,2) and (1,3), reveal that these only contribute

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at the beginning of the limit cycle development, before about 1.5 s into the simulation. For the time periodic portion of the simulation, their contribution approaches zero. The von Karman shift modes show a different behavior: Their activity has a peak around 0.9 seconds, and a minimum at about 1.5 seconds. It can be shown that their phase with respect to their physical mode (Mode (n,1)) changes at that time by about 180 degrees and as a result they add to the main mode’s activity for one portion of the simulation, while they decrease the activity for the other portion of the simulation. In combination with their different spatial distributions, the observed shift in the shedding location is thus modeled correctly by the vortex shedding and their shift modes. The physical mode together with its shift modes thus offer a possibility to develop for example a multi input single output controller that adjusts its feedback

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parameters in accordance with the vortex shedding location shift: By applying different gains to the mode and shift modes, the feedback can be tuned to the required phase at any time.

6 Sensor Based State Estimation The time histories of the temporal coefficients of the POD model are determined by introducing the spatial Eigenfunctions into the flow field data using the least squares technique. For closed-loop control, we are interested in a system that maps, in realtime, velocity measurements provided by the sensors onto the estimates of the fifteen temporal modes. The estimation scheme predicts the temporal amplitudes of the first fifteen POD modes from a finite set of velocity measurements obtained from the CFD solution of the “D” shaped cylinder wake. For each sensor configuration, 341 velocity measurements were used equally spaced at 0.01seconds apart. Of the 341 snapshots, the first 170 were used for training of the estimator, whereas, the final 171 snapshots were used for validation purposes. Only data concerning velocity components in the direction of the flow were used for the sensor placement and number studies reported in this effort. The main approach in this effort is the incorporation of a non-linear dynamic estimator. The decision was to look into universal approximators, such as artificial neural networks (ANN), for their inherent robustness and capability to approximate any non-linear function to any arbitrary degree of accuracy. The ANN employed in this effort, in conjunction with the ARX model is the mechanism with which the dynamic model is developed using the POD time-coefficients extracted from the high resolution CFD simulation. Non-linear optimization techniques, based on the back propagation method, are used to minimize the difference between the extracted POD time coefficients and the ANN while adjusting the weights of the model (Nørgaard et al., 2003). The main hypothesis is that the non-linearity and scaling characteristics of the temporal coefficients lead to numerical stability issues which undermine the development and analysis of effective estimation/control laws. In order to assure model stability, the ARX dynamic model structure is incorporated. This structure is widely used in the system identification community. A salient feature of the ARX predictor is that it is inherently stable even if the dynamic system to be modeled is unstable. This characteristic of ARX models often lends itself to successful modeling of unstable processes as described by Nelles (2001). The artificial neural network (ANN) has the following features: •

Input Layer: Eight sensor signals, n (n = 8), namely, U-Velocity at the given sensor locations. In addition to these readings, in order to obtain a strong representation of the dynamics of the system, the input layer includes 8 past inputs and 15 time delays. No past outputs were used. The eight sensors are placed on the maxima/minima of the first eight modes as described by the heuristic procedure developed by Cohen et al. (2006). The total number of inputs to the net is as follows: # inputs to ANN = # time delays *[ # past outputs + (# past inputs per sensor)] + bias

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By inserting the values chosen after a brief sensitivity study, we obtain: # inputs to ANN = 15*[0 + 8] + 1 = 121

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From the above relation, we can see that the number of inputs is 121 for all the cases irrespective of the number of sensors. Hidden Layer: One hidden layer consisting of 15 neurons. The activation function in the hidden layer is based on the non-linear tanh function. A single bias input has been added to the output from the hidden layer. Output Layer: Fifteen outputs, namely, the 15 states representing the temporal coefficients of the 15 mode POD reduced order model developed in the previous Section. The output layer has a linear activation function. Weighting Matrices: The weighting matrices between the input layer and the hidden layer (W1) and between the hidden layer and the output layer (W2) depend on the number of sensors. In this effort, W1 is of the order of [121*15] and W2 is of the order of [15*16]. These weighting matrices are initialized randomly. Training the ANN: Back propagation, based on the Levenberg-Marquardt algorithm, was used to train the ANN using Nørgaard et al’s (2003) toolbox. The training procedure converged near 350 iterations. Generally speaking, the training data fits well and this should not be very surprising. Results for the training data are enclosed in Fig. 7. Validating the ANN: The validation data is as described in Table 1. Results for the validation data are enclosed in Fig. 8. Table 1. RMS Errors [%] for ANNE Estimates

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RMS of Training Error [%] 0.12 0.84 0.47 3.0 2.4 1.8 2.3 4.1 16.2 11.2 2.5 11.4 9.7 27.0 67.2

RMS of Validation Error [%] 0.12 0.93 0.63 2.4 3.0 2.5 6.3 5.6 18.5 10.5 5.7 8.7 12.6 80.3 94.5

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7 Discussion We introduce Double Proper Orthogonal Decomposition (DPOD) as a means to derive POD spatial modes that span different flow conditions. We demonstrate its ability by applying it to a transient simulation of the development of the limit cycle of a D shaped cylinder wake.

8 Outlook The derivation of valid spatial modes and temporal coefficients constitutes the first step towards developing a set of low dimensional equations capturing the dynamic behavior of a flow field. Both linear and nonlinear techniques, such as system identification and Galerkin projection may be used to do so, and we will explore their efficiency for this task in the future. While we demonstrate the capabilities of DPOD by applying it to a transient flow field which in essence consitutes one parameter change, DPOD may also be used to model the effects of several different parameters on a flow field, like the impact of changes in actuation frequency and amplitude, or Reynolds number. Thus it is a generic technique that can be applied to problems of any degree of complexity, not just the relatively simple sample flow field that was used in this work.

Acknowledgments The authors would like to acknowledge funding by the Air Force Office of Scientific Research, LtCol Sharon Heise, Program Manager. We would also like to thank Dr. Jim Forsythe of Cobalt Solutions, LLC for their support. The authors appreciate the fruitful discussions with Dr. Young Sug-Shin on system identification using Artificial Neural Networks and for his very helpful insight.

References Noack, B. R.; Afanasiev, K.; Morzynski, M.; Tadmor, G., Thiele, F., 'A hierarchy of lowdimensional models for the transient and post-transient cylinder wake', J. Fluid Mechanics, 497, 2003 Luchtenburg, M., G. Tadmor, O. Lehmann, B.R. Noack, R. King, M. Morzyinski, 'Tuned POD Galerkin models for transient feedback regulation of the cylinder wake', 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, AIAA-2006-1407, 2006 M. Bergmann, L. Cordier, and J.-P. Brancher, 'Optimal rotary control of the cylinder wake using POD reduced order model', 2nd AIAA Flow Control Conference, AIAA 2004-2323, 2004 Siegel, S.; Cohen, K.; McLaughlin, T., 'Feedback Control Of A Circular Cylinder Wake In Experiment And Simulation (invited)', 33rd AIAA Fluid Dynamics Conference, Orlando, AIAA 2003-3569, 2003

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Siegel, S., Cohen, K., Seidel, J., McLaughlin, T., 'Short Time Proper Orthogonal Decomposition for State Estimation of Transient Flow Fields', 43rd AIAA Aerospace Sciences Meeting, Reno, AIAA2005-0296, 2005 Siegel, S., Cohen, K., Seidel, J., McLaughlin, T., 'Two Dimensional Simulations Of A Feedback Controlled D-Cylinder Wake', AIAA Fluid Dynamics Conf Toronto, ON, CA, AIAA 20055019, 2005 Holmes P., Lumley, J. L., Berkooz, G., 1996, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press Cohen, K., Siegel S., and McLaughlin T.,"A Heuristic Approach to Effective Sensor Placement for Modeling of a Cylinder Wake”, Computers and Fluids, Volume 35, Issue 1, January 2006, pp. 103-120. Nelles, O., Nonlinear System Identification, Springer-Verlag, Berlin, Germany, 2001, Chap. 11. Nørgaard, M., Ravn., O., Poulsen, N.K., Hansen, L.K., Neural Networks for Modeling and Control of Dynamic Systems, 3rd printing, Springer-Verlag, London, U.K., 2003, Chap. 2.

A Uniﬁed Feature Extraction Architecture Tino Weinkauf1 , Jan Sahner1 , Holger Theisel2 , Hans-Christian Hege1 , and Hans-Peter Seidel2 1

2

Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany {weinkauf,sahner,hege}@zib.de MPI Informatik Saarbr¨ucken, Stuhlsatzenhausweg 85, 66123 Saarbr¨ucken, Germany {theisel,hpseidel}@mpi-inf.mpg.de

Summary We present a uniﬁed feature extraction architecture consisting of only three core algorithms that allows to extract and track a rich variety of geometrically deﬁned, local and global features evolving in scalar and vector ﬁelds. The architecture builds upon the concepts of Feature Flow Fields and Connectors, which can be implemented using the three core algorithms ﬁnding zeros, integrating and intersecting stream objects. We apply our methods to extract and track the topology and vortex core lines both in steady and unsteady ﬂow ﬁelds.

1 Introduction As the resolution of numerical simulations as well as experimental measurements like PIV have evolved signiﬁcantly in the last years, the challenge of understanding the intricate structures within their massive result data sets has made automatic feature extraction schemes popular. Exploratory techniques alone do not sufﬁce to analyze massive result data sets. Due to the sheer size of the data they have to be complemented by automatic feature extraction schemes, which give a reliable basis for subsequent manual explorations. In this paper we focus on the treatment of ﬂow ﬁelds. They play a vital role in many research areas. Examples are combustion chambers, turbomachinery and aircraft design in industry as well as visualization and control of blood ﬂow in medicine. For this class of data, topological and vortical structures are among the features of interest. Extraction of those structures helps in understanding processes inherent to the ﬂow. This knowledge is the basis for manipulating those processes in terms of ﬂow control. While [13] gives an overview on ﬂow visualization techniques focusing on feature extraction approaches, we give a short introduction here. Topological methods have become a standard tool to visualize 2D and 3D vector ﬁelds because they offer to represent a complex ﬂow behavior by only a limited number of graphical primitives. [6] and [5] introduced them as a visualization tool by extracting critical points and classifying them into sources, sinks and saddles, and integrating certain stream lines called separatrices from the saddles in the directions of the eigenvectors of the Jacobian matrix. Later, topological methods have been extended to higher order critical points [16] [28], boundary switch points [3] and curves [27], closed separatrices [32] [23], and saddle R. King (Ed.): Active Flow Control, NNFM 95, pp. 119–133, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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connectors [22]. In addition, topological methods have been applied to simplify [3] [25] [30], smooth [31], compress [9] and design [19] vector ﬁelds. While they aim at the segmentation of a vector ﬁeld into areas of different ﬂow behavior, vortex oriented methods highlight turbulent regions of the ﬂow. Recently some work has been done to link these different areas: [4] [26] employ topological methods to analyze the phenomenon of vortex breakdown. Vortices play a major role due to their wanted or unwanted effects on the ﬂow. In turbomachinery design, vortices reduce efﬁciency, whereas in burning chambers, vortices have to be controlled to achieve optimal mixing of oxygen and fuel. In aircraft design, vortices can both increase and decrease lift. Algorithms for the treatment of vortical structures can be classiﬁed in two major categories: – Vortex region detection is based on scalar quantities that are used to deﬁne a vortex as a spatial region where the quantity exhibits a certain value range. We refer to them as vortex region quantities. Examples of this are regions of high magnitude of vorticity or negative λ2 -criterion [8]. In general, these measures are Galilean invariant, i.e., they are invariant under adding constant vector ﬁelds. This is due to the fact that their computation involves derivatives of the vector ﬁeld only. Isosurfaces or volume rendering are common approaches for visualizing these quantities, which requires the choice of thresholds and appropriate isovalues or transfer functions. As shown in [14], this can become a difﬁcult task for some settings. – Vortex core line extraction aims at extracting line type features that are regarded as centers of vortices. Different approaches exist. [18] [12] consider lines where the ﬂow exhibits a swirling motion around it. [1] extracts vorticity lines seeded at critical points and corrected towards pressure minima. [15] considers stream lines of zero torsion. All of these approaches include a Galilean variant part, i.e., they depend on a certain frame of reference. In contrast to vortex region detection described above, the extraction of those lines is parameter free in the sense that their deﬁnition does not refer to a range of values. This eliminates the need of choosing certain thresholds. In this paper we present an uniﬁed approach to extracting and tracking a variety of ﬂow features. Hereby we deﬁne the term feature as follows: – A feature is an n-dimensional geometrical structure embedded into a m-dimensional domain. – It is located inside the domain of the analyzed data. – It yields certain “insight” into the data. Finally, the actual deﬁnition of a feature depends on the application. In this paper, we mainly treat topological and vortical structures of ﬂow ﬁelds. The paper is organized as follows. Section 2 explains the uniﬁed feature extraction architecture, while sections 3 and 4 treat the main concepts behind it, namely Feature Flow Fields and Connectors. We apply our method in section 5 to a number of data sets and feature deﬁnitions. Conclusions are drawn in section 6.

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2 Uniﬁed Feature Extraction Architecture Almost every feature can be extracted and tracked using a combination of the following core algorithms: – Finding zeros – Integrating stream objects – Intersecting stream objects We show in section 3 how the ﬁrst two algorithms can be combined in the Feature Flow Field approach to extracting and tracking features that are deﬁned locally. The intersection of stream objects becomes necessary, where the features we are interested in have a global nature, like closed stream lines. Here the Connectors approach can be applied (section 4). We now brieﬂy comment on each of the above algorithms. 2.1 Finding Zeros We are interested in extracting isolated zeros of functions f : Rn → Rn . Several approaches exist, some depending on the interpolation scheme: – Newton-Raphson: use the ﬁrst derivative to repeatedly predict a zero [12]. – In piecewise linear ﬁelds, e.g. tetrahedral grids, the zeros can be computed explicitely. – In piecewise trilinear ﬁelds, e.g. regular grids, a component wise change-of-sign test is a necessary condition for a zero inside the grid cell. Based on this test, a recursive domain decomposition can be applied to the cell that converges to a zero. This method extends to ﬁnding zeros of functions f : Rn → Rm , is quick, robust, and easy to implement. So we favor this method over the Newton-Raphsonapproach for trilinear ﬁelds. 2.2 Integration of Stream Objects Given a ﬂow ﬁeld f : Rn → Rn we aim at constructing m + 1-dimensional stream objects from m-dimensional seeding structures. – For m = 0 we obtain a stream line or integral curve. – For m = 1 we obtain a stream surface by triangulating stream lines seeded equidistantly on the seeding line, see [7] for a thorough treatment of this topic. – Starting from seeding surfaces, we obtain ﬂow volumes, see [10] for implementation details. Naively integrating stream objects from a seeding structure might result in passing through the whole dataset for each stream object, a costly undertaking, when the dataset is too large to ﬁt into main memory and the number of stream objects is high. Nevertheless Weinkauf et al. showed in [29] that it is possible for a huge subclass of features to do all stream object integrations by sequentially loading the dataset only once, keeping just two consecutive time steps in memory at a time. Where section 3 shows that ﬁnding zeros and integration of stream objects sufﬁces for ﬁnding features that are locally deﬁned, stream object integration becomes necessary if such a deﬁnition is not at hand.

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2.3 Intersection of Stream Objects Given a ﬂow ﬁeld f : Rn → Rn and two m-dimensional (m > 1) stream objects R (integrated in forward direction) and A (integrated in backward direction) we aim at extracting the intersection of R and A, i.e., the m − 1-dimensional stream object that both R and A share. – For m = 2 we obtain a stream line which lies in both intersecting stream surfaces. – For m = 3 we obtain a stream surface which lies in both ﬂow volumes. Figure 2 shows an example for m = 2, where two stream surfaces share a common stream line.

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The concept of feature ﬂow ﬁelds was ﬁrst introduced in [21]. It follows a rather generic idea: Consider an arbitrary point x known to be part of a feature in a (scalar, vector, tensor) ﬁeld v. A feature ﬂow ﬁeld f is a well-deﬁned vector ﬁeld at x pointing into the direction where the feature continues. Thus, starting a stream line integration of f at x yields a curve where all points on this curve are part of the same feature as x. FFF have been used for a number of applications, but mainly for tracking features in time-dependent ﬁelds. Here, f describes the dynamic behavior of the features of v: for a time-dependent ﬁeld v with n spatial dimensions, f is a vector ﬁeld IRn+1 → IRn+1 . The temporal evolution of the features of v is described by the stream lines of f . In fact, tracking features over time is now carried out by tracing stream lines. The location of a feature at a certain time ti can be obtained by intersecting the stream lines with the time plane ti . Figure 1a gives an illustration. Depending on the dimensionality of the feature at a certain time ti , the feature tracking corresponds to a stream line, stream surface or even higher-dimensional stream object integration. The stream lines of f can also be used to detect events of the features: – A birth event occurs at a time tb , if the feature at this time is only described by one stream line of f , and this stream line touches the plane t = tb “from above” (i.e., the stream line in a neighborhood of the touching point is in the half-space t ≥ tb ). – A split occurs at a time ts , if one of the stream lines of f describing the feature touches the plane t = ts “from above”. – An exit event occurs if all stream lines of f describing the feature leave the spatial domain. The conditions for the reverse events (death, merge, entry) can be formulated in a similar way. Figure 1b illustrates the different events. Integrating the stream lines of f in forward direction does not necessarily mean to move forward in time. In general, those directions are unrelated. The direction in time may even change along the same stream line as it is shown in ﬁgure 1b. This situation is always linked to either a birth and a split event, or a merge and a death event. Even though we treated the concept of FFF in a rather abstract way, we can already formulate the basics of an algorithm to track all occurrences of a certain feature in a time-dependent ﬁeld:

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(a) Tracking features by tracing stream lines. (b) Events: at the time tb a new feature is born, Features at ti+1 can be observed by interat the time ts it splits into two features. secting these stream lines with the time plane t = ti+1 . Fig. 1. Feature tracking using feature ﬂow ﬁelds

Algorithm 1. General FFF-based tracking 1. Get seeding points/lines/structures such that the stream object integration of f guarantees to cover all paths of all features of v. 2. From the seeding structures: apply a numerical stream object integration of f in both forward and/or backward direction until it leaves the space-time domain. 3. If necessary: remove multiply integrated stream objects. Algorithm 1 is more or less an abstract template for a speciﬁc FFF-based tracking algorithm. Before showing how this template can be used to track critical points and extract and track vortex core lines in ﬂow ﬁelds in the applications section 5, we can already note, how the steps of algorithm 1 correlate to the core algorithms given in section 2: the seeding points are usually extracted as critical points of some ﬁelds. Then we use the stream object integration from section 2.2 to track the feature. But what can be done, if the feature of interest does not admit a local deﬁnition? Here the connectors approach comes into play.

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Given a ﬂow ﬁeld f : Rn → Rn and two m-dimensional (m > 1) stream objects R (integrated in forward direction) and A (integrated in backward direction) we aim at extracting the intersection of R and A, i.e., the m − 1-dimensional stream object that both R and A share. Figure 2 shows an example for m = 2, where two stream surfaces share a common stream line. Since the intersection of R and A always starts at the repelling seeding structure of R and ends at the attracting seeding structure of A, it is called a connector. A connector is a global feature, i.e., it can not be locally deﬁned.

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(a) Two stream surfaces starting from saddle points.

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An algorithm for the extraction of line-type connectors has been treated in [22]. To ﬁnd the intersection between a separation surface in forward integration and a separation surface in backward integration, we integrate both separation surfaces simultaneously until a ﬁrst intersection point p1 is found. After reﬁning this intersection point (see [22] for details), a stream line from p1 is integrated both forwards and backwards. This stream line is the connector. Figure 3 gives an illustration of this algorithm.

5 Applications In this section we apply the Feature Extraction Architecture to a variety of feature extraction settings. In section 5.1, topological features are focused while section 5.2 shows how to extract and track vortex core lines using the Feature Extraction Architecture.

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5.1 Topological Feature Extraction Critical points, i.e. isolated points at which the ﬂow vanishes, are perhaps the most important topological feature of vector ﬁelds. For static ﬁelds, their extraction and classiﬁcation is well-understood both in the 2D [6] and the 3D case [27]. Critical points also serve as the starting points of certain separatrices, i.e. stream lines or surfaces which divide the ﬁeld into areas of different ﬂow behavior. Where the direct visualization of those stream surfaces result in cluttered images, Theisel et al. showed in [22] how restricting the display to the intersection lines of those surfaces, called saddle connectors, increases the comprehensibility. This has been achieved by using the connectors approach. In [23] and [24] Theisel et al. showed how to extract and track closed stream lines using the connectors approach. Considering a stream line oriented topology of time-dependent vector ﬁelds, critical points smoothly change their location and orientation over time. In addition, certain bifurcations of critical points may occur. To capture the topological behavior of timedependent vector ﬁelds, it is necessary to capture the temporal behavior of the critical points. Theisel et al. introduced in [21] a FFF-based approach to track critical points, which matches algorithm 1. We now show, how the Feature Extraction Architecture can be applied to this setting.

Fig. 4. Tracking 2D critical points: all points on a stream line of f have the same value for v. Note that the depicted stream line is a tangent curve of the feature ﬂow ﬁeld f and not of the original velocity ﬁeld v.

Critical Point Tracking. Let v be a 3D time-dependent vector ﬁeld, which is given as ⎛ ⎞ u(x, y, z, t) v(x, y, z, t) = ⎝ v(x, y, z, t) ⎠ (1) w(x, y, z, t) in the 4D space-time domain D = [xmin , xmax ] × [ymin , ymax ] × [zmin , zmax ] × [tmin , tmax ]. We can construct a 4D vector ﬁeld f in D with the following properties: for any two points x0 and x1 on a stream line of f , it holds v(x0 ) = v(x1 ). This means that a stream line of f connects locations with the same values of v. Figure 4 gives an illustration in 2D. In particular, if x0 is a critical point in v, then the stream line of f describes the path of the critical point over time. To get f , we search for the direction in space-time in which both components of v locally remain constant. This is the direction perpendicular to the gradients of the three components of v:

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f ⊥ grad(u) = (ux , uy , uz , ut )T , f ⊥ grad(v) , f ⊥ grad(w). This gives a unique solution for f (except for scaling) ⎛ ⎞ + det(vy , vz , vt ) ⎜ − det(vz , vt , vx ) ⎟ ⎟ f (x, y, z, t) = ⎜ ⎝ + det(vt , vx , vy ) ⎠ . − det(vx , vy , vz )

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Theisel et al. showed in [24] that two classes of seeding points guarantee that all paths of critical points are captured: the intersections of the paths with the domain boundaries, i.e. critical points on the boundaries of the space time domain and fold bifurcations, locations where a pair of critical point emerges or vanishes. Fold bifurcations can be characterized by [ v(x) = (0, 0, 0)T , det(Jv (x)) = 0 ] .

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Applying the Feature Extraction Architecture, we do the following: – Extraction of seeding structures boils down to ﬁnding zeros in the following ﬂow ﬁelds: for intersections with the domain boundary, ﬁnd zeros of the 4 3D ﬂow ﬁelds v(x, y, z, tmin ) = 0 and v(x, y, z, tmax ) = 0 for the unknowns x, y, z, v(x, y, zmin , t) = 0 and v(x, y, zmax , t) = 0 for the unknowns x, y, t, v(x, ymin , z, t) = 0 and v(x, ymax , z, t) = 0 for the unknowns x, z, t, v(xmin , y, z, t) = 0 and v(xmax , y, z, t) = 0 for the unknowns y, z, t.

As mentioned above, also the fold bifurcation serve as seeding points. To extract those, a 4D zero extraction has to be applied to formula (3). – Trace out f from each of the seeding points to obtain the evolution paths of the critical points. In a postprocessing step remove all lines that are integrated twice, e.g. resulting from stream lines that leave the domain at two different locations. Example: Out-of-core tracking of critical points. Weinkauf et al. showed in [29] how to track critical points in 2D and 3D time-dependent vector ﬁelds in an effective out-of-core manner: in one sweep and by loading only two slices at once. We applied this algorithm to a random 2D time-dependent data set. Random vector ﬁelds are useful tools for a proof-of-concept of topological methods, since they contain a maximal amount of topological information. Figure 5 shows the execution of the tracking algorithm between two consecutive time steps ti and ti+1 . Example: Cavity. Figure 6 shows the visualization of a vector ﬁeld describing the ﬂow over a 2D cavity. This data set was kindly provided by Mo Samimy and Edgar

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Caraballo (both Ohio State University) [2] as well as Bernd R. Noack (TU Berlin). 1000 time steps have been simulated using the compressible Navier-Stokes equations; it exhibits a non-zero divergence inside the cavity, while outside the cavity the ﬂow tends to have a quasi-divergence-free behavior. The topological structures of the full data set visualized in Figure 6a elucidate the quasi-periodic nature of the ﬂow. Figures 6b-c show approximately one period – 100 time steps – of the full data set, while Figures 6d-e point out some topological details. Figures 6b-c both reveal the overall movement of the topological structures – the most dominating ones originating in or near the boundaries of the cavity itself. The quasi-divergence-free behavior outside the cavity is afﬁrmed by the fact that a high number of Hopf bifurcations has been found in this area. The tracked closed stream line in Figure 6d starts in a Hopf bifurcation and ends in another one – thereby enclosing a third Hopf. Figure 6e shows a detailed view of time step 22, where a saddle connection has been detected. In the front of this ﬁgure a sink is going to join and disappear with a saddle, which just happened to enter at the domain boundary. 5.2 Vortex Core Line Extraction and Tracking We apply the Feature Extraction Architecture to Vortex Core Line Extraction. While [12] gives a good overview of existing vortex core line deﬁnitions, we use the most prominent technique by Sujudi and Haimes [18]. Using the notation of [12] and denoting w1 := v, w2 := ∇v · v, we deﬁne a vortex core line as locations where w 1 w2 ,

(4)

where denotes vector parallelity and v is a time dependent ﬂow ﬁeld as in (1). In this setting, the Feature Extraction Architecture can solve different tasks: 1. Extract vortex core lines at some time step t0 . 2. Track a given vortex core line in time, i.e., given a vortex core line at some time t0 , compute the evolution path of this vortex core line in the 4D-spacetime domain. This will assemble a surface. 3. Extract the complete vortex core line surface from 2 at once and use it for vortex core line display and tracking in time.

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(b) Stream line oriented topology of the ﬁrst 100 time steps.

(c) Path line oriented topology of the ﬁrst 100 time steps.

(d) Tracked closed stream line starting and ending in a Hopf bifurcation.

(e) Detail view with a saddle connection and a fold bifurcation.

Fig. 6. 2D time-dependent ﬂow at a cavity. The datasets consists of 1000 time steps which have been visualized in (a). All other images show the ﬁrst 100 time steps.

Spatial Extraction of Vortex Core Lines. By (4), a point x is on a vortex core line, whenever ⎛ ⎞ k(x, y, z, t) (5) s(x, y, z, t) := ⎝ m(x, y, z, t) ⎠ := v × ∇v · v = w1 × w2 = 0. n(x, y, z, t) Given a point x0 = (x0 , y0 , z0 , t0 )T ∈ D = [xmin , xmax ] × [ymin , ymax ] × [zmin , zmax ] × [tmin , tmax ] on a vortex core line (i.e. s(x0 ) = 0), we can trace stream lines of the following feature ﬂow ﬁeld f from [20] to extract vortex core lines from the seed point x0 at time t0 :

A Uniﬁed Feature Extraction Architecture

⎛ ⎞ ⎛ ⎞ e det(sy , sz , a) f (x, y, z, t0 ) = ⎝ f ⎠ = ⎝ det(sz , sx , a) ⎠ . det(sx , sy , a) g

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For the choice of a we refer to [20]. In the notation of the Feature Extraction Architecture, the complete skeleton of vortex core lines at t0 can be extracted as follows: – From [20] we know, that all vortex core lines are either closed or cross the boundary. Therefore, we extract as starting points all intersections of the vortex core lines with the 2D spatial domain boundary [xmin , y, z, t0 ]∪[xmax , y, z, t0 ]∪[x, ymin , z, t0 ]∪ [x, ymax , z, t0 ] ∪ [x, y, zmin , t0 ] ∪ [x, y, zmax , t0 ]∪ at timestep t0 , e.g. at xmin : s(xmin , y, z, t0 ) = (0, 0, 0)T .

(7)

This is a function R2 → R3 with isolated zeros due to the dependencies of the components in the cross product (5). Closed vortex core lines can be detected by ﬁnding isolated zeros in the ﬁeld [ s(x) = (0, 0, 0)T , e(x) = 0 ],

(8)

a function R3 → R4 , see again [20] for details. – Given those seeding points, we can extract all vortex core lines at time step t0 by tracing stream lines of f . Tracking of Vortex Core Lines in Time. For any vortex core line at a given time t0 , [20] shows that stream lines of g seeded from the vortex core line tracks the temporal evolution of the vortex core line: h×f h×f g(x, y, z, t) = = (9) f 2 e2 + f 2 + g 2 with

⎞ det(sx , st , a) h(x, y, z, t) = ⎝ det(sy , st , a) ⎠ . det(sz , st , a) ⎛

(10)

A complete Vortex Core Line Skeleton. In the 4D space time domain D, the vortex core lines build surface structures. In [20] a detailed algorithm is given, how this surface structure can be extracted based on a bifurcation analysis of the above feature ﬂow ﬁeld. In the Feature Extraction Architecture notation, the algorithm reads as follows: – Compute the seeding structures: 1. Compute the intersection curves of the vortex core line surface with the spatial boundaries of D. This can be done by spatial extraction of vortex core lines as explained above. 2. Extract all local bifurcations introduced in [20] by ﬁnding zeros of some function R4 → R4 .

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(a) Shortly before.

(b) The event.

(c) Shortly after.

Fig. 7. Example of the visualization of vortex core line surfaces. Shown is a saddle bifurcation of vortex core lines. The surfaces are displayed bright for future, dark for past times.

3. Extract closed vortex core lines at the times t = tmin and t = tmax respectively, pick a point on each extracted closed line, and apply a stream line integration of g starting from them. 4. Start a stream line integration of g from all inﬂow boundary bifurcations (zeros of some function R3 → R3 ). – Extract and visualize the vortex core line surface for a time interval [t0 , t1 ] with tmin ≤ t0 ≤ t1 ≤ tmax ): 1. Load the seeding lines obtained above. 2. Identify all parts of the seeding lines with t-values between t0 and t1 . 3. Starting from these seeding lines, apply a stream surface integration of f until it leaves D or returns to its starting point. 4. Visualize the stream surfaces obtained in 3. As projecting the complete vortex core surface to space leads to self-intersections already in quite simple settings, we use the following approach to visualize the evolution of vortex core line structures: at a given time we draw the vortex core lines as solid tubes inside the vortex core surface that is displayed only for a certain time range for future and past. At the boundary of the space domain the corresponding seeding lines are given for a larger time interval. Both the surfaces and the seeding lines fade out away from the current time. We use color coding to indicate past (dark) and future (bright). Figure 7 shows the evolution of a speciﬁc inner bifurcation called saddle bifurcation. Note that the width of the surface in ﬁgures 7a and 7c conﬁrms the intuition that the most drastic movements of the vortex core line over time takes place near the bifurcation points. Example: Flow behind a Circular Cylinder. Figures 8 and 9 demonstrate results of the Uniﬁed Feature Extraction Architecture of vortex core line extraction in a ﬂow behind a circular cylinder. The data set was derived by Bernd R. Noack (TU Berlin) from a direct numerical Navier Stokes simulation by Gerd Mutschke (FZ Rossendorf). It resolves the so called ‘mode B’ of the 3D cylinder wake at a Reynolds number of 300 and a spanwise wavelength of 1 diameter. The data is provided on a 265 × 337 × 65 curvilinear grid as a low-dimensional Galerkin model [11] [33]. The examined time

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Fig. 8. Flow behind a circular cylinder. Shown are vortex core lines in a certain frame of reference. Their evolution over time is tracked by our algorithm and depicted using transparent surfaces. Dark color encodes the past while bright shows the future.

Fig. 9. Flow behind a circular cylinder. The extracted seeding lines elucidate the alternating evolution of the vortical structures in transverse direction.

range is [0, 2π]. The ﬂow exhibits periodic vortex shedding leading to the well known von K´arm´an vortex street. This phenomenon plays an important role in many industrial applications, like mixing in heat exchangers or mass ﬂow measurements with vortex counters. However, this vortex shedding can lead to undesirable periodic forces on obstacles, like chimneys, buildings, bridges and submarine towers.

6 Conclusions In this paper we exempliﬁed that a rich variety of ﬂow features can be extracted and tracked by a combination of only three core algorithms, namely ﬁnding zeros, integrating and intersecting stream objects. The so-deﬁned Uniﬁed Feature Extraction Architecture builds upon the concepts of Feature Flow Fields and Connectors.

Acknowledgments We thank Bernd R. Noack for the fruitful discussions and providing the Galerkin model of the cylinder data set. All visualizations in this paper have been created using A MIRA – a system for advanced visual data analysis [17] (see http://amira.zib.de/).

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References [1] D.C. Banks and B.A. Singer. A predictor-corrector technique for visualizing unsteady ﬂow. IEEE Transactions on Visualization and Computer Graphics, 1(2):151–163, 1995. [2] E. Caraballo, M. Samimy, and DeBonis J. Low dimensional modeling of ﬂow for closedloop ﬂow control. AIAA Paper 2003-0059. [3] W. de Leeuw and R. van Liere. Collapsing ﬂow topology using area metrics. In Proc. IEEE Visualization ’99, pages 149–354, 1999. [4] C. Garth, X. Tricoche, and G. Scheuermann. Tracking of vector ﬁeld singularities in unstructured 3D time-dependent datasets. In Proc. IEEE Visualization 2004, pages 329–336, 2004. [5] A. Globus, C. Levit, and T. Lasinski. A tool for visualizing the topology of threedimensional vector ﬁelds. In Proc. IEEE Visualization ’91, pages 33–40, 1991. [6] J. Helman and L. Hesselink. Representation and display of vector ﬁeld topology in ﬂuid ﬂow data sets. IEEE Computer, 22(8):27–36, August 1989. [7] J. Hultquist. Constructing stream surfaces in steady 3D vector ﬁelds. In Proc. IEEE Visualization ’92, pages 171–177, 1992. [8] J. Jeong and F. Hussain. On the identiﬁcation of a vortex. J. Fluid Mechanics, 285:69–94, 1995. [9] S.K. Lodha, J.C. Renteria, and K.M. Roskin. Topology preserving compression of 2D vector ﬁelds. In Proc. IEEE Visualization 2000, pages 343–350, 2000. [10] N. Max, B. Becker, and R. Crawﬁs. Flow volumes for interactive vector ﬁeld visualization. In Proc. Visualization 93, pages 19–24, 1993. [11] B.R. Noack and H. Eckelmann. A low-dimensional galerkin method for the threedimensional ﬂow around a circular cylinder. Phys. Fluids, 6:124–143, 1994. [12] R. Peikert and M. Roth. The parallel vectors operator - a vector ﬁeld visualization primitive. In Proc. IEEE Visualization 99, pages 263–270, 1999. [13] F.H. Post, B. Vrolijk, H. Hauser, R.S. Laramee, and H. Doleisch. Feature extraction and visualisation of ﬂow ﬁelds. In Proc. Eurographics 2002, State of the Art Reports, pages 69–100, 2002. [14] M. Roth and R. Peikert. Flow visualization for turbomachinery design. In Proc. Visualization 96, pages 381–384, 1996. [15] M. Roth and R. Peikert. A higher-order method for ﬁnding vortex core lines. In D. Ebert, H. Hagen, and H. Rushmeier, editors, Proc. IEEE Visualization ’98, pages 143–150, Los Alamitos, 1998. IEEE Computer Society Press. [16] G. Scheuermann, H. Kr¨uger, M. Menzel, and A. Rockwood. Visualizing non-linear vector ﬁeld topology. IEEE Transactions on Visualization and Computer Graphics, 4(2):109–116, 1998. [17] D. Stalling, M. Westerhoff, and H.-C. Hege. Amira: A highly interactive system for visual data analysis. The Visualization Handbook, pages 749–767, 2005. [18] D. Sujudi and R. Haimes. Identiﬁcation of swirling ﬂow in 3D vector ﬁelds. Technical report, Department of Aeronautics and Astronautics, MIT, 1995. AIAA Paper 95-1715. [19] H. Theisel. Designing 2D vector ﬁelds of arbitrary topology. Computer Graphics Forum (Eurographics 2002), 21(3):595–604, 2002. [20] H. Theisel, J. Sahner, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Extraction of parallel vector surfaces in 3d time-dependent ﬁelds and application to vortex core line tracking. In Proc. IEEE Visualization 2005, pages 631–638, 2005. [21] H. Theisel and H.-P. Seidel. Feature ﬂow ﬁelds. In Data Visualization 2003. Proc. VisSym 03, pages 141–148, 2003.

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[22] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Saddle connectors - an approach to visualizing the topological skeleton of complex 3D vector ﬁelds. In Proc. IEEE Visualization 2003, pages 225–232, 2003. [23] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Grid-independent detection of closed stream lines in 2D vector ﬁelds. In Proc. Vision, Modeling and Visualization 2004, 2004. [24] H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Topological methods for 2D timedependent vector ﬁelds based on stream lines and path lines. IEEE Transactions on Visualization and Computer Graphics, 11(4):383–394, 2005. [25] X. Tricoche, G. Scheuermann, and H. Hagen. Continuous topology simpliﬁcation of planar vector ﬁelds. In Proc. Visualization 01, pages 159 – 166, 2001. [26] Xavier Tricoche, Christoph Garth, Gordon Kindlmann, Eduard Deines, Gerik Scheuermann, Markus Ruetten, and Charles Hansen. Visualization of intricate ﬂow structures for vortex breakdown analysis. In Proc. IEEE Visualization 2004, pages 187–194, 2004. [27] T. Weinkauf, H. Theisel, H.-C. Hege, and H.-P. Seidel. Boundary switch connectors for topological visualization of complex 3D vector ﬁelds. In Data Visualization 2004. Proc. VisSym 04, pages 183–192, 2004. [28] T. Weinkauf, H. Theisel, H.-C. Hege, and H.-P. Seidel. Topological construction and visualization of higher order 3D vector ﬁelds. Computer Graphics Forum (Eurographics 2004), 23(3):469–478, 2004. [29] T. Weinkauf, H. Theisel, H.-C. Hege, and H.-P. Seidel. Feature ﬂow ﬁelds in out-of-core settings. In Proc. Topo-In-Vis 2005, Budmerice, Slovakia, 2005. [30] T. Weinkauf, H. Theisel, K. Shi, H.-C. Hege, and H.-P. Seidel. Topological simpliﬁcation of 3d vector ﬁelds by extracting higher order critical points. In Proc. IEEE Visualization 2005, pages 559–566, 2005. [31] R. Westermann, C. Johnson, and T. Ertl. Topology-preserving smoothing of vector ﬁelds. IEEE Transactions on Visualization and Computer Graphics, 7(3):222–229, 2001. [32] T. Wischgoll and G. Scheuermann. Detection and visualization of closed streamlines in planar ﬂows. IEEE Transactions on Visualization and Computer Graphics, 7(2):165–172, 2001. [33] H.-Q. Zhang, U. Fey, B.R. Noack, M. K¨onig, and H. Eckelmann. On the transition of the cylinder wake. Phys. Fluids, 7(4):779–795, 1995.

Control of Wing Vortices I. Gursul, E. Vardaki, P. Margaris, and Z. Wang Department of Mechanical Engineering University of Bath Bath, BA2 7AY, United Kingdom

Summary Vortex control concepts employed for slender, nonslender and high aspect ratio wings were reviewed. For slender delta wings, control of vortex breakdown has been the most important objective, which is achieved by modifications to swirl level and pressure gradient. Delay of vortex breakdown with the use of control surfaces, blowing, suction, high-frequency and low-frequency excitation, and feedback control was reviewed. For nonslender delta wings, flow reattachment is the most important aspect for flow control methods. For high aspect ratio wings, vortex control concepts are diverse, ranging from drag reduction to attenuation of wake hazard and noise, which can be achieved by modifications to the vortex location, strength, and structure, and generation of multiple vortices.

1 Introduction Controlling vortical flows over wings may have various benefits, such as enhancement of lift force, generation of forces and moments for flight control, attenuation of buffeting, reduction of drag, and attenuation of noise due to vortex/blade interaction. Control methods include manipulation of one or more of the following flow phenomena: flow separation from the wing, separated shear layer, vortex formation, flow reattachment on the wing surface, and vortex breakdown. The occurrence and relative importance of these phenomena strongly depend on the wing sweep angle. For delta wings, flow reattachment and vortex breakdown are two important phenomena, which determine the effective flow control strategies. The reattachment location on the wing surface moves inboard with increasing angle of attack, and reaches the wing centreline at a particular incidence. Beyond this limiting angle of attack αFR, flow reattachment to the wing surface is not possible. The variation of predictions [1] of αFR with wing sweep angle is shown in Figure 1. This prediction is only valid for slender wings, and the only experimental observation [2] for a nonslender wing (by visualization of the reattachment line for a sweep angle of Λ=50°) is also given in Figure 1. For slender wings, it is seen that the reattachment incidence decreases with increasing wing sweep angle. Therefore, on highly swept wings, reattachment does not occur beyond small angles of attack. Vortex breakdown appears on the wing with increasing angle of attack and crosses the trailing-edge at a particular incidence αBD. The variation of this angle of attack R. King (Ed.): Active Flow Control, NNFM 95, pp. 137–151, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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[3] with wing sweep angle is also shown in Figure 1. It is seen that this angle of attack increases with increasing wing sweep angle. On nonslender wings, vortex breakdown appears at very small incidences. As the vortex control concepts become increasingly diverse, new actuators and closed-loop control strategies are being developed. It is useful to consider the flow physics and dominant mechanisms as these determine which flow control methods are effective. The main objective of this paper is to review the vortex control concepts, which mainly depend on the wing sweep.

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2 Slender Delta Wings 2.1 Overview of Flow Physics The flow over a delta wing is characterised by a pair of counter-rotating leading-edge vortices that are formed by the roll-up of vortex sheets. The time-averaged axial velocity is jet-like at low and moderate incidences. The large axial velocities in the vortex core are due to very low pressures, which also generate additional suction and lift force, known as vortex lift, on the delta wings. Vortex lift contribution increases with wing sweep angle [4]. At a sufficiently high angle of attack, the vortices undergo a sudden expansion known as vortex breakdown. The axial flow downstream becomes wake-like with very low velocities. For slender wings (defined as Λ ≥ 65° in this paper), vortex breakdown is the dominant flow mechanism that is responsible for decreased lift. It is also the dominant source of unsteadiness that causes wing and fin buffeting [5]. Hence control of vortex breakdown has been the subject of many investigations [6]. There are two important parameters affecting the occurrence and movement of vortex breakdown: swirl level and pressure gradient affecting the vortex core. An increase in the magnitude of either parameter promotes the earlier occurrence of breakdown. Very early experiments [7] demonstrated that vortex breakdown moves upstream over delta wings when the magnitude of either parameter is increased.

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More recently, it was shown [8] that the minimum swirl level required for breakdown decrease with increasing magnitude of adverse pressure gradient. Naturally, flow control methods for the delay of vortex breakdown rely on modification of these two parameters. Active flow control methods generally rely on unsteady excitation in flow control applications. Unsteady forcing has been used for the control of vortex and breakdown in some cases. There are various sources of unsteadiness [5]: shear layer instabilities, vortex wandering, helical mode instability of vortex breakdown, oscillations of breakdown, vortex interaction, and vortex shedding. The frequency spectrum of the unsteady flow phenomena that exist over stationary wings is very wide, which is one of the challenges in numerical simulations of these flows. Vortex breakdown, vortex interactions, and vortex shedding, either alone or in combination, play an important role in wing and fin buffeting, although vortex breakdown is the main source of buffeting over slender wings. Flow control approaches for vortex breakdown are reviewed next. 2.2 Control Surfaces Various control surfaces [6] have been investigated to control the formation, location, strength, and breakdown of the leading-edge vortices: canards, strakes, leading-edge flaps, apex flap, and variable-sweep wings. It is well known that canards can provide substantial delays [9] in vortex breakdown by affecting the external pressure gradient acting on the vortex core. Since all of the vorticity of leading-edge vortices originates from the separation point along the leading-edge, leading-edge flaps are particularly attractive tools that can be used to influence the strength and structure of these vortices. Leading-edge flaps that are deflected downward have been found to reduce drag and improve the lift-to-drag ratio [10]. On the other hand, the flap deflection in the upward direction causes an increase in lift as well as drag. This type of vortex management can be used for landing or aerodynamic manoeuvres [10]. It has been found that the flaps deflected upward generate a stronger vortex lift at low and moderate angles of attack. However, flaps may also induce vortex breakdown [11]. Leading-edge flaps modify the strength and location of the vortices, thereby affecting the parameters that control vortex breakdown. It was shown [11] that breakdown location and its sensitivity strongly depend on incidence and flap angle. For large angles of attack, the variation of breakdown location is not monotonic, and therefore, not suitable for control purposes. A variable leading-edge extension [12] that effectively varies the sweep angle has been used to control leading-edge vortices and breakdown. The advantage of this method is that the variation of breakdown with sweep angle is monotonic, hence suitable for active control purposes. Because most of the vorticity within the vortex core originates from a small region near the apex of the wing, an apex flap can be an effective control surface. It was shown [13] that a drooping apex flap could delay vortex breakdown.

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2.3 Blowing and Suction Blowing and suction applied at various locations are commonly used as flow control methods for leading-edge vortices and breakdown. The most widely used versions include: (a) leading-edge suction/blowing, (b) trailing-edge blowing, (c) along-thecore blowing. These are discussed in detail below. Since the vorticity of the leading-edge vortices originates from the separation line along the leading-edge, control of separation characteristics or shear layer can be used to influence the strength and location of the vortices as well as the location of vortex breakdown. Steady blowing [14, 15] and suction [15, 16] at the leading-edge has been employed, but these methods differ in terms of their effect on swirl level. While blowing, in particular tangential blowing, may increase the swirl level, suction reduces the strength and swirl level due to removal of some of the vorticity shed from the leading-edge. Figure 2(a) shows how the location of shear layer and vortex is modified upstream of breakdown when suction is applied. Figure 2(b) shows the axial velocity contours at the trailing-edge, which show the change from wake-like to jet-like velocity as a result of the delay of vortex breakdown. Detailed measurements [16] show that maximum swirl angle in the core and circulation decrease with suction, which causes the vortex breakdown location to move downstream. It is also worth noting that the leading-edge suction technique does not require thick rounded leading edges. Control of vortices can be achieved without the use of the Coanda jet effect.

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The effect of trailing-edge jets on wing vortices and vortex breakdown has been investigated in several studies [17-20]. Blowing at the trailing-edge also modifies the external pressure gradient and causes delay of vortex breakdown. Favourable effect of a trailing-edge jet [17] could be observed even in the presence of a fin, which produces a strong adverse pressure gradient for a leading-edge vortex. It was shown that fin-induced vortex breakdown can be delayed even for the head-on collision of the leading-edge vortex with the fin [19]. Hence the adverse pressure gradient caused by the presence of the fin could be overcome with a trailing-edge jet. The

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effectiveness of a trailing-edge depends on the wing sweep angle [20]. It appears that it becomes more difficult to delay vortex breakdown with decreasing sweep angle. Along-the-core blowing [21, 22] accelerates the axial flow in the core, and modifies the pressure gradient favourably. Figure 3 shows the effectiveness of various blowing/suction methods from various studies published in the literature. Here the effectiveness is defined as (ǻxbd/c)/Cȝ, where ǻxbd is the change in breakdown location (positive corresponding to delay), c is the chord length and Cµ is the momentum coefficient. Figure 3 shows that along-the-core blowing is the most effective method in terms of delaying vortex breakdown. This can be attributed to the importance of the pressure gradient affecting the vortex core. It is also seen that the effectiveness of blowing and suction is nearly the same. The effectiveness of trailingedge blowing is the lowest among all blowing methods considered. 2

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2.4 Unsteady Control There have been various attempts to control vortices and breakdown by using unsteady excitation. These include small and large amplitude oscillations of leadingedge flaps, periodic variations of sweep angle, periodic suction or blowing, and combined use of leading-edge flaps and intermittent trailing-edge blowing, which are summarized in Reference [5]. These studies fall into two categories: (i) highfrequency excitation, St = fc/U∞ = O(1), (where f is the frequency and U∞ is the free stream velocity) and (ii) low-frequency excitation, St = O(0.1). For the high-frequency excitation, Gad-el-Hak and Blackwelder [23] applied periodic perturbations of injection and suction along the leading-edge of a delta wing (Λ = 60°). They found maximum changes in the evolution of the shear layer when the frequency of perturbations (St = 5.5) is the subharmonic of the frequency of KelvinHelmholtz instability. However, no results were reported regarding the structure of the main vortex and the effect on vortex breakdown. Gu et al. [24] applied periodic suction-blowing in the tangential direction along the leading-edge of the wing (Λ = 75°) and reported a delay of vortex breakdown. The most effective period of the alternate suction-blowing corresponded to fc/U∞ = 1.3. For a less slender wing

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(Λ = 60°), it was shown that oscillatory blowing at the leading edge can enhance the lift at high angles of attack [25], and optimum reduced frequency varied in the range of fc/U∞ = 1 to 2. For small amplitude flap oscillations, the strength of the vortices was larger than that of the quasi-steady case [26], when the excitation frequency was St = 1.2. This range of effective frequencies presumably corresponds to subharmonics of the Kelvin-Helmholtz instability due to vortex pairing. For the low-frequency excitation, there have been reports [5] of increased vortex circulation and delay of vortex breakdown for oscillating flaps, delay of vortex breakdown for harmonic variations of sweep angle for a variable sweep wing, and delay of breakdown for combined flaps and unsteady trailing-edge blowing. When the spectra of unsteady flow phenomena [5] are considered, it is less clear which unsteady phenomena are exploited for the low-frequency excitation. However, it is suggested [27] that the variations in the external pressure gradient generated by unsteady excitation plays a major role. 2.5 Feedback Control The pressure fluctuations induced by the helical mode instability of vortex breakdown can be measured and used as a feedback signal for active control. In such a approach, the rms value of pressure is chosen as the control variable and a feedback control strategy is considered [28]. The monotonic variation of the amplitude of the pressure fluctuations with vortex breakdown location makes the feedback control possible. In identifying a suitable flow controller, several methods were considered, including blowing, suction, and flaps. However, the relationship between the vortex breakdown location and control parameter is unknown or undesirable (i.e., not monotonic) for these methods. A desirable controller should have a monotonic relationship between the control parameter and breakdown location. It was shown [28] that it was feasible to use a variable sweep angle mechanism as a means of controlling the breakdown location by influencing the circulation of the leading-edge vortex. The system was idealized as a first-order system, and integral control was used. The feedback control of breakdown was demonstrated for stationary as well as pitching delta wings.

3 Nonslender Delta Wings 3.1 Overview of Flow Physics Vortical flow over nonslender delta wings (Λ ≤ 55°) has recently become a topic of increased interest in the literature. While the flow topology over more slender wings, typically Λ ≥ 65°, has been extensively studied and is now reasonably well understood, the flow over lower sweep wings has only recently attracted more attention [29]. Vortical flows develop at very low angles of attack, and form close to the wing surface. One of the distinct features of nonslender wings is that reattachment of the separated flow is possible even after breakdown reaches the apex of the wing. However, at large angles of attack in the post-stall region, reattachment

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is not possible, and completely stalled flow occurs on the wing. Active and passive control of reattachment may be beneficial for lift enhancement in the post-stall region. According to Polhamus’ leading-edge suction analogy [4], the vortex lift contribution becomes a smaller portion of the total lift as the sweep angle decreases. Vortex breakdown occurs over the wing even at small incidences, and there is no obvious correlation between the onset of vortex breakdown over nonslender wings and the change of the lift coefficient. Hence, vortex breakdown is not a limiting phenomenon as far as the lift force is concerned for nonslender wings. On the contrary, flow reattachment is key to any flow control strategy as suggested in Figure 1. 3.2 Passive Control with Wing Flexibility Passive lift enhancement for flexible delta wings has been demonstrated as a potential method for the control of vortex-dominated wing flows [30]. Force measurements over a range of nonslender delta wings (with sweep angles Λ = 40° to 55°) have demonstrated the ability of a flexible wing to enhance lift and delay stall compared with a rigid wing of similar geometry. An example for Λ = 40° is shown in Figure 4a. This recently discovered phenomenon appears to be a feature of nonslender wings. Flow visualization, PIV and LDV measurements show that flow reattachment takes place on the flexible wings in the post-stall region of the rigid wings. The lift increase in the post-stall region is accompanied with large self-excited vibrations of the wings as shown in Figure 4b. The dominant frequency of the vibrations of various nonslender delta wings is St = O(1). These vibrations promote reattachment of the shear layer, which results in the lift enhancement. Various measurements including wing-tip accelerations, rms rolling moment, and hot-wire measurements confirm that the dominant mode of vibrations occur in the second antisymmetric structural mode in the lift enhancement region. The self-excited vibrations are not observed for a halfwing model, hence passive flow control for a flexible wing occurs only in the anti-symmetric mode. 1.5

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Spectral analysis of velocity fluctuations [31] along the shear layer showed large sharp peaks, corresponding to the wing vibrations. There are also broad dominant peaks in the spectra of velocity fluctuations in the range of St = 1 to 5 for the poststall incidences, and these correspond to the shear layer instabilities. The center frequency of these peaks decreases with streamwise distance as the shear layer vortices shed conically. There is also a decrease in the spanwise direction due to the vortex pairing process. The frequency of the structural vibrations is in the same range as these natural frequencies [31]. 3.3 Active Control of Reattachment Rigid delta wings undergoing small amplitude oscillations [32] in the post-stall region exhibit many similarities to flexible wings, including reattachment in the post-stall region (see Figure 5). For simple delta wings and cropped delta wings [31] with Λ = 50°, 40°, and 30°, an optimum frequency around St = 1 was identified for which the reattachment is observed. Note that these dramatic changes are observed with unsteady forcing in the post-stall region, whereas there is little effect in the pre-stall region. Hence active control methods in the form of leading-edge oscillations or blowing can be effective. An important parameter is the wing sweep angle. The effect of excitation on a swept wing is similar to the response of the flow over a backward-facing step [33] to the periodic excitation. However, for zero sweep angle, formation of a closed separation bubble at high angles of attack in the post-stall region is not possible. It seems that moderate sweep angles (around 50°) help the formation of semi-open separation bubbles, hence the wing sweep is beneficial in flow reattachment. However, there is a lower limit of sweep angle below which the beneficial effect of wing sweep will diminish. This lower limit of sweep angle is around Λ = 20°. Symmetric perturbations in the form of small amplitude pitching oscillations (1° amplitude) were studied for Λ = 50° simple delta wing. The results show that symmetric perturbations also promote reattachment and vortex re-formation. Hence, St=0

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for active control purposes, both symmetric and anti-symmetric excitations are effective. However, passive control for a flexible wing occurs only in the antisymmetric mode. 3.4 Vortex Re-formation Within the reattachment region, axial flow may develop, resulting in re-formation of the leading-edge vortices. Figure 6 shows an example of vortex re-formation [32], which occurs when the amplitude of forcing is sufficiently large and the frequency of excitation is near an optimum value. The mean breakdown location becomes a maximum at an optimum frequency as shown in Figure 7.

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The time-averaged vorticity flux increases due to the oscillating leading edge, which leads to increased circulation. Although the leading edge vortices become stronger due to the leading edge motion, vortex breakdown is delayed for the oscillating wing compared to the stationary wing for which breakdown is at the apex. This appears to be in contrast to the well-known studies of vortex breakdown, which indicate that increased strength of vortices should cause premature, rather than

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delayed, breakdown. This result suggests that streamwise pressure gradient might be modified favourably due to the wing motion.

4 High Aspect Ratio Wings 4.1 Vortex Control Concepts Control of formation and development of tip vortices in both near-wake and far-wake has potential benefits in a variety of aerodynamic problems. Tip vortices form in a similar way to the leading-edge vortices over low aspect ratio wings and the roll-up process becomes complete within a few chord-lengths downstream of the trailingedge. There appear to be three main applications of flow control approaches with regard to the tip vortices: 1) reduction of induced drag, 2) attenuation of vortex wake hazard on following aircraft, 3) reduction of helicopter noise due to the blade-vortex interaction. These will be briefly reviewed with regard to vortex control concepts. For reduction of induced drag, various methods were considered [34], including planform shape and span, tip shape, winglets, fences and tip sails. Wing tip devices are used to redistribute the vorticity near the wing tip. Similarly, any flow control method that can displace the tip vortices in the outboard direction can be used to reduce the drag as it is inversely proportional to the square of the spanwise distance between the vortices. In order to attenuate wake vortex hazard on following aircraft, the core size of the vortices can be increased by turbulence injection into the core. Various wing tip devices were tested, which were found to be not much effective in the far-wake. Those that seemed to decrease the vortex hazard (such as decelerating chutes and splines placed behind the trailing-edge) had an unacceptable drag penalty [35]. It was noted [36] that passive devices in the form of turbulence generators could increase the size of the vortex core and also promote cooperative instabilities as an additional benefit. A second and more promising method is to rely on the long–wave instabilities occurring in a system of several vortices [37]. Compared to the alleviation of a single vortex by turbulent diffusion, excitation of the long-wave cooperative instabilities of multiple vortices has the potential for much faster destruction of vortex wakes. Hence deliberate creation of multiple vortices and excitation by oscillating surfaces or oscillatory blowing may be a promising strategy. One of the sources of helicopter noise is the interaction of rotor blades with the vortices shed from preceding blades. Noise and vibration caused by this interaction can be reduced by i) increasing the distance between the rotor blade and tip vortex, and (ii) by increasing the size of the vortex core and decreasing the maximum tangential velocity. 4.2 Tip Blowing Most of the intended modifications to the tip region (such as wing tip devices), to the vortex location, strength, and structure can be achieved without the use of passive devices. Active flow control using wing tip blowing can achieve multiple tasks

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during different flight regimes. It was shown that the strength, location, core structure, and number of vortices can be effectively manipulated by tip blowing [38]. The effect of continuous blowing using high-aspect ratio jets is very sensitive to the blowing direction. The location and strength of the vortices generated by the jet and their interaction with the tip vortex lead to different flow configurations. Blowing in the upward direction produces additional co-rotating vortices in the nearwake as shown in Figure 8. These strong vortices forming on the wing surface can be used to increase the lift in certain applications such as hovering rotorcraft. Figure 9 shows that a diffused vortex is obtained with spanwise blowing near the pressure surface. Also, downward blowing produces diffused vortices. For these cases, blowing appears to add a substantial amount of turbulence in the vortex core, resulting in diffused trailing vortices. For downward blowing, it was possible to displace the vortices in the outboard direction, which should result in an induced drag

Fig. 8. Vorticity distribution for upward blowing

Fig. 9. Vorticity distribution for spanwise blowing near the pressure surface

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(b)

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Fig. 10. (a) Velocity and vorticity for the synthetic jet (with no freestream); (b) variation of maximum tangential velocity at x/c=1.0 as a function of dimensionless frequency

reduction. Spanwise blowing also results in a lift augmentation [39] as the downwash decreases with blowing. Drag reduction and lift augmentation are essentially inviscid phenomena due to an effective increase in span and aspect ratio of the wing. Synthetic jets can be useful for wing tip blowing as the air supply problem is avoided. Figure 10 shows that a synthetic jet is beneficial in terms of diffusing the trailing vortex [40]. Synthetic jets produce large turbulence and much larger vortex wandering. For the blowing coefficients used, the effect of excitation frequency was minor. While there are substantial changes in the circulation of the trailing vortex for continuous jets, the circulation remains virtually the same for synthetic jets. The intermittent and diffused jet vortices are weak and do not appear to affect the total circulation.

5 Conclusions For slender delta wings, control of vortex breakdown has been the primary goal of many investigations. Delay of vortex breakdown is possible with the modifications to the swirl level and pressure gradient. The use of control surfaces such as leadingedge flaps makes it possible to control the location, strength, and structure of the vortices. Blowing and suction at the leading-edge, trailing-edge, or along the core have differences in terms of their effects on swirl level and pressure gradient affecting the vortex core. Along-the-core blowing is the most effective method for delaying vortex breakdown. Active flow control using high-frequency excitation targets the Kelvin-Helmholtz instability of the separated shear layers. In contrast, low-frequency excitation appears to modify the external pressure gradient only. For nonslender delta wings, flow reattachment is the most important aspect for flow control methods. Passive lift enhancement on flexible wings is due to the selfexcited wing vibrations, which promote flow reattachment in the post-stall region. The frequency of the wing vibrations (St = O(1)) is in the same range as the natural frequencies of the shear layer instabilities. Rigid delta wings undergoing small amplitude oscillations in the post-stall region exhibit many similarities to flexible wings, including reattachment and vortex re-formation. Moderate sweep angles help the formation of semi-open separation bubbles, hence the wing sweep is beneficial.

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For high aspect ratio wings, vortex control concepts are diverse, ranging from drag reduction to attenuation of wake hazard and noise. Modifications to the vortex location, strength, and structure are common ideas in these applications. Active flow control using wing tip blowing is shown to have potential for various objectives. Depending on the blowing configuration and direction, more diffused or stronger vortices, and also, multiple vortices, that move inboard or outboard can be generated. The deliberate creation of multiple vortices and excitation of vortex instabilities may be a promising strategy for effective control in the far-wake.

Acknowledgements The authors acknowledge the financial support of the Air Force Office of Scientific Research (AFOSR), Engineering and Physical Sciences Research Council (EPSRC) and the Ministry of Defence in the UK.

References [1] Mangler, K.W. and Smith, J.H.B., “A Theory of the Flow Past a Slender Delta Wing with Leading Edge Separation”, Proceedings of Royal Society, A, vol. 251, 1959, pp. 200-217. [2] Taylor, G. and Gursul, I., “Buffeting Flows over a Low Sweep Delta Wing”, AIAA Journal, vol. 42, no. 9, September 2004, pp. 1737-1745. [3] Erickson, G.E., “Water-Tunnel Studies of Leading-Edge Vortices”, Journal of Aircraft, vol. 19, No. 6, June 1982, pp. 442-448. [4] Polhamus, E. C., ‘Predictions of vortex lift characteristics by a leading edge suction analogy’, Journal of Aircraft, Vol. 8, No. 4, pp. 193-199, 1971. [5] Gursul, I., “Review of Unsteady Vortex Flows over Slender Delta Wings”, Journal of Aircraft, vol. 42, no. 2, March-April 2005, pp. 299-319. [6] Mitchell, A.M. and Delery, J., “Research into Vortex Breakdown Control”, Progress in Aerospace Sciences”, vol. 37, 2001, pp. 385-418. [7] Lambourne, N.C. and Bryer, D.W., “The Bursting of Leading Edge Vortices: Some Observation and Discussion of the Phenomenon”, Aeronautical Research Council, R&M 3282, 1962. [8] Delery, J., Horowitz, E., Leuchter, O., and Solignac, J.L., “Etudes Fondamentales Sur Les Ecoulements Tourbillonnaires”, La Recherche Aerospatiale, no. 2, 1984, pp. 81-104. [9] Myose, R.Y., Hayashibara, S., Yeong, P.C. and Miller, L.S., “Effects of Canards on Delta Wing Vortex Breakdown During Dynamic Pitching”, Journal of Aircraft, vol. 34, no. 2, March-April 1997, pp. 168-173. [10] Lamar, J.E. and Campbell, J.F., “Vortex Flaps – Advanced Control Devices for Supercruise Fighters”, Aerospace America, January 1984, pp. 95-99. [11] Deng, Q. and Gursul, I., “Effect of Leading-Edge Flaps on Vortices and Vortex Breakdown”, Journal of Aircraft, vol. 33, no. 6, November-December 1996, pp. 10791086. [12] Yang, H. and Gursul, I., “Vortex Breakdown over Unsteady Delta Wings and its Control”, AIAA Journal, vol. 35, no. 3, 1997, pp. 571-574. [13] Klute, S.M., Rediniotis, O.K., and Telionis, D.P., “Flow Control over a Maneuvering Delta Wing at High Angle of Attack”, AIAA Journal, vol. 34, no. 4, 1996, pp. 662-668.

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[14] Wood, N.J., Roberts, L. and Celik, Z., “Control of Asymmetric Vortical Flows over Delta Wings at High Angles of Attack”, Journal of Aircraft, vol. 27, no. 5, May 1990, pp. 429435. [15] Gu, W., Robinson, O. and Rockwell, D., “Control of Vortices on a Delta Wing by Leading-Edge Injection”, AIAA Journal, vol. 31, no. 7, July 1993, pp. 1177-1186. [16] McCormick, S. and Gursul, I., “Effect of Shear Layer Control on Leading Edge Vortices”, Journal of Aircraft, vol. 33, no. 6, November-December 1996, pp. 1087-1093. [17] Helin, H.E. and Watry, C.W., “Effects of Trailing-Edge Jet Entrainment on Delta Wing Vortices”, AIAA Journal, vol. 32, no. 4, 1994, pp. 802-804. [18] Shih, C. and Ding, Z., “Trailing-Edge Jet Control of Leading-Edge Vortices of a Delta Wing”, AIAA Journal, vol 34, no(7), 1996, 1447-1457. [19] Phillips, S., Lambert, C., and Gursul, I., “Effect of a Trailing-Edge Jet on Fin Buffeting”, Journal of Aircraft, vol. 40, no. 3, 2003, pp. 590-599. [20] Wang, Z. and Gursul, I., “Effects of Jet/Vortex Interaction on Delta Wing Aerodynamics”, 1st International Conference on Innovation and Integration in Aerospace Sciences, 4-5 August 2005, Queen’s University Belfast, UK. [21] Guillot, S., Gutmark, E.J., and Garrison, T.J., “Delay of Vortex Breakdown over a Delta Wing via Near-Core Blowing”, AIAA 98-0315, 36th Aerospace Sciences Meeting and Exhibit, January 12-15, 1998, Reno, NV. [22] Mitchell, A.M., Barberis, D., Molton, P., and Delery, J., “Oscillation of Vortex Breakdown Location and Blowing Control of Time-Averaged Location”, AIAA Journal, vol. 38, no. 5, May 2000, pp. 793-803. [23] Gad-el-Hak, M. and Blackwelder, R.F., “Control of the Discrete Vortices from a Delta Wing”, AIAA Journal, vol. 25, no. 8, 1987, pp. 1042-1049. [24] Gu, W., Robinson, O. and Rockwell, D., “Control of Vortices on a Delta Wing by Leading-Edge Injection”, AIAA Journal, vol. 31, no. 7, July 1993, pp. 1177-1186. [25] Margalit, S., Greenblatt, D., Seifert, A. and Wygnanski, I., “Delta Wing Stall and Roll Control Using Segmented Piezoelectric Fluidic Actuators”, Journal of Aircraft, vol. 42, no. 3, 2005, pp. 698-709. [26] Deng, Q. and Gursul, I., “Effect of Oscillating Leading-Edge Flaps on Vortices over a Delta Wing”, AIAA 97-1972, 28th AIAA Fluid Dynamics Conference, June 29 – July 2, 1997, Snowmass Village, CO. [27] Yang, H. and Gursul, I., “Vortex Breakdown over Unsteady Delta Wings and Its Control”, AIAA Journal, vol. 35, no. 3, 1997, pp. 571-574. [28] Gursul, I., Srinivas, S. and Batta, G., “Active Control of Vortex Breakdown over a Delta Wing”, AIAA Journal, vol. 33, no. 9, 1995, pp. 1743-1745. [29] Gursul, I., Gordnier, R., and Visbal, M., “Unsteady Aerodynamics of Nonslender Delta Wings”, Progress in Aerospace Sciences, vol. 41, 2005, pp. 515-557. [30] Taylor, G., Kroker, A. and Gursul, I., “Passive Flow Control over Flexible Nonslender Delta Wings”, AIAA-2005-0865,43rd Aerospace Sciences Meeting and Exhibit Conference,10-13 January 2005,Reno,NV. [31] Gursul, I., Vardaki, E. and Wang, Z., “Active and Passive Control of Reattachment on Various Low-Sweep Wings”, AIAA-2006-506, 44th AIAA Aerospace Sciences Meeting and Exhibit, 9-12 January 2006, Reno, NV. [32] Vardaki, E., Gursul, I. and Taylor, G., “Physical Mechanisms of Lift Enhancement for Flexible Delta Wings”, AIAA-2005-0867, 43rd Aerospace Sciences Meeting and Exhibit, 10-13 January 2005, Reno, NV.

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[33] Roos, F.W. and Kegelman, J.T., “Control of Coherent Structures in Reattaching Laminar and Turbulent Shear Layers”, AIAA Journal, vol. 24, no. 12, December 1986, pp. 19561963. [34] Kroo, I., “Drag Due to Lift: Concepts for Prediction and Reduction”, Annual Review of Fluid Mechanics, vol. 33, 2001, pp. 587-617. [35] Spalart, P.R., “Airplane Trailing Vortices”, Annual Review of Fluid Mechanics, vol. 30, 1998, pp. 107-138. [36] Coustols, E., Stumpf, E., Jacquin, L., Moens, F., Vollmers, H., Gerz, T., “Minimised Wake: a Collaborative Research Programme on Aircraft Wake Vortices”, AIAA 2003-0938, 41st Aerospace Sciences Meeting and Exhibit, 6-9 January 2003, Reno, NV. [37] Jacquin, L., Fabre, D., Sipp, D., Theofilis, V., and Vollmers, H., “Instability and Unsteadiness of Aircraft Wake Vortices”, Aerospace Science and Technology, vol. 7, 2003, pp. 577-593. [38] Margaris, P., Gursul, I., “Effect of Steady Blowing on Wing Tip Flowfield,” AIAA 20042619. 2nd Flow Control Conference, Portland, Oregon, USA. June-July 2004. [39] Tavella, D. A., Wood, N J., Lee, C. S., Roberts, L., “Lift Modulation with Lateral WingTip Blowing,” Journal of Aircraft, Vol. 25, No. 4, 1988, pp 311-316. [40] Margaris, P., Gursul, I., “Wing Tip Vortex Control Using Synthetic Jets”, CEAS/KATnet Conference on Key Aerodynamic Technologies, 20-22 June 2005, Bremen, Germany.

Towards Active Control of Leading Edge Stall by Means of Pneumatic Actuators C.J. K¨ ahler, P. Scholz, J. Ortmanns, and R. Radespiel Technische Universit¨ at Braunschweig, Institut f¨ ur Str¨ omungsmechanik, Bienroder Weg 3, 38106 Braunschweig, Germany [email protected]

Summary This contribution summarizes the ﬂow control research results obtained at TU Braunschweig and their implication for control on high-lift devices. The superordinate aim of the examination is the control of leading-edge stall on a two-element airfoil by means of dynamic 3D actuators. This is of great practical interest in order to increase the maximum angle of attack and/or the lift coeﬃcient and to alter the drag coeﬃcient in takeoﬀ and landing conﬁguration of future aircrafts. To reach this aim, several pneumatic actuators were designed and systematically tested to determine their characteristics and impulse response on the input signal at ﬁrst. Secondly, their potential for active ﬂow control was investigated in a small wind tunnel. Thirdly, the interaction of promising actuator concepts was studied in detail to examine the beneﬁt of actuator arrays. Finally, the experiences were combined to examine the potential of the actuators in delaying leading edge separation on a generic airfoil. Therefore, an appropriate airfoil was designed, built and equipped with the developed actuator technology and investigated in a large wind tunnel. The results illuminate the potential of dynamic actuators for high-lift applications and the eﬀect of diﬀerent actuator parameters (actuator design, orientation, spacing and position as well as amplitude, frequency and duty-cycle). Furthermore, practical actuator designs and operation rules are deduced from the examination. In the future they may be assistant to assign the results on real aircraft conﬁgurations.

1

Introduction and Motivation

The active control of leading-edge stall by means of dynamic actuators is of great industrial interest because of the ability to increase the maximum angle of attack and thus the lift and to alter the drag in takeoﬀ and landing conﬁguration. Hence, it becomes possible to reduce the number of gaps in high-lift conﬁguration (slat-less wing). This reduces the noise during takeoﬀ and landing and decreases the drag during the climb ﬂight period. In addition it becomes possible to establish new low noise takeoﬀ and landing approach procedures. Furthermore, the slat-less wing may enable laminar ﬂow conditions during cruise ﬂight which is important to reduce fuel consumption. Thus the potential of active control by R. King (Ed.): Active Flow Control, NNFM 95, pp. 152–172, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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means of dynamic actuators can lead to transport aircrafts with reduced design, development, fabrication and maintenance costs and with less impact on the environment. The active control by means of pneumatic actuators has been examined extensively [5]. Basically there are two control strategies, namely control by stimulating natural instabilities[21] and control by enhancing turbulent mixing. The stimulation of instabilities is quite eﬃcient, e.g. in preventing stall on the ﬂap of a high-lift conﬁguration [18]. Unfortunately, this approach fails when the ﬂow state is stable or the excitation frequency unmatched to the frequency of the instable modes. More robust is the transfer high momentum ﬂuid from the outer part of the boundary layer toward the wall by mixing. Another advantage is the fact, that for the stimulation of instabilities dynamic pulsing is an indispensable part of the mechanism, whereas the turbulent mixing does work with either dynamic or static blowing. The actuation can be done directly, by transferring momentum into the desired region by means of tangential blowing [1], or indirectly. The ﬁrst method is not applicable at cruise ﬂight conditions because the eﬃciency decreases with increasing base ﬂow velocity. In the second case, the blockage of the actuator jet is exploited to enhance the mixing of the base ﬂow, similar to mechanical vortex generators [11]. In the past many attempts have been made to replace the stationary blowing actuators in favour of dynamic ones [6,13]. The motivation for applying these actuators is the eﬃciency gain, due to the reduced mass ﬂux, and the excitation of span-wise oriented vortices, similar to starting vortices, which promote the mixing. An increase of approximately 30% of peak vorticity and a penetration of 50 % farther into the boundary layer can be achieved according to [9]. However, due to the large number of control parameters the optimization is diﬃcult. Compton and Johnston varied the skew angle of a 45◦ inclined round jet to demonstrate that a skew angle of 90◦ is an optimal conﬁguration to produce a strong longitudinal vortex [2]. McManus et al. showed that the ratio between the the boundary layer thickness and the actuator diameter should be around four for an eﬃcient control [13]. Nagib et al. concluded that the reduced frequency must be approximately one [15]. In addition they describe that the most eﬀective location for unsteady forcing is near but upstream of the point of separation. Unfortunately, the mutual dependency of the inﬂuencing factors is an open question along with the universality and assignability of the ﬁndings because of the diﬀerent boundary conditititons. This will be examined systematically by determining the 1. 2. 3. 4.

impulse response of dynamic actuators working principle of single actuators in boundary layers synergy eﬀects resulting from the interaction of actuator arrays aerodynamic potential for separation control on high-lift conﬁgurations.

2

Actuator Design and Optimization

To assess the potential of pneumatic actuators for ﬂow control applications their impulse response in terms of amplitude pV , frequency f , duty cycle ∆, waveform,

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settling chamber volume V and feed line length lS was examined at ﬁrst. This is important for characterizing the actuators and the experimental boundary conditions. The dynamical switching was realized with a FESTO MH2 electromechanical fast-switching valve. This device allows to vary frequency and amplitude independently and it can operate at reasonably high frequencies (fmax ≈ 150 Hz) and adequate feeding pressures (pmax ≈ 8.0 bar). The input-signal was provided by a 16 channel custom built frequency generator. Each channel phasing and duty-cycle can be adjusted independently from the other channels. The generator is based on a 16 MHz EEPROM and is conﬁgurable via a dot matrix display or a serial port (RS232). The frequency generator delivers square-wave signals whose amplitude can be adjusted to the speciﬁcations of the magnetic valves by using two custom-built 8-channel ampliﬁers. To characterize the impulse response of the actuators on the input signal a hot-wire probe and a phased-locked stereoscopic and time resolved PIV system were applied [16]. In a ﬁrst set of experiments the conﬁguration of the pressure supply system was analyzed, especially regarding the question how the temporal development of the actuator exit velocity relates to the signal from the frequency generator. Beside the geometry and the size of the exit area S, the volume of the settling chamber V and the length of the feed line lS were of primary importance. For the investigation a modular construction that supports inlets with diﬀerent size, geometry and orientation and a continuous variation of the volume V and the tube length lS was designed [16]. A round actuator-exit with a 1 and 2 mm diameter drill and a rectangular one with a variable aspect ratio in the range b = 5 - 20 mm (length) and h = 0 - 2 mm (width) was examined. In summary it can be stated that the feed line length must be minimized to avoid undesired pressure losses. However, a decreasing feed line length is associated with an ampliﬁcation of pressure oscillations, which travel in the feeding lines with the speed of sound [16]. These oscillations, which are clearly visible in the passive part of the actuator process (actuator oﬀ), result in a signiﬁcant disturbance at the actuator exit and thus in a reduced performance of the dynamic actuation process. At low frequencies the amplitude of the oscillations can reach 20% of the primary actuator exit signal (actuator on). To damp these disturbances, the volume and geometry of the settling chamber between the valve and the actuator exit must be properly designed. The volume must be suﬃciently large to avoid pressure oscillations and disturbances due to the piston characteristics, but small enough to avoid a signiﬁcant damping or modiﬁcation of the exit velocity signal due to additional losses or secondary ﬂows in the settling chamber. In a second series of experiments an actuator array was designed, based on the results of the single actuator examination, and optimized to obtain a similar impulse response at each outlet port. Therefore, the settling chamber was used to distribute the air from a single valve to multiple actuator exits. The essential result of this examination is the fact that the generation of a homogeneous squared wave output signal of identical amplitude at each port requires that the feed line connection is orthogonal to the actuator axis. Otherwise the kinetic

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energy of the jet entering the settling chamber is partly superimposed to the exit velocity of the actuators located adjacent to the feed line outlet. This causes a large scale modulation of the actuator array in span-wise direction which would further complicate any systematic analysis.

3

Investigation of Single Actuators

The geometry and orientation of the actuator exit is probably one of the most important part of the design process. One frequently applied variant are circular holes, which can be pitched and skewed according to ﬁgure 1. The angle between the surface and the blowing direction is called the pitch angle α, the skew angle β is deﬁned as the angle between the blowing axis and the bulk ﬂow direction. A second common variant are rectangular, skewed slots. Other possibilities, such as pitched slots or elliptic holes, are less popular, as problems related with their fabrication are believed to be disproportionate to their possible advantages. z

z

z x

x y

y

x

y

α β

U

β

8

U

8

8

U

Fig. 1. Left: symmetric actuator (α = 90◦ , β not deﬁned); Center: asymmetric actuator (α and β variable); Right: slot actuator (α = 90◦ and β variable)

3.1

Flow Field Analysis

To determine the most suitable actuator geometry and orientation various concepts were systematically studied in a low-speed wind tunnel by means of phasedlocked stereoscopic PIV [16,19]. The diﬀerent actuators were installed 300 mm behind the elliptical leading edge of a ﬂat plate. The boundary layer thickness was about δ ≈ 14 mm at the actuator position and the free stream velocity was set to U∞ = 14 m/s. The orientation of the light-sheet plane was perpendicular to the main ﬂow direction (yz-plane) to visualize the eﬀect of longitudinal vortices. The vector ﬁelds displayed in ﬁgure 2 illustrate the mean ﬂow ﬁeld generated 10 mm behind the actuator at the phase angle of φ = 180◦ (just before closing the valve) for the skew angle α = 60◦ , 30◦ and 90◦ (top to bottom). It can be seen that one important parameter regarding the eﬀectiveness of the actuators is the strength and position of the longitudinal vortices, induced by the actuator, because these ﬂow structures transfer high momentum ﬂuid from the outer part of the boundary layer towards the wall. A second important parameter is the size and location of the blockage domain, which is represented by red color in the lower result. This low momentum region must be small or far away from the wall as any blockage near the wall will promote ﬂow separation [20].

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Fig. 2. Eﬀect of the orientation of a circular actuator on the ﬂow ﬁeld (the out-of-plane velocity component is color coded). Top: α = 60◦ ; Center: α = 30◦ ; Bottom: α = 90◦ .

To visualize the change of stream-wise momentum caused by the actuation, the undisturbed boundary layer was subtracted in ﬁgure 3. In case of the circular actuator (left column) it can be seen that without a blowing component in spanwise direction (upper left image) no high momentum ﬂuid is transferred into the near-wall region. However, when the blowing direction is rotated around the z axis the ﬂow ﬁeld becomes asymmetric and high momentum ﬂuid is transferred indirectly towards the wall. The right column of ﬁgure 3 shows the performance for a rectangular actuator with b = 10 mm and h = 0.314 mm. Far away from the wall the stream-wise momentum is enhanced but in the near-wall region, right

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Fig. 3. Momentum variation relative to the undisturbed ﬂow for the round actuator with 1 mm exit diameter (left) and the rectangular one with b = 10 mm and h = 0.314 mm (right). Recording parameter: f = 100 Hz, ∆ = 50%, φ = 180◦ . Left: α = β = 90◦ ; α = 45◦ , β = 90◦ ; α = 45◦ , β = 60◦ (top to bottom). Right: α = β = 90◦ ; α = 90◦ , β = 60◦ ; α = 90◦ , β = 30◦ (top to bottom).

behind the actuator, the ﬂow is strongly delayed when the actuator is symmetric (upper right image). This implies that this actuator orientation is not suited to prevent ﬂow separation. However, when β is altered to 60◦ and 30◦ , as shown in the lower two images, an eﬃcient transfer of stream-wise momentum toward the wall takes place. This comparison indicates the signiﬁcance of the geometry and orientation of the actuator on the gain in momentum close to the wall. 3.2

Eﬀect of Actuator Geometry and Orientation

To judge the eﬃciency of various conﬁgurations in downstream direction the following assessment factor was deﬁned to characterize the gain in momentum closed to the wall.

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∆I = b

∆u · |∆u| · dy

(1)

This factor can be calculated from PIV velocity ﬁelds by integrating the signed square of the change in stream-wise velocity ∆u(x, y, z) along a span-wise line with constant distance to the surface. The upper result of ﬁgure 4 displays the spatio-temporal momentum contour generated by an inclined slot actuator with β = 45◦ and Λ = Uexit /U∞ = 5.0 and the lower image shows the corresponding assessment factor as a function of the wall distance z. The result implies that the wall distance of the integration line is not important for estimating the eﬃciency of the actuator, provided the position is reasonably closed to the wall and below the lowest vortex core. Thus the gain in momentum according to equation (1) is a well suited measure to quantify the performance of the actuators for active ﬂow control.

Fig. 4. Top: Spatio-temporal distribution of the stream-wise velocity component relative to the undisturbed ﬂow for Λ = 5.0. Blue color indicate regions with increased stream-wise velocity. Bottom: Spatio-temporal distribution of the assessment factor deﬁned by equation (1). Red color indicate a gain in momentum.

To determine an appropriate actuator design for leading edge separation control the assessment factor was calculated for diﬀerent actuator geometries in dependency of the orientation. Figure 5 indicates the superior performance of the circular actuator with α = 30◦ . However, due to the fact that this actuator design is very sensitive on the Reynolds number and the ﬂow direction it can be

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0.2 slot actuator circular hole α=45° circular hole α=30

0.1

∆I/b

[N/m]

0.15

0.05

0 0

20

40

60

β

80

100

120

[°]

Fig. 5. Assessment factor ∆I/b for diﬀerent actuator exit geometries; calculated x = 70 mm donstream of exit; static, Λ = 5.0

concluded that the inclined slot with β = 45◦ − 55◦ is better suited for industrial applications with changing boundary conditions. 3.3

Eﬀect of Amplitude, Frequency and Duty-Cycle

To access the signiﬁcance of amplitude, frequency and duty-cycle of the actuation the assessment factor was determined from wall-parallel PIV results measured suﬃciently close to the wall according to the forgoing discussion. The left result in ﬁgure 6 visualizes that the velocity ratio Λ = Uexit /U∞ of the actuator exit velocity and the free stream velocity has quite a strong eﬀect on the actuator performance. For small Λ the mixing process is ineﬃcient due to the small strength of the longitudinal vortices. However, with increasing Λ the eﬀectiveness of the actuation increases gradually with increasing amplitude so that an eﬀective control of the far ﬁeld can be expected. Only for very large amplitudes the performance decreases again because of the increasing distance of the longitudinal vortex from the wall. In contrast the central result of ﬁgure 6 shows that the frequency has only a very little eﬀect on the gain of momentum in the near wall region along the x coordinate. The duty-cycle on the other hand, shown in the right graph, has a more signiﬁcant eﬀect on the velocity distribution closed to the wall. For ∆ = 50% the curve progression is almost equal to the steady case (which results in a doubling of eﬃciency because only half the input energy is needed), ∆ = 75 % even raises up to higher values. Another important result that can be deduced from the results is the loss of momentum close behind the actuators. This is caused by the blockage of the actuator jet and the fact that the transfer of high-momentum ﬂuid toward the wall by means of a vortex takes some time or domain. For this reason a certain spacing is required between the position of actuation and the position of separation. The exact distance depends on the strength and position of the stream-wise vortex and thus mainly on the actuator geometry and orientation.

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4

Investigation of Actuator Arrays

As the span-wise area that can be controlled by a single actuator is limited, arrays of actuators must be implemented to prevent separation globally. The interactions between actuators have rarely been studied [22]. However, preliminary investigations indicated a promising ampliﬁcation behavior due to non-linear interaction [19,17]. Two methods of combination are possible: Simple staggering of the slots with identical orientation (co-rotating vortices) or alternating staggering, which results in inversely orientated vortex-pairs. The interactions of corotating vortices are very hard to utilize, because the vortices merge quickly or they stay unaﬀected by their neighbors according to [17]. Alternating staggering of two actuators can be structured into diverging or converging conﬁgurations. If two slots are diverging a counter-rotating vortex pair is generated with a downward motion in between. Following the nomenclature of [17] this conﬁguration can be referred to as the common-ﬂow-down conﬁguration. Converging slots on the other hand are enforcing an up-wash in the middle and overspeed areas beside it, the so called common-ﬂow-up conﬁguration. The result displayed in ﬁgure 7 reveals the assessment factor for the commonﬂow-up and common-ﬂow-down conﬁguration for various spacings between the actuators. Due to the retarded area behind the actuators the eﬃciency is negative in the near ﬁeld as already discussed. Approximately 25 mm behind the actuator the overspeed starts to dominate and the eﬃciency becomes positive. The isolated actuator has a broad peak at x = 65 mm with ∆I/b ≈ 0.1. Further downstream the peak gradually decreases due to a deterioration of the vortex’ circulation. In contrast the eﬃciency of the arrays is much greater, but it should be noted that equation (1) does not take the amount of input energy into account. Two actuators actually require twice the ﬂow-rate in the supply system, therefore they should deliver at least twice the gain of momentum! However, they can deliver more: Two slots with ∆y = 20 mm create a very pronounced peak at x ≈ 40 mm due to the conﬂated and therefore understated retardation

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areas and the higher peak velocity. After that initial peak the eﬃciency decreases and at x > 100 mm it is always approximately twice the eﬃciency of the isolated actuator, which means that there is no more ampliﬁcation, only a superposition of the eﬀect of two vortices. For the array ∆y = 30 mm the initial peak is less pronounced, but further downstream the eﬃciency is a little more than twice the one of the isolated actuator. And ﬁnally the arrays with ∆y = 40 and 50 mm do have a pronounced initial peak, then start to decrease in eﬃciency and afterwards feature a constant rise of eﬃciency in the far downstream part of the ﬂow ﬁeld, although the circulation of the vortices should decrease due to viscosity. This behavior can be explained by a beneﬁcial interaction of the vortices [20]. The lower result displayed in ﬁgure 7 displays the eﬃciency of the commonﬂow-down conﬁgurations. The small spacing ∆y = 20 mm is not very beneﬁcial, because the outstanding high initial peak ∆I/b ≈ 0.5 is counteracted by a huge loss of eﬃciency. Much more convenient is a spacing of ∆y = 30 mm, where the vortices are suﬃciently close to amplify the downwash, but the viscosity does not retard the circulation of them – although the eﬃciency of this conﬁguration diminishes quicker than that of the isolated actuator. With greater spacings the distribution changes its character: The initial peak is less pronounced and it takes some time until this combination of actuators starts to be eﬃcient. But

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in the far ﬁeld the greater spacings are actually the best choice – they do not feature high velocities, but their overspeed areas spread out broad. Maximum interaction takes places approximately x = 175 mm behind the actuator, where the common-ﬂow-down conﬁguration features three times the eﬃciency with only twice the supply-energy. By comparing the eﬃciency of common-ﬂow-down and common-ﬂow-up conﬁguration it can be noticed that in principle the common-ﬂow-down arrangement delivers more eﬃciency – at least in the displayed part of the ﬂow ﬁeld. Anyway, a choice between both alternatives is somehow insigniﬁcant, because in order to manipulate a large area obviously more than two actuators will have to be used, which is only possible with alternate orientation. Thus the investigations rather focused on the question of the spacings for the individual conﬁgurations.

5

Airfoil Design, Actuator Integration and Experimental Setup

To validate the potential of the actuators under realistic conditions a singleelement airfoil was designed with the typical leading-edge separation characteristics of a modern slat-less two-element airfoil in high-lift conﬁguration. The advantage of this approach is that a single-element airfoil is less complicated to build and, when testing, it has no uncertainties related with the correct ﬂap angle, gap and overlap, which quickly superimpose the leading edge stall behavior. The disadvantage on the other hand are the diﬀerent pressure distributions. The ﬂap on a two-element airfoil lowers the static pressure at the trailing-edge. Hence the rise in static pressure from the suction spike to the trailing edge is reduced and therefore the tendency to separate. To simulate this eﬀect with a one-element airfoil it was necessary to ﬁrst copy the pressure distribution of the two-element airfoil, then subtract a constant ∆cp so that the rise in pressure between cp,min and the trailing edge equals that one of the two-element airfoil, but the trailing edge is at cp ≈ 0. Finally the pressure gradients had to be designed, so that the boundary layers of the two element airfoil and that one of the design become as comparable as possible. The design was performed for a Re number of 1.5 · 106 with the codes XFoil (for one-element conﬁgurations) and MSES (for multielement conﬁgurations), both developed by Drela et al [3,4]. Figure 8 indicates that the one-element design matches the reference pressure distribution and the gradients quite well, when a constant ∆cp is added - the symbols highlight the pressure distribution and the pressure gradients of the reference conﬁguration. Even the value of cp,min at the suction spike and the small bump due to the laminar-turbulent transition at x/c ≈ 0.01 agree quite well. So the remaining diﬀerence between the two conﬁgurations is the fact, that of course the boundary layer for both elements starts in the stagnation point (cp = 1) and therefore has a slightly longer drop of pressure for the two-element conﬁguration. Figure 9 highlights the main boundary layer parameters ue /Uinf , δ ,Θ,Hk and ReΘ . The ﬁrst 20% of the chord length, which are important for the leading edge ﬂow control devices, agree very well. Just the ue /Uinf is of

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course on a higher level for the two-element conﬁguration. Since the state of the boundary layer dominates the formation of separation it can be expected that the one element airfoil stalls similarly to the two element airfoil. For the experiments a wind-tunnel model with a wingspan of 1.3 m and a chord length of 0.4 m was built using classical carbon ﬁber reinforced plastic manufacturing methods. The actuator system inside the model consists of the valves, which are feed by a master pressure supply line. 20 of these valves and the master supply line are integrated in the space between the spar, the shells and the trailing edge reinforcement. From the valves the air is guided through the spar into the nose of the model. This nose is CNC-milled from a solid aluminumblock and holds 20 settling chambers and the corresponding connections to the valve system. Each valve feeds one individual settling chamber. The chambers are closed with inlays, which hold the actuator oriﬁces. Each settling chamber supports one inlet with four slots. The slots are oriented with a skew angle of 45◦ . The inlays are fabricated using stereo-lithography, they are simply glued into a pocket in the aluminum nose and grinded into the airfoils contour, so they are ﬂush with the model surface. The airfoil model is equipped with 31 static ports which are connected to a PSI pressure system. Thin airfoils at small Reynolds numbers often feature a laminar separation bubble shortly behind the pressure spike. The typical leading edge stall is then dominated by the burst of this bubble, which is a common problem when analyzing airfoil stall behavior at lower Re-numbers. This laminar leading edge

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stall is not the kind of stall that is supposed to appear on the single-element airfoil, which is designed to exhibit leading edge stall even in the absence of the laminar separation bubble. To remove the laminar separation bubble a 0.1 mm thick and 2mm wide transition trip was properly placed just behind the pressure minimum. The experiments were performed in the low-speed wind tunnel MUB which is a closed return atmospheric tunnel with a closed 1.3 x 1.3 m test section. The maximum Reynolds number is Remax = 1.3·106 and the turbulence level is about 0.2 % at 40 m/s [8]. The facility is equipped with a total-pressure rake behind the trailing edge to capture the wake of the model. Forces and coeﬃcients can be determined by integrating the pressure distributions following Jones’ approach. However, in the following only the normal force coeﬃcient cnp will be highlighted, which is very closely connected to the lift coeﬃcient cl .

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Control of Leading Edge Separation

Figure 10(a) displays the diﬀerent characteristics of the cases with free and ﬁxed transition. Both show a distinct hysteresis, as it is typical for a low-Re leading edge stall. In the case with free transition the laminar bubble bursts at α = 10.2◦ . With ﬁxed transition the laminar bubble is destroyed and consequently can not burst, hence greater normal force coeﬃcients can be achieved (cnp,max ≈ 1.24). 1.4

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The testing-matrix covered three excitation frequencies (f = 50 Hz, 75 Hz and 100 Hz), two amplitudes (speciﬁed by a certain supply pressure: pV = 1.0 bar and 1.5 bar) four diﬀerent duty cycles (∆ =12 %, 25 %, 50 % and 75%) and additionally two static cases with diﬀerent amplitudes (pV = 1.0 bar and 1,5 bar). Figure 10(b) revals the measured cnp - α-curves for f = 75 Hz and pV = 1.5 bar. The sensitivity on the duty cycle is highlighted by diﬀerent linestyles. It can be concluded that short the duty cycles is more eﬀective than the longer ones. Only at very high angles of attack the slightly longer duty cycle ∆ =25 % can outreach the ∆ =12 %-case. This behavior is in fact characteristic for all the measured cnp curves and independent on the frequency f or the amplitude pV . Secondly all the curves show a small local maximum near α = 12.5◦ and a local minimum near α = 14.0◦. In ﬁgure 11(a) six diﬀerent, selected conﬁgurations are shown. The three dashed lines and the three solid lines each represent one conﬁguration with just a change in frequency. Generally it can be stated that the sensitivity on the excitation frequency is weak compared to a variation of the duty-cycle. Only slight diﬀerences can be measured with changing frequencies. Figure 11 (b) highlights the eﬀect of the amplitude for two diﬀerent conﬁgurations. The results indicate

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that a small amplitude (dashed lines) is better in the beginning of the stall (α ≈ 12◦ ), but with increasing angle the higher amplitude cases (solid lines) overcome the smaller ones. This again is consistent throughout the measured cnp -curves and not dependent on the frequency. Figure 12 displays the pressure distributions for two cases in order to highlight the eﬀect of a variation of the duty cycle ∆. The left side displays a case at α = 14.5◦ just before stall. The clean conﬁguration develops a distinct pressure spike. When the actuators are active the pressure spike degrades - for some reason it degrades most for the shortest duty cycle ∆ = 12 %. Exactly this conﬁguration on the other hand develops a lowered cp in the region x ≈ 40 mm· · · 200 mm. The case with higher angle of attack α = 16.0◦ does show quite a similar behavior. The clean airfoil is stalled completely, but the conﬁgurations with actuation are able to maintain a pressure spike. Comparing the diﬀerent duty cycles again the shortest duty cycle shows the greatest cp,min , but again develops the region of lower cp further downstream. To access the physical mechanism of the actuation process extended PIV measurements have been performed for many diﬀerent conﬁgurations. Three exempliﬁed results at α = 19.0◦ are shown in ﬁgure 13, along with the measured pressure distributions. The left row shows the basic case without actuation, where the ﬂow is separated from the very leading edge. The point of separation is at the nose at x ≈ 2 mm (which equals x/c ≈ 0.005). The turbulent kinetic energy level is low, since the shear layer has relative static strength and position. The other two rows display conﬁgurations with actuation (refer to above ﬁgures for the normal fore coeﬃcients of these conﬁgurations). With actuation the airfoil produces a reasonably high cnp coeﬃcient, however it appears to be

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separated as well. Nevertheless, with actuation the ﬂow turning at the nose is noticeably greater in both actuated conﬁgurations. With dynamic actuation (ﬁgure 13(b)) separation occurs slightly later than in the clean conﬁguration, hence a lower nose-pressure can be abided, which is coherent to the measured pressure distributions. The turbulent kinetic energy level is high. With static actuation the ability to turn the ﬂow around the leading edge is even greater, resulting in a nose-pressure of cp = −4.0, which is quite remarkable at α = 19.0◦ . In the contour plots a noticeable smaller separation area is visible as well as a level of the turbulent kinetic energy, which roughly equals that of the dynamic actuation. This is quite remarkable, since the displayed velocity and energy ﬁelds are mean values and not phase locked. So although for the dynamic actuation the turbulence level actually contains the unsteadiness of the actuators, the static actuation results in the same amount of turbulent kinetic energy. The reason for this is not really understood, however, the static actuation results in lower cnp , since the dynamic one develops the lowered cp in the region further downstream (as discussed in Fig. 12). Although the actuators cannot prevent separation totally, they have a very advantageous inﬂuence on the separation, as the lift is maintained far beyond the reference αmax of the clean conﬁguration. This behavior explains the local maximum at α = 12.5◦ and the local minimum at α = 14.0◦ in the cnp -curves discussed above. These local extrema are due to a slow transition from a nonseparated (for α < 12.5◦) into a completely separated state (for α > 14.0◦). Since the separation is inﬂuenced by the ﬂow control system this “stall” is not associated with a great loss of cnp , but only with a small “bump”, after which cnp rises again. As a matter of fact the overall behavior that was found with the airfoil model is not very consistent to the sensitivities that were found during the optimization process—for example the system was expected to be most eﬀective with long

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Fig. 13. Result of the PIV-measurements at high angle of attack α = 19.0◦ . Velocity magnitude |V | in m/s, turbulent kinetic energy Ekin in [m2 /s2 ]. (a) clean airfoil (no actuation). (b) dynamic actuation f = 75 Hz, ∆ = 25 %, pV = 1.5 bar. (c) static actuation pV = 1.5.

duty cycles and almost independent of the excitation frequency (see Fig. 6). It has to be concluded that whether the amplitude of excitation is too low or the actuators are positioned too far downstream, so that the separation line is in the unaﬀective area shortly behind the actuator, where ∆I/b is negative. Both facts can be circumvented by positioning the actuators further upstream. Since the stall is eﬀectively at the leading edge this means, that the actuators have to be positioned at the lower side, in between the stagnation point and the leading edge. Due to the relatively slow ﬂow velocity a much higher velocity ratio λ can be realized at the same supply pressure pV or exit velocity vj . This variation was performed by simply turning the aluminum nose. Although the nose is not perfectly symmetric the diﬀerence in the pressure distribution is negligible. Fig. 14 displays a variation of amplitude pV and a variation of the duty cycle ∆. As can be seen the cnp,max was again raised by about 0.1,

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but with this turned-nose the cnp -curve was extended almost linearly. With this conﬁguration the dependencies, that were found during the optimization process, are again present, namely greater duty cycle or greater supply pressure leading to better inﬂuence. The excitation frequency on the other hand has no eﬀect on the displayed cnp -curves. The new cnp,max ≈ 1.4 is a somehow natural limit of this conﬁguration, since none of the actuations were able to go further, neither by increasing the duty cycle nor by increasing the supply pressure.

7

Conclusions

Various pneumatic actuators were systematically examined to examine their eﬃciency and potential for active ﬂow control on high-lift devices. The investigation indicates the signiﬁcance of actuator geometry and orientation and the eﬀect of the amplitude, frequency and duty-cycle for active ﬂow control. In addition the advantage of actuator arrays was outlined and their potential for leading edge separation control could be estimated. The following main results can be deduced from the analysis: 1. To optimize pneumatic actuators the strength and position of the longitudinal vortices and the size and location of the blockage domain has to be properly aligned. This can be done by means of the geometrical and dynamical actuator parameter. 2. Pneumatic actuators become most eﬃcient when the orientation is not symmetrical to the main ﬂow direction, because a strong large-scale secondary ﬂow is generated that transfers high momentum ﬂuid in the near wall region.

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3. The circular actuator with α = 30◦ and β = 55◦ shows a very good mixing characteristics. However, as this actuator design is very sensitive on the Reynolds number and the ﬂow direction the inclined slot with β = 45◦ − 55◦ is a better choice for industrial applications with changing boundary conditions. 4. A certain spacing between the position of actuation and the position of separation is required for eﬃcient ﬂow control, because of the blockage of the actuator jet and the fact that the transfer of high-momentum ﬂuid towards the wall by means of a vortex requires some spatial distance. The exact distance depends on the strength and position of the stream-wise vortex and thus mainly on the actuator geometry and orientation. 5. The ratio between the actuator exit velocity and the free stream velocity has a strong eﬀect on the actuator performance. For small values the mixing process is ineﬃcient due to the small strength of the longitudinal vortices. With increasing actuator amplitude the eﬀectiveness of the actuation increases gradually and an eﬀective control of the far ﬁeld is possible. 6. The frequency has a minor eﬀect on the gain of momentum close to the wall in downstream direction. 7. The duty cycle has a strong eﬀect on the gain of momentum close to the wall in downstream direction. Generally, large duty cycles are well suited to control separation in the far ﬁeld. For near ﬁeld control small duty cycles should be applied. 8. The eﬃciency of actuator arrays relatively large compared to the single actuators when the actuator spacing and orientation is properly selected. 9. The eﬃciency of common-ﬂow-down actuator is superior to common-ﬂow-up array conﬁgurations – at least in the displayed part of the ﬂow ﬁeld. 10. For leading edge stall separation control small actuator amplitudes and small duty cycles are better in the beginning of the stall (α ≈ 12◦ ). With increasing angle the higher amplitude case overcomes the smaller one as trailing edge stall appears gradually when the leading edge stall is suppressed. 11. Generally the conﬁgurations with actuation are able to maintain a pressure spike. The shortest duty cycle shows the greatest cp,min , but develops the region of lower cp further downstream. 12. Although a single span-wise actuator array cannot fully prevent the separation on the entire airfoil, they maintain the lift far beyond the αmax without actuation. 13. By displacing the actuation position further upstream the eﬃciency scales with the introduced momentum and due to the relatively slow mean ﬂow velocity Uedge at the upstream position a much higher velocity ratio Λ = Uexit /Uedge can be realized at the same supply pressure pV or exit velocity Uexit . 14. At the upstream actuator position the turning of the base ﬂow on the suction side caused by the jets is avoided and thus the local dip of the lift. In eﬀect the cnp -curve increases quite linearly with increasing angle of attack.

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Acknowledgements The project is part of BMWA’s Lufo III “Innovative Hochauftriebskonﬁgurationen” and is being dealt with as participant of the joint research project “Dynamische 3D Str¨ omungskontrolle”.

References [1] Chang P. K.; Control of Flow Separation; Hemisphere Publishing Corporation, McGraw-Hill Book Company, 1976 [2] Compton, D. A. and Johnston, J. P.; Streamwise Vortex Production by Pitched and Skewed Jets in a Turbulent Boundary Layer; AIAA Journal, Vol. 30, No. 3, pp. 640-647, 1992 [3] Drela, M.; Youngren, H.; XFoil 6.94 User Guide; MIT, Aero & Astro, 2001 [4] Drela, M.: A User’s Guide to MSES 2.95; MIT, Computational Aerospace Sciences Laboratory; 1996 [5] Fiedler, H.E.; Fernholz, H.-H.; On Management and Control of Turbulent Shear Flows; Prog. Aerospace Sci., Vol. 27, Pergamon Press, 1990 [6] Gad-el-Hak, M.; Flow Control: The Future; Journal of Aircraft, Vol. 38, No. 3, pp. 402-418, 2001 [7] http://www.tu-braunschweig.de/ism/institut/wkanlagen [8] http://www.tu-bs.de/ism/institut/wkanlagen/mub [9] Johari, H. and Rixon, G. S.; ﬀects of Pulsing on the Vortex Generator Jet; AIAA Journal, Vol. 41, No. 12, pp. 2309-2315, 2003 [10] Johari, H., Pacheco-Tougas M. and Hermanson, J. C.; Penetration and Mixing of Fully Modulated Turbulent Jets in Crossﬂow; AIAA Journal, Vol. 37, No. 7, pp. 842-850, 1999 [11] Johnston, J. P. and Nishi, M.; Vortex Generator Jets - Means for Flow Separation Control; AIAA Journal, Vol. 26, No. 6, pp. 989-994, 1990 [12] K¨ ahler, C.J. and Scholz, U.; Investigation of laser-induced ﬂow structures with time-resolved PIV, BOS and IR technology; 5th International Symposium on Particle Image Velocimety, Busan, Korea, September 22-24, Paper 3223, 2003 [13] McManus, K., Ducharme, A., Goldey, C. and Magill, J.; Pulsed Jet Actuators for Suppressing Flow Separation; AIAA Paper 96-0442, 1996 [14] McManus, K. R., Joshi, P. B., Legner, H. H. and Davis S. J.; Active Control of Aerodynamic Stall using Pulsed Jet Actuators; AIAA Paper 95-2187; 1995 [15] Nagib, H., Kiedaisch, J., Greenblatt D., Wygnanski, I. and Hassan, A.; Eﬀective Flow Control for Rotorcraft Applications at Flight Mach Number; AIAA Paper 2001-2974, 2001 [16] Ortmanns, J.; K¨ ahler, C.J.: Investigation of Pulsed Actuators for Active Flow Control Using Phase Locked Stereoscopic Particle Image Velocimetry; Proceedings of the 12th Int. Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, 2004 [17] Pauley, W.R.; Eaton, J.K.; Experimental Study of the Development of Longitudinal Vortex Pairs Embedded in a Turbulent Boundary Layer; AIAA Journal, Vol. 26, No. 7, American Institute of Aeronautics and Astronautics, 1988 [18] Schatz, M.; Thiele, F.; Petz, R.; Nitsche, W.; Separation Control by Periodic Excitation and its Application to a High Lift Conﬁguration; AIAA-2000-42507, American Institute of Aeronautics and Astronautics, 2004

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[19] Scholz, P.; Ortmanns, J.; K¨ ahler, C.J.; Radespiel, R.: Inﬂuencing the Mixing Process in a Turbulent Boundary Layer by Pulsed Jet Actuators; 12.DGLRsymposium of AG STAB, Bremen, 2004; (to be published in: “Notes on Numerical Fluid Mechanics and Multidisciplinary Design”, Springer-Verlag, 2005) [20] Scholz, P.; Ortmanns, J.; K¨ ahler, C.J.; Radespiel, R.: Performance Optimization of Jet Actuator Arrays for Active Flow Control; Proceedings of the CEAS/KATnet Conference, Bremen, Germany, 2005 [21] Seifert, A.; Bachar, T.; Wygnanski, I.; Koss, D.; Shepshelovich, M.; Oscillatory Blowing, A Tool to Delay Boundary Layer Separation; AIAA Journal, Vol. 31, No. 11, pp. 2052-2060, 1999 [22] Watson, M.; Jaworski, A.J.; Wood, N.J.; Contribution to the Understanding of Flow Interactions Between Multiple Synthetic Jets; AIAA Journal, Vol. 41, No. 4, American Institute of Aeronautics and Astronautics, 2003

Computational Investigation of Separation Control for High-Lift Airfoil Flows Markus Schatz, Bert G¨unther, and Frank Thiele Hermann-F¨ottinger-Institute of Fluid Mechanics, M¨uller-Breslau-Str. 8, 10623 Berlin, Germany [email protected] http://www.cfd.tu-berlin.de

Summary This paper gives an overview of numerical ﬂow control investigations for high-lift airfoil ﬂows carried out by the authors. Two conﬁgurations at stall conditions, a generic two-element setup with single ﬂap and a second conﬁguration with slat and ﬂap of more practical relevance are investigated by simulations based on the Reynolds-averaged Navier-Stokes equations and eddy-viscosity turbulence models. For both cases ﬂow separation can be delayed by periodic vertical suction and blowing through a slot close to the leading edge of the ﬂap. By simulating different excitation modes, frequencies and intensities optimum control parameters could be identiﬁed. Comparison of aerodynamic forces computed and ﬂow visualisations to experiments allows a detailed analysis of the dominant structures in the ﬂow ﬁeld and the effect of ﬂow control on these. The mean aerodynamic lift can be signiﬁcantly enhanced by the active ﬂow control concepts suggested here.

Nomenclature c, cclean , ck c l , cd Cµ f, F + H Re F, St u0 , ua α, δf , δs ∆t

chord length of main airfoil, clean airfoil, ﬂap lift coefﬁcient, drag coefﬁcient 2 momentum coefﬁcient, Cµ = 2 Hc uua0 excitation frequency, non-dimensional frequency F + = f ck /u0 slot width (H = 0.004ck ) Reynolds number based on chord length vortex-shedding frequency, Strouhal number based on ﬂap length inﬂow velocity, excitation velocity ua in the slot angle of attack of the main airfoil, ﬂap and slat deﬂection angle time stepping

1 Introduction Modern commercial airplanes are equipped with sophisticated multi-element high-lift devices consisting of slat and single or multiple ﬂaps that must generate a tremendous R. King (Ed.): Active Flow Control, NNFM 95, pp. 173–189, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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amount of lift during take-off and landing in order to reduce ground speeds and runway lengths. As such elements tend to be complex, heavy and expensive, aerodynamic research has traditionally aimed at simpliﬁcation of such elements without loosing efﬁciency. Experimental and numerical investigations showed that effectivity can significantly be improved by delaying ﬂow separation on the ﬂap in the case of high ﬂap deﬂections for airfoil conﬁgurations at low and high Reynolds numbers [1,2,3]. In the last decades a large number of experimental and numerical studies showed the general effectiveness of ﬂow control for single airfoils. In most investigations, leading edge suction is applied for transition delay [4]; nonetheless, jet ﬂaps are also employed for lift increase and manoeuvering. Surface suction/blowing can be used to rapidly change lift and drag on rotary wing aircraft [5]. However, most control techniques considered in the past showed low or negative effectiveness. In further investigations oscillatory suction and blowing was found to be much more efﬁcient with respect to lift than steady blowing. The process becomes very efﬁcient if the excitation frequencies correspond to the most unstable frequencies of the free shear layer, generating arrays of spanwise vortices that are convected downstream and continue to mix across the shear layer. Suction and blowing can be applied tangential to the airfoil surface [6], rectangular [7] or with cyclic vortical oscillation. In order to create an effective and efﬁcient control method previous studies have been primarily focused on the parameters of the excitation apparatus itself. Overviews are given by Wygnanski and Gad-el-Hak [8,9].

NACA4412

SCCH

NA

CA

441

5

Fig. 1. Active ﬂow control by means of vertical suction and blowing at the ﬂap, applied to; left: generic tow-element high-lift conﬁguration; right: three-element conﬁguration

In the investigations surveyed in this paper, periodic excitation is applied to delay ﬂow separation on the ﬂap of multi-element high-lift conﬁgurations resulting in enhanced lift and reduced drag. The objective of this investigation is to better understand the functionality of the periodic excitation on pressure-driven separated ﬂow over multielement high-lift conﬁgurations. All presented investigations are obtained numerically. In wide parts however, the study includes results from cooperations with experimentalists using Particle Image Velocimetry (PIV) measurements. Additional beneﬁt can be obtained from the combination of those results and the presented relatively low-cost simulations that allow an easy overview of the entire ﬂow ﬁeld in order to capture the effect of ﬂow control techniques.

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2 Computational Method All numerical investigations are based on the ELAN code that was developed at the Hermann-F¨ottinger-Institute of Fluid Mechanics at the Berlin University of Technology. The method is based on a Finite-Volume solver for the incompressible Navier-Stokes equations. The method is fully implicit and of second order in space and time. The SIMPLE pressure correction algorithm is applied based on a collocated storage arrangement for all quantities. Convective ﬂuxes are approximated by a TVD-MUSCL-scheme. 2.1 Turbulence Modelling The simulation program can be run in URANS mode, solving the Unsteady Reynoldsaveraged Navier-Stokes equations using statistical turbulence models as well as in a mode for Large-Eddy Simulation (LES) or combinations of both. In previous URANS investigations with a large variety of different one- and two-equation turbulence models as well as Explicit Algebraic Reynolds-stress Models (EARSM) the SST k-ω model by Menter [10] and the LLR k-ω model by Rung [11] exhibited the best overall performance for steady and unsteady airfoil ﬂows with large separation [12]. The latter represents an improved two-equation eddy-viscosity model formulated especially with respect to the realizability conditions. Beside these two, three other eddy-viscosity turbulence models are applied in comparison: the Spalart-Allmaras (SA) and SALSA oneequation models [13,14] and the Wilcox k-ω model [15]. 2.2 Boundary Conditions At the wind tunnel entry all ﬂow quantities including the velocity components and turbulent properties are prescribed. The level of turbulence at the inﬂow is set to T u = 1 2 1/2 = 0.1% and the turbulent viscosity µt /µ = 0.1. At the outﬂow a convecu0 ( 3 k) tive boundary condition is used that allows unsteady ﬂow structures to be transported outside the domain. All surfaces of airfoil, slat and ﬂap are modeled as a non-slip boundary condition. As the resolution is very ﬁne, a low-Re formulation is applied. The wind tunnel walls however, are considered as frictionless walls. To model the excitation apparatus, a suction/blowing type boundary condition is used. The perturbation to the ﬂow ﬁeld is introduced through the inlet velocity to the small chamber representing the excitation slot. In the case of zero-net-mass excitation it is given by: u0 + 1 c Cµ · cos t · 2π F (1) uexc (t) = u0 2H ck where Cµ is the non-dimensional steady momentum blowing coefﬁcient and H is the slot width. The other excitation modes discussed later are based on similar formulation.

3 Two-Component High-Lift Conﬁguration The ﬁrst part of the study is related to experimental investigations by Tinapp [1] and extensions of Petz [16]. The test model is a generic two element high-lift conﬁguration,

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Fig. 2. Sketch of the two-component conﬁguration in the wind tunnel and in the simulation with ﬂow control by periodic suction and blowing through a slot on the ﬂap

consisting of a NACA 4412 main airfoil and a NACA 4415 ﬂap with ck /c = 0.4 relative chord length. Both proﬁles have bluff trailing edges. A previous study [17] of a similar test case showed that due to strong blocking the effect of the tunnel walls is very important and needs to be considered. The main airfoil is mounted at 52% of the tunnel height (h = 7.8 c), whereas the ﬂap is situated at a ﬁxed position underneath the trailing edge of the main airfoil, thus forming a gap of F G = 0.078c with an overlap of F O = 0.027c (ﬁgure 2). In the numerical study, the angle of attack is ﬁxed at α = 3◦ for the main airfoil and β = α + δf = 40◦ for the ﬂap. Under this condition the ﬂow is characterized by the onset of stall on the ﬂap. According to the experiments the freestream velocity corresponds to a ReynoldsNumber of Re = 1.6 · 105 based on the main-airfoil chord. Transition is ﬁxed at the positions of turbulator strips at 4.5% chord on the main airfoil and 2.8% chord on the ﬂap according to the experimental setup. In the experiments [1] periodic oscillating pressure pulses are generated externally by an electrodynamic shaker driving a small piston. This results in an oscillating jet emanating perpendicular to the chord from the narrow slot 4% chord behind the ﬂap leading edge. The slot width is given by H = 0.004 ck . Due to the experimental setup this excitation is presumed to be completely two-dimensional and therefore all computations are correspondingly 2d. The computational domain starts 4 chords upstream and ends 8 chords downstream of the conﬁguration. The computational c-type mesh provides 202 chordwise cells around the main airfoil and 196 around the ﬂap resulting in 37,000 cells in total. The non-dimensional wall-distance of the ﬁrst cell center remains below Y + = 1 on the complete surface for an attached steady case. The excitation slot is resolved by 20 × 20 cells. Additional simulations use a 64,000 cells mesh to evaluate the measure of mesh dependency (for more details see [2]). 3.1 Unexcited Flow First the ﬂow around the conﬁguration is simulated without excitation. In an early stage of the investigations Franke et al. [17] could obtain convincing results by using the LLR

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k-ω model of Rung and Thiele [11] that are signiﬁcantly better compared to standard kε and k-ω formulations for similar conditions. Their simulation results strongly depend on the transition position on the ﬂap as well as on the turbulence model. In the fully turbulent case, the ﬂow remains attached to the ﬂap surface, whereas tripped transition leads to ﬂow separation. 6 -1

Exp. Nitsche/Tinapp

5

LLR k-ω Wilcox k-ω Spalart-Allmaras

10

LLR k-ω Spalart-Allmaras

4

amplitude of cl

Wilcox k-ω 3 -cP 2 1

-2

10

-3

10 0 -1

-4

0.0

0.5

1.0 x/c

10

0.0

0.5

1.0 1.5 2.0 Strouhal number

2.5

3.0

Fig. 3. left: Pressure distribution on main airfoil and ﬂap for different turbulence models without excitation in comparison to experiments [18]; right: Spectrum of lift coefﬁcient for different models without excitation

In the present investigation the ﬂow is characterized by massive separation which is predicted well by all turbulence models. The prediction of the ﬂow topology agrees qualitively with experimental results as can be seen in the pressure distribution in ﬁgure 3 (left) that is available for the upper surfaces of main airfoil and ﬂap. Compared to the experimental results the suction peak is overpredicted in the numerical simulation. This is caused by the direction of the ﬂow-vector behind the main airfoil trailing edge. In the experiments strong three-dimensional effects appear that are neglected in the simulation. The most promising results were obtained by using the k-ω models (ﬁgure 3 left). The separation point for this case is located in the laminar part of the ﬂap boundary-layer slightly upstream of the excitation slot, such that a large reverse ﬂow region forms downstream (ﬁgure 4). The two-equation models provide a correct prediction of the ﬂap separation. In the case of the Spalart-Allmaras model however, the ﬂap separation position is located too close to the leading edge, resulting in slightly underpredicted pressure in the reverse ﬂow region. Here the ﬂow is almost steady and compared to the lift coefﬁcient spectra of the two-equation models, no dominant frequencies, that might indicate vortex-shedding, occur (ﬁgure 3 right). Results of the two-equation models, however, show a strong amplitude for a Strouhal number based k ≈ 0.5. These results show the typical behaviour of the on the ﬂap chord St = F uc∞ SA-model for unsteady airfoil ﬂows reporting a very low unsteadyness. In the majority of following investigations the LLR k-ω model is used. To correctly capture all unsteady ﬂow features a proper time-resolution is required and the simulation has to last at least several periods to avoid launching effects. In the present case, all simulations consider an unsteady ﬂow ﬁeld with a non-dimensional

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Fig. 4. Iso-contour plot of the mean velocity obtained by hot wire measurements of Tinapp/Petz [16] (left ﬁgure) and a numerical simulation (right ﬁgure) for the unexcited ﬂow at a low Reynolds number of Re = 160, 000 k time stepping of ∆t = 0.01 uc∞ . One period of vortex-shedding is resolved by 230 time steps in this case. Results of a ﬁner time-stepping with 595 steps per period (∆t = k ) do not show signiﬁcant differences to the present results. 0.004 uc∞ The results of the URANS simulation are plotted in ﬁgure 4 right, compared to mean velocities obtained by traversing hot wire measurements in ﬁgure 4 left. Flow patterns in both results are in good agreement and show that leading edge separation on the ﬂap occurs coupled with a large recirculation region downstream. The ﬂow passing through the slot between main airfoil and ﬂap behaves similar to a free jet rather than following the ﬂap surface. Operating the conﬁguration under these conditions is not effective and its performance must be improved in order to become applicable for airplanes. The effect of the Reynolds number on the mean lift coefﬁcient was investigated by Petz [3]. A signiﬁcant increase in lift can be observed for higher Reynolds numbers, however, the overall structure of the mean ﬂow remains more or less similar to the lowest Reynolds number in ﬁgure 4. Dominant structures with a single frequency have been identiﬁed in URANS computations as well as in experiments. These correspond to a Kelvin-Helmholtz type of ﬂuctuation with a Strouhal number that increases for higher Reynolds numbers.

3.2 Active Flow Control Simulations In the following investigations ﬂow control mechanisms are applied. All ﬂow control computations start from the baseline case solution as the initial ﬂow condition. The k is used. LLR k-ω model and a time stepping of ∆t = 0.01 uc∞ Excitation modes. Figure 5 gives an overview of the investigated excitation modes. Very simple approaches of ﬂow control are steady suction and blowing like the modes c) and d) in ﬁgure 5. First this kind of excitation is investigated and results are summarised in ﬁgure 5 right. The blowing/suction intensity is set to Cµ = 100 · 10−5 . The results indicate, that only steady suction is effective. Flap separation can be delayed and the mean lift increases by 2% while the drag drops. It is also remarkable

excitation

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a)

Case b)

c)

d)

mean mean ﬂow Strouhal lift drag sep. number c¯l c¯d xd /ck St

without excitation 2.69 0.20 4.4% steady blowing c) 2.63 0.20 3.6% steady suction d) 2.74 0.17 14.6%

0.44 0.42 0.53

time

Fig. 5. left: Time evolution of excitation for different modes; right: Resulting ﬂow properties for steady suction and blowing

that the frequency of detaching vortices signiﬁcantly increases from St = 0.44 in the baseline simulation to St = 0.53 for steady suction. Steady blowing does not give any positive effect. For the majority of investigations a zero-net-mass-ﬂux kind of excitation like mode a) in ﬁgure 5 is applied. In the case of excitation mode b) which represents a kind of periodic suction an enhancement of the mean lift coefﬁcient appears (ﬁgure 9 centre). However, the effect always remains below that of excitation mode a) for comparable amplitudes. If not particularly mentioned, all following investigations are based on mode a) excitation. Periodic excitation with different frequencies. Experimental investigations show that different excitation frequencies require different intensities to get the same kind of ﬂow control in post-stall cases [1,19]. In order to ﬁnd an optimum excitation, simulations with different frequencies and Cµ = 50 · 10−5 are performed. The largest lift can be found at a frequency of F + = 0.62 (ﬁgure 6 right). In this case the lift coefﬁcient can be enhanced by 14% compared to the baseline simulation. The optimum frequency is slightly larger than the frequency of detaching vortices without excitation (F + ≈ 0.5). At the same time the mean separation position moves from less than 5% chord downstream to 15.5% (ﬁgure 7 left) and drag drops from cd = 0.2 in the baseline case down to cd = 0.11 for frequencies of 0.4 < F + < 1.0 (ﬁgure 7 right). The differences in drag and separation position, however, are very small over a wide range of frequencies. Visualizations of the ﬂowﬁeld indicate the relevance of the detaching vortices for the effectivity of periodic excitation. This structure is dominated by the excitation frequency and turns out to be more important than the instability frequency of the free shear layer in the baseline case. The periodic excitation behaves like a periodic suction that always moves the free shear layer close to the ﬂap surface. At very low frequencies more than one vortex can detach during one excitation period (ﬁgure 8 left). In the case of high frequency excitation, however, the time between two suction events is too short to form a complete vortex (ﬁgure 8 right). Consequently only a part of a complete vortex can detach during one period and each vortex is devided into subvortices. Both cases are less

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mean lift coefficient cl

lift coefficient cl

Cµ=50x10

3.00

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2.8

2.6 +

2.90

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+

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2.4

F = 1.54 0

10 20 30 non-dimensional time t u0/c

40

2.60 0.0

0.5

1.0 1.5 + excitation frequency F

2.0

Fig. 6. left: Unsteady lift coefﬁcient over the non-dimensional time for three different excitation frequencies; right: Mean lift coefﬁcient for different excitation intensities 0.20 -5

0.15

mean drag coefficient cd

detachment position xd/ck

Cµ=50x10

0.10

0.05

0.00 0.0

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Cµ=50x10

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1.0 1.5 + excitation frequency F

2.0

0.25

0.20

0.15

0.10 0.0

0.5

1.0 1.5 + excitation frequency F

2.0

Fig. 7. left: Mean separation position for different excitation frequencies and Cµ = 50 · 10−5 intensity; right: Mean drag coefﬁcient for different excitation frequencies

effective than excitation with a frequency that allows one complete vortex after the other to detach (ﬁgure 8 centre). The same effect appears in the time evolution of the lift coefﬁcient in ﬁgure 6 left: in the case of low frequency excitation higher harmonics occur whereas for high frequency excitation a low frequency signal is superimposed on the main frequency. Only at F + = 0.62, where a clear sinusoidal signal can be seen, the lift coefﬁcient reaches a maximum. In the numerical study frequencies around F + ≈ 0.62 appear to form an optimum excitation. In the experiments the range of frequencies with most promising results is 0.25 < F + < 0.5 [3]. The discrepancy might result from the 2d modelling. Periodic excitation with different intensity. To assess the effectiveness of oscillatory blowing and suction, various blowing coefﬁcients Cµ are tested at two different excitation frequencies of F + = 0.62 and F + = 1.03.

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Fig. 8. Flowﬁeld around ﬂap represented by vorticity distribution for different excitation frequencies left: F + = 0.26; centre: F + = 0.62; right: F + = 1.54

3.10

+

0.05

F =1.03 + F =0.62 -5

-5

40x10 80x10 excitation intensity Cµ

2.90 2.80 2.70 2.60

+

F =1.03, excitation type a) +

2.50 2.40 -5 0x10

drag coefficient cd

0.10

0.00 -5 0x10

+

3.00

0.15

lift coefficient cl

detachment position xd/ck

0.20

F =1.03 + F =0.62

0.25 0.20 0.15

F =0.62, excitation type a) +

F =1.03, excitation type b) -5

-5

40x10 80x10 excitation Cµ

0.10 -5 0x10

-5

-5

40x10 80x10 excitation intensity Cµ

Fig. 9. left: Mean separation position for different excitation intensities; centre: Mean lift coefﬁcient for different intensities; right: Mean drag coefﬁcient for different intensities

Compared to the separated ﬂow in the baseline simulations at Cµ = 0, excitation with low intensity does not increase the lift or delay separation (ﬁgure 9 left and centre). If Cµ becomes larger than 25 · 10−5 , however, lift continously climbs. This effect is perfectly in line with the experiments, but the maximum achievable intensity there was limited to Cµ = 65 · 10−5 . The present study shows that further increasing intensity might bring another gain in lift. In former investigations [6] ﬂow control by tangential suction and blowing with stronger excitation also results in further increase of lift. In the present case of vertical excitation, separation position and drag coefﬁcient, however, do not improve for extremely strong suction and blowing. The drag coefﬁcient however, has a minimum at Cµ = 50 · 10−5 (see ﬁgure 9 right). All effects are documented for both excitation frequencies F + = 0.62 and F + = 1.03.

4

Three-Component High-Lift Conﬁguration

The second test model represents the SCCH (Swept Constant Chord Half - model) high-lift conﬁguration with practical relevance that has already been used for several experimental studies targeting passive ﬂow and noise control concepts [20,21,22]. The typical three-component setup consists of a main airfoil equipped with an extended slat with 0.158 cclean relative chord length and an extended ﬂap which has a relative chord length of ck = 0.254 cclean (ﬁgure 10). As seen in the ﬁrst test model

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M. Schatz, B. G¨unther, and F. Thiele

SCCH δS = 26.5°

δ F = 32°

Fig. 10. Sketch of the SCCH high-lift-conﬁguration

all proﬁles have blunt trailing edges. The ﬂap is situated at a ﬁxed position underneath the trailing edge, forming a gap of 0.0202 cclean and an overlap of 0.0075 cclean (ﬁgure 11). In the experiments, the three-dimensional wing has a sweep angle of Φ = 30◦ and a constant wing depth in spanwise direction. This three-dimensionality, however, is neglected in the ﬁrst part of the numerical investigations in order to reduce the computational costs. In all investigations the freestream velocity corresponds to a Reynolds number of Re = 106 , based on the chord of the clean conﬁguration (with retracted high-lift devices). The transition positions for each element have not been reported from the experimental measurements.

0.075 c k δs ; δf

δ F = 32°

gap

gap/cclean overlap/cclean

slat

25.5◦

1.66%

−0.41%

ﬂap

32.0◦

2.02%

0.75%

overlap

Fig. 11. Details of the geometry between main airfoil and ﬂap

4.1 2d-Simulation of the Proﬁle The dimensions of the computational domain are 15 chords forward, above and below the conﬁguration and 25 chords behind. Around the main airfoil 309 chordwise cells are located, 116 around the slat and 118 around the ﬂap. The computational c-type mesh consists of 75,000 cells in total. The non-dimensional wall-distance of the ﬁrst cell centre again remains below Y + = 1 on the complete surface. The position of an effective perturbation is deﬁned by the detachment position on the upper surface of the ﬂap without excitation. For this reason the perturbation is introduced through a slot located at 7.5% chord behind the ﬂap leading edge (ﬁgure 11) with a width of H = 0.0016 ck , and it is directed perpendicular to the ﬂap surface. This ﬁrst excitation appears to be two-dimensional and therfore all computations with excitation are limited to 2d.

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4.2 Unexcited Flow In the beginning, two-dimensional, unsteady investigations without excitation have been carried out. The angle of attack was ﬁxed at zero degrees for the whole conﬁguration. Results of all URANS-simulations were obtained by using the LLR k-ω model and considering fully turbulent ﬂow on all elements. The time-dependent resolution of k , resolving the ﬂow ﬁeld was realised by a non-dimensional time step of ∆ t = 0.005 uc∞ each period of non-excitated vortex-shedding by 458 time steps.

-1

without excitation

amplitude of cl

10

10

-2

10

10

-3

-4

-5

10 0

0.5

1.0

1.5

2.0

Strouhal number

2.5

3.0

Fig. 12. left: Flow ﬁeld around ﬂap represented by vorticity distribution and iso-contours of the mean velocity with vectors for the unexcited ﬂow at a Reynolds number of Re = 1, 000, 000; right: Spectrum of lift coefﬁcient without excitation

Similar to the two-component conﬁguration the ﬂow ﬁeld of the SCCH-conﬁguration without excitation is characterized by massive separation above the upper surface of the ﬂap. The mean separation point is located at 7.5% chord behind the ﬂap leading edge, and downstream a large recirculation region occurs. The unsteady behaviour of separated ﬂow is mainly governed by large vortices shedding from the ﬂap trailing edge and interacting with the vortices generated in the shear-layer between the recirculation region and the ﬂow passing through the slot between main airfoil and ﬂap nose (ﬁgure 12 left). The spectrum of the lift coefﬁcient in ﬁgure 12 (right) shows a dominant k = 0.44, amplitude for a Strouhal number formed with the ﬂap chord of St = F uc∞ mainly produced by this vortex-shedding. The above described base ﬂow conﬁguration involving massive separation over the ﬂap is not desirable and needs to be improved by active ﬂow control. 4.3 Excited Flow In order to reduce the computational costs, all simulations with active ﬂow control consider only a two-dimensional conﬁguration neglecting the effect of the swept wing. Based on the case of the unexcited ﬂow, computational investigations with periodic, active ﬂow control followed. Therefore simulations with a ﬁxed frequency F + = 0.44 and varied intensities Cµ have been carried out. The selection of this excitation frequency

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is based on the vortex shedding frequency in the unexcited ﬂow, as it is supposed that the excitation of such ﬂows is particularly effective at this characteristic frequency. Periodic excitation with different intensity. In the cases of excited ﬂow periodic perturbations with an intensity of Cµ = 50 · 10−5 . . . 300 · 10−5 have been introduced. Compared to the detached, unexcited ﬂow the excited lift could be continuously increased with growing intensity. No local maximum of lift can be identiﬁed, so that the largest lift occurred for the strongest excitation of Cµ = 300 · 10−5 . In this case the lift coefﬁcient can be enhanced by 17% compared to the baseline simulation (ﬁgure 14 left). 3

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Fig. 13. left: Pressure distribution for unexcited ﬂow and two different excitation intensities; right: Mean separation position on the ﬂap for different excitation intensities 0.14

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Fig. 14. left: Mean lift coefﬁcient for different excitation intensities; right: Mean drag coefﬁcient for different excitation intensities

The gain in lift by the excitation is mainly based on a change of ﬂow direction at the trailing edge of the main airfoil. With the excited condition of ﬂow above the ﬂap the trailing edge angle is increased and the pressure distribution above the main airfoil could be enhanced (ﬁgure 13 left). The excited ﬂow ﬁeld above the ﬂap is characterised by a decreased recirculation region. With this effect the mean separation position moves from less than 10% chord downstream to more than 15% (ﬁgure 13 right). At the same time the displacement of the detachment position is saturated with continously increased intensity.

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The distribution of the mean drag coefﬁcient shows a similar behaviour. The drag drops from cd = 0.13 in the unexcited case down to cd ≈ 0.10 for the excited cases. At the same time the drag coefﬁcient, however, has a minimum at Cµ = 100 · 10−5 (see ﬁgure 14 right) and also shows a saturated behaviour for higher intensities. 4.4 Quasi-3d-Simulation of a Inﬁnite Swept Wing The three-dimensional mesh is based on an expansion of the two-dimensional mesh into the third direction. For consideration of an inﬁnite swept wing 8 layers of the two-dimensional mesh are combined to resolve a three-dimensional wing section by 530,000 cells in total. The inﬁnite character is simulated by means of periodic boundary conditions. In order to provide separated ﬂow conditions that can be controlled by active methods, different ﬂap settings have been tested. For four different angles of deﬂection, starting from the baseline conﬁguration (δf = 32◦ ), steady computational investigations have been performed. Figure 15 shows the effect of ﬂap deﬂection on the ﬂow ﬁeld for δf = 32◦ and δf = 37◦ .

flap deflection angle δf

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Fig. 15. left: Different ﬂap angles δf and corresponding ﬂow ﬁelds represented by iso-contours of u-velocity; right: Steady separation position for different ﬂap deﬂection angle

Iso-contours in ﬁgure 15 (left) give an idea of the expanding separated region with recirculation above the ﬂap by increased ﬂap deﬂection. As a result of this expanded separation, the detachment position moves from 13% relative chord at δf = 32◦ to 6% relative chord at δf = 37◦ (see ﬁgure 15, right). The larger recirculation region behaves like a ﬂap with less effective camber and produces a lower suction peak. The upwash-effect reduced the velocity near the wall of the main trailing edge. Thereby the ﬂow above the main trailing edge is subject to separation. If the angle of deﬂection is increased beyond δf = 37◦ , the detachment position remains unchanged. The following investigations with active ﬂow control will be conducted on an inﬁnite swept airfoil with a sweep angle of Φ = 30◦ . The effect of sweep is demonstrated in ﬁgure 16 for a steady computation of the baseline conﬁguration with α = 6◦ incidence. The streamlines on the surface of the main airfoil show the three-dimensional component of the ﬂow produced by the local pressure gradient in spanwise direction. On the

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Fig. 16. Surface streamlines on an inﬁnite swept airfoil segment

upper surface oft the ﬂap ﬂow separation becomes visible by the direction of surface streamlines. The huge recirculation region downstream of the detachment line is based on near-wall ﬂow in upstream direction with a dominant component in spanwise direction. This component is also produced by the sweep and is strongly developed in the slow detached ﬂow.

5 Discussion The important features of the ﬂow ﬁeld and the effects of excitation on its properties can be discussed for both conﬁgurations together. Flow control by periodic excitation is dominated by the development of spanwise vortices in these cases. One of the results from simulations with different excitation frequencies and intensities is the difference in the size of these detaching vortices. Compared to both baseline cases without excitation the size of structures in the wake becomes smaller with increasing excitation frequency and high intensity. An optimum lift combined with minimum drag corresponds to smaller vortices. As the predominant part of the lift of a high-lift conﬁguration is generated by the main airfoil, the most important effect of periodic excitation is to change the ﬂow direction at the main airfoil trailing edge. By delaying separation on the ﬂap the mean ﬂow direction behind the trailing edge is changed (see ﬁgure 8 and ﬁgure 17). Vortices are generated and transported downstream and interact with those vortices detaching from the main airfoil. The surface pressure and the lift coefﬁcient oscillate with the excitation frequency. Flow control with lower intensity means smaller vortices, which are able to

Fig. 17. Flow ﬁeld around ﬂap represented by vorticity distribution for different excitation intensities left: Cµ = 50 × 10−5 ; centre: Cµ = 100 × 10−5 ; right: Cµ = 200 × 10−5

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penetrate the ﬂap boundary layer and the shear layer between the freestream and the reverse ﬂow. However, larger vortices move away from the ﬂap surface and are less effective. This may explain the small effect of low intensity excitation. The ﬂow separation is located in the turbulent part of the ﬂow. However, turbulence intensity is very low at the separation point. One effect of periodic excitation is to transfer energy from the potential ﬂow region into the boundary layer. Steady simulations of turbulent ﬂow with high turbulence intensity predict attached ﬂow for high ﬂap angles [2] stressing that high turbulence intensity can avoid ﬂow separation on the ﬂap.

6 Conclusion Two different high-lift conﬁguration were studied for the application of active ﬂow control by means of periodic suction and blowing in the ﬂap boundary layer. First a twoelement conﬁguration and later a setup with main airfoil, slat and ﬂat were simulated by unsteady RANS simulations. The effect of vertical zero-net-mass excitation was studied. The results computed were compared with available wind tunnel test results to determine the prediction capability of the computational method. For the baseline simulations without excitation, the inﬂuence of mesh resolution and time-stepping was studied and reasonable results could be obtained with the LLR and SST k-ω turbulence models and correct transition ﬁxing. Both conﬁgurations were adjusted to provide separated ﬂow conditions on the ﬂap. This desired behaviour with a large recirculation region over the ﬂaps was predicted and the results agree fairly well with the experiments. For the active control cases, steady suction and blowing do not show satisfactory effects. To study ﬂow control by periodic excitation, various blowing coefﬁcients were investigated at a given excitation frequency. In general, the results for both cases show that the lift increases if the intensity exceeds a certain limit that is about Cµ > 25 · 10−5 in the present cases. However, for very large intensities with Cµ > 50·10−5 the positive effects become increasingly smaller. By changing the excitation frequency at a constant intensity an optimum frequency can be identiﬁed for the two-component conﬁguration. It corresponds to a sinusoidal lift and detaching vortices of a suitable size. The mean lift coefﬁcient increases by up to 30% compared to the baseline simulation, and separation can be delayed. The effect of Reynolds number turned out not to be crucial for the range 105 < Re < 106 investigated, and the ﬂow control concept could successfully be applied. The behaviour of two completely different conﬁgurations showed the same principle inﬂuence of the excitation parameters on the resulting ﬂow ﬁeld. In both cases lift, drag and the mean ﬂap separation could be controlled by parameters that do not differ substantially.

7 Outlook In the near future the investigations will be extended to simulations of ﬂow control for the SCCH conﬁguration including the 3d effects of the swept wing. The threedimensional model also allows investigations of excitation modes that vary in the

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spanwise direction and that might have a beneﬁcial effect. Due to the uncertainties of URANS for simulations at higher Reynolds numbers the upcoming computations will be based on a Detached-Eddy Simulation (DES). Further investigations will be focused on a coupling of the ﬂow control apparatus to an advanced active feed-back control system.

Acknowledgement This research is funded by the German National Science Foundation (Deutsche Forschungsgemeinschaft, DFG) under the umbrella of the Collaborative Research Center (Sonderforschungsbereich, Sfb 557, ’Kontrolle komplexer turbulenter Scherstr¨omungen’) at the Berlin University of Technology. The simulations were performed on the IBM pSeries 690 supercomputer at the North German Cooperation for High-Performance Computing (HLRN). This support is gratefully acknowledged by the authors.

References [1] F.H. Tinapp. Aktive Kontrolle der Str¨omungsabl¨osung an einer HochauftriebsKonﬁguration. PhD thesis, Technische Universit¨at Berlin, 2001. [2] M. Schatz and F. Thiele. Numerical study of high-lift ﬂow with separation control by periodic excitation. AIAA Paper 2001-0296, 2001. [3] M. Schatz, F. Thiele, R. Petz, and W. Nitsche. Separation control by periodic excitation and its application to a high lift conﬁguration. AIAA Paper 2004-2507, 2004. [4] D.V. Maddalon, F.S. Collier, L.C. Montoya, and C.K. Land. Transition ﬂight experiments on a swept wing with suction. AIAA Paper 89-1893, 1989. [5] A.A. Hassan and R.D. Janakiram. Effects of zero-mass synthetic jets on the aerodynamics of the NACA-0012 airfoil. AIAA Paper 97-2326, 1997. [6] S.S. Ravindran. Active control of ﬂow separation over an airfoil. TM-1999-209838, NASA, Langley, 1999. [7] J.F. Donovan, L.D. Kral, and A.W. Cary. Active ﬂow control applied to an airfoil. AIAA Paper 98-0210, 1998. [8] I. Wygnanski. The variables affecting the control separation by periodic excitation. AIAA Paper 2004-2505, 2004. [9] M. Gad-el Hak. Flow control: The future. Journal of Aircraft, 38(3), 2001. [10] F.R. Menter. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8):1598–1605, 1994. [11] T. Rung and F. Thiele. Computational modelling of complex boundary-layer ﬂows. In 9th Int. Symp. on Transport Phenomena in Thermal-Fluid Engineering, Singapore, 1996. [12] M. Schatz. Numerische Simulation der Beeinﬂussung instation¨arer Str¨omungsabl¨osung durch frei bewegliche R¨uckstromklappen auf Tragﬂ¨ugeln. PhD thesis, Technische Universit¨at Berlin, 2003. [13] P.R. Spalart and S.R. Allmaras. A one-equation turbulence model for aerodynamic ﬂows. AIAA Paper 92-0439, 1992. [14] T. Rung, U. Bunge, M. Schatz, and F. Thiele. Restatement of the spalart-allmaras eddyviscosity model in a strain-adaptive formulation. AIAA Journal, 41(7):1396–1399, 2003. [15] D.C. Wilcox. Turbulence Modeling for CFD. DCW Industries, Inc., La Ca˜nada, 1993.

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[16] R. Petz, W. Nitsche, M. Schatz, and F. Thiele. Increasing lift by means of active ﬂow control on the ﬂap of generic high-lift conﬁguration. In Proc. 14th DGLR-Fach-Symposium der AG STAB, Bremen, Germany, 2004. [17] M. Franke, T. Rung, M. Schatz, and F. Thiele. Numerical simulation of high-lift ﬂows employing improved turbulence modelling. In ECCOMAS 2000, Barcelona, September 11-14, 2000. [18] F. Tinapp and W. Nitsche. On active control of high-lift ﬂow. In W. Rodi and D. Laurence, editors, Proc. 4th Int. Symposium on Engineering Turbulence Modelling and Measurements, Corsica. Elsevier Science, 1999. [19] A. Seifert and L.G. Pack. Oscillatory excitation of unsteady compressible ﬂows over airfoils at ﬂight reynolds numbers. AIAA Paper 99-0925, 1999. [20] R. Meyer and D.W. Bechert. Beeinﬂussung von Str¨omungsabl¨osungen an Tragﬂ¨ugeln. Abschlussbericht, Hermann-F¨ottinger-Institut, TU Berlin, 1998. [21] L. Koop. Aktive und Passive Str¨omungsbeeinﬂussung zur Reduzierung der Schallabstrahlung an Hinterkanntenklappen von Tragﬂ¨ugeln. PhD thesis, Technische Universit¨at Berlin, 2005. [22] K. Kaepernick, L. Koop, and K. Ehrenfried. Investigation of the unsteady ﬂow ﬁeld inside a leading edge slat cove. In 11th AIAA/CEAS Aeroacoustics Conference (26th Aeroacoustics Conference), Monterey, CA, USA, 2005.

Steady and Oscillatory Flow Control Tests for Tilt Rotor Aircraft M. Schmalzel, P. Varghese, and I. Wygnanski The Aerospace and Mechanical Engineering Department The University of Arizona, Tucson, AZ. 85721

Nomenclature AFC AF c CD CDo Cdp Cdf CF CL CP CQ cµ Cµ D DL f F+

Active Flow Control Active Flap chord length drag coefficient: D / q c Where CD = Cdp for a particular Cµ form drag coefficient: ³ (p − p∞) dy / q c skin friction coefficient: ³ τ dy / q c integrated force coefficient: (CL2 + Cdp2)1/2 lift coefficient: L / q c pressure coefficient: (p − p∞) / q steady volume flow coefficient: Q / SU∞ Combined oscillatory momentum coefficient: Cµ+ steady momentum coefficient: (2h/c)(USlot/U∞) 2 oscillatory momentum coefficient: (h/c)(USlotMax /U∞) 2 drag download force frequency of excitation non-dimensional frequency: (f xc / U∞)

h L NK PK q Re T xc

slot height lift No Kruger flap Passive Kruger flap dynamic pressure: ½ ρU2∞ Reynolds Number: U∞ c / ν Thrust distance from slot to trailing edge angle of attack flap angle of attack

ηD ηL

Figure of Merit for decrease in drag per unit of momentum: ∆CD/Cµ Figure of Merit for increase in lift per unit of momentum: ∆CL/Cµ

α δf

R. King (Ed.): Active Flow Control, NNFM 95, pp. 190–207, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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Abstract The purpose of this manuscript is to address one of the many questions plaguing the application of fluidic active flow control for performance enhancement over wings and airplanes. Specifically, what mode of Active Flow Control (AFC) is most effective; steady suction, steady blowing, or a periodic variation of both? The tilt rotor model is chosen because it represents very demanding requirements over a wide range of incidence angles, α, varying from –90o<α<+20o and flap deflections 0o<δf<85o as it transitions from hover to cruise. Measurements were carried out on a V-22 airfoil that has a simple flap (contrary to the flap being used currently on the airplane) with AFC emanating from a single slot carved in the flap for the purpose of download alleviation. The same slot was used to improve the performance of the airfoil in cruise and determine its dependence on the method of flow control. Levels of momentum input and the frequency of the periodic actuation were investigated as well. Steady and oscillatory suction and pure (zero mass flux) periodic perturbations proved to be very effective at low momentum coefficients while steady blowing required a large threshold value be exceeded before proving its effectiveness. These observations apply in cruise and hover alike. Download measurements were also carried out on a three-dimensional, 1/10th scale model of the airplane whereupon the two dimensional parametric studies were confirmed. The use of suction reduced the download on the model by 30% while weak periodic excitation reduced it by 16%.

1 Introduction Prandtl[1] carried out the earliest experiment that was successful in delaying flow separation from the surface of a circular cylinder through suction. He was followed by Schrenk[2] who investigated slot suction from thick airfoils that were otherwise plagued with separation. However as the quest for speed increased, and the wing sections became thinner, suction gave way to blowing because compressed air could be conveyed in small diameter pipes. The ready availability of compressed air on jetpropelled aircraft also favored blowing, which was adopted as the standard technique for lift augmentation in excess of what could be provided by deflected flaps and slats. Until 1947 mass flux removed by suction or injected by blowing, was considered the primary parameter affecting the flow. After Poisson-Quinton[3] suggested that momentum flux is the correct variable to be used in conjunction with circulation control experiments, most of the investigations that followed used the momentum coefficient, Cµ, as the primary parameter defining boundary layer control by blowing. Steady suction is still characterized by a mass flow coefficient, CQ, since its primary application is to stabilize the laminar boundary layer and delay transition to turbulence. It is generally believed that lift enhancement attained by blowing encompasses two effects considered to be sequentially dependent on the level of Cµ . At low Cµ, a wall jet helps to prevent flow separation, while in excess of a threshold value of Cµcrit, blowing increases the circulation because a strong jet may act as a fluid flap that extends beyond the trailing edge of the wing. A typical value of Cµcrit varies between

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2% and 5% depending on the specific application. This implies that for a moderately efficient airplane whose L/D≈20, the entire thrust at cruise has to be implemented in boundary layer and circulation control. No wonder there was a “disconnect” between the research community that used large momentum coefficients (Cµ>10%) to demonstrate the effects of blowing, and the engineering community that could not afford momentum levels exceeding Cµ>1%1. The introduction of periodic suction or blowing as well as pure excitation through a slot requires the consideration of oscillatory momentum input that is generally low and further blurs the criteria used to separate boundary layer control and circulation control. Further complications arise from the addition of periodicity because an effective excitation may trigger a suite of instabilities dormant in the flow that amplify and generate coherent structures that periodically scour the wing surface and change the state of the flow. Since there are many articles[4,5,6] describing and reviewing experiments that provide lift augmentation on flapped airfoils they will not be discussed presently. Less is known about the download forces on tilt rotor wings in hover and their alleviation, so a few words describing the problem and its control, are in order. A tilt-rotor airplane in hover experiences download resulting from the impingement of the rotors’ wakes on top of the wings and the fuselage. The download drag coefficient is Cdp≈1 and it amounts to approximately 12% of the V-22’s thrust, which exceeds the L/D ratio of a typical fixed wing airplane. A highly deflected trailing edge flap reduces the download until the flow separates from its surface. The ability to prevent separation from the flap’s surface at deflection angles exceeding the naturally attached-flow limit, further reduces the download. This capability is provided by active flow control. The effectiveness of AFC was successfully demonstrated in the laboratory and on a full scale XV-15 aircraft[7,8]. In these tests periodic excitation was used because the effectiveness of this method was well established[9]. The aerodynamic test procedure started by examining the two-dimensional wing/flap configuration for the most effective oscillatory momentum coefficients, at the best non-dimensional frequency, F+. In these tests, the flap deflection could be increased by an additional 15o without incurring flow separation. This delay was accompanied by download alleviation of approximately 30% in two dimensions. It was also seen that the greatest reduction in download was achieved at reasonably low momentum coefficients, ≈ 0.5%, and that further increases in had a minor effect on download alleviation. The development of an optimal control strategy is a daunting task that requires a good understanding of the aerodynamic effects involved. Periodic excitation is often compared with steady blowing, but what are the effects of periodic suction or blowing on download? Although finite mass flux requires interior ducting, such a system may be simpler, lighter, and perhaps more effective to operate. From a practical point of view one could define a Figure of Merit that provides a given increase in lift or reduction in drag per unit of momentum input: ηL=(∆CL/Cµ)

ηD=(∆CD/Cµ)

Without concern of whether this Cµ attached the flow or merely increased the circulation of an airfoil. Presently, an effort is made to provide criteria for the 1

Very few exceptions in military aviation are noted, the most significant perhaps is the Lockheed F-104 “Star Fighter” of which more than 1700 airplanes were built.

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effectiveness of various flow control strategies in cruise and in hover. This ongoing research will provide quantitative comparisons that define the efficacy of the methods employed and will improve our ability to control, manipulate, and classify the leading parameters affecting any flow of interest. The data presented was obtained at moderately low Reynolds numbers and incompressible Mach numbers.

2 Description of the Test Setups An airfoil, similar to the one used on the Osprey tilt-rotor aircraft, was tested in hover and in normal cruise and loiter modes (Figure 1). It had a simple flap hinged along the chord line and could accommodate interior actuators. When the flap was not deflected, the profile of the original V-22 airfoil was recovered. The airfoil was mounted vertically in a wind tunnel whose test section is 24” high and 40” wide. The solid side walls of the tunnel were retained when the airfoil was tested in cruise mode (0o<α<20o), but they were partially removed for hover tests (α≈-90o) in order to avoid large blockage and better simulate the finite width of a jet impinging on the wing as in the real three dimensional configuration.

Uinf

25 pressure ports on Main element 15 ports on flap Flap slot located@20% flap chord.

Fig. 1. The test airfoil used

Exterior actuation could be introduced independently to the airfoil and flap through openings in the ceiling and floor of the wind tunnel. Specially machined parts were made to connect the airfoil to a compressed air supply, a vacuum source or to sources of periodic actuation. Consequently, the slots machined on the upper surface of the flap and airfoil could be used for the injection or removal of fluid from the boundary layer. A 1/10th scale V-22 model was installed in a high bay laboratory room. It was mounted on an “A” frame in an inverted position, directing the rotor downwash up toward the ceiling that was approximately eight rotor diameters high. This allowed the model to be removed of any ground effect. The model was attached to a thick aluminum plate that did not protrude into the flow, and was connected to three load cells on a tripod suspended from the A-frame. The wings of the model were machined aluminum and hollowed for internal actuators. Sealed access hatches for internal actuators were located on the bottom surface of the wings while ducting was provided through the fuselage for steady blowing or suction. The rotors were mounted on stacks bolted to the floor, which were not coupled to the model, making the evaluation

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of the download on the model straightforward. Their collective angle was adjustable and driven by a 40 hp electric motor equipped with a variable frequency RPM controller that was located outside a protective safety cage. The torque applied to each rotor was monitored as well as its RPM. Each rotor was 4 feet in diameter and was mounted on top of a balance that measured thrust. Attention was paid to avoid the blade passage natural frequencies with accelerometers continuously monitoring the stack vibrations. The principle operating speed was chosen to be 2200 RPM. This speed was selected because it gave a representative downwash with a peak velocity of about 26 m/s at the wing at the 0.75 r/R position when the pitch collective angle was 16o. The accuracy of the calibrations are at best ±10%, the accuracy of the steady component of the momentum coefficient is better because the mass flow was independently measured and it is estimated to be ±5%. For a two-dimensional bluff body like the airfoil at α= -85o it was assumed that the total drag is determined by the pressure distribution around the airfoil. This assumption is not valid in cruise conditions when the portion of the skin friction drag is not negligible as is the momentum added or removed through transpiration. In this case, the total drag is measured by traversing the wake. In three dimensions the download is measured directly using a static calibration of the balances and load cells used.

3 Experiments 3.1 Two Dimensional Tests in Cruise and Loiter Configuration A schematic drawing of the airfoil and its flap is shown in figure 1. The flap has a 0.75mm wide slot milled on its upper surface at 20% of its chord, consequently when δf=0o the slot is located at x/c= 0.77 and it is not obstructed by the main element’s cove. Therefore, the same slot can be used for download alleviation in hover and separation control in loiter. The slot is inclined at 30o to the downstream surface. This inclination may cause a small bubble during blowing and it may also enhance the mixing with the external stream. The airfoil has two roughness strips on its upper surface and one on its lower loft. The roughness strips degraded the airfoil performance at low Reynolds numbers but they desensitized its dependence on Re, thus the results presented at Re=266,000 were equally valid at Re>106. The maximum lift generated without flap deflection, CLmax=1.2 which occurs at αmax=12o. The airfoil stalls very gently maintaining its CLmax over 6o of incidence. Flap deflection of 10o increases CLmax to 1.35 while δf=20o generates CLmax=1.45 at αmax=9o (Figure 2). The increase in flap deflection increases the drag and consequently does not increase the maximum L/D which remains constant at L/D=50 for 0o<δf<20o. Nevertheless the deflected flap is very useful since it enables the airplane to loiter at a lower speed. The application of steady suction at Cµ=2% increases the lift at α < αmax, but does not increase CLmax as long as δf<15o. At δf=20o suction increases CLmax from 1.45 to 1.6 while concomitantly decreasing αmax to 5o (Figure 2). This level of suction has a major effect on L/D to facilitate increases from 50 to 280 at CLmax≈1.5 (Figure 3). Steady blowing at Cµ=2% increases the lift more effectively than suction; particularly

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Fig. 3. L/Dt vs. CL for Cµ=2%

when flap deflection results in separation. Thus for δf=10o this kind of blowing increases CLmax from 1.35 to 1.55, while for δf=20o the increase in CLmax is more dramatic (CLmax increases from 1.45 to 1.9, (Figure 2). Blowing increases the drag yielding an L/Dmax of 125 relative to suction’s 280 for the same value of momentum coefficient, (Figure 3).

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The consequence of Cµ on the efficacy of steady blowing or suction was tested at α=2o and δf=15o where the unperturbed base flow is separated from the upper loft of the flap and entirely attached to the main element. Suction at very low values of Cµ is very effective in reducing CD, in this case the total drag was halved when Cµ≈0.1%. A further increase in Cµ generated a more moderate result, nonetheless the drag coefficient measured at Cµ<2% was equal to ¼ of its original value (Figure 4(b)). Blowing is much less effective in reducing the drag, requiring a Cµ=1% to halve the original value and more than 3% to quarter it. Nevertheless for Cµ>3%, blowing becomes more effective than suction in its ability to reduce drag. Suction is also more effective than blowing in increasing CL at low levels of Cµ (Figure 4(a)), but in this instance the cross over occurs at Cµ≈0.7%. 1.7 CL

0.035

CD

1.6 Steady Suction Steady Blowing

1.5

0.030 0.025

1.4

0.020

1.3

0.015 0.010

1.2

0.005

1.1 0.01

0.1 1 Log10[C (%)]

10

0.01

0.1

1

10

Log10[C (%)]

Fig. 4. Effect of Steady Suction/Blowing Cµ on (a) CL and (b) CD for α=20,δflap=150

It is interesting to compare the pressure distributions over the airfoil and the velocity deficits in the wakes for various values of Cµ. This provides a better understanding of the different effects that suction and blowing has on general performance. The comparisons are carried out at α=2o and δf=15o also. For Cµ≈0.25% suction reduced the drag from CD≈0.031 to CD≈0.015 increasing the performance of the airfoil to L/D≈80 without a substantial increase in lift, only ∆CL≈4%. The pressure distribution over the flap (Figure 5(a)) suggests that the flow is separated, because the prevailing CP over most of its chord remained unchanged by either suction or blowing. The CP at the suction slot is lowest and its upstream effect lingers on, although it is not substantially lower than the CP attained without active intervention or the one attained by using the same level of blowing. Suction however, generated the smallest velocity deficit in the wake of the airfoil (Figure 5(b)), therefore generating the lowest drag. At this low level of Cµ, the profiles showing the velocity defect for the basic airfoil wake and for the blowing case are symmetric with respect to the wake’s centerline, but when suction was used the symmetry of the velocity profile was broken.

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Pressure distributions at Cµ =0.25%, 0

α =2 ,δ flap =15

-1.3

0

Wake profiles @Cµ =0.25%,α =20,δflap =150 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

Cȝ=0%,CL=1.174 Steady suction,CL=1.23 Steady blowing,CL=1.2

Cp -0.84

0 Y (mm)

-0.38

0.5

0.6

0.7

0.8

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U/Uw Baseline Steady suction Steady Blowing

75

1

150 225

0.08

x/C

300

0.54

375 450

1 Pressure distributions at Cµ =2%, 0

0

Wake profiles @Cµ =2%,α =2 ,δ flap =15

0

α=2 ,δ flap=15

0.86 0.88

Cȝ=0%,CL=1.174 Steady Suction,CL=1.37

-2.5

Cp -2

0.9

U/Uw

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-1.5

150

-1

225

0.2

0.4

0.6

0.8

1

Baseline Steady suction Steady Blowing

300

x/C

0

1

0

Y (mm)

Steady Blowing,CL=1.46

-0.5 0

0

0.92 0.94 0.96 0.98

0.5

375 450

1

Fig. 5. {(a)-(d)} Pressure (Cp) and Wake deficit velocity (U/Uw) distributions for 0.25%
0

D=2 ,G flap=15

0

Wake profiles @CP =2%,D =2 ,G flap =15

Cȝ=0%,CL=1.174 Steady Suction,CL=1.37

-2.5

Cp -2

Steady Blowing,CL=1.46

-1.5

0

0.2

0.4

0.6

0.8

0.9

0

0.92 0.94 0.96 0.98

1

0

Y (mm) 75 150

-1 -0.5 0

0.86 0.88

Baseline Steady suction Steady Blowing

U/Uw

225

1

x/C

300

0.5

375

1

450

Fig. 5.{(e)-(f)} Pressure (Cp) and Wake deficit velocity (U/Uw) distributions for 0.25%
Increasing Cµ to 2% enabled the blowing to attach the flow over the flap resulting in a CP≈0 at the trailing edge while for equivalent suction the CpTE≈-0.2. The pressure distribution over the flap suggests that suction was unable to attach the flow over half its chord-length (Figure 5(c)). The minimum CP measured in the vicinity of the slot was approximately equal for suction and for blowing, as was the pressure distribution over the upper surface of most of the airfoil. The difference in lift [CLsuction=1.37 vs. CLblowing=1.46] is attributed to the differences in circulation around the flap only,

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where the inclined blowing seems to have enclosed a bubble on the upper surface near the slot while the larger pressure recovery at the trailing edge increased the CP on the flap’s lower surface. The highest CP upstream of the suction or blowing slot was –0.7, enabling a pressure recovery of ∆CP=1.7 over 55% of the airfoil’s chord. The pressure distribution over the unperturbed airfoil (Figure 5(c)) indicates that the flow was attached2 over the upper surface of the main element (0<x/c<0.7) enabling the same pressure recovery, ∆CP, which was obtained by using suction or blowing. Except that the flow was separated over the entire flap generating a base pressure CP=-0.3. Steady suction at Cµ≈2% attached the flow over 40% of the flap’s chord reducing the negative base CP to –0.2. A more significant effect of suction is upstream of the slot; where it acts as a two-dimensional sink lowering the CP in its vicinity. Since the boundary layer can withstand a prescribed pressure recovery over the airfoil’s upper surface, the lower CP at x/c=0.7 affects the entire pressure distribution over the upper loft of the airfoil. This is the major contribution of suction to the enhancement of lift on this airfoil, since it does not manage to attach the flow to the flap. 0

0

Drag vs. Cµ for Steady Suction & Blowing @ α=2 ,δflap=15 Steady Blowing, Cdp CD Steady Suction, Cdp CD

0.06 CD,Cdp 0.05 0.04 0.03 0.02 0.01 0.00 0.01

0.1 1 Log10[Cµ(%)]

10

Fig. 6. Effect of Cµ on CD and CDp

Steady blowing at approximately the same level of Cµ≈2% as the suction, did not decrease the pressure upstream of the slot any better, thus resulting in identical CP distributions between 0.2<x/c<0.7. The difference in the Kutta condition over the flap (in one case the flow is attached over the entire flap while for suction the flow is separated from x/c=0.9), increased the circulation around the airfoil that, in turn, lowered the CP at x/c<0.2. A high velocity wall jet entrains fluid from its surroundings and this entrainment is particularly strong near the nozzle. The upstream effect of the jet may be represented by an array of sinks of decreasing strength along its path, and so, steady blowing generates an upstream effect that is similar to suction. 2

The through in the pressure distribution located between 0.1<x/c<0.2 is due to the roughness strip that interferes with the two pressure taps in its immediate vicinity.

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The wake traverses corresponding to Cµ=0 and to Cµ=±2% are shown in Figure 5(d). Clearly there is a major reduction in drag for both suction and blowing. The center of wake corresponding to Cµ=±2% is located below the center position corresponding to Cµ=0, reflecting the increase in lift resulting from the use of blowing or suction. However, the velocity deficit associated with blowing does not uniformly approach the free stream velocity, as do the other two velocity profiles. It reveals a wide secondary defect region that is situated above the upper loft of the airfoil. This contribution to drag stems from the losses associated with the entrainment or engulfment of the outer potential flow by the large eddies characterizing the jet. The entrained fluid looses momentum due to the turbulent mixing that is not recovered far downstream where the static pressure in the wake returns to its ambient value. A further increase in Cµ to Cµ=±3.35% attached the flow over the entire flap for both suction and blowing. The CP just upstream of the slot was hardly affected when compared to Cµ=±2%, however downstream of the slot, blowing reduced the minimum CP much more effectively than suction. The pressure recovery, ∆CP, over the blown flap (0.8<x/c<1) became equal to the ∆CP experienced over almost the entire main element (0.1<x/c<0.74). Blowing decreased the minimum CP over the flap much more than suction, it affected the entire circulation over the airfoil almost doubling the lift increment that suction generated for this flap deflection and incidence (Figure 5(e)). Most interesting is the velocity distribution in the wake that corresponds Cµ=+3.35% (Figure 5(f)). The velocity deficit associated with the flow over the upper surface of the airfoil is weak but broad, suggesting that blowing enhances the mixing with the ambient fluid. However remnants of the boundary layer generated over the upper surface upstream of the blowing slot persevere in the wake over long distances. Similarly residues of the jet momentum in excess of the free stream velocity are observed below the deficit region in the wake. The overall velocity distribution starts resembling the wake of a self-propelled body. It is surprising that the upstream history of the wake prevails over 3 chord-lengths downstream of the trailing edge. We often assume that the total drag of an airfoil has two components: 1. Skin friction drag generated by attached boundary layers on the upper and lower surfaces of the airfoil 2. Pressure drag (form drag) that results from the inability of the flow to reach stagnation pressure at the trailing edge of the airfoil. This component of drag stems from the inadequacy of the ideal flow model that was historically used to predict the pressure distribution on airfoils, and is still embedded in most computational programs used in preliminary airfoil design. It is often stated that form drag, which is more readily available in experiments[10,11], may represent the total drag as well. This representation is put to test in the context of active flow control as shown in Figure 6. Take the conditions considered above, α=2o and δf=15o that generate a total CD=0.03, where 2/3 of it is attributed to form drag Cdp=0.02. At very low levels of suction, Cµ=0.03%, the boundary layer momentum deficit was essentially eliminated reducing CD to the value of Cdp measured in the absence of flow control. However, Cdp increased with increasing Cµ and around Cµ=1% was 3 times larger than CD. The rise in Cdp was

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referred to as a “sink drag” and it was predicted by Goldschmied[12] who assumed that suction has a drag penalty associated with its application. It appears that “penalty” applies to form drag only since the total drag continues to decrease. Blowing requires Cµ=0.4% for Cdp to be equal to CD, but while Cdp∝Cµ ,CD decreased and for Cµ>3% Cdp≈10*CD. One may be tempted to assume that Cdp≈CDo + Cµ while CD ≈ CDo - Cµ. In this case a simple sink model does not apply although jet entrainment had been modeled and the model predicts the generation of Cdp. In fact one may question the validity of the normalization by the free stream dynamic head (½ρU2∞) when substantial jet momentum is involved. The effect of steady and oscillatory flow control (at a reduced frequency of F+=0.7 and a combined cµ whose phase locked component, ≈ 0.5 Cµ) on airfoil performance is shown in figure 7. At low levels of Cµқ, periodic suction attains the same CD as the steady case, after which steady suction produces lower drag. Periodic suction provides the same CL as the steady case for Cµ~0.3%. The effect on CL is however minor at for small levels of Cµ. In the instance of blowing, for 0.03%< Cµ<0.3% periodic case produces a higher CL. Further increase of Cµ to 1% enhances CL for steady blowing compared to oscillatory blowing. For 0.1%
1.35 CL 1.30

CD

Steady Blowing Steady Suction + Osc. Blowing, F =0.7 + Osc. Suction, F =0.7

0.030 0.025

1.25

0.020 0.015

1.20

0.010

1.15

0.005

0.01

0.1 Log10[CP(%)]

1

0.01

0.1

1

Log10[CP(%)]

Fig. 7. Effect of Steady and Oscillatory Blowing/Suction Cµ on CL and CD for α=20, δflap=150

3.2 The Application of Suction, Blowing and Periodic Excitation to the Airfoil in Hover An airfoil of a tilt rotor airplane in hover is in the wake of the rotor at a typical incidence α≈-90o and it is therefore a “bluff body”. On its upper surface the stream comes to stagnation while the pressure over its lower surface is highly negative (Figure 8). The pressure difference pulls the airfoil downward and the only way to lower the download force is to deflect the flap thus reducing the surface area over

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Fig. 8. Hover Cp distribution at α=-850, δflap=700, Steady Suction, Cµ=1.4%

which the pressure difference is integrated. Active Flow Control enables the flow to remain attached to the flap at larger flap deflections resulting in a low pressure over the upper flap surface and a slightly higher (less negative) base pressure over the entire airfoil. This of course alleviates the download, but it creates lift that exceeds 1/3 of the download in magnitude (in hover this is called ‘push back’ force because if left uncompensated it will result in the airplane drifting backward). This is of no practical significance on the airplane since a small tilt forward of the rotor overcomes the push back automatically. In this case the wake is so broad that the total drag could not be accurately measured by surveying the wake, however when a comparison was made, pressure drag was approximately equal to the total drag. Therefore, reattachment of the flow to a highly deflected flap does not only reduce the download, but also narrows the wake. Since the experiments were carried out at low Reynolds numbers two transition trip-strips (using No 60 grit equivalent sand roughness) were affixed to its surface to achieve a virtual Reynolds number independence resulting in an almost constant Cdpmin (1.23>Cdpmin>1.20) between 1.5x105≈2% where the additional flap deflection required is approximately ∆δf=15o. Wherever the flow is naturally attached, (i.e. at δf <50o), the addition of periodic excitation increases slightly the download by thickening periodically the boundary layer over the flap. This effect was noticed in other applications and it is not deemed as being significant.

202

M. Schmalzel, P. Varghese, and I. Wygnanski Baselines,α=-850,NK,20% flap slot open

Cdp 1.6 1.5 1.4 1.3 1.2

Clean Re=170,000 w/ roughness Re=250,000 -3% error +3% error Re=340,000

1.1 1 0.9

δflap

0.8 40

45

50

55

60

65

70

75

80

85

Fig. 9. Baseline download vs. δflap for different Re 0

Cdp

+

Periodic actuation,α =-85 , Uinf=15m/s, F =0.7

1.6 1.5 1.4 1.3 1.2 1.1 1

Baseline w/ roughness =2%

0.9

δ flap

0.8 40

45

50

55

60

65

70

75

80

85

Fig. 10. Effect of periodic actuation on download

The dependence of Cdpmin on is highly non linear as was already observed during the XV-15 investigation. It was then realized that increasing the amplitude for a given flap and slot settings beyond a prescribed threshold value is of little benefit. The same holds true for the present experiments. The dependence of the download on for a slot located at x/c=20% when periodic excitation at a frequency of 100Hz (corresponding to F+=0.57) was used to control it at δf=75o, is shown in Figure 11. In the absence of actuation Cdp≈1.5, and it decreased to Cdp=0.92 at the low amplitude of =0.25%. A further increase of by an order of magnitude results in an almost linear but meager decrease in Cdp of 0.04, to Cdp=0.88. Consequently the efficacy of AFC should be judged by the download reduction at the threshold . Steady blowing at Cµ=4.09% reduced the download by ∆Cdp=0.25 (i.e. to Cdp=0.95). Measurements at intermediate values of Cµ suggest that the download alleviation for steady blowing is almost linearly dependent on Cµ. In fact a threshold

Steady and Oscillatory Flow Control Tests for Tilt Rotor Aircraft o

203

0

F+=0.57(100Hz), Amplitude Sweep at δflap=75 ,α=-85 ,Uinf=15m/s

1.6 Cdp

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.0

0.5

1.0 1.5

2.0

2.5

Fig. 11. Cµ sweep Blowing Comparison,20% flap slot open,NK,α=-850,Uinf=15m/s 1.6

Cdp 1.5 1.4 1.3 1.2 1.1

Baseline w/ roughness Cȝ=1.4% Cȝ=4.09% Cȝ=(0.22%;0.11%) F+=0.32 Cȝ=(0.22%;0.11%) F+=0.64 Cȝ=(0.63%;0.31%) F+=0.64

1 0.9 0.8

δflap 40

45

50

55

60

65

70

75

80

85

Fig. 12. Steady and periodic blowing effects on download

value of blowing Cµ has to be surpassed before Cdp will be reduced. Oscillatory blowing (i.e. blowing with minimum velocity of 0 m/s but with a mean mass flow that contributes to a RMS =0.31%) yielded the almost as good a result at a combined cµ=Cµ+=(0.63%, 0.31%)as the steady blowing at Cµ=4.09% (Figure 12). The reduced frequency used in this case was F+=0.64. The first number in the “complex” definition of cµ indicates the DC component while the second the RMS variation about the mean. If efficacy is measured by momentum input, it may be stated that pure periodic excitation (at zero mass flux and the appropriate threshold level) is more effective than oscillatory blowing and the latter is more effective than steady blowing in reducing the download on this airfoil. Steady suction proved to be much more effective than blowing in download alleviation. A suction of Cµ=1.4% generated a ∆Cdp=0.35 (Figure 13) while the equivalent blowing achieved only 0.1. Even a much stronger blowing did not do as

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0

Suction Comparison,20% flap slot open,NK,α=-85 ,Uinf=15m/s

Cdp 1.6 1.5 1.4 1.3 1.2 1.1 Baseline w/ Roughness Cȝ=1.4% Cȝ=(0.22%;0.11%) F+=0.32 Cȝ=(0.22%;0.11%) F+=0.64 Cȝ=(0.08%;0.04%) F+=0.64

1 0.9

δflap

0.8 40

45

50

55

60

65

70

75

80

85

Fig. 13. Steady and periodic suction effects on download

Fig. 14. Highlight of flap and wing system on 3D V-22 model

well as the suction, however periodic excitation at approximately ½ the did almost as well. Oscillatory suction proved to be most effective in reducing the download reaching Cdpmin=0.91 for combined cµ=(0.22%; 0.11%). Periodic suction is therefore as effective as periodic excitation. At higher level of periodic suction the download reduction even surpassed the ∆Cd=0.35 value. 3.3 Three Dimensional Tests in Hover The 3D results confirm the observations made in two-dimensions realizing a minimum download at a slightly larger deflection angle (δf min.=57o vs 53o) than on the two dimensional airfoil configuration, (Figure 14). The results yield a download, DL, normalized by the total thrust, T, as having a minimum value of 0.122, implying that at least 12% of the overall thrust is absorbed by the download. The most effective use of periodic excitation on the flap reduced to (DL/T)min. =0.102 or ∆(DL/T)=0.019 that was approximately equivalent to a 15% download reduction. By replacing the actuators with steady suction that was comparable in magnitude to the 2D results (i.e. Cµ=1.2%), the download was reduced to DL/T=0.089 while

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Baseline 3D V-22 Active Flap, 16o Collective, 2200 RPM 0.15

NK/AF, AFC, Cu=0.23%, F+=0.71 NK/AF, Suction, Cu=0.23% NK/AF, Suction, Cu=1.2%

0.14

DL/T

0.13 0.12 0.11 0.10 0.09 0.08 0.07 50

55

60

65

70

δflap

75

80

85

Fig. 15. Download alleviation comparing suction, blowing and periodic perturbations at two values of Cµ = 0.23% & 1.2%

Fig. 16. Download alleviation as a function of Cµ when steady blowing or suction is used

increasing the optimal flap deflection to δf=77.5o. Decreasing the level of suction to a Cµ=0.23% reduced the download to DL/T=0.105. The result is comparable to the one obtained using periodic excitation. Steady blowing at the high level of Cµ=1.2% lowered the download to DL/T=0.114 while suction was able to lower it to DL/T =0.089 (Figure 15). These results confirm the 2D observations that also favor suction over blowing as the method of choice in download alleviation. A comparison of suction and blowing was carried out for various levels of momentum coefficient, Cµ<1.2%. The comparison was carried out at different flap deflection angles (Figure 16). The angles selected were the ones for which the chosen method was most

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effective at Cµ=1.2% (i.e. δf=75o for suction and δf=60o for blowing). Weak blowing at Cµ<0.5% proved to be deleterious while weak suction at comparable levels reduced the download significantly. In fact 93% of the asymptotic download alleviation, attained at high levels of suction was achieved at Cµ<0.5%.

4 Conclusions The aerodynamic effects of active flow control on a variant of the V-22 flap system were considered. The investigation started with a detailed study of the airfoil in both hover and cruise and proceeded to the 1/10th scale model of the airplane. The change in flap geometry results in two aerodynamic benefits: (i) the minimum download occurs at smaller flap deflections thus requiring less actuation, (ii) there is a slight (2.5%) reduction in the download force. The addition of a 20% chord Kruger flap reduces the download by 12%, while the elimination of the protruding landing gear doors may reduce the download by 6.5%. The most effective methods for download alleviation are periodic suction and periodic excitation. Both generate a large effect at small Cµ, while steady blowing requires momentum coefficients that are not realizable unless propulsion and lift generation are integrated into a single system. Suction at Cµ~1% reduces the download on the model by approximately 30% when the flap is deflected to its optimal value. At Cµ<0.25% a reduction of 16-18% is attained by using either suction or periodic excitation. Larger reductions in download are possible by adding a passive Kruger flap, a reduction of almost 38%. There was not enough suction available to provide adequate control over the Kruger and the flap simultaneously. Observations made on the 2D airfoil were able to predict the general behavior of the flow over the 3D model. Thus, optimal flap angles, slot locations, and necessary Cµ values can all be estimated on the basis of the 2D measurements.

References [1] Prandtl, L. Journal of the Royal Aeronautical Society, Vol. 31, 1927, p. 735. [2] Schrenck, (edited by: Lachmann, G.V.) Boundary Layer and Flow Control: Its Principles and Application. Pergamon Press, New York, 1961. [3] Poisson-Quinton, Ph. “Recherches Theoriques et Experimentales sur le controle de couche limites” VII International Congress of Applied Mechanics, London 1948. [4] Stratford, B.S. “Early Thoughts on the Jet Flap”. Aeronautical Quarterly, Vol. VII, p. 45, February, 1956. [5] Seifert, A., Bachar, T., Koss, D., Shepshelovich, M., and Wygnanski, I., “Oscillatory Blowing: A Tool to Delay Boundary-Layer Separation”, AIAA Journal, Vol. 31, No. 11, 1993. [6] Nishri B. and Wygnanski I., “Effects of Periodic Excitation on Turbulent Flow Seperation from a Flap”. AIAA Journal, Vol. 36, No. 4, 1998. [7] McVeigh M.A., Nagib H., Wood T., Kiedaisch J., Stalker A., Wygnanski I., “Model & Full Scale Tiltrotor Download Reduction Tests Using Active Flow Control” Presented at the AHS 60th annual forum, Baltimore MD, June 7-10, 2004

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[8] Nagib H., Kiedaisch J., Stalker A., Wygnanski I., McVeigh M.A., Wood T., “First-InFlight Full-Scale Application of Active Flow Control: The XV-15 Tiltrotor Download Reduction” Proceedings of NATO R&T Specialists’ meeting Prague, 2004 [9] Greenblatt D. & Wygnanski I., “The control of flow separation by periodic excitation”, Progress in Aerospace Sciences 36, 487, 2001. [10] Squire, H.B. and Young, A.D. The calculation of the profile drag of airfoils”. Rep. Memor. Aero. Res. Coun., London, 1938. [11] Williams, J. An Analysis of data on blowing over trailing edge flaps for increasing lift. Curr. Pap. Aero. Res. Coun. London, 1955. [12] Goldschmied, F.R., Integrated Hull Design, Boundary Layer Control and Propulsion of Submerged Bodies. J. Hydrodynamics. Vol. 1, p.2, 1967.

Reduced-Order Model-Based Feedback Control of Subsonic Cavity Flows – An Experimental Approach M. Samimy1, M. Debiasi1, E. Caraballo1, A. Serrani2, X. Yuan2, J. Little1, and J.H. Myatt3 Collaborative Center for Control Science The Ohio State University, Columbus, Ohio 43235 USA 1

Gas Dynamics and Turbulence Laboratory; Department of Mechanical Engineering 2 Department of Electrical and Computer Engineering 3 Air Force Research Laboratory – Air Vehicle Directorate, Wright-Patterson AFB

Summary The results of an ongoing research activity in the development and implementation of reduced-order model-based feedback control of subsonic cavity flows are presented and discussed. Particle image velocimetry data and the proper orthogonal decomposition technique are used to extract the most energetic flow features or POD eigenmodes. The Galerkin projection of the Navier-Stokes equations onto these modes is used to derive a set of ordinary nonlinear differential equations, which govern the time evolution of the modes, for the controller design. Stochastic estimation is used to correlate surface pressure data with flow field data and dynamic surface pressure measurements are used for real-time state estimation of the flow model. Three sets of PIV snapshots of a Mach 0.3 cavity flow were used to derive three reduced-order models for controller design: (1) snapshots from the baseline (no control) flow, (2) snapshots from an open-loop forced flow, and (3) combined snapshots from the cases 1 and 2. Linear-quadratic optimal controllers based on all three models were designed and tested experimentally. Real-time implementation shows a remarkable attenuation of the resonant tone and a redistribution of the energy into various modes with much lower energy levels.

1 Introduction Flow control can be divided into two general categories: passive and active. In the former, which is much easier to implement and has wide-spread applications, control is accomplished by geometrical modifications to the flow system. In the latter, energy in some form is added to the flow. Active control is divided into open-loop and closed-loop. In open-loop control, actuation takes place based on an operator’s command or a predetermined schedule. In the closed-loop case, information from one or more sensors in the flow along with a flow model guides the actuation process. Successful application of feedback control is widespread in areas such as robotics, aerospace, telecommunication, transportation systems, manufacturing systems, and R. King (Ed.): Active Flow Control, NNFM 95, pp. 211–229, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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chemical processes. Only in recent years feedback control of aerodynamic flows has received focused attention, [1]-[11]. Open-loop flow control, which can be quite useful in many applications, lacks the responsiveness and flexibility needed for application in dynamic flight environments. In contrast, closed-loop flow control is well-suited to the successful management of such cases since it allows adaptability to variable conditions. In addition, closed-loop control shows the potential to significantly reduce power requirements in comparison to open-loop control strategies [1]. Unfortunately, the tools of classical control systems theory are not directly applicable to aerodynamics flows since such systems display spatial continuity and nonlinear behavior while also posing formidable modeling challenges due to their infinite dimensionality, a complexity introduced by the Navier-Stokes equations. In order to design and successfully implement a closed-loop control strategy, it is necessary to obtain a reduced-order dynamical model of the system, which can capture the important dynamic characteristics of the flow and actuation while remaining sufficiently simple to allow its use in model-based feedback control design. The flow over a shallow cavity - a configuration present in many practical applications that has been extensively studied in the literature – was selected for the present study. This flow is characterized by a strong coupling between the flow dynamics and the flow-generated acoustic field that can produce self-sustained resonance known to cause, among other problems, store damage and airframe structural fatigue in weapons bays. A comprehensive review of this phenomenon and of various control and actuation strategies developed for its suppression is given in [6]. Rossiter [12] first developed an empirical formula, which was later modified and improved by Heller and Bliss [13], for predicting the frequencies of cavity-flow resonance, today referred to as Rossiter frequencies or modes. Rossiter also investigated the concept of a dominant mode of oscillation that was later observed by others to coincide with the longitudinal cavity acoustic mode [14]. In such a condition a strong single-mode resonance occurs [3]; otherwise multiple modes exist in the flow. A similar interaction could also occur between Rossiter modes and the transversal cavity acoustic modes, [15] and [16]. Recent theoretical models of the cavity acoustic resonance based on edge scattering processes, [17] and [18], formulate these behaviors. Rapid switching between modes has been observed in multi-mode conditions [19], [3] and [20]. The random switching between multiple modes on a rapid time scale places large bandwidth and fast time response requirements on the actuation scheme and feedback control algorithm. Extensive work has been carried out to control the flow over a cavity. Different open-loop control strategies have been used in recent years with varying degrees of success, [16], [21]-[23]. There have also been significant efforts to investigate closedloop control approaches, [1], [3], [4], [24]-[30]. The results of these closed-loop endeavors are encouraging, but also indicate that many issues remain to be resolved. While we have examined other control approaches in recent years ([20], [31] and [32]), our primary objective from the onset has been the development of control techniques based on a reduced-order model of the cavity flow [33], [34], [29] and [30]. The approach we have followed in the development of this model is based on the proper orthogonal decomposition (POD). This technique relies on the energycontaining eddies in the flow, which can be extracted using the spatial correlation

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tensor of the velocity field, in the form of spatial eigenmodes called POD modes. These structures are the most dominant features in the flow and are perhaps the only entities that can effectively be controlled. The dynamics of the flow are obtained when these modes are modulated by mode amplitudes obtained by projecting the NavierStoke equations governing the flow onto the POD basis. This results in a set of nonlinear, ordinary differential equations that are used for controller design. The equations are autonomous and not useful for controller design purposes. Consequently, they must be recast in a form expressing the control input explicitly so that a feedback controller can be designed using the tools of control theory, [35] and [29]. In the next section we will introduce the flow facility used in this study. Section 4 will present the POD and Galerkin methods adopted for deriving the reduced-order model, and the stochastic estimation approach used for real-time estimation of the flow model variables directly from dynamic surface pressure measurements. This is followed in Section 5 by a discussion of the flow characteristics and of the reducedorder model results, and in Section 6 by the design and implementation of the linearquadratic controller. We will present and discuss the control results in Section 7, followed by concluding remarks in Section 8.

2 Experimental Facility and Techniques Details of the experimental facility and the experimental techniques used can be found in [16] and [36]. The facility, located at the Gas Dynamics and Turbulence Laboratory of The Ohio State University, is a small scale blow-down wind tunnel with a cavity recessed in the floor and spanning the width of the test section. The air is conditioned in a settling chamber to minimize free stream turbulence and directed to the 50.8 mm (2 in) by 50.8 mm (2 in) test section, Figure 1, through a contoured converging nozzle before exhausting into a large exhaust pipe taking it to the atmosphere. The facility can be operated in the Mach number range 0.20 to 0.70. The focus of this work is on a shallow cavity with length 50.8 mm and depth of 12.7 mm (0.5 in) corresponding to a cavity aspect ratio, L/D, of 4. For Mach 0.30 flow, the Reynolds number based on this cavity depth is approximately 105. Optical quality windows surround the test section and allow laser-based flow diagnostics from 15 mm upstream to 25 mm downstream of the cavity. The output of a Selenium D3300Ti compression driver is channeled to the cavity leading edge where it exits at an angle of 30o with respect to the main flow through a 2-D slot of 1 mm height that spans the cavity width. This arrangement provides zero net mass, non-zero net momentum flow for actuation, similar to that of a synthetic jet. Actuation can be achieved in the frequency range of 1-20 kHz. The actuator flow to main flow momentum ratio is in the range of 10-4 to 10-6. The planar snapshots of the velocity field are acquired using a two-component LaVision particle image velocimetry (PIV) system. The flow is seeded with oil particles by using a 4-jet atomizer upstream of the stagnation chamber. This location allows homogenous dispersion of the particles throughout the test section. The PIV provides a velocity vector grid of 128 by 128 over the approximate measurement domain of 50.8 mm (2 in) by 50.8 mm (2 in), which translates to velocity vectors separated by approximately 0.4 mm.

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Flow

W

Actuator output L

Kulite pressure transducer Compression driver

Fig. 1. Scaled drawing of the experimental set up

Fig. 2. Location and numbering of Kulite pressure transducers in the test section

Six flush-mounted Kulite transducers are placed at various locations on the walls of the test section for dynamic pressure measurements, Figure 2. The transducers have a flat frequency response up to about 50 kHz, are powered by a dedicated signal conditioner, and their signals are band-pass filtered between 100 Hz and 10 kHz. Recordings of 218 points are acquired at 200 kHz and converted to non-dimensional pressure referenced to the commonly used value of 20 µPa. Short-time Fourier transform (STFT) is utilized to provide information on the time evolution of the frequency content of the unsteady pressure signal and spectra are obtained by averaging the corresponding spectrograms, [36]. For state estimation, dynamic pressure measurements are recorded simultaneously with the PIV measurements. In the current study, 1000 PIV snapshots are recorded for each flow/actuation condition explored. The PIV snapshots with a sampling rate of a few Hz are time-uncorrelated. For each PIV snapshot, 128 samples from the laser Qswitch signal and from each of the pressure transducers of Figure 2 are acquired at 50 kHz. The simultaneous sampling of the laser Q-switch signal and the pressure signals allows the identification of the section of pressure time traces corresponding to the instantaneous velocity field. For closed-loop control of the flow, a dSPACE 1103 DSP board connected to a Dell Precision Workstation 650 is used. This system utilizes four independent, 16-bit A/D converters each with 4 multiplexed input channels and allows simultaneous acquisition and control processing of 4 signals and almost simultaneous, due to multiplexing, acquisition and processing of additional signals at a rate up to 50 kHz per channel to produce at the same rate a control signal from a 14-bit output channel.

3 Reduced-Order Modeling Procedure Development of tools and procedures for feedback control based on reduced-order models has been our primary goal from the onset of this research program [33], [37] and [34]. Our recent work has focused on deriving the reduced-order models of the cavity flow from PIV and surface pressure measurements [29] and [30]. The technique combines three separate tools and procedures to obtain and implement a controller. First, the POD method is used to obtain spatial eigenmodes or POD modes

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of the flow. Second, the Navier-Stokes equations are projected onto the POD modes using the Galerkin projection method to obtain the flow model, which consists of a set of ordinary nonlinear differential equations. These equations govern time evolution of the POD modes. In the third and final step, stochastic estimation is used to correlate the flow velocity field to surface pressure data and to provide real-time state estimation of the model derived in step 2. Each one of these three steps will briefly be discussed in this section. 3.1 Proper Orthogonal Decomposition Technique The POD method was introduced to the fluid dynamics community by Lumley [38] as an objective tool to extract energy-containing large scale structures in a turbulent flow. Implementation of POD technique requires detailed flow data, which could be obtained using either numerical simulations or laser-based planar flow measurements. The original derivation, however, favored time-resolved data over a long time period at a few spatial locations (hot-wire type data, e.g. [39]). More details of the method can be found in [40] and [41]. Later, Sirovich [42] developed the snapshot POD method, which favors spatially-resolved, but time-uncorrelated snapshots of the flow field. Such data can be obtained using advanced laser-based planar flow diagnostics or numerical simulations. We are currently using PIV data with the snapshot method. Details of the snapshot method and its application in the present context are given in [33] and [37]. The POD method uses M snapshots of the flow and casts the fluctuations in the flow in terms of N spatial orthonormal modes (N<M) or POD modes, ϕ i ( x ) , and time varying amplitudes for these modes, ai(t). Equation (3.1) is for streamwise velocity fluctuations that contain major portion of the kinetic energy in the flow. u ' (x , t ) ≅

N

¦

i=1

a

i

(t )ϕ i ( x )

.

(3.1)

For each one of the several flow conditions explored in this work, 1000 PIV snapshots of the flow field were acquired [36]. Each snapshot contains twocomponent of instantaneous velocity on a grid of 128 by 128 over an approximate measurement domain of 50.8 mm (2 in) by 50.8 mm (2 in) on the x-y plane passing through center of the cavity. The results indicate that the mean turbulence kinetic energy is converged by using approximately 700 snapshots [30]. Consequently all 1000 snapshots were used to obtain the POD modes. 3.2 Galerkin Projection The second step in the process of deriving a reduced-order model is the projection of the Navier-Stokes equations onto the POD modes, ϕ i ( x ) , using the Galerkin projection method. The result of this procedure is a set of nonlinear ordinary differential equations for the mode amplitudes in equation (3.1), ai(t) [i=1 to N]. The compressible Navier-Stokes equations used are those derived in [43] and details of the procedure are given in [33] and [37].

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In the Galerkin projection method, each instantaneous flow variable is decomposed into its mean and fluctuating components. Then, the POD expansion in equation (3.1) is written for each of the fluctuating components. Finally, the flow variables in the Navier-Stokes equations are replaced by the expanded expressions of mean and fluctuating components. The new form of the governing equations is then projected onto the POD modes by taking the inner product of each term with the POD modes according to the vector norm defined in [43]. This procedure yields a set of ordinary differential equations for the mode amplitudes in equation (3.1). These equations are autonomous and are not useful for controller design. In order to derive a model where the control input appears explicitly in the equations, a few methods are currently being explored. The one used in the present work is based on spatial sub-domain separation, [35] and [32], which yields the following set of ordinary differential equations: ª a T (t )H 1 a (t ) º « » a (t ) = F + Ga (t ) + « # » + B Γ (t ) + T N « a (t )H a (t )» ¬ ¼

(

)

(

)

ª B 1 Γ (t ) T a (t ) º » « # » « « B N Γ (t ) T a (t )» »¼ «¬ .

(3.2)

where the matrices of constant coefficients F, G, Hi, B and B i , i=1,…N, are obtained from the Galerkin projection, and Γ(t) is the control input applied at the forcing location, [34]. Equation (3.2) represents a model of the cavity flow in terms of the mode amplitudes ai(t) obtained with the POD method from M time uncorrelated PIV snapshots. By using a finite number (N) of modes to describe the flow, one not only filters out smaller flow structures, but also fails to account for the energy transfer process between the N retained modes and the neglected ones. Therefore, an additional viscous term, the modal eddy viscosity [44] was added to the model to maintain the overall energy balance and to compensate for the truncated modes. This additional term is added to the viscous term in the Navier-Stokes equations, and is obtained by a modal energy balance [44].

3.3 Stochastic Estimation Design of a controller based on the reduced-order model of equation (3.2) will be presented and discussed in Section 4. In implementing the controller in the experiment, the variables involved in the reduced-order model must be linked to the variables that can be measured experimentally in real-time. A similar situation will also arise in any practical application. The real-time experimental data could only be obtained via surface measurements (e.g. surface pressure or surface shear stress measurements) in any realistic setting. We used surface pressure measurements and stochastic estimation (SE) to correlate these measurements with the flow velocity data obtained via PIV. Stochastic estimation was developed by Adrian [45]. The technique estimates flow variables at any location by using statistical information about the flow at a limited number (L) of locations. In the current work, quadratic stochastic estimation was employed to estimate the mode amplitudes of the flow model, equation (3.2), directly from real-time

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measurements of surface pressure fluctuations at a small number (L) of locations. The estimates of the mode amplitudes can be written in the following form: aˆ i ( t ) = C ik p k′ ( t ) + D ikl p k′ ( t ) p l′ ( t )

,

i=1…N,

k, l=1…L.

(3.3)

where C and D are the matrices of the estimation coefficients obtained by minimizing the average mean square error between the values of ai (tr) obtained with equation (4.2) from the snapshots and the estimated ones aˆ i (t r ) at the same times. To calculate the matrices in equation (3.3), surface pressure measurements at L locations taken simultaneously with the PIV snapshots were used. A total of 1000 simultaneous PIV-surface pressure measurements were acquired. The procedure to obtain the estimation matrices is described in more detail in [37]. In our experimental setup, real-time measurements of the surface pressure were obtained at L = 6 locations in the cavity test section, Figure 2. It was observed in [37] that retaining both the linear and the quadratic terms in equation (3.3) significantly improved the results. Both terms are retained in the current work as well.

4 Flow Characteristics and Reduced-Order Model Results 4.1 Flow Characteristics

The Mach number of the flow varied between Mach 0.20 and 0.70 using Mach number increments of 0.01 to explore the cavity flow characteristics. For each of these flows, surface dynamic pressure and the sound pressure level (SPL) spectra were obtained at the center of the cavity floor. Figure 3 shows the SPL intensity level and frequency as a function of the Mach number. In the same figure are also shown the lines corresponding to the first four Rossiter modes (R1-R4) predicted by the modified Rossiter formula [13], the 1st longitudinal acoustic mode based on the cavity length (L1), and the 1st and 2nd transversal acoustic modes (T1, T2). Strong resonant tones are observed near the intersections of the predicted Rossiter modes with both the transversal and the longitudinal acoustic modes. The observation of the interaction

R4

R2 L1 T1

SPL (dB)

R3 T2

R1

Fig. 3. Frequency and SPL amplitude for various Mach numbers of pressure fluctuations at the center of the cavity floor

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between Rossiter and transversal acoustic modes is similar to that of [15]. The work of [12], [14] and [3] all examined cavity flows for which the tunnel vertical dimension was significantly larger than the cavity length. Therefore, they observed the interaction of Rossiter modes with only the longitudinal cavity acoustic mode, as the transverse acoustics mode had much lower frequency in their cases. Based on these and previous observations [16], we have used the Mach 0.30 flow as our reference baseline case because it shows a single tone at about 2900 Hz, which is near with the intersection of the 3rd Rossiter mode and the 1st transversal acoustic mode. In addition, at this Mach number the actuator has enough authority and realtime feedback control is practical. In the current work we have explored this baseline flow along with several controlled cases. Figure 4 presents the SPL spectrum of the surface pressure measured at the test section side wall at the center of the cavity (sensor # 5 in Figure 2) for the baseline Mach 0.3 cavity flow resonating at the third Rossiter mode, Figure 3.

Fig. 4. SPL spectrum of baseline Mach 0.30 cavity flow from surface pressure at the center of the cavity floor

Figure 5 presents the instantaneous planar flow visualization images of the baseline Mach 0.3 cavity flow and two open-loop forced cases. Three coherent large scale structures are clearly visible in the baseline case, consistent with the spectrum in Figure 4 and for a flow resonating at the 3rd Rossiter mode. Figure 5(b) is for the same flow excited at 1830 Hz (i.e. at the second Rossiter mode). As a result, two large coherent structures are clearly visible in the shear layer. This shows that forcing at this frequency weakens or eliminates the natural feedback mechanism for the third Rossiter mode, but excites the flow at this lower mode. Figures 5(c) presents the Mach 0.30 flow forced at 3920 Hz. This frequency is close to the fourth Rossiter mode and as a result four large-scale structures are visible in the instantaneous image. The phase-locked images of these three flows also clearly show the existence of these coherent structures [36]. These results confirm the authority of the actuators to control the flow to resonate at various Rossiter modes and to significantly affect the flow field. Interestingly, detailed PIV results show no significant changes in the ensembleaveraged flow characteristics for these three quite different flows [36]. Three reducedorder models using PIV snapshots of the baseline flow, the same flow forced at 3920 Hz, and from both cases are derived and used for controller design.

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a) Baseline

b) forced at 1830 H

219

c) forced at 3920 Hz

Fig. 5. Instantaneous planar images of the baseline Mach 0.30 cavity flow and two controlled flows. Flow is from left to right.

4.2 Reduced-Order Modeling Results

Figure 6 shows the first two POD modes for the v-component of velocity fluctuations of the baseline Mach 0.30 flow. There are three structures in the flow consistent with the flow visualizations results shown in Figure 5 (a) and with the surface pressure spectrum of Figures 3 and 4, which shows that the cavity resonates at the third Rossiter mode. The first four modes with energy content of 40% were used in the design of the controller. Phase-average v-component of velocity fluctuations from PIV data show a structure pattern very much similar to the 2nd POD mode shown in Figure 6 [36].

Fig. 6. First two POD modes for the v-component of velocity fluctuations of the baseline flow

In an earlier work, we used a logic based open-loop control and showed that at certain forcing frequencies the cavity fluctuations are significantly reduced [16]. One such frequency is 3920 Hz (StD~0.5), which is close to the 4th Rossiter mode. Forcing at this frequency changes the resonance to multi-mode regime and reduces the peak at the third Rossiter modes by about 20 dB, but adds a smaller peak (10 to 20 dB lower depending on the sensor location) at the forcing frequency. It is interesting to note, that one of the multiple modes is a sub-harmonic of the forcing frequency. This mode seems to dominate over the other modes. The first two POD modes for this case are shown in Figure 7. It is obvious that the forcing has disrupted the natural resonance at the 3rd Rossiter mode, but established a resonance at the 2nd Rossiter mode with two clearly defined structures in the flow. The energy contained in the first four modes is about 37%.

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Fig. 7. First two POD modes for the v-component of velocity fluctuations of the flow forced at 3920 Hz

The POD modes from the combined PIV snapshots of the two cases discussed above with a total of 2000 snapshots take some characteristics from each. For example, the first and second modes with two dominant structures and the third and fourth modes with three structures resemble those of the forced and the baseline cases, respectively. The set of nonlinear ordinary differential equations in (4.2), obtained by Galerkin projection of the Navier-Stokes equations onto the POD modes, was solved to check the evolution and convergence of the mode amplitudes. The mode amplitudes for the baseline cavity flow converged using a number of modes N between 4 and 10. After an initial transient period, the coefficient oscillates close to zero, as expected. The mode amplitudes for the baseline case from the PIV snapshots were obtained using equation (4.1): a i (t ) =

³ u ' (x , t )ϕ i ( x ) d x .

(4.1)

D

The qualitative comparison between the results from the solution of equation (3.2) and values obtained with equation (4.1) from the PIV measurements is quite good. Using FFT analysis the frequency of oscillation of the mode amplitude for the first mode was found to be about 2417 Hz, a somewhat lower value than the experimental one (about 2840 Hz). It was observed that the system trajectories converged to the same behavior, irrespective of the initial condition of the mode amplitude used for the solution of equation (3.2), showing the existence of a stable limit cycle. Similar results were obtained for the mode amplitudes of the other three modes of the baseline Mach 0.3 flow, and the forced case, and the combined forced and baseline cases. Also, in evaluating the stochastic estimation technique results, discussed in Section 3.3, the mode amplitudes obtained via quadratic stochastic estimation, equation (3.3), were compared with those obtained via PIV snapshots and equation (4.1). Again, the qualitative comparison was good, and the results from the solution of equation (4.3) fell within the range obtained experimentally from PIV results and equation (4.1) [29] and [30].

5 Controller Design and Implementation In this section, we present and discuss the design of the model-based controller. The control design approach has been presented in details by the authors in previous

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works [34] and [29], and thus it will be only outlined here. The design procedure includes equilibrium computation, coordinates transformation, linear approximation of the Galerkin system, and linear-quadratic state feedback control design. Three reduced-order flow models obtained from POD methods have been investigated in this work: (1) the baseline Mach 0.3 flow; (2) the same flow forced at 3920 Hz using open-loop; (3) a flow model obtained from POD modes derived combining PIV snapshots from cases 1 and 2. The reduced-order flow models for all three cases for control design are similar nonlinear state space model given by equation (3.2), with N = 4, with different numerical values of the model parameters. 5.1 Equilibrium Analysis and Model Simplification

In performing equilibrium analysis and coordinate transformation on the model, equation (3.2), the constant term F is removed from the model, shifting the origin of the coordinates to the equilibrium point. The resulting simplified state space model in the new set of coordinates a~ = a − a 0 becomes ª a~ ~ ~ « ~ a = G a + « « a~ ¬

T

T

(

ª B a~ º « ~ » # » + B Γ + « « B H 4 a~ »¼ ¬« H

1

(

Γ

)

# 4 Γ

)

1

T

T

a~ º » » a~ »» ¼

, (5.1)

where a 0 is the equilibrium point computed for the model, equation (3.2), and

(

ª a 0T H ~ « G = G + « « a 0T H ¬

(

1

4

+ (H # + (H

1

4

)T )T

)º , » » » ¼

)

ª(B ~ « B = B + « «(B ¬

)T a

0

# 4 )T a

0

1

º » » » ¼

. (5.2)

Clearly, the modified model has an equilibrium point at the origin, which is more convenient for controller design and stability analysis. The reader is referred to [29] for a detailed description of the model simplification techniques. 5.2 Linear Quadratic State Feedback Control

A linear approximation of equation (5.1) at the origin is readily obtained as ~ ~ a~ = G a~ + B Γ

.

(5.3)

The eigenvalues of the matrix of the unforced system, equation (5.3) have been computed for the three cases as

( )

~ λ G1

ª 1596.6 + 7023.1i « 1596.6 - 7023.1i = « « - 3652 « - 879.9 ¬

º » » » » ¼

,

ª 1567.2 + 6880.9i « 1567.2 - 6880.9i ~ λ G2 = « « - 4030 « - 524.6 ¬

( )

º » » » » ¼

,

ª 1397 + 7061.7i « 1397 - 7061.7i ~ λ G3 = « « - 2870.9 « - 697.2 ¬

( )

º, » » » » ¼

(5.4)

where the subscripts 1, 2, 3 correspond to the model based on the baseline flow, the model based on the same flow with open-loop sinusoidal forcing at 3920 Hz, and the model based on the combination of these, respectively. All the fourth-order Galerkin

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systems for the three cases exhibit the same qualitative features (2 unstable complex conjugate eigenvalues plus 2 stable real eigenvalues) and as well as quantitative similarities. The presence of two unstable complex conjugate eigenvalues implies, as expected, that the flow corresponding to the equilibrium a 0 is an unstable solution for the Galerkin system, equation (4.2). Since the pairs ( G~ , B~ ) for all cases are controllable, linear state-feedback design based on the linearized model, equation (5.3), offers a simple approach to the design of a controller for the nonlinear model, equation (5.1). Recall that the stochastic estimation method provides a way to estimate the mode amplitudes of the Galerkin system from real-time surface pressure measurements, equation (3.3). The availability of real-time estimates of the state of the Galerkin model, equation (5.1), allows the use of linear state-feedback control to globally stabilize the origin of equation (5.3). This, in turn, yields a controller that locally stabilizes the origin of the nonlinear system, equation (5.1). A convenient and well-established methodology for the statefeedback controller design is offered by linear-quadratic (LQ) optimal control. The LQ design computes the gain matrix K such that the state-feedback law Γ ( t ) = − K a~ ( t ) ,

(5.5)

minimizes the quadratic cost function

J

c

( a~ , Γ ) =

³ (a~ ∞

T

0

W

a~

a~ + W

Γ

Γ

2

) dt

,

(5.6)

where Wa~ > 0 and W Γ > 0 are positive definite weighting functions for the state vector and the control signal, respectively. Minimization of J c results in asymptotic stabilization of the origin, while the control energy is kept small. In our design, the weights have been chosen as W a~ = I 4× 4 and W Γ = 1 for all three models, and the corresponding control gains with respect to the three flows read as 1

=

K

2

=

K

3

=

K

[− 56 [52.5 [17.6

.2

3.9

− 12 . 8 ] ,

− 417 . 2

8 .8

- 168

208 . 8

- 102

],

(5.7)

− 146 . 8 ] .

11 . 6

Applying the state feedback control, equation (5.5), to the linearized system, equation (5.3), results in mirroring all the right-half plane eigenvalues of the matrix ~ G to the left half plane. Figure 8 shows the simulation results obtained by applying the state feedback control, equation (5.5), to the finite-dimensional nonlinear model, equation (3.2), which indicate that the closed-loop state trajectories a(t ) converge to the corresponding equilibrium points in each case, given by a 01 = [ − 0.5036 0.2788 − 0.1930 0.4980]T , a 0 2 = [ − 0 . 3081

a 0 3 = [ − 0 .3261

0 . 1483

− 0 . 2083

0 . 4895 ] T

0 .2158

− 0 .2598

T

0 .4753 ]

,

(5.8)

.

It can be concluded that, in principle, the LQ controller, equation (5.5), designed for the linear approximation, equation (5.3), succeeds in stabilizing the equilibrium of the four modes nonlinear Galerkin system, equation (3.2).

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(a)

(b)

223

(c)

Fig. 8. Time coefficient solutions of the closed-loop simulation results. (a) baseline flow model, (b) open-loop forced flow model with forcing frequency of 3920 Hz, (c) combined flow model of the above two cases.

6 Feedback Control Results and Discussion 6.1 Modeling Results

Before presenting the results of the experimental implementation of the controller, it is worth summarizing the structure of the model-based controller derived in Section 5. As depicted in Figure 9, the model-based controller includes a stochastic estimation subsystem and a feedback from the estimated states. The estimate a~ˆ of the deviation from the equilibrium of the mode amplitudes of the Galerkin model, required to

Fig. 9. Diagram of the closed loop system with LQ state feedback control

implement the feedback law, equation (5.5), may be in principle obtained by means of stochastic estimation by first estimating aˆ (t ) from raw pressure measurements using equation (3.3), and then subtracting the equilibrium value a 0 computed from the model data. However, in implementing the controller, subtracting the equilibrium values from the estimated ones is not required, since the DC values have been removed from the pressure measurements by means of high-pass filtering. That is, equation (3.3) naturally produces the values of a~ˆ from the experimental measurements. It is important to point out that, to prevent any damage to the actuator, the control input signal is limited to the range ±10V. Since the gains of the LQ

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control, equation (5.7), are quite large, constant saturations of the actuator were observed during closed-loop experiments for all cases under investigation. Therefore, it was necessary to introduce a scaling factor 0<Į<1 in the state-feedback to keep the actuator below the saturation limit. The largest possible scaling factors have been found to be α 1 = 0.265 , α 2 = 0.35 and α 3 = 0.5 for each of the three flow models considered, and the corresponding scaled control is in the form Γ α ( t ) = − α K a~ˆ ( t )

.

(6.1)

The scaled LQ controls, equation (6.1), have also been simulated on corresponding nonlinear models, equation (3.2), as depicted in Figure 10. It is evident that, though the scaled LQ control for the given values of α is not able to asymptotically stabilize the origin of the nonlinear model, equation (3.2), it nevertheless provides a significant reduction of the amplitude of the stable limit cycle in all three cases. This result is in agreement with a mathematical analysis carried out on the nonlinear finitedimensional Galerkin model, equation (5.1), which predicts a reduction of the amplitude of limit cycle (corresponding to the fundamental cavity tone) as the gain Į increases from 0 to 0.5, with complete suppression of the oscillation only possible for Į >0.5.

(a)

(b)

(c)

Fig. 10. Closed loop responses at sensor 5 with different scaling factor α. (a) baseline flow model, (b) open-loop forced flow model with forcing frequency of 3920 Hz, (c) combined flow model of the above two cases.

6.2 Experimental Results

The performance of the scaled control law, equation (6.1), has been tested experimentally. We now discuss the results obtained for Mach 0.3 cavity flow. Specifically, we present the closed-loop sound pressure level spectra from sensor 5 located on the cavity wall in Figure 2 (sensor 6 exhibit similar results) obtained in closed-loop with each LQ controller designed on the basis of the three flow models discussed in Section 6. In addition, a comparison is made with the results obtained using open-loop sinusoidal excitation at 3920 Hz (optimal forcing frequency), and combined open-loop forcing with closed-loop LQ control.

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(a)

(b)

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(c)

Fig. 11. SPL spectra obtained from sensor 5 in closed-loop experiments with LQ design based on: baseline flow model (a), forced flow model (b), and combined flow model (c)

The results for closed-loop LQ control, shown in Figure 11, show a considerable attenuation of the resonance peak in sensor location 5 (with a similar behavior at sensor location 6), and a redistribution of the energy into various modes, especially lower frequency modes, with much lower energy level. It is worth noting that the results do not differ significantly when a forced flow model (Figure 10 (b), (c)), is considered in place of the baseline flow (Figure 10 (a)), although a slightly more uniform attenuation of the SPL can be noted in the third case. This is somewhat surprising, as one would expect the presence of forcing to improve the fidelity of the model in closed-loop conditions, and ultimately to provide a “richer” model, capable of delivering better results. The lack of significant improvement is probably related to the particular technique for control separation that has been employed to render the presence of the control input explicit in the model [35]. A better resolution of the effect of external forcing may be obtained resorting to a method of control separation that makes use of “actuation modes” directly at the level of POD modeling. This is a current area of investigation. A comparison with the results obtained under optimal open-loop forcing shown in Figure 12 (a) reveals that, while both control cases forced the flow to multi-mode regime, overall the closed-loop control performs better than the open-loop control. Note specially the presence of a significant peak (in excess of 120 dB) at the subharmonic of the open-loop forcing frequency and lack thereof any such significant peak in the feedback control case. This behavior, intrinsic in open-loop forcing, may be somehow alleviated combining optimal open-loop forcing with LQ feedback control, in which the open-loop sinusoidal forcing plays the role of a feedforward control. From the point of view of the finite-dimensional modeling and control separation methodology adopted here, this situation is indeed preferable to using only LQ feedback control derived on the basis of forced flow models, as the addition of sinusoidal forcing resembles the conditions under which the forced model has been derived. Also, one may hope to combine the beneficial effects of feedback control in terms of a more uniform attenuation of the resonance with the sharper results obtained by open-loop forcing. The results reported in Figure 12 (b)-(c) seem to validate this conjecture only in part, and further investigation is needed to clarify the appropriateness of this approach.

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(a)

(b)

(c)

Fig. 12. SPL spectra obtained from sensor 5 with open-loop forcing (a), and under combined open-loop and feedback control designed on the forced flow model (b), and combined openloop and feedback control designed on the combined flow model (c)

7 Concluding Remarks The work presented and discussed in this paper is part of our ongoing research activities in the development and implementation of reduced-order models-based feedback control of subsonic cavity flows. The cavity is shallow with L/D of 4 and spans the width of the wind tunnel test section. The facility can be operated continuously between Mach 0.2 and 0.7, but the majority of the work was carried out around Mach 0.3 with a Reynolds number based on the cavity depth of approximately 105. The output of a compression driver is channeled to the cavity leading edge where it exits at an angle of 30o with respect to the main flow through a 2-D slot of 1 mm height that spans the cavity width. This arrangement provides zero net mass, non-zero net momentum flow for actuation, similar to that of a synthetic jet. Actuation can be achieved in the frequency range of 1-20 kHz. The actuator flow to main flow momentum ratio is in the range of 10-4 to 10-6. With open-loop forcing, we can force the cavity to operate in a single-mode and lock onto various Rossiter modes or to operate in multi-mode with rapid switching between modes. The work includes using various laser based flow diagnostics to understand flow physics and to obtain detailed data for the derivation of reduced-order models for controller design. PIV data and the snapshot-based POD technique are used to extract the most energetic flow features or POD eigenmodes. For each flow case, 1000 timeuncorrelated PIV snapshots are used. The Galerkin projection of the Navier-Stokes equations onto the POD modes is used to derive a set of nonlinear ordinary differential equations, which govern the time evolution of the modes, and to use for the controller design. Quadratic stochastic estimation is used to correlate PIV data to surface pressure data thus enabling a real-time estimation of the state of the model based on time-resolved pressure measurements. Three sets of PIV snapshots of a Mach 0.3 cavity flow were used to derive three reduced-order models for the controller design: (1) snapshots from the baseline (no control) single-mode flow, (2) snapshots from the same flow open-loop forced at 3920 Hz, which produces multi-mode cavity resonance, and (3) combined snapshots from the cases 1 and 2. A linear-quadratic optimal controller based on each of the

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three models was designed to reduce cavity flow resonance and tested in the experiments. The results obtained for all three flow models showed improvement over those obtained using open-loop control, indicating that feedback control strategies based on reduced-order flow models represent a compelling approach to subsonic cavity flow control. Notwithstanding the encouraging results reported and discussed in this work, further investigation is needed to understand how to incorporate more effectively the presence of actuation in reduced-order POD models of the flow system that can capture more closely the behavior of forced flows, as well as to clarify the interplay between feedforward and feedback control.

Acknowledgments This work is supported by the AFRL/VA and AFOSR through the Collaborative Center of Control Science. The authors would like to thank Hitay Özbay, Chris Camphouse, David Williams, Bernd Noack, Lou Cattafesta, Kelly Cohen, and Stefan Siegel for help and fruitful discussions.

References [1] Cattafesta III, L.N., Garg, S., Choudhari, M., and Li, F., “Active Control of Flow-Induced Cavity Resonance”, AIAA Paper 97-1804, June 1997. [2] Gad-el-Hak, M., Flow Control – Passive, Active, and Reactive Flow Management, Cambridge University Press, New York, NY, 2000. [3] Williams, D., Fabris, D., and Morrow, J., “Experiments on Controlling Multiple Acoustic Modes in Cavities,” AIAA Paper 2000-1903, 2000. [4] Kegerise, M., Cattafesta, L., and Ha, C., “Adaptive Identification and Control of FlowInduced Cavity Oscillations,” AIAA Paper 2002-3158, 2002. [5] Rowley, C., and Williams, D., “Control of Forced and Self-Sustained Oscillations in the Flow Past a Cavity,” AIAA Paper 2003-0008, 2003. [6] Cattafesta III, L. N., Williams, D. R., Rowley, C. W., and Alvi, F. S., “Review of Active Control of Flow-Induced Cavity Resonance,” AIAA Paper 2003-3567, June 2003. [7] Samimy, M., Debiasi, M., Caraballo, E., Özbay, H., Efe, M.O., Yuan, X., DeBonis, J., and Myatt, J.H., “Closed-Loop Active Flow Control: A Collaborative Approach,” AIAA Paper 2003-0058, 2003. [8] Siegel, S., Cohen, K., Seidel, J., and McLaughlin, T., “Feedback Control of a Circular Cylinder Wake in Experiments and Simulations (Invited),” AIAA Paper 2003-3569, June 2003. [9] Gerhard, J., Pastoor, M., King, R., Noack, B., Dillmann, A., Morzynski, M., and Tadmor, G., “Model-Based Control of Vortex Shedding using Low-Dimensional Galerkin Models,” AIAA Paper 2003-4262, June 2003 [10] Glauser, M. N., Higuchi, H., Ausseur, J., and Pinier, J., “Feedback Control of Separated Flows (Invited),” AIAA Paper 2004-2521, June 2004. [11] Tadmor, G., Noack, B., Morzynski, M., and Siegel, S., “Low-Dimensional Models for Feedback Flow Control. Part II: Control Design and Dynamical Estimation,” AIAA Paper 2004-2409, June 2004.

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[12] Rossiter, J.E., “Wind Tunnel Experiments on the Flow Over Rectangular Cavities at Subsonic and Transonic Speeds”, RAE Tech. Rep. 64037, 1964 and Aeronautical Research Council Reports and Memoranda No. 3438, Oct. 1964. [13] Heller, H. H., and Bliss, D. B., “The Physical Mechanisms of Flow-Induced Pressure Fluctuations in Cavities and Concepts for their Suppression,” AIAA Paper 75-491, March 1975. [14] Rockwell, D., and Naudascher, E., “Review—Self-Sustaining Oscillations of Flow past Cavities,” Journal of Fluids Engineering, Vol. 100, 1978, pp. 152-165. [15] Ziada, S., Ng, H., and Blake, C., “Flow Excited Resonance of a Confined Shallow Cavity in Low Mach Number Flow and its Control,” Journal of Fluids and Structures, 18, 2003, p. 79-82. [16] Debiasi, M. and Samimy, M., “Logic-Based Active Control of Subsonic Cavity Flow Resonance,” AIAA Journal, Vol. 42, No. 9, pp. 1901-1909, September 2004. [17] Kerschen, E., and Tumin, A., “A Theoretical Model of Cavity Acoustic Resonances Based on Edge Scattering Processes,” AIAA Paper 2003-0175, 2003. [18] Alvarez, J., Kerschen, E., and Tumin, A., “A Theoretical Model for Cavity Acoustic Resonances in Subsonic Flows,” AIAA Paper 2004-2845, 2004. [19] Cattafesta, L., Garg, S., Kegerise, M. and Jones, G., “Experiments on Compressible Flow-Induced Cavity Resonance,” AIAA Paper 1998-2912, 1998. [20] Debiasi, M., Little, J., Malone, J., Samimy, M., Yan, P., and Özbay, H., “An Experimental Study of Subsonic Cavity Flow – Physical Understanding and Control,” AIAA Paper 2004-2123, June 2004. [21] Stanek, M.J., G., Kibens, V., Ross, J.A., Odedra, J., and Peto J.W., “High Frequency Acoustic Suppression – The Mystery of the Rod-in-Crossflow Revealed,” AIAA Paper 2003-0007, January 2003. [22] Shaw, L., “Active Control for Cavity Acoustics”, AIAA Paper 98-2347, June 1998. [23] Grove, J., Leugers, J., and Akroyd, G., “USAF/RAAF F-111 Flight Test with Active Separation Control”, AIAA Paper 2003-0009, January 2003. [24] Shaw, L., and Northcraft, S., “Closed Loop Active Control for Cavity Resonance”, AIAA Paper 99-1902, May 1999. [25] Cattafesta, L., Shukla, D., Garg, S., and Ross, J., “Development of an Adaptive WeaponsBay Suppression System,” AIAA Paper 1999-1901, 1999. [26] Williams, D.R., Rowley, C., Colonius, T., Murray, R., MacMartin, D., Fabris, D., and Albertson, J., “Model-Based Control of Cavity Oscillations Part I: Experiments,” AIAA Paper 2002-0971, 2002. [27] Rowley, C. W., Williams, D. R., Colonius, T., Murray, R. M., MacMartin, D. G., and Fabris, D., “Model-Based Control of Cavity Oscillations Part II: System Identification and Analysis,” AIAA Paper 2002-0972, January 2002. [28] Cabell, R. H., Kegerise, M. A., Cox, D. E., and Gibbs, G. P., “Experimental Feedback Control of Flow Induced Cavity Tones,” AIAA Paper 2002-2497, June 2002. [29] Caraballo, E., Yuan, X., Little, J., Debiasi, M., Yan, P. Serrani, A., Myatt, J., and Samimy, M., “Feedback Control of Cavity Flow Using Experimental Based Reduced Order Model,” AIAA Paper 2005-5269, June 2005. [30] Caraballo, E., Yuan, X., Little, J., Debiasi, M., Serrani, A., Myatt, J., and Samimy, M., “Further Development of Feedback Control of Cavity Flow Using Experimental Based Reduced Order Model,” AIAA Paper 2006-1405, January 2006. [31] Efe, M., Debiasi, M., Yan, P., Özbay, H., and Samimy, M., “Control of Subsonic Cavity Flows by Neural Networks-Analytical Models and experimental Validation,” AIAA Paper 2005-0294, January 2005.

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[32] Yan, P., Debiasi, D. Yuan, X., Little, J., Özbay, H., and Samimy, M., “Closed-loop Linear Control of Cavity Flow,” AIAA Journal, Vol. 44, No. 5, pp. 929-938, May 2006. [33] Samimy, M., Debiasi, M., Caraballo, E., Malone, J., Little, J., Özbay, H., Efe, M. Ö., Yan, P., Yuan, X., DeBonis, J., Myatt, J. H., and Camphouse, R. C., “Exploring Strategies for Closed-Loop Cavity Flow Control,” AIAA Paper 2004-0576, January 2004. [34] Yuan, X. , Caraballo, E., Yan, P., Özbay, H., Serrani, A., DeBonis, J., Myatt, J. H. and Samimy, M., “Reduced-Order Model-Based Feedback Controller Design for Subsonic Cavity Flows”, AIAA Paper 2005-0293, January 2005. [35] Efe, M.Ö., and Özbay, H., “Proper Orthogonal Decomposition for Reduced Order Modeling: 2D Heat Flow,” IEEE Int. Conf. on Control Applications (CCA'2003), June 23-25, Istanbul, Turkey, pp. 1273-1278, 2003 [36] Little, J., Debiasi, M., and Samimy, M., “Flow Structure in Controlled and Baseline Subsonic Cavity Flows,” AIAA Paper 2006-0480, January 2006. [37] Caraballo, E., Malone, J., Samimy, M., and DeBonis, J., “A Study of Subsonic Cavity Flows - Low Dimensional Modeling,” AIAA Paper 2004-2124, June 2004. [38] Lumley, J. “The Structure of Inhomogeneous Turbulent Flows”, Atmospheric Turbulence and wave propagation. Nauca, Moscow. 1967 166-176. [39] Glauser, M., Eaton, E., Taylor, J., Cole, D., Ukeiley, L., Citrinity, J., George, W and Stokes, S., “Low-Dimensional descriptions of Turbulent Flows: Experiment and Modeling.” AIAA Paper 1999-3699, June- July 1999 [40] Holmes, P., Lumley, J.L., and Berkooz, G., “Turbulence, Coherent Structures, Dynamical System, and Symmetry,” Cambridge University Press, Cambridge, 1996. [41] Delville, J., Cordier, L. and Bonnet, J.P., “Large-Scale-Structure Identification and Control in Turbulent Shear Flows,” In Flow Control: Fundamentals and Practice, edited by Gad-el-Hak, M., Pollard A. and Bonnet, J., Springer-Verlag, 1998, pp. 199-273. [42] Sirovich, L. “Turbulence and the Dynamics of Coherent Structures”, Quarterly of Applied Math. Vol. XLV, N. 3, 1987, pp. 561-590. [43] Rowley, C. W., “Modeling, Simulation and Control of Cavity flow Oscillations”, Ph.D. thesis, California Institute of Technology. 2002. [44] Noack, B., Tadmor, G., and Morzynski, M. ., “Low-Dimensional Models for Feedback Flow Control. Part I: Empirical Galerkin Models,” AIAA Paper 2004-2408, June 2004. [45] Adrian, R. J., “On the Role of Conditional Averages in Turbulent Theory,” Turbulence in Liquids, Science Press, Princeton, 1979.

Supersonic Cavity Response to Open-Loop Forcing David R. Williams1 , Daniel Cornelius1 , and Clarence W. Rowley2 1

Illinois Institute of Technology, Chicago IL 60616, USA [email protected] http://fdrc.iit.edu/people/williams.php 2 Princeton University, Princeton, NJ 08544, USA

Summary The response of a supersonic cavity to open-loop forcing with a pulsed-blowing actuator is explored experimentally. It is shown that excitation at frequencies near the Rossiter modes are ampliﬁed, while frequencies between the ﬁrst two Rossiter modes are attenuated. The results clearly demonstrate that the Rossiter modes in the supersonic cavity are weakly damped (not self-excited) modes. The supersonic modes are not saturated, and do not show the kind of nonlinear interactions with the forcing modes observed in subsonic ﬂow. These differences between supersonic and subsonic ﬂows are consistent with previously developed models of cavity oscillations, and the results suggest that linear techniques for the design of closed-loop controllers may be particulary effective for supersonic ﬂows. For the ﬂow regime studied, the oscillatory component of openloop forcing does not play a signiﬁcant role in the suppression mechanism in supersonic cavity ﬂows.

1 Introduction The interest in controlling resonant acoustic tones in cavities with modern active ﬂow control techniques has both practical and fundamental motivations. The ability to suppress resonant acoustic tones in open aircraft cavities would reduce the possibility of structural fatigue and component damage related to the high acoustic loads [1,2]. Resonant tones are correlated with high drag on the cavity [3]. Current practice uses passive devices, such as spoilers, to suppress the tones. This approach leads to an increase in drag, and does not always work at off-design conditions. Active ﬂow control offers the possibility of tone suppression that can adapt to changing ﬂight conditions. On the fundamental side, the gross features of the acoustic resonance mechanism are reasonably well understood, but the details are not. Finding a suitable ﬂow model and control architecture remains a challenge. The ”cavity problem” has become a canonical test case for both ﬂuid dynamics and control theory. Major challenges for ﬂuid dynamicists are to design effective actuators, determine appropriate scaling relations between the forcing and the ﬂow ﬁeld, and develop sufﬁciently accurate low-dimensional models of the cavity system. Control theorists search for robust and adaptive algorithms capable of suppressing tones under changing ﬂight conditions. The mechanism for acoustic resonance in cavities was identiﬁed by Rossiter [4]. There are four elements in the mechanism, 1) downstream propagation of vorticity R. King (Ed.): Active Flow Control, NNFM 95, pp. 230–243, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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waves in the cavity shear layer; 2) production of an acoustic wave by interaction of the shear layer with the downstream edge of the cavity; 3) upstream acoustic wave propagation; 4)receptivity at the leading edge, i.e., conversion of the pressure ﬂuctuation to a vorticity wave. When the feedback of the acoustic wave to the leading edge of the cavity reinforces the vorticity wave amplitude in the shear layer, then the timing for resonance is correct. Rossiter derived an empirical formula for this process, which was slightly modiﬁed by Heller, et al. [5, 6] as shown below. St =

fL = √ U

m−α M

2 1+ γ−1 2 M

+

1 κ

(1)

In this equation ”m” is the integer mode number, α is a phase delay factor , and κ is the shear layer wave speed normalized by the freestream speed. Typical values for the adjustable constants are α= 0.25 and κ = 0.57, although substantial variations in both have been observed in experiments. Many active ﬂow control techniques have demonstrated the ability to suppress tones, particularly at subsonic speeds. The key strategy in all cases is to disrupt the Rossiter feedback mechanism. Passive, active open-loop, and closed-loop control approaches have shown varying degrees of success. Cattafesta, et al. [7] and Colonius [8] provide extensive reviews of active ﬂow control techniques used, and discussions of the physical mechanisms involved in controlling cavity tones. Active control with open-loop forcing of the shear layer attempts to suppress tones by forcing at a non-resonant frequency. Sarno and Franke [9], Shaw [10,11], Samimy, et al. [12], and Cattafesta, et al. [13] have shown the ability to suppress cavity tones with the open loop approach. Cattafesta, et al. [13] compared suppression by closed-loop ﬂow control to the open-loop case, and demonstrated that the closed-loop approach used an order of magnitude less power. The mechanism by which open-loop forcing of the shear layer suppresses the resonant tones is not understood. Why, for example, when the shear layer is excited by a non-resonant frequency, would not the forcing frequency simply superpose on the baseline spectrum? Apparently some type of nonlinear interaction occurs between the base ﬂow state and the forcing ﬁeld, which interferes with the resonance mechanism. At least ﬁve different arguments for the open-loop suppression mechanism can be found in the literature, and have been itemized below. The sixth mechanism in the list refers to linear wave cancelation that can only occur with a closed-loop control system. 1. Lifting the shear layer which changes the downstream reattachment point [14, 15] - modiﬁcation of mean shear proﬁle combined with lifting [16] 2. Change of shear layer stability characteristics by thickening the shear layer [8, 17] 3. Low-frequency excitation of the shear layer at off-resonance condition [9, 15, 13, 18, 19, 20] 4. High-frequency (hifex) excitation [17, 20]- accelerated energy cascade in inertial range ”starves” lower frequency modes [21] - mean ﬂow alteration, which changes stability characteristics [22] 5. Oblique shock ﬂow deﬂection and reduction of longitudinal ﬂow speed [23] 6. Cancelation of feedback acoustic wave [13, 25]

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The suppression mechanisms listed above are primarily intuitive, and do not offer much predictive capability. Progress toward developing a predictive model of the effect of shear layer thickening and the change in stability characteristics is discussed in detail by Colonius [8]. Sahoo, et al. [23] developed a physics based model to explain the mechanism by which micro-jets suppress cavity resonance in a supersonic ﬂow. By considering the effect of an oblique shock formed at the leading edge of the cavity by the micro-jets, they were able to estimate the ﬂow deﬂection angle and speed reduction effects. Their model correlated very well with the experimental data. The objective of this experiment was to get a better understanding of the cavity response to open-loop forcing by systematically varying the forcing frequencies and amplitudes. The effect of dynamic pressure could be studied by changing the wind tunnel stagnation pressure. Our initial expectations were to ﬁnd nonlinear interactions between the forcing ﬁeld and the base state resonant modes, similar to the subsonic case, but this did not happen. The following sections describe the calibration of the pulsed-blowing actuator used for the forcing, and the pressure measurements of the cavity response when the forcing amplitude and frequency and dynamic pressure were varied.

2

Experimental Setup

The experiments were conducted in the supersonic wind tunnel at the Illinois Institute of Technology Fluid Dynamics Research Center. The facility is a blow down type wind tunnel with a variable throat area. The test section is 102 mm wide and 114 mm high. The Mach number was ﬁxed at M=1.86 for this study and U∞ =629 ± 19m/s. Wind tunnel stagnation pressures (absolute) ranged from 0.31 MPa to 0.72 MPa. Operating the wind tunnel at different stagnation pressures made it possible to change the static and dynamic pressure in the test section, while keeping the Mach number ﬁxed. The stagnation temperature in the wind tunnel was 290 K. The unit Reynolds number at M=1.86 was 49x106 per meter. The cavity model was machined into the ﬂoor of the test section as shown in Fig.1. The sidewall of the wind tunnel was removed for this photograph to expose the details of the cavity and actuator nozzle block. The pulsed-blowing air from the siren valve enters through the side of the wind tunnel and enters the plenum of the nozzle block seen on the left side of the cavity. The cavity is 152 mm long, 102 mm wide and 30.5 mm deep, giving it L/D = 5 and L/W = 1.5. Pressure ﬂuctuations inside the cavity were recorded with two Kulite XCS093 transducers located in the center span of the cavity ﬂoor at 8.25 mm and 144 mm from the upstream cavity wall. The boundary layer thickness approaching the leading edge of the cavity was estimated from schlieren images and a boundary layer rake of total pressure probes to be δ = 8 mm. The pulsed-blowing actuator consisted of a compressed air supply, siren valve and nozzle block. The siren valve manufactured by Honeywell was connected to the side of the nozzle plenum with a 75 mm long tube, giving a bandwidth of approximately 1.5 kHz. Two interchangeable nozzle blocks were constructed with different exit angles,

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Fig. 1. Photograph of cavity model in supersonic wind tunnel. The nozzle block is visible on the left (upstream) side of the cavity.

one exiting parallel to the ﬂow direction and the other at 45o relative to the downstream direction. For both nozzles the exit spanned the width of the cavity and had a height of 3.2 mm. Pulsed-blowing actuators require some ﬂow through the system in order to produce oscillations, but with careful tuning of the plumbing system it is possible to generate oscillation amplitudes larger than the mean ﬂow speed, producing instantaneously reversed ﬂow. The actuator performance was documented using both a hot-wire anemometer to measure velocity at the slot exit, and a Kulite pressure transducer to measure the instantaneous pressure in the slot exit of the actuator nozzle. The face of the Kulite pressure sensor was oriented directly at the exit of the actuator to record the instantaneous total pressure. Time series traces of the velocity and pressure at 750 Hz forcing frequency with an actuator supply pressure of 124 kPa (absolute) are shown in Fig.2. Velocity measurements of the mean velocity and root mean square (r.m.s.) velocity at the exit of the actuator are shown in Fig.3a. The forcing frequency was set at 750 Hz, while the supply pressure was varied from 101 kPa to 240 kPa. The velocity oscillation amplitude saturates as the supply pressure to the actuator is increased, while the mean velocity increases monotonically. This type of behavior is common for pulsed-blowing actuators. Attempts to increase oscillation amplitude by increasing the supply pressure often do more to increase the mean ﬂow than the oscillatory component of velocity. The corresponding mean and r.m.s. pressure values at the actuator exit are shown in Fig.3b. The mean pressure steadily increases with supply pressure, while the r.m.s. pressure grows at a much slower rate. The mean ﬂow through the actuator is expressed as a blowing coefﬁcient, Bc as deﬁned in the following equation. Following Zhuang, et al. [24] the reference area is deﬁned as the cavity length times cavity width. Bc =

ρjet Ajet Ujet ρ∞ Aref U∞

(2)

The output from the pulsed-blowing actuator was strongly dependent of the forcing frequency. To document the frequency response of the actuator, the siren valve frequency was varied from 400 Hz to 2500 Hz, while maintaining a nominally constant input pressure of 138 kPa. Figure 4a shows a sharp cutoff in the r.m.s. velocity

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90 Speed (m/s) Pressure (Pa x 40)

80 70 60 50 40 30 20 10 0

0

0.002

0.004

0.006

0.008

0.01

Time (s)

Fig. 2. Velocity (solid line) and pressure (dashed line) time series at actuator exit plane. Actuator supply pressure = 124 kPa, frequency = 750 Hz.

ﬂuctuation amplitude near 1500 Hz. The r.m.s. pressure shown in Fig. 4b has a more gradual decay in amplitude with frequency.

3

Results

The ﬂuctuating pressure was recorded with transducers in the ﬂoor of the cavity. The baseline response without forcing is presented in Sect.3.1. The response of the cavity to the open-loop forcing at different frequencies and amplitudes is described in Sect.3.2. 3.1 Baseline Cavity Behavior – No Forcing The supersonic cavity control experiments by Zhuang, et al. [24] measured a linear dependence of the overall sound pressure level with wind tunnel stagnation pressure. A similar linear dependence was found in this experiment as shown in Fig.5 for supersonic ﬂow. At stagnation pressures below 300 kPa the ﬂow was subsonic in the wind tunnel. Pressure spectra measured by the upstream pressure sensor are shown in Fig.6 for different wind tunnel stagnation pressures. Without forcing six identiﬁable Rossiter modes can be found in the spectra. The best ﬁt of the Rossiter equation (1) to the data was obtained using α=0.2 and κ=0.4. The predicted mode frequencies are indicated by the vertical lines in Fig.6. A close look at the ﬁgures shows that increasing the wind tunnel stagnation pressure does not affect the resonant frequencies, but does increase the amplitude of the spectral peaks. 3.2 Cavity Response to Periodic Forcing The performance of pulsed-blowing actuators is dependent on the pressure difference between the supply pressure and the pressure at the actuator nozzle exit. Increasing the

Supersonic Cavity Response to Open-Loop Forcing (a)

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(b)

70

5 Mean Rms

Mean Rms

4.5

60 4 3.5 Exit Pressure (kPa)

Velocity (m/s)

50

40

30

3 2.5 2 1.5

20

1 10 0.5 0 100

0 100

150 200 250 Actuator Pressure (kPa)

150 200 250 Actuator Pressure (kPa)

Fig. 3. Supply pressure dependence of the mean and r.m.s. pressures measured at the actuator exit plane with 750 Hz forcing frequency (a)

(b)

40

4 Mean Rms

35

3 Exit Pressure (kPa)

Velocity (m/s)

30 25 20 15

2.5 2 1.5 1

10 5

Mean Rms

3.5

0.5

0

1000 2000 Frequency (Hz)

3000

0

0

1000 2000 Frequency (Hz)

3000

Fig. 4. Frequency dependence of the mean and r.m.s. velocities measured at the actuator exit plane with a 138 kPa input pressure (Bc = .0010)

wind tunnel stagnation pressure increases the static pressure in the test section, which may reduce the effectiveness of the pulsed-blowing actuator. A plot of 800 Hz peak amplitude against the wind tunnel stagnation pressure at ﬁxed forcing amplitude is shown in Fig.7a. Above a stagnation pressure of 450 kPa the cavity response decreases with the dynamic pressure. Similarly, the dependence of the 800 Hz peak amplitude on the actuator supply pressure is shown in Fig.7b. Initially the peak growth is proportional to actuator supply pressure, then the cavity response begins to saturate at actuator pressures above 140 kPa. This behavior is consistent with the saturation of the ﬂuctuating

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Overall Sound Pressure Level (dB)

161 160 159 158 157 156 155 154

supersonic

subsonic

153 152 100

200

300

400 500 600 Stagnation Pressure (kPa)

700

800

Fig. 5. Overall sound pressure level increases linearly with stagnation pressure in the supersonic ﬂow regime 180

308 kPa 377 kPa 406 kPa 514 kPa 583 kPa 652 kPa 721 kPa

160

SPL (dB)

140

120

100

80

60

0

1000

2000

3000 4000 5000 Frequency (Hz)

6000

7000

Fig. 6. Pressure spectra with no forcing are superposed. Each spectrum increases in amplitude as the wind tunnel stagnation pressure (values shown in the legend) is increased.

velocity levels seen in Fig.3. It can be shown that the actuator response scales with the pressure difference across the actuator. The pulsed-blowing actuator was set to a frequency of 1000Hz and a supply pressure of 170kPa. The wind tunnel stagnation pressure was ﬁxed at 584 kPa, giving a static pressure in the test section close to the calibration conditions. The pressure spectrum measured before the wind tunnel was started is shown in Fig.8 as the dashed line. The spectrum obtained with the wind tunnel running at M=1.86 is superposed in the ﬁgure as a solid line. The input from the actuator was ampliﬁed 25 dB above the no-ﬂow condition by the cavity.

142

146

141

144

140

142

Sound Pressure level (dB)

Sound Pressure Level (dB)

Supersonic Cavity Response to Open-Loop Forcing

139 138 137

237

140 138 136

136

134

135

132

134 300 400 500 600 700 Tunnel Stagnation Pressure (kPa)

130 100

150 200 250 Forcing Pressure (kPa)

Fig. 7. Decay and growth of the 800 Hz spectral peak amplitude with: a) changing wind tunnel stagnation pressure (actuator supply pressure = 101.6 kPa); b) changing actuator supply pressure (wind tunnel stagnation pressure = 584 kPa). Actuator frequency = 800 Hz.

150 M=1.86 M=0 baseline

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There was some concern that the sharp peak in the spectrum at the forcing frequency was not a ﬂuid dynamic response of the cavity, but possibly an acoustic signature of the actuator, such as, a simple superposition of the forcing ﬁeld. To check this, the forcing frequency and amplitude were varied, and the response of the cavity measured to get a better understanding of the nature of the forcing peak.

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The actuator frequency was changed to 800 Hz, slightly below the ﬁrst Rossiter mode at 880 Hz. The amplitude of the pulsed-blowing actuator was changed by adjusting the supply pressure. The pressure spectra are superposed in Fig.9a along with the baseline (no-forcing) case. Each spectrum corresponds to a different supply pressure to the actuator. The growth of the unsteady forcing peak with increasing supply pressure can be seen. We also found that nonlinear mode interactions (combination modes) do not appear, which is signiﬁcantly different behavior than the subsonic ﬂow case. The 800 Hz peak amplitude with supersonic ﬂow is plotted against the forcing amplitude in the quiescent wind tunnel in Fig.9b. The dashed line has a slope of 1.0, which implies a linear relationship between the forcing and response amplitudes. The data appear to be close to displaying a linear relationship. Next the forcing frequency was changed to 1300 Hz, which was between the ﬁrst and second Rossiter mode. Figure10 compares the baseline spectrum (quiescent wind tunnel) with the forced case. At this frequency the cavity response is lower than the acoustic forcing level without ﬂow in the tunnel, indicating that the cavity system is attenuating the disturbance. The forcing frequency was varied from 500 Hz to 2400 Hz in 100 Hz increments, while maintaining a constant input pressure to the actuator of 170 kPa. The measured spectra are superposed in Fig.11a. The response contains both the frequency response of the actuator and that of the cavity system. At the lower forcing frequencies near the ﬁrst Rossiter mode, the actuator frequency response is reasonably constant (see Fig.4), and the response of the cavity follows the peak seen in the unforced spectrum. As the

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frequency is increased toward the second Rossiter mode, the amplitudes of the cavity response decrease for two reasons. First the actuator frequency response decreases (Fig.4), and second, as shown in the previous ﬁgure, the cavity system is attenuating disturbances between the Rossiter modes relative to the input disturbance amplitude. To isolate the cavity system dynamics from these measurements it is necessary to account for the actuator frequency response. To do this, we measured the peak amplitude at each forcing frequency in the quiescent wind tunnel, M=0. The “gain” was deﬁned as the difference between the dB level of the peak amplitude with the tunnel running and the quiescent tunnel measurement. The gain is shown in Fig.11b. Positive gain is seen around the ﬁrst two Rossiter mode frequencies, and negative values corresponding to attenuation are located between the Rossiter modes. The corresponding phase between the actuator oscillations and the oscillating pressure ﬁeld is plotted in Fig.11c.

4 Discussion The results of the previous section strongly suggest that for the ﬂow regime studied, the cavity ﬂow behaves as a linear ampliﬁer, amplifying the actuator signal at its resonant frequencies, and attenuating it at other frequencies. This linear relationship is supported both by the linear scaling with amplitude in Fig. 9, as well as the response to the forcing at different frequencies in Fig.11, in which the pressure spectrum is altered only at the frequency of forcing. Previous experiments [12] have shown that for some subsonic ﬂows, a much more complicated nonlinear interaction of frequencies occurs. No such coupling was observed in the present experiment. The differences between the supersonic results here and previous subsonic results may be explained by models such as those described in [26]. These models consider the cavity ﬂow as a dynamical system with a ﬁxed point (i.e., a steady solution of the

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Navier-Stokes equation, which is close to the mean ﬂow), which may be either stable or unstable, depending on the total loop gain present in the Rossiter mechanism. For many ﬂow regimes, the ﬂow acts like a stable, but lightly-damped oscillator, amplifying disturbances at resonant frequencies. If the loop gain increases (for instance, in a ﬂow with a thinner boundary layer), then the ﬁxed point can become unstable, a stable limit cycle is formed, and nonlinearities determine the amplitude of the oscillations. Experiments by Rowley, et al. [26,27] with subsonic cavity ﬂows determined that some of the Rossiter modes were not self-excited (depending on Mach number), but instead were weakly damped. Kerschen, et al. [28] and Alvarez, et al. [29] developed an analytic model for the cavity resonance mechanism, which was capable of predicting growth rates of the unstable modes, and also found many modes to be weakly damped. In the case of weakly damped modes, the cavity will amplify (or attenuate) external disturbances to a level dependent on the initial amplitude of the disturbance, growth rate through the shear layer, and receptivity. The results of the previous section are consistent with the existence of a stable ﬁxed point, and a linear mechanism for ampliﬁcation. Supersonic ﬂows may be more likely than subsonic ﬂows to operate in this linear regime, because compressibility reduces the ampliﬁcation of instabilities in the free shear layer (i.e., the ampliﬁcation of KelvinHelmholtz modes decreases as Mach number increases), and so one would expect the total loop gain to decrease as Mach number increases. Of course, other factors such as boundary layer thickness also inﬂuence the total shear layer ampliﬁcation, so Mach number is clearly not the only parameter relevant for determining the stability of the ﬂow. A careful study of the ﬂow regimes and scaling laws determining the regions of stability of the cavity ﬂow would be valuable for understanding the dynamics of these ﬂows. When a system is in a self-excited limit cycle, then the ﬁnal amplitude is determined by nonlinear saturation. No evidence of nonlinear behavior was observed in the spectra

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obtained in this experiment. Sum and difference combination modes between the forcing and the Rossiter modes did not appear, and the existing peaks in the spectra did not appear to be affected by the forcing. The linear behavior of the shear layer suggests that linear control approaches, such as those developed by Becker, et al. [30] may be useful for supersonic cavity control. Finally, a nonlinear response mechanism is necessary for the cavity to be modiﬁed by the oscillatory forcing at frequencies different from the forcing frequency. In a linear system, excitation at a single frequency cannot suppress more than one Rossiter tone simultaneously. However, many other investigators have been able to suppress the tones with actuators that add mass or momentum to the system. To examine the effects of steady ﬂow injection, the siren valve was removed from the actuator and steady blowing was applied to the nozzle block with a 45o exit angle. The results are shown in Fig.12. With a 409 kPa forcing pressure and a blowing coefﬁcient Bc = 0.036, the Rossiter tone peaks are signiﬁcantly reduced in amplitude. This blowing coefﬁcient is two orders of magnitude larger than the values used in identifying the response of the cavity to openloop forcing described earlier in the paper. This result strongly suggests that the tone suppression is accomplished purely by the steady ﬂow mass addition, and the oscillatory component does not play a signiﬁcant role in the suppression.

5 Conclusion Open-loop forcing experiments with an L/D = 5 cavity have been conducted at M=1.86. A pulsed-blowing type actuator was used to provide controlled inputs of mean and oscillating ﬂow at the upstream edge of the cavity. The objective was to systematically vary the forcing frequency and amplitude, and the dynamic pressure in the wind tunnel to obtain a better understanding of the cavity system response to open-loop forcing. The behavior of the supersonic cavity was signiﬁcantly different than has been

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observed in subsonic ﬂow experiments. Nonlinear interactions between the forcing and the naturally occurring Rossiter modes was not observed in this experiment. However, strong ampliﬁcation of the input disturbances occurred when the forcing frequency was close to a Rossiter mode, and attenuation occurred when the forcing frequency was between Rossiter modes. In regions of ampliﬁcation, the increase in the cavity response amplitude was proportional to the input disturbance amplitude, which suggests the overall system behaves linearly. The Rossiter modes were clearly not in a nonlinearly saturated state, because their amplitude could be increased by 20dB with small amplitude inputs of the external forcing.

Acknowledgements The many helpful suggestions from Tim Colonius and Lou Cattafesta are gratefully acknowledged. The work by C. W. Rowley and D. R. Williams was supported by AFOSR under grants F49620-03-1-0081 and F49620-03-1-0074 with program managers Sharon Heise and John Schmisseur.

References [1] J.E Grove, M. A. Pinney, and M.J. Stanek: ”A Cooperative Response To Future Weapons Integration Needs”. Applied Vehicle Technology Panel, Symposium on Aircraft Weapon System Compatibility and Integration, NATO- R&T Organization, Sept. 1998, pp 24-1 24-12. [2] J. Grove, L. Shaw, J. Leugers and G. Akroyd: ”USAF/RAAF F-111 Flight Test with Active Separation Control”. AIAA 2003-0009, Jan. 2003. [3] O.W. McGregor and R. A. White: ”Drag of Rectangular Cavities in Supersonic and Transonic Flow Including the Effects of Cavity Resonance”. AIAA Journal, 8, 1970, pp. 19591964. [4] J. E. Rossiter: ”Wind-Tunnel Experiments on the Flow over Rectangular Cavities at Subsonic and Transonic Speeds”. Aeronautical Research Council Reports and Memoranda No. 3438, London, Oct. 1964. [5] H.H.Heller, D.G. Holmes, and E.E. Covert: ”Flow Induced Pressure Oscillations in Shallow Cavities”. J. of Sound and Vibration, 18, No. (4), 1971, pp.545-553. [6] H.H. Heller and D. Bliss: ”The Physical Mechanism of Flow-Induced Pressure Fluctuations in Cavities and Concepts for their Suppression”. AIAA Paper 75-491, March 1975. [7] L. Cattafesta, D.R. Williams, C. Rowley, and F. Alvi: ”Review of Active Control of FlowInduced Cavity Resonance”. AIAA 2003-3567, June 2003. [8] T. Colonius: ”An overview of simulation, modeling, and active control of ﬂow acoustic resonance in open cavities”. AIAA 2001-0076, Jan. 2001. [9] R.L. Sarno and M.E. Franke: ”Suppression of Flow-Induced Pressure Oscillations in Cavities”. J. of Aircraft, 31, No. 1, 1994, pp. 90-96. [10] L. Shaw: ”Active Control for Cavity Acoustics”. AIAA 98-2347, June 1998. [11] L. Shaw, and S. Northcraft: ”Closed Loop Active Control for Cavity Acoustics”. AIAA Paper 99-1902, June 1999. [12] M. Samimy, M. Debiasi, O. Efe, H. Ozbay, J. Myatt, and C. Camphouse: ”Exploring Strategies for Closed-Loop Cavity Flow Control”. AIAA 2004-0576, Jan. 2004.

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[13] L.N. Cattafesta, S. Garg, M. Choudhari and F. Li: ”Active Control of Flow-Induced Cavity Resonance”. AIAA 97-1804, June 1997. [14] L.S. Ukeiley, M.K. Ponton, J.S. Seiner and B. Jansen: ”Suppression of Pressure Loads in Cavity Flows”. AIAA 2002-0661, Jan. 2002. [15] P.C. Bueno, .H. nalmis, N.T. Clemens and D.S. Dolling: ”The Effects of Upstream Mass Injection on a Mach 2 Cavity Flow”. AIAA 2002-0663, Jan 2002. [16] L.S. Ukeiley, M.K. Ponton, J.M. Seiner and B. Jansen: ”Suppression of Pressure Loads in Cavity Flows”. AIAA J., 42, No. 1, Jan 2004, pp. 70-79. [17] L.S. Ukeiley, M.K. Ponton, J.M. Seiner, and B. Jansen: ”Suppression of Pressure Loads in Resonating Cavities Through Blowing”. AIAA 2003-0181, Jan. 2003. [18] D. Fabris and D.R. Williams: ”Experimental Measurements of Cavity and Shear Layer Response to Unsteady Bleed Forcing”. AIAA 99-0606, Jan. 1999. [19] G. Raman, S. Raghu and T.J. Bencic: ”Cavity Resonance Suppression using Miniature Fluidic Oscillators”. AIAA 99-1900, May 1999. [20] R.F. Schmit, D.R. Schwartz, V. Kibens, G. Raman, J.A. Ross: ”High and Low Frequency Actuation Comparison for a Weapons Bay Cavity”. AIAA 2005-0795, Jan. 2005. [21] M. Stanek, G. Raman, V. Kibens, J. Ross, J. Odedra and J. Peto: ”Control of Cavity Resonance through Very High Frequency Forcing”. AIAA 2000-1905, June 2000. [22] M.J.Stanek, G. Raman, J.A. Ross, J. Odedra, J. Peto, F. Alvi, and V. Kibens: ”High Frequency Acoustic Suppression - The Mystery of the Rod-in-Crossﬂow Revealed”. AIAA 2003-0007, Jan. 2003. [23] D. Sahoo, A. Annaswamy, N. Zhuang, and F. Alvi: ”Control of Cavity Tones in Supersonic Flow”. AIAA 2005-0793, Jan. 2005. [24] N. Zhuang, F.S. Alvi, M.B. Alkislar, C. Shih, D. Sahoo and A.M. Annaswamy: ”Aeroacoustic Properties of Supersonic Cavity Flows and Their Control”. AIAA 2003-3101, 9th AIAA/CEAS Aeroacoustic Conf., Hilton Head South Carolina, May 2003. [25] L.N.III Cattafesta, D. Shukla, S. Garg and J.A. Ross: ”Development of an Adaptive Weapons-Bay Suppression System”. AIAA 99-1901, May 1999. [26] C.W. Rowley, D.R. Williams, T. Colonius, R.M. Murray and D.G. MacMynowsky: ”Linear models for control of cavity ﬂow oscillations”. J. Fluid Mech., 547, Jan 2006, pp.317-330. [27] C.W. Rowley and D.R. Williams: ”Dynamics and Control of High-Reynolds Number Flow over Open Cavities”. Annual Review of Fluid Mechanics, 38, 2006, pp. 251-276. [28] E.J.Kerschen and A. Tumin: ”A Theoretical Model of Cavity Acoustic Resonance Based on Edge Scattering Processes”. AIAA 2003-0175, Jan. 2003. [29] J.O.Alvarez, E.J.Kerschen and A. Tumin: ”A Theoretical Model for Cavity Acoustic Resonance in Subsonic Flow”. AIAA 2004-2845, June 2004. [30] R. Becker, M. Garwon and R. King: ”Development of model-based sensors and their use for closed-loop control of separated shear ﬂows”. ECC, 2003.

Active Drag Control for a Generic Car Model A. Brunn1 , E. Wassen2 , D. Sperber1 , W. Nitsche1 , and F. Thiele2 1

Institute of Aeronautics and Astronautics Technical University Berlin, Marchstr. 12, 10587 Berlin, Germany [email protected] 2 Institute of Fluid Mechanics and Engineering Acoustics Technical University Berlin, Mueller–Breslau–Str.8, 10623 Berlin, Germany [email protected]

Summary Experimental and numerical investigations were carried out aiming at the reduction of the total aerodynamic drag of a generic car model by means of active separation control. For two different conﬁgurations separate control approaches were tested, taking into account the differences in the wake topology of the models. The targeted excitation of the respective dominant structures in the wake region leads to their effective attenuation. The experiments as well as the numerical simulations showed that a weakening of a spanwise vortex in the separated ﬂow over the slant is strongly coupled with the occurrence of stronger streamwise vortices along the slant edges and vice versa.

1 Introduction Flow control over a bluff body for drag and noise reduction purposes is considered to be one of the major issues in aerodynamics. The pressure difference between the front and the rear end of a bluff body is the main contributor to the overall drag, this difference being primarily due to the ﬂow separation at the rear end of the body [4, 8]. Looking at the generic car model introduced by Ahmed et al. [1] (Fig. 1, left), the ﬂow ﬁeld in the wake is highly three-dimensional, unsteady, and it depends strongly on the rear slant angle ϕ [1, 7]. Longitudinal vortices occur at slant angles up to 30◦ , leading to a dramatic increase in pressure drag (Fig. 1, right). With increasing slant angles these vortices burst, and spanwise vortices dominate the wake region, causing a reduction in drag. Active methods of ﬂow control can be applied to avoid or reduce this type of separation-induced performance loss. This fact was demonstrated on a two-dimensional Ahmed body with an slant angle of ϕ = 35◦ in [2]. The ﬂow ﬁeld in the near wake is dominated in this case by two-dimensional spanwise vortex structures, and therefore a control approach for this kind of vortex was applied. Actuators generating periodic perturbations were used to excite the shear layer separating from the rear slant of the car model. Forcing frequencies in the range of the initial shear layer instability and the vortex shedding were used to test the receptivity of the ﬂow. The excitation by periodic perturbations at the slant edge led to increased velocity ﬂuctuations in the shear layers, while the momentum transfer between the recirculation region and the outer ﬂow was R. King (Ed.): Active Flow Control, NNFM 95, pp. 247–259, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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Fig. 1. Ahmed car model [1] with illustrated vortex structures behind the rear end (left), and the total drag coefﬁcient for different slant angles (right)

signiﬁcantly intensiﬁed due to forcing of the vortex-shedding frequency. The most effective frequency for drag control was observed for a corresponding Strouhal number StH = 0.2, based on the model height. A total drag reduction of 27 % was achieved for this forcing case. The ampliﬁed large scale vortices connected to the vortex shedding process are the key to reducing separation and consequently to reducing the pressure drag of the Ahmed body. In other studies attempts were made to control the ﬂow for the original threedimensional model as introduced by Ahmed et al. [1]. In numerical simulations using a RANS (Reynolds Averaged Navier Stokes) approach for a slant angle of ϕ = 35◦ , both constant blowing and suction were separately investigated as means of reducing the drag of the vehicle [10]. Varying the blowing/suction velocity as well as the direction, it was shown that in almost all cases the pressure drag was increased. When applying constant suction, the ﬂow over the slant could be forced to re-attach, reducing the size of the wake signiﬁcantly. At the same time, however, vortices were generated at the slant edges, which in turn caused a drag increase. Other simulations for the 25◦ case using a Lattice Boltzmann approach applied constant suction only [11]. Similar to the investigations for the 35◦ case, the separated region on the slant could be reduced, while in turn the strength of the vortices near the slant edges was increased. In this study, an overall drag reduction of 13% was achieved. In the present paper, ﬁrst results of an ongoing investigation are presented for active drag control approaches for a three-dimensional Ahmed model, using two different slant angles, ϕ = 35◦ and ϕ = 25◦ . Considering the differences in the wake topology of these two cases, separate control techniques were applied. For ϕ = 35◦ a periodic excitation by blowing/suction was investigated experimentally. A combined numerical and experimental study was carried out for ϕ = 25◦ . In order to inﬂuence the steady longitudinal side edge vortices, constant blowing was used in this case. The experimental and numerical setup is described in section 2, and the results for the different ﬂow control approaches are presented in section 3.

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The shape of the generic car model investigated here was based on the original geometry of the Ahmed body [1]. In the experiments all dimensions (Fig. 1) were scaled to a

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quarter of the original size considering the given test section geometry. The Reynolds number based on the inﬂow velocity and the model length was set at ReL = 0.5 · 106 for both the experiments and the numerical simulations. 2.1 Experiments The experimental investigations were conducted in two different ﬂow channels: a wind tunnel with a closed test section and a closed water channel in order to investigate the ﬂow ﬁeld in the near wake of the Ahmed body using Particle Image Velocimetry (PIV) and digital ﬂow visualization methods (Fig. 2, left). The PIV system consists of a frequency-doubled Nd:YAG laser, two CCD-Cross-Correlation-Cameras and a Synchronization Unit. A force balance in the low speed wind tunnel consisting of a sensitive strain gauge sensor located below the test section was used in order to measure the effect of the applied ﬂow control methods on the total drag of the Ahmed body (Fig. 2, right). The periodic pressure perturbations in the water channel experiments were generated by a water pump connected to a rotating valve and injected into a cavity-slit-system, resulting in an oscillating wall jet without net mass ﬂux [2]. The perturbations imposed on the ﬂow around the Ahmed body with a slant angle of ϕ = 35◦ were aimed at amplifying the spanwise vortex structures in the separated shear layer to increase the growth of the vortices and intensify the entrainment process (Fig. 3, left). For ϕ = 25◦ , steady blowing cavities were integrated into the slant corners of the Ahmed body (Fig. 3, right). The steady blowing forcing intensity was adjustable within in the range of 0 ≤ cµ ≤ 7 · 10−3 , where 2 AS cS · , (1) cµ = AB u ¯∞ and u ¯∞ is the average velocity at the inﬂow, cS is the perturbation velocity at the slit exhaust, and AB and AS are the cross section of the Ahmed body and the active actuator area, respectively. The synthetic jet was directed normal to the mean ﬂow direction to prevent an interaction between the thrust of the jet and the drag measurements.

Fig. 2. Experimental setup of the Ahmed body investigations in the water channel (left) and the force balance of the low speed wind tunnel test section (right)

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Fig. 3. Actuator conﬁgurations for the two different slant angles: periodic forcing (left) and steady blowing (right)

2.2 Simulations Large–Eddy Simulations (LES) were performed to calculate the ﬂow around the model vehicle for a slant angle of ϕ = 25◦ . The computational domain had a width of 1.791L and a height of 1.341L. In the streamwise direction it extended 2.014L upstream from the front of the body and 5.793L downstream from the rear. At the inﬂow boundary, a constant velocity of uin = u¯∞ was imposed. The sides and the top of the ﬂow domain were treated with a symmetry boundary condition (“slip wall”). It should be noted that for the simulation of a fully turbulent ﬂow with LES or DNS the use of a symmetry boundary condition is usually not advisable, since it suppresses turbulent ﬂuctuations normal to the boundary. In the current case, however, turbulence is only generated near the solid walls of the ﬂoor and the body, as well as in the wake region. Therefore, far away from the model to the side and to the top the ﬂow is essentially laminar, thus justifying the application of symmetry boundary conditions in these areas. At the outﬂow, a convective boundary condition was used, and all solid walls were treated with a no-slip Stokes condition. Simulations were performed for the base ﬂow, i.e. without ﬂow control, and for a control case applying active blowing. For the latter, a small part of the model’s slanted rear surface was re-deﬁned as an inﬂow boundary. The solution algorithm used here is based on a ﬁnite-volume discretization and is of second order accuracy in space and time. For the LES, the subgrid stresses were modelled using a Smagorinsky subgrid-scale model [9] with a model constant of 0.1. The numerical grid consisted of 17 million cells.

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In the present study, experiments as well as numerical simulations were carried out for two different slant angles of the Ahmed body using two different drag control approaches. 3.1 Ahmed Body with a Slant Angle of ϕ = 35◦ Base ﬂow The unforced base ﬂow in the near wake region of the Ahmed body was documented ﬁrst by means of time-averaged PIV-measurements (Fig. 4b–d). Here, an ensemble of

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around 200 instantaneous PIV images for three different spanwise positions (Fig. 4a) are depicted. Some signiﬁcant differences in the velocity ﬁelds at different spanwise positions could be observed. The ﬂow structures typically occurring behind bluff bodies are visible (e.g. [6]): Two counter rotating vortex structures and a free stagnation point that closes the separation region. The overall separation length in this case was measured to be 35% of the model length behind the rear end, which agrees very well with the experimental data from Lienhart (33% of the model length) [7]. The velocity ﬁeld in the side plane (Fig. 4d) shows no recirculation and the effect of the separation bubble is limited to the model width in spanwise direction. Spanwise ﬂow structures appear in the near wake region of the Ahmed body. These structures are comparable to the ones found in investigations on the two-dimensional car conﬁguration [2], which is why the present experiments focused on a similar approach to drag reduction in terms of active separation control.

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Forced ﬂow All drag control experiments for this conﬁguration were performed with a forcing intensity of cµ = 4 · 10−3 , in the view of the results from earlier investigations on the two-dimensional Ahmed body [2] and the power limits of the actuator used. A signiﬁcant reduction of the turbulent car wake separation length was achieved for all forcing Strouhal numbers investigated (based on the inﬂow velocity and the car model height H) in the range of 0.1 ≤ StH ≤ 0.9, but without noticeable differences in the resulting (mean) ﬂow ﬁeld (Fig. 5). An excitation according to the vortex shedding frequency (StH = 0.2, Fig. 5a), observed to be the most effective in the two-dimensional conﬁguration [2], shows a drastic reduction of the recirculation area in the symmetry plane compared to the base ﬂow (Fig. 4). But periodic forcing with higher frequencies shown in Fig. 5, corresponding to the shear layer instability, results

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in similar velocity ﬁelds. The reason for this frequency insensitivity is shown in Fig. 5b, where the velocity ﬁeld in the side plane is shown for these forcing cases: A longitudinal vortex structure starting at the slant corner is clearly visible. This kind of structure is usually expected for smaller slant angles, and the ﬂow ﬁeld is very similar to the results of Ahmed et al. for a slant conﬁguration of ϕ ≤ 30◦ [1]. Obviously the recirculation region above the slant surface is reduced due to the periodic forcing in the frequency range tested here. In addition, a small reduction of the separation bubble leads to ﬂow conditions (decreased pressure on the slant surface) which cause streamwise vortices to appear. The presence of these strong longitudinal vortices prevents a lateral expansion of the separation bubble on the slant, and this effect is much stronger than that of the periodic forcing. The logical conclusion of this result is that an effective approach for drag control of the Ahmed body has to control both the separation over the slant and the longitudinal vortices. The latter are structures that are typically found in the base ﬂow for ϕ < 30◦ , so this case was investigated next. 3.2 Ahmed Body with a Slant Angle of ϕ = 25◦ A combined experimental and numerical study was done for the model with a slant angle of ϕ = 25◦ . It should be noted beforehand, that the results from the current experiments and simulations cannot be expected to match exactly. For one, the wind tunnel cross section with respect to the model size was different from the cross section of the numerical ﬂow domain, resulting in different blockage ratios. Furthermore, in the wind tunnel the Ahmed body was ﬁxed on four stilts, in addition to which there were

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also two tubes feeding the control ﬂow through the wind tunnel ﬂoor and the model underside into the body. The stilts and the tubes were omitted in the LES, since they would have required a mesh reﬁnement at the respective locations, which would have increased the computational cost signiﬁcantly. Base ﬂow In the wind tunnel, the overall drag coefﬁcient was determined by force balance measurements to be cD = 0.41, whereas in the LES it was cD = 0.374. The difference can be attributed to the different setup as described above. In Figs. 6–8 streamlines of the time averaged base ﬂow (no control) are shown in the wake of the body. The ﬂow has been visualized using the LIC (Line Integral Convolution) technique in the three different planes indicated in Fig. 4. At the center plane (Fig. 6), the separation and re-attachment of the ﬂow on the slant can be seen clearly in both simulation and experiment. By visual comparison the sizes of the separated region agrees fairly well, even though PIV measurements near solid surfaces are somewhat difﬁcult to make. In both cases the ﬂow appears to re-attach on the slant at a distance of about 70–80% of the slant length, measured from the upper edge of the body. Looking at the ﬂow behind the rear plane, two major vortical structures can be observed, a larger upper one, and a smaller lower one. In the LES, the lower vortex seems to consist of two to three even smaller vortical structures. While the overall topology of the ﬂow ﬁeld agrees well between the LES and the PIV measurements, there are differences in the sizes and locations of the vortices. The vortex center of the upper vortex has a distance of ∆xvc,sim = 0.039L from the rear plane, whereas in the experiment this distance is larger, ∆xvc,exp = 0.047L. However, the vertical distance of the vortex center from the underside of the body is identical at ∆yvc,sim = yvc,exp = 0.099L. The most obvious difference between the simulation and the experiment is the size and location of the lower vortex, which is much larger in the experiment. This also results in a different location of the free rear stagnation point. The position of the stagnation point is ∆xsp,sim = 0.122L and ∆ysp,sim = 0.016L in the LES, and ∆xsp,exp = 0.184L and ∆ysp,exp = 0.041L in the experiment. As mentioned above, this different behavior is to be expected, since in the wind tunnel there are stilts and tubes underneath the car model, which cause a higher turbulence intensity in the ﬂow in this area. Looking at an intermediate plane between the center and the side of the model (Fig. 7), the three-dimensionality of the wake becomes obvious. The separated region on the slant is smaller in the LES (compared to the center plane), and in the experimental results it is virtually not visible anymore, which can most probably be attributed to uncertainties in the PIV measurements. Comparing the ﬂow ﬁeld behind the model, again there are clear differences between simulation and measurement, but the overall behavior in comparison to the center plane agrees very well in both cases. The upper vortex decreases in size, its center moves downstream to about twice the distance from the rear plane, while staying almost at the same height. The locations are ∆xvc,sim = 0.068L, ∆yvc,sim = 0.105L and ∆xvc,exp = 0.099L, ∆yvc,exp = 0.109L, respectively. Along with the changes of vortex sizes and positions, also the stagnation point can be observed at a different location. In the LES the new location is ∆xsp,sim = 0.153L,

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Fig. 6. Base ﬂow streamlines at the symmetry plane (ϕ = 25◦ ); left: LES, right: experiment

Fig. 7. Base ﬂow streamlines at an intermediate plane (ϕ = 25◦ ); left: LES, right: experiment

Fig. 8. Base ﬂow streamlines at the side plane (ϕ = 25◦ ); left: LES, right: experiment

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∆ysp,sim = 0.043L, and in the wind tunnel experiment it is found at ∆xsp,exp = 0.175L, ∆ysp,exp = 0.058L. The ﬂow at the side plane (Fig. 8) exhibits no visible inﬂuence of the wake behind the body. Only a deﬂection of the ﬂow can be observed above the slant. This effect is caused by the strong vortex along the slant edge, which is typical for slant angles below 30◦ . Again, the qualitative agreement between the numerical simulation and the PIV measurements is fairly good. Forced ﬂow Longitudinal vortices are driven by the low pressure area over the slanted surface. To reduce the intensity of these structures two approaches can be imagined. The ﬁrst is to increase the static pressure over the slanted surface’s recirculation area. However, while this would reduce the strength of the streamwise vortices, it also increases the aerodynamic drag by increasing the size of the separation region. In this case the wake structure would be more similar to the ﬂow around the Ahmed body with a slant angle of ϕ = 35◦ . On the other hand, actuators in the core of the developing side edge vortices could modify their intensity. Hence, the idea of the present numerical and experimental investigations was the reduction of the aerodynamic drag by means of blowing a steady jet into the center of the longitudinal vortex. The balance measurements, depicted in Fig. 9 in form of cµ -sweep, show a minor sensitivity of the vortex structure to steady blowing. A drag reduction of 2.5% was achieved only for intensities of 0.0015 ≤ cµ ≤ 0.003, for higher intensities of cµ > 0.003 the drag coefﬁcient is increased. This makes clear that the inﬂuence of the steady blowing on the overall drag is relatively small, even though a considerable inﬂuence of the forcing on the longitudinal vortices could be observed. In Fig. 10 it can be seen that there is indeed a measurable effect of the blowing on the vortex near the slant edge. In this ﬁgure, velocity vectors

Fig. 9. Drag force measurements in the low speed wind tunnel for different blowing intensities (ϕ = 25◦ )

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b

Fig. 10. Base ﬂow (left) and forced ﬂow (right) in the near wake of an Ahmed body (experiment, ϕ = 25◦ )

Fig. 11. Vortices (iso-surface λ2 = −3000) of the time-averaged ﬂow ﬁeld; left: base ﬂow, right: forced ﬂow (LES, ϕ = 25◦ )

obtained from the PIV measurements are shown at the center plane and the side plane for the base ﬂow (left) and one case of forced ﬂow (right). For the latter, the ﬂow perturbation at the side plane near the upper corner is clearly visible. Further downstream it appears that the deﬂection of the ﬂow is less strong compared to the unforced case, indicating that the longitudinal vortex has been weakened by the perturbation. Looking at the center plane, however, one can observe that for the ﬂow control case the separation region on the slant is more pronounced than for the base ﬂow. This result conﬁrms earlier ﬁndings [2, 10, 11] that there is a strong coupling between the side vortices and the separated ﬂow on the slant. If one of these two structures gets weaker, the other one gets stronger and vice versa. This ﬁnding is also supported by the LES results. In the simulations, constant blowing was applied through two short slits along the upper rear edge of the model next to

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Fig. 12. Time-averaged pressure distribution on the rear of the Ahmed body; left: base ﬂow, right: forced ﬂow (LES, ϕ = 25◦ )

the corners. The length of each slit was 0.019L, the width was b = 0.0028L, and the total momentum coefﬁcient was cµ = 0.0154. The direction of blowing had an angle of 45◦ with respect to the roof without any yaw angle. Figure 11 shows visualizations of vortex structures of the time averaged ﬂow ﬁelds using the λ2 technique [5]. For the base ﬂow (left), the strong longitudinal vortex along the slant edge can be seen. In the case of constant blowing (right) this structure has almost completely disappeared. It is important to note, that this does not imply that the vortex itself has disappeared. But the fact that it is not visible anymore at this iso-level of λ2 indicates, that it has at least weakened as a result of the blowing. Similar to the experiments, however, the weakening of the vortex had no noticeable effect on the drag of the Ahmed body. The largest contribution to the total drag of the model comes from the pressure drag, i.e. the difference between the pressure on the forebody and the pressure on the rear. u2∞ In Fig. 12 the simulated distribution of the pressure coefﬁcient cp = 2(p − p∞ )/ρ¯ on the rear of the body is shown. It can be seen that there are two distinct regions on the slant where the pressure is particularly low: The area beneath the side edge vortex and the area beneath the separated ﬂow downstream of the upper rear edge. For the blowing case (right), the “footprint” of the longitudinal vortex is somewhat wider, and also the pressure increases slightly faster in the downstream direction within this area. This is consistent with the above results showing that the vortical structure has become weaker, thus having a smaller contribution to the pressure drag. On the other hand, there is a more extended region of very low pressure immediately downstream of the blowing slit, and also the overall pressure in the separated region towards the central part of the slant is lower. So while the drag contribution of the edge vortex decreases, at the same time the contribution of other regions increases. This fact emphasizes once more that

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there is a strong coupling between the different structures and areas in the wake of the Ahmed body, and changing one of the structures by means of ﬂow control in many cases has an adverse effect on the other structures.

4 Conclusions Experimental and numerical investigations on active separation control were performed to reduce the total drag of a generic car model (Ahmed body) for two different slant conﬁgurations. For a slant angle of 35◦ , actuators generating periodic perturbations were used to excite the shear layer separating from the rear slant of the car model, where spanwise vortex structures are dominant in the near wake. This control approach achieved a reduction of the reverse ﬂow region for a wide forcing frequency range of 0.1 ≤ StH ≤ 0.9. At the same time, however, the development of streamwise vortices generated at the slant side edges was observed, a ﬂow feature that is usually found at slant angles below 30◦ . For the slant angle of 25◦ , constant blowing near the slant corners was investigated in order to control the longitudinal vortices originating at these corners. It could be shown experimentally as well as numerically that the vortices can be weakened by this approach. Despite this fact, the overall change in total drag was very small, because a weakening of the slant edge vortices went along with an increase of the separated ﬂow region over the slant, compensating the possible positive effect on the drag. As mentioned above, this study is still in progress. In order to achieve an actual reduction in drag it appears to be necessary to apply a combined ﬂow control approach, inﬂuencing both the longitudinal vortices and the separation over the slant.

Acknowledgments This research was funded by the German Science Foundation (DFG) within the scope of the Collaborative Research Center SFB 557. The simulations were performed on the IBM pSeries 690 supercomputer at the North German Cooperation for HighPerformance Computing (HLRN). This support is gratefully acknowledged by the authors. The numerical grid is based on a geometry that has been provided by Sinisa Krajnovic, Chalmers University, Gothenburg, Sweden.

References [1] A HMED S.R., R AMM R. & FALTIN G. (1984). Some Salient Features of the TimeAveraged Ground Vehicle Wake. SAE-Techn. Paper Series, 840300 [2] B RUNN A. & N ITSCHE W. (2005). Active Control of Turbulent Separated Flows by Means of Large Scale Vortex Excitation. In: RODI W. and M ULAS M. (Eds.) Engineering Turbulence Modelling and Experiments 6, Elsevier Science Ltd., 555–564 [3] G ILLIERON P. & C HOMETON F. (1999). Modelling of stationary three-dimensional separated air ﬂows around an Ahmed reference model. In: 3rd Int Workshop on Vortex, ESAIM 7, 173–182

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[4] H UCHO W.-H (2002). Aerodynamik der stumpfen K¨orper - Physikalische Grundlagen und Anwendung in der Praxis, Vieweg-Verlag [5] J EONG J. & H USSAIN F. (1995). On the identiﬁcation of a vortex. J. Fluid Mech. 285, 69–94 [6] L EDER A (1992). Abgel¨oste Str¨omungen - Physikalische Grundlagen, Vieweg-Verlag [7] L IENHART H., S TOOTS C. & B ECKER S. (2002). Flow and Turbulence Structures in the Wake of a Simpliﬁed Car Model (Ahmed model). In: Notes on Numerical Fluid Mechanics 77, Springer, 323–330 [8] M OREL T. (1978). The Effect of Base Slant Angle on the Flow Pattern and Drag of ThreeDimensional Bodies with Blunt Ends. In: Proc. of Symp. Aerod. Drag Mechanisms of Bluff Bodies and Road Vehicles. Plenum Press, New York, 191–226 [9] S MAGORINSKY J. (1963). General circulation experiments with the primitive equations. Monthly Weather Review 91, 99–164 [10] G UILMINEAU E. & D UVIGNEAU R. (2005). Drag reduction by ﬂow control for the Ahmed body. In: Proc. of 4th Symposium on Bluff Body Wakes and Vortex–Induced Vibrations, Santorini, Greece, 21–24 June 2005, 247–250 [11] ROUMEAS M. & G ILLIERON P. & KOURTA A. (2005). Analyze and control of the near– wake ﬂow around a simpliﬁed car geometry. In: Proc. of 4th Symposium on Bluff Body Wakes and Vortex–Induced Vibrations, Santorini, Greece, 21–24 June 2005, 251–254

Continuous Mode Interpolation for Control-Oriented Models of Fluid Flow Marek Morzy´nski1,2, Witold Stankiewicz2 , Bernd R. Noack1 , Rudibert King3 , Frank Thiele1 , and Gilead Tadmor4 1

Institute of Fluid Dynamics and Technical Acoustics, Berlin University of Technology, Straße des 17. Juni 135, D-10623 Berlin, Germany [email protected], [email protected] 2 Institute of Combustion Engines and Transportation, Poznan University of Technology, Piotrowo 3, 60-965 Poznan, Poland [email protected], [email protected] 3 Measurement and Control Group, Berlin University of Technology, Hardenbergstraße 36a, D-10623 Berlin, Germany [email protected] 4 Department of Electrical and Computer Engineering, Northeastern University, 440 Dana Research Building, Boston, MA 02115, USA [email protected]

Summary In the current study, a hierarchy of control-oriented Galerkin models is proposed targeting least-dimensional representations at different operating conditions. These models are employed for passive as well as active actuation. In passive control, a linearised model is shown to reproduce a wake stabilization experiment of Strykowski & Sreenivasan (1990). In particular, the effect and optimal position of control wires are accurately predicted. In active closed-loop control, focus is also placed on experiment. Here, control design requires a model which has on the hand a sufﬁciently broad dynamic range and is on the other hand low-dimensional enough for online computation. POD Galerkin models have a desirable mathematical structure and dimension for an online capable control design but tend to be over-optimised for the reference conditions. The resulting limited dynamic bandwidth is associated with the underlying expansion modes which change their shape at different operating conditions. To increase that model bandwidth, a novel continuous mode interpolation technique is proposed. The mode interpolation smoothly connects not only different operating conditions, but also stability and POD modes and even ﬂows at different boundary conditions. In addition, the extrapolation of modes outside the design conditions is illustrated. The interpolated modes enable ’least-order’ Galerkin models keeping the dimension from a single operating condition but resolving several states. These models are well suited for control design. The mode interpolation technique is demonstrated for three benchmark problems, the ﬂow around circular cylinder, a NACA airfoil and an Ahmed body. R. King (Ed.): Active Flow Control, NNFM 95, pp. 260–278, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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1 Introduction Flow control is a key technology to improve the performance of transport systems, like cars, trains and airplanes. Small actuators are employed to change the ﬂow and transcend the limitations of aerodynamic design [1][2]. Typically, the actuation may be operated in i) passive, ii) active open-loop and iii) active closed-loop mode. The latter variant may be considered as the most general type of control. The effect of a passive device may be mimicked by an active actuation, like blowing or suction. An open-loop actuation, i.e. blindness to ﬂow changes, is the trivial form of the closedloop operation. The potential of closed-loop ﬂow control is increasingly realised and exploited [3]. The design of actuators opens myriads of possibilities including the choice of the type of actuator, its location, and its amplitude and frequency range. Engineering wisdom based on a good understanding of this ﬂow can guide a number of these choices. That understanding includes how the actuator changes coherent structures and their dynamics in a desirable manner, i.e. riblets suppress streamwise vortices and splitter plates in wakes delay vortex shedding. Eventually, the actuator shall demonstrate its anticipated effect in experiment. In wind- or water-tunnel, however, a parametric study of actuator and sensor types and locations may become expensive and beneﬁt from optimisation with computational ﬂuid mechanical models. However, high-ﬁdelity simulations may be equally limited due to the associated large computational costs. Hence, reduced models may be useful for less expensive exploration of control opportunities. In addition, closed-loop control in experiment requires robust and online-capable feedback laws. Robustness and simplicity imply sufﬁciently low-dimensional models, which are on the one hand accurate enough to resolve the most relevant ﬂow structures but have on the other hand no ”superﬂuous” degree of freedom. A frequent observation is that observers and controllers derived from higher-ﬁdelity models are less robust. Each new phasespace direction for a secondary ﬂow effect may be a direction in which the observed state gets ’lost’ or the controller transcends the ROM’s region of validity [4][5]. In analogy [6], it is much easier to look for and play with an animal in a foggy English forest if it lives on the ground (2 dimensions) than if it can also climb up trees (3 dimensions). In the current study, we pursue reduced models which predict the effect of actuation on one or few coherent structures. The manuscript is organised as follows. First (§2), the model hierarchy is outlined. This hierarchy includes global stability analysis describing the linearised dynamics of the most unstable eigenmodes as well as POD Galerkin models. The dimension of the hybrid models with stability and POD eigenmodes [7] are further reduced by a novel continuous interpolation technique for the modes. In §3, the optimisation of passive control with linear models is demonstrated. In §4 the mode interpolation is employed for the cylinder wake. In §5, a predictive effect of mode interpolation is demonstrated for wakes behind a NACA wing and an Ahmed body. The proposed interpolated Galerkin models can be employed for ﬂow control design as shown in Ref. [8]. Finally (§6), the main results are summarised.

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2 Hierarchy of Control-Oriented Models This section outlines a hierarchy of models for ﬂow control design. First (§2.1), reducedorder models (ROM) based on POD modes are discussed. The need for a mode interpolation is elucidated. In §2.2, linearised Navier-Stokes models are described. In analogy to POD-based ROM, these linear models are interpolated at different operating conditions and complemented with an evolution equation for the interpolation parameter. Finally (§2.3), a technique is proposed for continuous mode interpolation between different operating conditions — even between different modes and different conﬁgurations. The proposed modelling elements generalise mean-ﬁeld [9][10], POD [11], and hybrid models [7]. Results will be presented in the following sections (§3, §4, §5). 2.1 Reduced-Order Model Based on POD Modes In the current study, we consider two-dimensional incompressible ﬂows around stationary cylinders with size D placed in a uniform stream with velocity U . Thus, the ﬂow can be deﬁned in a stationary domain Ω with time-independent boundary conditions. The velocity and pressure ﬁeld are denoted by u and p and depend on location x ∈ Ω and time t. The 2D region is described with cartesian coordinates x = (x, y), such that the origin is centred in the obstacle and the x-axis points in the direction of the ﬂow. In the following, all variables are assumed to be non-dimensionalised with velocity U and length D. The ﬂow is characterised by the Reynolds number Re = U D/ν, ν being the kinematic viscosity of the ﬂuid. As actuators, Nb local volume forces with amplitudes bl and carrier ﬁelds gl (x), γ = 1, . . . , Nb , are employed. A volume force may represent a Lorentz force, a bouyancy term, a control cylinder (penalty method), or mimic the effect of a local acoustic actuator [5]. Thus, the Navier-Stokes equation reads b 1 u + bl (t) gl (x). Re

N

∂t u + ∇ · (u ⊗ u) = −∇p +

(1)

l=1

Following classical books of traditional Galerkin methods [12][13][11], the velocity ﬁeld is decomposed in a base ﬂow u0 and a ﬂuctuation u . The ﬂuctuation is approximated in the Hilbert space L2 (Ω) of square-integrable solenoidal vector ﬁelds. In this space, v is deﬁned by the inner product between two velocity ﬁelds u and (u, v)Ω := Ω dx u · v. The associated norm is given by u Ω := u, u)Ω . In any ∞ Hilbert space, there exist an complete orthonormal basis {ui }i=1 . This basis allows for a Galerkin approximation of a velocity ﬁeld u(x, t), u[0..N ] (x, t) = u0 (x) +

N

ai (t) ui (x),

(2)

i=1

where ai = (u − u0 , ui )Ω . The approximation converges against the ﬂow in L2 sense as N → ∞. Furthermore, Eq. (2) satisﬁes the incompressibility condition and the boundary condition for arbitrary Fourier coefﬁcients ai . This property is the prerequisite for a Galerkin projection onto the Navier-Stokes equation.

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Targeting least-dimensional representations, we sacriﬁce completeness of the modes for reduced dimension by employing a POD reduction of the ﬂow data. Following [8][14] small deformations of a ﬂow structure during transients shall be incorporated in interpolated modes uκi where the change of operating condition is parametrised by κ. This gives rise to a generalised Galerkin approximation

uκ (x, t) =

N

aκi (t) uκi (x),

(3)

i=0

where aκ0 ≡ 1 for reasons of simplicity. The superscript κ refers to the operating condition which this expansion shall approximate. Traditional Galerkin models do not include the mode adjustments for time-varying operating condition. However, modes varying with stationary operating conditions have been utilised before [15][16]. The resulting low-dimensional Galerkin model can be derived from (1) and (3) in a straightforward manner [17] assuming a slowly varying κ, Nb N N 1 κ κ d κ κ ai = lij aj + qijk aκj aκk + gilκ bl , dt Re j=0 j,k=0

(4a)

l=1

d κ = F (κ, aκ , b). dt

(4b)

The Fourier coefﬁcients and control amplitudes are summarised in vectors aκ := (aκ1 , aκ2 , . . . , aκN ) and b := (b1 , b2 , . . . , bNb ), respectively. The Galerkin system coκ κ efﬁcients lij and qijk depend on the operating condition κ. During a natural transient of an oscillatory ﬂow, uκ0 may be the phase-averaged velocity and κ may be identiﬁed with the shift-mode amplitude [7]. Moreover, (4b) is derived from the Navier-Stokes equation ﬁltered with the phase average and projected onto the mean-ﬁeld correction. During an actuated transient, κ may be identiﬁed with a forcing amplitude or frequency. In this case, (4b) is replaced by a prescribed dynamics of κ. In a simple case, the Galerkin system coefﬁcients may be linearly interpolated between two close operating conditions identiﬁed by κ = 0 and κ = 1: κ 0 1 0 = lij + κ (lij − lij ), lij

κ 0 1 0 qijk = qijk + κ (qijk − qijk ).

(5)

The accompanying interpolation of the modes is discussed in §2.3. ROM of the form (4) are easily accessible to nonlinear control design. For the cylinder wake benchmark, a large class of design strategy was found to have comparable performance [18]. The performance may, however, deteriorate in the high-ﬁdelity model, when model-based control does not respect the range of validity of the ROM [19][20].

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2.2 Reduced-Order Model Based on the Linearised Navier-Stokes Equation In the preceeding section, the modes need to be empirically derived from ﬂow data of the transient between two operating conditions. Close to the steady Navier-Stokes solution, the modes may be derived from global stability analysis, i.e. from ﬁrst principles. The velocity of the ﬂow u is decomposed into the steady solution us and a disturbance u : u = us + u . (6) Linearising (1) around the steady solution as the target state leads to b 1 u + bl (t) gl (x). Re

N

∂t u + ∇ · (us ⊗ u + u ⊗ us ) = −∇p +

(7)

l=1

Any spatial discretisation of (7) yields a ﬁnite-dimensional evolution equation of the form d a = A a + B b, (8) dt where a is a vector of state variables, for instance, the velocity components at the grid nodes. In the current study, the ﬁnite-element method (FEM) is chosen as discretisation technique. In principle, a traditional Galerkin method with mathematical modes [21][15] or Stokes modes [22][23] may be used as well. By construction, (8) is an accurate evolution equation for inﬁnitesimal perturbations. However, ﬁnite perturbations and nonlinear effects introduce errors away from the target state. One nonlinear effect is amplitude saturation, which such a linear model cannot resolve in principle. However, a number of studies suggest that a local linearisation around the base ﬂow may be suitable for the described natural transients as well as for control design [24]. Variants of the mean-ﬁeld model [7] suggest that a local linearisation of the ﬂow can be employed to describe frequency and amplitude dynamics. In the sequel, we assume that the unstable steady state us is associated with κ = 0 and the mean ﬂow u0 of the stable limit cycle with κ = 1. The interpolation parameter κ may be a normalised shift-mode amplitude. The locally linearised pendant of (4) reads d a = Aκ a + Bκ b dt d κ = F (κ, a, b). dt

(9a) (9b)

It may be worthwhile to note that the FEM discretised ﬂow a is independent of κ. The function F is derived like in §2.1. Variants of (9b) have been investigated by [25] and [26]. The κ dependence of (9a) may be approximated by a pendent of (5), Aκ = A0 + κ A1 − A0 , Bκ = B0 + κ B1 − B0 , (10) where the superscript ’0’ refers to the steady solution and ’1’ to the averaged ﬂow. POD model variants of the form (9,10) have been exploited for wake stabilisation studies [4][27]. Here, κ was identiﬁed with the shift-mode amplitude.

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The solution of the linear FEM equation is much faster than the full nonlinear system. Moreover, the stability eigenmodes can be utilised as basis for a ROM and for control design. Control design of linear systems is a well-established discipline. A proportional controller b = K a , for instance, gives rise to the forced dynamics d a = Ac a, dt

(11)

with the modiﬁed linear term Ac = A + B K. Here, the design parameter K may be used to stabilise or destabilise the system. Besides volume forces, also models with wall-mounted actuators can be written in the form (8) [28]. In the penalty method of CFD, obstacles are modelled by time-dependent volume forces which lead to vanishing velocity in the obstacle. This corresponds to a local actuation term Bb and a strongly stabilising controller b = K a. Thus, the actuation term of (11) may model passive control with control wires, splitter plates, riblets or other devices. Here, Ac describes the effect of the device and optimisation of a passive device may be guided by global stability analysis. An example is provided in §3. 2.3 Continuous Mode Interpolation In this section, an interpolation technique for a reduced-basis representation of ﬂows at different operating conditions is proposed. High-ﬁdelity ﬂow models, like a FEM discretisation of the Navier-Stokes equation, are designed to describe a rich kaleidoscope of ﬂow states with large number of local constant expansion modes. The goal of lowdimensional modelling is to resolve the key coherent structures with a small number of modes. In contrast to FEM expansions, these modes need to be ’ﬂexible’ and change at different operating conditions [8][14]. This mode change shall be modelled. Interpolation problem. In order to simplify the discussion, we consider periodic cylinder wakes at two operating conditions κ = 0, 1 which are approximated by uκ (x, t) = aκ1 (t) uκ1 (x) + aκ2 (t) uκ2 (x),

κ = 0, 1.

(12)

The resulting vortex streets look similar but may have slightly different wavenumbers, frequencies, and ﬂuctuation envelopes. The states may, for instance, be near the onset of vortex shedding (κ = 0) and the asymptotic state (κ = 1) [7][14]. Other pairs of states are the natural and forced wake [8], or the natural ﬂow at two different Reynolds number [29]. Deane et al. [29] demonstrate that intermediate ﬂow states are inadequately represented by POD bases due to small wavenumber and other changes. This difﬁculty may be elucidated with two travelling waves at κ = 0, 1 having slightly different frequencies and wavenumbers: κ uκ (x, t) = cos 1 + (x − t) = aκ1 (t) uκ1 (x) + aκ2 (t) uκ2 (x), κ = 0, 1. (13) 5

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Exact representations are achieved for κ 1+ t , 5 κ t , aκ2 = sin 1 + 5

aκ1 = cos

κ uκ1 = cos 1 + x , 5 κ uκ2 = sin 1 + x . 5

(14a) (14b)

Obviously, an intermediate travelling wave of the same family with wavenumber and circular frequency 11/10 (κ = 0.5) cannot be obtained by averaging (13) at κ = 0, 1: u0 (x, t) + u1 (x, t) 1 11 = cos (x − t) cos (x − t) . 2 10 10

(15)

Averaging leads to a beat phenomenon in the amplitude, both in the solution as well as in the modes. This beat phenomenon is evidenced in any additive interpolation procedure, like the POD method based on the union of two snapshot ensembles [30], the union of two POD bases [31], and sequential POD [32].1 In the following, we propose interpolation techniques which will yield modes with intermediate wavenumbers and frequencies. Continuous interpolation between stability eigenmode bases. One interpolation technique can be motivated by our 1D travelling wave example. The intermediate modes uκ1,2 are eigenmodes of the interpolated eigenproblem associated with the travelling wave equation ∂t u + (1 + κ/5)∂x u = 0. Similarly, the most ampliﬁed stability eigenmodes of linearised Navier-Stokes equation at two base ﬂows can be interpolated. We assume one dominant oscillatory eigenmode, like for the ﬂow around a bluff cylinder [34]. Then, the associated two sets of modes constitute complex eigenmodes f κ = uκ1 + ıuκ2 of the eigenvalue problem Aκ f κ = λκ f κ ,

κ = 0, 1,

(16)

where λκ are the associated eigenvalues. Here, Aκ may be the linearised Navier-Stokes operator or the FEM discretisation as stability matrix. The stability matrix may be linearly interpolated between κ = 0, 1, Aκ = A0 + κ (A1 − A0 ),

(17)

thus giving rise the intermediate eigenproblems of the form (16) and intermediate complex eigenmodes f κ = uκ1 + ıuκ2 . A smooth connection between both eigenmodes is not guaranteed per se [35] but a common observation for the most unstable stability modes of shear ﬂows. 1

Additive interpolation procedures may, however, be useful for distinct processes, e.g. 2D and 3D shedding states, von K´arm´an vortex shedding and the shear-layer vortices. POD from natural and forced transient ﬂows [33] may resolve intermediate states at the large price of arriving at a number of modes which is one order of magnitude larger than for a single state.

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Continuous interpolation between POD bases. POD bases are interpolated in analogy to stability modes by referring to the eigenproblem in space domain. The i-th POD mode satisﬁes the Fredholm eigenproblem dy R(x, y) ui (y) = λi ui (x) (18) Ω

with the autocorrelation function R(x, y) = u(x, t) ⊗ u(y, t) =

∞

λi ui (x) ⊗ ui (y).

i=1

The brackets denote an ensemble average. The eigenproblem (18) is cleaned from higher-order modes by introducing a modiﬁed autocorrelation function: Rκ (x, y) := uκ1 (x) ⊗ uκ1 (y) + uκ2 (x) ⊗ uκ2 (y). With that Kernel, the modiﬁed Fredholm eigenproblem (18) dy Rκ (x, y) uκi (y) = λκi uκi (x)

(19)

(20)

Ω

has only two non-trivial solutions uκ1 , uκ2 with eigenvalues of unity, λκ1,2 = 1, at both operating conditions. The Fredholm kernel may be interpolated in κ ∈ [0, 1] like in (17), (21) Rκ (x, y) = R0 (x, y) + κ R1 (x, y) − R0 (x, y) . The interpolated Fredholm eigenproblem will give rise to the original modes as four eigensolutions if all four modes {uκi : κ = 0, 1, i = 1, 2}, are orthogonal to each other. In this case, no continuous interpolation is possible. A smooth interpolation of the modes may be possible, if there is sufﬁcient overlap between the ﬁrst and second mode set, i.e. if the mode pairs are not mutually orthogonal to each other. As a rule of thumb, at least 50% of energy of the ﬁrst mode set should be resolved by the second set and vice versa. The results shown in the following are based on this latter scenario. Continuous interpolation between POD and stability eigenmode basis. The FEM discretisations of the stability and POD eigenproblem on the same grid lead to matrices of same dimension. Hence, both eigenproblems could be interpolated in the spirit of (17), where one matrix represents the linearised Navier-Stokes equation and the other matrix the discretised Fredholm kernel. The interpolation between the symmetric Fredholm kernel with real eigenvalues and non-normal stability matrix with complex eigenvalues may give rise to numerous complications. Hence, we propose to orthonormalise the eigenmodes and treat them like POD modes at one operating condition in (19, 20, 21). Foreshadowing the results described below, Fig. 1 elucidates how stability, POD, and the shift mode are connected in the phase space. Generalisations of the proposed interpolation technique to higher-dimensional bases are straightforward.

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Fig. 1. Principal sketch of the wake dynamics connecting the stability analysis of the unstable steady solution and the POD approximation of the stable limit cycle. The left side displays the mean ﬂow (top), shift-mode (middle) and steady solution (bottom). The right side illustrates interpolated vortex streets on the mean-ﬁeld paraboloid (middle column). The planes are spanned by a pair of stability (bottom) and POD modes (top). The curves illustrate a characteristic trajectory from a Galerkin projected dynamics onto this plane.

3

Wake Stabilisation Using Model-Based Passive Control

Passive suppression of vortex shedding may be achieved with splitter plates [36,37] or small control cylinders [38]. In this case, the Ac matrix of (11) is computed in terms of parameters of passive actuators. The no-slip conditions on the surface of a control cylinder or a splitter plate can be imposed with penalisation of the surface velocities with time- and space-dependent volume forces as described in §2.2. Thus, we can study how the geometry of passive actuators is related to the ﬂow stability varying the parameters of the actuator (like position or size). In this way, the linear model can be employed to optimise passive control. This approach is demonstrated for the Strykowski [38] control cylinder. The critical Reynolds number is determined at different locations of the control cylinder with global nonparallel ﬂow stability analysis (Fig. 2). The

(a)

(b)

Fig. 2. Real (a) and imaginary (b) part of the most unstable eigenmode of the passively manipulated cylinder wake at Re = 100. The control cylinder (d/D = 0.1) is placed at x = 1.2, y = 1.2. The cylinder is indicated by the solid circle and the ﬂow ﬁeld is visualised with streamlines. Thick (thin) lines correspond to positive (negative) stream-function values.

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(a) One control cylinder

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(b) Two control cylinders

Fig. 3. Optimisation of passive control with one and two Strykowski wires employing global ﬂow stability analysis. The dots and numbers in the ﬁeld denote the control cylinder position and respective critical Reynolds numbers. The second control cylinder in subﬁgure (b) is placed symmetric with respect to the x-axis.

eigenanalysis (see Fig. 3) yields the optimal position for one and two symmetrically placed control cylinders. Isolines in that ﬁgure denote the critical Reynolds number values found experimentally in [38], small circles denote analysed positions of the control cylinder, and the numbers represent critical values obtained by global stability analysis. The critical Reynolds number is the largest in a narrow region behind the cylinder. Global stability analysis predicts also that any non-symmetrical conﬁguration is less effective for stabilisation than the optimal symmetrical one. The passive control optimisation with Strykowski wires depicted in Fig. 3 shows good agreement with experiment. Below the critical Reynolds number, the steady solution is at least locally stable. The agreement of the largest achievable critical Reynolds number in experiment and in the current study suggests the domain of attraction of the steady solution contains at least the periodic shedding state and may contain all ﬂows.

4 Continuous Mode Interpolation Between Different Wake States In this section, the interpolation techniques presented in §2.3 are demonstrated for stability eigenmodes and POD modes of the ﬂow around circular cylinder (§4.1, §4.2, §4.3). Thereafter, more complex problems like extrapolation of modes outside the known interval (§4.4) and the approximation of POD modes with physical modes (§4.5) are addressed. 4.1 Interpolation Between Stability Eigenmode Bases The stability eigenproblem for ﬂow states at two different Reynolds numbers is interpolated with (17) from §2.3. The interpolation parameter κ is linearly related to Re, Re = 50 + κ 50,

κ = 0, 1.

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(a) Re = 50

(b) Re = 75, computed

(c) Re = 100

(d) Re = 75, interpolated

Fig. 4. Interpolation between eigenmodes at different Re numbers. See text for explanation.

The procedure is demonstrated in Fig. 4. The eigenmode at Re = 75 is computed with global stability analysis at κ = 0.5 interpolating between the most unstable eigenmodes at Re = 50 and Re = 100 (see Fig. 4, left). The ﬂows at the right-hand side of that ﬁgure shows a good agreement between computed and interpolated modes. 4.2 Interpolation Between POD Bases In §2.3, the method of continuous mode interpolation between POD bases is deﬁned by (21). Interpolated modes are obtained by solving the associated eigenvalue problem (see Fig. 5). The POD modes at Re = 75 (right side of that ﬁgure) are computed from the snapshots of unsteady ﬂow simulation and interpolated (κ = 0.50) from the most energetic ones at Re = 50 and Re = 100.

(a) Re = 50

(b) Re = 75, computed

(c) Re = 100

(d) Re = 75, interpolated

Fig. 5. Interpolation between POD modes at different Re numbers

4.3 Interpolation Between POD and Stability Eigenmode Basis We interpolate between POD and stability eigenmode bases following the algorithm described in §2.3. An example of such an interpolation between the ﬁrst POD mode and

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(a) POD mode, κ = 0

(b) κ = 0.25

(c) κ = 0.50

(d) κ = 0.75

(e) Eigenmode, κ = 1 Fig. 6. Interpolation between POD modes and eigenmodes at Re = 100. Left) streamlines; right) ﬂuctuation energy q = 12 u2 along the x-axis.

the most unstable stability mode for the wake behind a circular cylinder at Re = 100 is shown in Fig. 6. On the right-hand side of the ﬁgure, the energy q = 12 u 2 on the centreline is depicted. The modes change smoothly from one operating state to another. Thereby, the maximum of ﬂuctuation level moves to the outﬂow. A similar observation has been made by Lehmann et al. [39] for the ﬂow subjected to increasingly aggressive stabilising control.

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4.4 Extrapolation of POD Bases In previous sections, the construction of an interpolated stability eigenmode (Fig. 4) is investigated. The interpolation Reynolds number is midway between reference values of that example. Fig. 5 has illustrated an analogous interpolation between POD modes. The idea of continuous mode interpolation between the modes of the same kind has been further extended to interpolation between physical and empirical ones shown in Fig. 6. In all these computations, the interpolated modes and the ones determined directly from eigenanalysis or POD method agree well. This raises the question if it is possible to extrapolate the operating conditions by chosing κ > 1 or κ < 0. Thus, one can determine the approximation of the ﬂow ﬁeld at conditions outside the investigated range without new computations. This attractive feature of the approach is tested as follows. The POD modes at Re = 200 and Re = 150 were applied to predict the conditions at a lower Reynolds number Re = 100 (see Fig. 7). The righthand side of the ﬁgure shows almost indistinguishable stream-function ﬁelds for the computed POD mode and the extrapolated one. Also the ﬂuctuation energy plot along x-axis of the domain (see Fig. 8) shows negligible differences between the computed and interpolated distributions.

(a) Re = 200

(b) Re = 100, computed

(c) Re = 150

(d) Re = 100, extrapolated

Fig. 7. Extrapolation of POD mode at Re = 100 from POD modes both at higher Reynolds numbers Re = 150 and 200

Fig. 8. Same as Fig.7(b,d), but illustrating the ﬂuctuation distribution q = x-axis for the computed (left) and extrapolated (right) POD mode

1 2

u2 along the

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4.5 Approximation of a POD Basis Using Stability Eigenmodes The minimal control-oriented Galerkin model consists of two POD modes representing vortex shedding and the shift mode describing the mean-ﬁeld direction [4]. In this section, POD and shift modes are both approximated from ﬁrst principles. Thus, Galerkin models of oscillatory ﬂow processes may be constructed without the need for empirical ﬂow data. The empirical POD mode is approximated by extrapolating suitable stability eigenmodes. Starting points are eigenmodes computed from a global stability analysis of the steady solution and of time-averaged ﬂow. The POD mode shall be expressed in terms of these physical modes (stability eigenmodes) only. The real parts of the most unstable eigenvectors at Re = 100 are depicted in left part of Fig. 9. With these stability modes the POD-mode approximation is constructed. The result is shown in right bottom part of Fig. 9 and compared with the POD mode at Re = 100 computed from the snapshots of unsteady ﬂow simulation. In the near wake, the extrapolated ﬁeld is in very good agreement with the POD computation. As the near wake is particularly important for ﬂow control, the results of the a priori mode are encouraging. The shift-mode is derived from the Reynolds equation. Figure 10 shows the empirical and an a priori shift mode in virtually good agreement. It may be worthwhile to note

(a) Eigenmode of steady solution

(b) POD mode, computed

(c) Eigenmode of mean ﬂow

(d) POD mode, approximated (κ = 1.44)

Fig. 9. Approximation of POD mode extrapolating from two eigenmodes at Re = 100. For details see text.

(a)

(b)

Fig. 10. Empirical (left) and a priori (right) shift mode. The empirical mode is computed from the mean ﬂow and steady solution as described in [7]. The a priori shift mode is derived from the Reynolds equation using the POD representation of the Reynolds stress tensor.

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that higher-order modes can be derived from the non-linear Navier-Stokes term and utilizing harmonic expansions [25], considering each POD mode pair as one harmonic. Hence, reﬁned a priori models are in reach, too.

5 Continuous Mode Interpolation Between Different Conﬁgurations In the previous sections, interpolation and extrapolation of reduced-basis representations between various operating conditions and between empirical and physical modes has been successfully demonstrated. Here, the application of continuous mode interpolation to ﬂows with different boundary conditions is shown. The practical beneﬁts of interpolation and extrapolation of modes and ﬂow ﬁelds at changing boundary conditions can be signiﬁcant. Costly unsteady aero-elastic computations may thus be partially replaced by the proposed interpolation techniques. Here, computed and interpolated POD modes of the NACA 0012 airfoil having different angles of attack and the Ahmed body at different distance from the ground are presented.

(a) α = 30o

(b) α = 37.5o , computed

(c) α = 45o

(d) α = 37.5o , interpolated

Fig. 11. First POD modes of the NACA airfoil at different angles of attack and Re = 100. Interpolation from α = 30◦ and α = 45◦ , The interpolation parameter for the intermediate angle α = 37.5◦ is κ = 0.40.

In Fig. 11, the vorticity of the ﬁrst POD mode at the angle of attack α = 30◦ and α = 45◦ is shown. Employing interpolation, the ﬂow at intermediate angle α = 37.5◦ is predicted. The result is compared to the POD mode computed from the snapshots of an unsteady simulation at this angle of attack. The comparison, depicted in the right part of the Fig. 11 shows again the applicability of the interpolating approach. Further interpolation experiments are shown in Fig. 12. Here, the Ahmed body is placed at several distances from the ground. When a bluff body (2D Ahmed body) is placed in a ﬂow stream far from the ground, the most unstable eigenmodes are related to von K´arm´an vortices. As the distance from the ground is reduced, the modes are deformed due to the inﬂuence of solid boundary. Finally, near the ground, the dominant

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(a) d = 4.5

(b) d = 2.5

(c) d = 1.5

(d) d = 2.5, interpolated, κ = 0.42 Fig. 12. Eigenmodes of Ahmed body at different distances from ground d and Re = 150. Interpolation from d = 4.5 and d = 1.5 to d = 2.5.

mode is related to Kelvin-Helmholtz vortices of the shear layer. The schematic sketches in right part of the Fig. 12 show the position of the Ahmed body above the ground while the domain used for interpolation is adjusted to the lowest position. The upper three pictures illustrate the eigenmodes computed at different distances between the body and the ground. The bottom picture, below the line, shows the interpolated mode approximating the second ﬂow from top. For both conﬁgurations, NACA airfoil and Ahmed body, the interpolation yiels good approximations of the operating conditions due to changing geometry. Thus, interpolation is suitable for more complex geometries and unsymmetrical ﬂows.

6 Conclusions Control-oriented reduced-order modelling strategies for single stationary state are generalized for multiple operating conditions including transients. In particular, a timedependent reduced-basis representation of coherent structures is included. This generalization is motivated by the frequent observation that coherent structures change continuously with the operating condition, e.g. the Reynolds number or the forcing parameters. One example is transient or forced vortex streets behind obstacles. Hence,

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accurate reduced-order representations designed for one ﬂow state may be inaccurate at another state. In the current study, a continuous mode interpolation between two reference states is proposed to ensure low dimension and uniform accuracy over the interval of operating conditions between these states. The key enabler is a suitably interpolated eigenproblem from global stability analysis and from the Fredholm equation with the autocorrelation function as kernel. The stability analysis comes with the advantage of providing the growth-rate of the most unstable eigenmode. Thus, the linear effect of passive control devices can be modelled. This capability is demonstrated for a wake stabilisation experiment using a control wire [38]. The effect and optimal locations of control wires are accurately predicted. On a more kinematic level, POD mode interpolation is successfully demonstrated for the ﬂow around a circular cylinder, around an Ahmed body as well as around an inclined airfoil. In particular, interpolation is carried out for one kind of modes at different operating conditions, different kinds of modes, and ﬁnally even for different ﬂow conﬁgurations. Furthermore, extrapolation of the modes outside the reference states is shown to be possible. Interpolated modes provide a least-dimensional Galerkin approximation for several operating conditions, thus enabling resulting Galerkin models for control design. For wake ﬂows, the most energetic POD modes are approximated by stability modes and the shift-mode is derived from a mean-ﬁeld consideration. Thus, robust low-dimensional models can be derived from ﬁrst principles without empirical input. The construction of these Galerkin models is currently pursued by the group.

Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via the Collaborative Research Center (Sfb 557) ”Control of complex turbulent shear ﬂows” at the Berlin University of Technology and the grant NO 258/1-1. Support of US National Science Foundation grants numbers CNS-0410246 and ECS-0136404 is acknowledged. We acknowledge stimulating discussions with Pierre Comte, Laurent Cordier, Jo¨el Delville, Mark Glauser, Stefan Siegel, Tino Weinkauf, Jose-Eduardo Wesfreid, and the TU Berlin team (Oliver Lehmann, Weichen Liu, Mark Luchtenburg, Mark Pastoor, Michael Schlegel and Jon Scouten). We thank the referees for helpful comments.

References [1] Gad-el-Hak, M.: Flow Control: Passive, Active and Reactive Flow Management. Cambridge University Press (2000) [2] Fiedler, H. & Fernholz, H.H.: On the management and control of turbulent shear ﬂows. Progr. Aeronaut. Sci. 27 (1990) 305–387 [3] Becker, R., Garwon, M., Gutknecht, C., B¨arwolff, G., King, R.: Regelung aerodynamischer Str¨omungen am Beispiel einer r¨uckw¨artsgewandten Stufe (transl.: Control of aerodynamics ﬂows exampliﬁed for the backward-facing step). Automatisierungstechnik 50 (2002) 79–86

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Active Management of Entrainment and Streamwise Vortices in an Incompressible Jet D. Greenblatt, Y. Singh, Y. Kastantin, C.N. Nayeri, and C.O. Paschereit Technical University of Berlin, Hermann Foettinger Institute of Fluid Mechanics 8 Mueller-Breslau Street, D-10623 Berlin, Germany [email protected]

Summary A pilot experimental investigation was conducted to study the active generation and management of streamwise vortices in an incompressible jet flow. The lip of the jet was equipped with small flaps (flaplets) deflected away from the stream at 30°, that incorporated flow control slots through which steady suction and zero mass-flux perturbations were introduced. Control via suction, reduced the pressure on the flaplets, thereby entraining fluid from the surrounding fluid and generating streamwsie vortices. Flaplet length had very little effect on the vortex formation but exerted a profound influence on the dissipation of the vortices further downstream. Jet momentum could also be increased by up to 25%. Preliminary experiments using zero mass-flux control indicated that entrainment of the surrounding fluid was combined with a reduction of momentum in the jet.

1 Introduction Control of instabilities and mixing in laminar, transitional and turbulent flows is of great importance in many engineering applications. For example, in certain combustors it is important to suppress or eliminate symmetric and/or helical thermoacoustic instabilities, where these instabilities are driven by large-scale structures. A number of active and passive techniques have been developed, that are designed to alter the vortical structure in the flow. Jet noise is another example where rapid mixing of the exhaust gas with the atmosphere or a co-flowing jet can bring about reduced low frequency contributions. Control of mixing jets is achieved using three main approaches: passive control, active steady control and periodic active control. Passive control is by far the most common and is achieved by one or other geometric modifications, such as swirlers (e.g Naughton and Settles [1]), tabs or vortex generators (e.g. Swithenbank et al. [2]), lobed or corrugated mixers (Presz et al. [3]), chevrons (e.g. Bridges and Brown [4]) and non-circular jet geometries (Gutmark and Grinstein [5]). Significant effort has been expended in optimizing chevron and chevron/tab combinations in recent years (Bridges and Brown [4]; Stone et al. [10]). A negative consequence of successful noise control invariably results in some loss of thrust at cruise conditions. Thus recent work is aimed at applications oriented research using, for example, shape memory alloys to minimize the losses at cruise conditions (Calkins and Butler [11]). R. King (Ed.): Active Flow Control, NNFM 95, pp. 281–292, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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Active methods generally require either periodic forcing or the introduction of a high-speed jet or the removal of fluid via steady suction (e.g. Strykowski et al. [6]). The control of jets (and shear layers) by periodic perturbations can be classified into two main approaches. The first involves the introduction of low amplitude perturbations into a base flow that are consistent with linear amplification criteria (e.g. Oster and Wygnanski [7]; Paschereit et al. [8]); the second involves high amplitude forcing that can be on the order of the base flow itself, at frequencies that are not necessarily related to the base flow stability characteristics (e.g. Parekh et al. [9]). For active control using periodic perturbations, low amplitude excitation has been employed extensively to understand the stability characteristics of shear layers and reasonable agreement with stability theory has been observed (Gaster, et al. [12]; Ho and Huerre [13]). Observations made with low amplitude excitation clearly demonstrated the process of linear amplification and the onset of vortex rollup and nonlinear effects. Shear layers and jets can also be excited by multiple frequencies or perturbations at multiple locations (e.g. Long and Petersen [14]). As a general rule, the low amplitude perturbation approach limits control authority because it relies on the natural amplification of perturbations. Furthermore, the large vortical, or so-called coherent, structures that result do not efficiently enhance small-scale turbulent mixing, which is important for combustion and noise control. In high amplitude forcing, the actuator inputs to the flow are far in excess of the linear perturbation limits (Parekh et al. [9]). In these studies, zero mass-flux jets were used to force the controlled jet at frequencies that were one order of magnitude larger than the most amplified frequency. Amplitude modulation was also used to excite the jet at frequencies below the resonance frequency of the actuator. Wiltse and Glezer [15] extended this work to study supersonic jets and the method has been used to demonstrate effective mixing on a full-scale jet engine. The principal drawback of the method is that it requires high amplitude perturbations, possibly incorporating additional plumbing for the excitation jets. Pack & Seifert [27] studied incompressible jetvectoring by introducing periodic forcing at the junction of a diffuser attached to a jet. Local forcing enhanced the mixing and promoted attachment of the jet shear layer to the diffuser wall. The global objective of the present investigation is to actively control entrainment and streamwise vortices in a jet by deploying finite-span, controlled flaplets (small flaps) around the perimeter of the jet. The specific task addressed here is that of studying and quantifying the effect of control in conjunction with different-sized flaplets. Measurements included flaplet surface pressures, flowfield measurements using laser Doppler anemometry (LDA) and particle image velocimetry (PIV). Control is achieved by zero mass-flux oscillatory blowing and steady suction.

2 Experimental Setup Experiments were performed on a 90mm diameter (D=2R) round air jet in the Reynolds number range 16,000ReŁU0D/ν48,000. Three interchangeable flaplets of chord lengths L/D=0.167, 0.33 and 0.5 (flaplets 1, 2 and 3 respectively), each with a divergence of 12° were mounted at the exit of the slot. A schematic of the setup for flaplet 1 is shown in fig. 1. The flaplets were deflected away from the jet at δf=30°

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and incorporated a 1mm wide flow control slot extending 70% of the span, 1mm downstream of the jet exit. Each slot was equipped with a small plenum for the purpose of introducing zero or non-zero mass flux jets. The flaplet were also equipped with surface pressure ports. 30° flaplet

U0

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Fig. 1. (a) Side and (b) front and side view schematics of the jet and flaplet setup; (c-e) relative flaplet sizes showing pressure port locations on flaplet 3

Suction was achieved using a small vacuum pump; volumetric suction rates (Q) were measured by means of a rotameter and quantified as CQ=Q/U0A, where A is the jet exit area. Excitation was produced via an acoustic driver, where the excitation frequency (f) was non-dimensionalized in the standard manner as F+ = fX/U0, where X is the distance from the slot to the flaplet trailing edge [25]. A hot wire anemometer was used to calibrate the slot amplitude and expressed as the momentum coefficient: Cµ =Jslot/ Jjet [25]. Flowfield data was acquired by means of two-dimensional laser Doppler anemometry and two-dimensional particle image velocimerty (PIV) in the y-z plane (see fig. 1 above). The LDA was setup to provide u-v data and the PIV was setup to provide v-w data from which streamwise vorticity

ω x = ∂w / ∂y − ∂v / ∂z

(1)

was calculated. Reflections from the flaplet surface were minimized by rotating the jet/flaplet assembly counterclockwise through 90°. Measurements were performed on the flaplet at x/L=0.09 downstream of the slot and at x/L locations downstream of the flaplets. Flaplet 3 was equipped with pressure ports and measurements were performed using a pressure scanner.

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3 Principle of Operation and Parameters The principle of operation for actively enhancing mixing and generating streamwise vortices is as follows: A control input in the form of a perturbation or suction is introduced at the control slot. The relatively low pressure on the flaplet surface that results from control draws fluid from the ambient air into the jet. This has a number of effects. Firstly, the jet momentum is increased due to the forced entrainment of ambient fluid. Secondly, jet flow that has deflected closer to the flaplet surface, and possibly attached thereto, is expelled from the jet to the surroundings. Thirdly, due to the finite nature of the flaplets, and the pressure difference that exists across the flaplet, streamwise vortices are formed at the edges and may bear some similarity to controlled vortices trailing finite wings (see Greenblatt et al [23,24]). If multiple control flaplets are deployed, the induced velocity of adjacent vortices will alternately entrain and detrain fluid into and out-of the jet. Jet entrainment or the characteristics of isolated vortices such as strength, size and location, are functions of many parameters. For example, with control using steady suction, the vortex strength:

Γ = Γ (L/D, AR, CQ, δf)

(2)

with additional parameters such as flaplet planform shape, surface curvature, and slot location and orientation. For multiple flaplets, additional parameters such as number of flaplets Nf and peripheral flaplet spacing l/D are important.

4 Discussion of Results 4.1 Steady Suction

Initial trials were conducted by measuring the upper surface pressure as a function of steady suction CQ on flaplet 3 at Re=21,000 and 44,000 (figs. 2a-2d and 3). Data are presented in terms of the pressure coefficient defined here as C p = ( p − ps ) / 1 2 ρU 02 , where ps is the static pressure in the core of the jet. With no control applied to the flaplet (CQ =0), the flow is fully separated and no pressure difference can be discerned. As the suction rate increases (fig. 2b) a localized low pressure region is generated just downstream of the control slot, while towards the tailing edge of the flaplet virtually no pressure change is generated. As the suction rate is increased further (fig. 2c and 2d), further reductions in the pressure downstream of the flaplet are observed while simultaneously higher pressures are generated near the side edges of the flaplet. Near the trailing-edge, however, the pressures remain largely unaffected. This pressure distribution is symmetric with respect the flaplet centerline and was also independent of Reynolds number as shown in fig. 3. The strong pressure gradient across the span of the flaplet just downstream of the control slot is consistent with a strong crossflow from the flaplet edges toward the flaplet centerline. Increases in CQ at different Re resulted in continuously deceasing pressures downstream of the flaplet (fig. 3), although the suction effectiveness, as expressed by dCp/dCQ at the centerline, decreases gradually with increasing CQ. With sufficient control authority, the

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Fig. 2. Effect of steady suction on flaplet 3 surface pressures 0.4 0.2 Re = 21000 y/L = -0.2 Re = 21000 y/L = 0 Re = 21000 y/L = 0.2 Re = 4000 y/L = -0.2 Re = 44000 y/L = 0 Re = 44000 y/L = 0.2

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2

Fig. 3. Flaplet 3 surface pressures at the centerline (z/L=0) and edges (z/L=±0.2) downstream of the control slot at x/R=1.1

minimum pressure attainable will depend on the flaplet angle deflection. Hence potentially stronger pressure gradients can be expected with larger flaplet deflection. Dimensionless streamwise vorticity ωxR/U0 measured on flaplets 1, 2 and 3 respectively for CQ=3.22% and Re=21,000 at x/L =0.09 are shown in fig. 4a-4c respectively. Increasing the flaplet size, while maintaining the same suction rate, has the effect of

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Fig. 4. Non-dimensionalized streamwise vorticity ωxR/U0 measured on flaplets 1, 2 and 3 respectively for CQ=3.22% and Re=21,000 at x/L corresponding to pressure port locations at x/R=1.1

generating slightly elongated vortices that are closer to the flaplet centerline. However, the strength of the individual vortices does not appear to change significantly and the peak absolute values of ωxR/U0 do not vary by more than 5% for the different flaplet sizes. This observation is broadly consistent with the pressure data (fig. 2) where it was shown that the strong pressure gradients are generated close to the slot. Thus, while the flaplet should be long enough to sustain a pressure difference, its length does not play a defining role in the initial formation of the vortices. The downstream development of the vortices for flaplets 1 and 3 are shown at x/L=1.17 (corresponding to the location just downstream of flaplet 3) in fig. 5b and 5c. Unlike trailing vortices from wing tips and flaps, the vortices do not retain an approximately symmetric shape, but rather become oval shaped with the long axis corresponding to the radial direction. We will refer to these as the primary vortices. This is due to radial outflow near the center of the flaplet that generates high negative and positive ∂w / ∂y above and below the flaplet respectively. Although the length of the flaplet does not play a large role regarding the initial formation of the vortices, it certainly does play a role in the dissipation of the vortices downstream. Here the peak vorticity reduction for flaplets 1 and 3 are 12% and 30% respectively, although the

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Fig. 5. Non-dimensionalized streamwise vorticity ωxR/U0 measured at x/L= 1.514 for (a) baseline, (b) downstream of flaplet 1 and (c) downstream of flaplet 3, where control is at CQ=2.27% and Re=21,000

vortices for the two different flaplets have approximately the same size and shape. In addition to the primary vortices, secondary vortices are generated at the locations corresponding an annular region in the vicinity of the jet radius, r≈R. The secondary vortices at r >R adjacent to the flap, above and below, each have the same sign as the primary vortices generated on the same side of the flap. At r
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Fig. 6. Non-dimensionalized streamwise vorticity ωxR/U0 measured at x/R= 0.44 downstream of flaplet 1 for CQ=2.27% and Re=21,000

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Namely, consider the development of the vortices associated with the shortest flaplet when CQ=2.27%. Over the relatively short axial distance ∆x/R=0.73 the peak vortex strength has diminished by a factor of 4. Axial velocity profile traverses using an LDA are shown in figs. 7a-7d for the baseline case at successively increasing suction rates. For all control cases, an increase in overall jet momentum is apparent. The increase in momentum is due to entrainment; this was confirmed using smoke flow visualization. In addition, axial momentum is not reduced within the core of the jet. With increasing suction rates the effect on axial momentum appears to saturate (cf. figs.7c and 7d), even though flaplet surface pressures continue to decrease downstream of the control slot (see figs. 2 and 3). This is probably due to the flow being fully attached to the flaplet and thus further increases in suction rate make very little difference to the flaplet flow. The momentum added to the jet using control was quantified by calculating

J = ρ u 2 dA

³

(3)

for each case, and then comparing the ratios. Clearly this is not representative of the entire jet flow, but it is representative of the momentum ratio expected with multiple controlled flaplets around the jet circumference. For the data presented in fig. 7: at CQ=0.54% the jet momentum was increased by 10%; at CQ=1.37% and 3.37% the increase by more than 25%. In principle, momentum could be further increased by increasing the flaplet deflection, providing that the flow could be attached to the flaplet. As expected, large crossstream velocity fluctuations w’ are associated with the high shear ∂u / ∂x (see fig. 8). 1

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Fig. 8. Cross stream velocity fluctuations (w’) measured using an LDA for the baseline case and three control cases, corresponding to fig. 7 (Re=21,000)

4.2 Outlook for Control Via Excitation

Perturbations normal to the flaplet surface, as in the present setup, were not ideal for separation control [27]. Nevertheless, for perturbations at F+=0.38 and Cµ ~0.2%, a negative pressure of Cp=−0.05 was generated on the flaplet downstream of the slot. Corresponding LDA data showed that zero mass flux perturbations have a fundamentally different effect on the local shear layer above the flaplet in that a loss of momentum in the core of the jet accompanies the local increase in jet spreading (figs. 9a and 9b). Mixing at this control frequency is therefore achieved by two mechanisms: the low pressure on the flaplet draws stagnant ambient fluid towards the jet while, simultaneously, the fluid in the jet is detrained from the jet to the region above the flaplet. Thus mixing of the jet and ambient fluid is above the flaplet. Excitation generally produces flap-spanwise vortices in two-dimensional flows and it is therefore assumed that a finite region of spanwise vortices will exist on the flaplet. On the other hand, the ambient fluid is entrained towards the low pressure region of the flaplet and manifests as streamwise vortices. Thus streamwise vortices, entrained around the side edges of the flaplet, will be superimposed on spanwise vortices that scale with some fraction of the flaplet length. The jet control methodology discussed here is different in principle to methods that excite instabilities in jets (e.g. Long and Peterson [14]) as well as those that use high amplitude perturbations to bring about jet mixing (e.g. Parekh et al. [9]). For zero-mass flux control, the instability mechanism that is being exploited is the KelvinHelmholtz instability that exists in the shear layer above the flaplet. Based on experience with two-dimensional flows, it is surmised that disturbances are amplified for a

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very short distance <
5 Future Work The data presented here are representative of the initial phase of this work. Future work will entail involve a parametric study where flaplet angle and aspect ratio are varied. In addition, a new flaplet with excitation in the direction of the jet is being developed with a view to fully exploiting the benefits of zero mass-flux control by introducing perturbation parallel to the jet flow direction. Acoustic drivers and piezoelectric actuators are envisaged for this purpose. The present jet has also been modified to be equipped with multiple flaplets to facilitate the investigation of control on adjacent flaplets. In addition to the traditional excitation methods, excitation using plasma actuators is also being considered. Due to the problem of jet noise, future work will also involve extending these principles to high-speed jets.

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References [1] Naughton, J.W. and Settles G.S., “Experiments on the enhancement of compressible mixing via streamwise vorticity. Part I – optical measurements,” AIAA Paper 92-3549. [2] Swithenbank, J., Eames, I., Chin, S., Ewan, B., and Yang Z., “Turbulence mixing in supersonic combustion systems,” AIAA Paper 89-0260, 1989. [3] Presz,W.M., Gousy, R. and Morin, B.L., “Forced mixer lobes in ejector designs,” AIAA Paper 86-1614, 1986. [4] Bridges J. and Brown C.A., “Parametric Testing of Chevrons on Single Flow Hot Jets,” NASA/TM 2004-213107, AIAA Paper 2004-2824. [5] Gutmark, E. and Grinstein, F., “Mixing in non-circular jets,” Annual Review of Fluid Mechanics, Vol. 31, pp. 239-272, 1999. [6] Strykowski, P.J. Krothapalli, A. and Forliti, D.J. “Counterflow thrust vectoring of supersonic jets,” AIAA Journal, Vol. 34, No. 11, 1996, pp. 2306-2314. [7] Oster, D. and Wygnanski, I., “The forced mixing layer between parallel streams,” J. Fluid Mech., Vol. 123, p. 91, 1982. [8] Paschereit, C.O., Wygnanski, I. and Fiedler, H.E., “Experimental investigation of subharmonic resonance in an axisymmetric jet,” J. Fluid Mech., Vol. 283, pp. 365-407, 1995. [9] Parekh D.E., Kibens V., Glezer A., Wiltse J.M. and Smith D.M., “Innovative jet flow control - mixing enhancement experiments,” AIAA Paper 96-0308, AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV Jan 15-18, 1996. [10] Stone, J.R. Krejsa, E.A. and Clark B.J., “Jet Noise Modeling for Coannular Nozzles Including the ffects of Chevrons,” NASA/CR 2003-212522. [11] Calkins, F.T. and Butler, G.W., “Subsonic Jet Noise Reduction Variable Geometry Chevron,”AIAA Paper 2004-190, 42nd Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 5-8, 2004. [12] Gaster, M., Kit E. and Wygnanski, I. “Large scale structures in a forced turbulent mixing layer,” J. of Fluid Mech., Vol. 150, No. 23, 1985. [13] Ho, C.M. and Huerre, P., “Perturbed free shear layers,” Annual Review of Fluid Mechanics, Vol. 16, pp. 365-424, 1984. [14] Long, T.A. and Petersen, R.A., “Controlled interactions in a forced axisymmetric jet. Part 1. The distortion of the mean flow, J. Fluid Mech., Vol. 235, Feb. 1992, pp. 37-55. [15] Wiltse, J.M. and Glezer, A. “Manipulation of free shear layers using piezoelectric actuators,” J. Fluid Mech., 1993, vol. 249, pp. 261-285. [16] Paschereit, C.O., Oster, D., Long, T.A., Fiedler, H.E. and Wygnanski I., “Flow visualization of interactions among large coherent structures in an axisymmetric jet,” Experiments in Fluids, Vol. 12, 1992, pp. 189-199. [17] Paschereit, C.O., Gutmark, E. and Weisenstein, W., “Coherent structures in swirling flows and their role in acoustic combustion control,” Physics of Fluids, Vol. 9, pp. 26672678, 1999. [18] Paschereit, C.O., Gutmark, E. and Weisenstein, W., “Structure and control of thermoacoustic instabilities in a gas turbine combustor,” Combust. Sci. and Tech., Vol. 138, pp. 213-232, 1998. [19] Paschereit, C.O., Gutmark, E. and Weisenstein, W., Excitation of thermoacoustic instabilities by the interaction of acoustics and unstable swirling flow,” AIAA Journal, Vol. 38, pp. 1025-1034, 2000. [20] Paschereit, C.O. and Gutmark, E., “Application of vortex generators in low-emission gasturbine combustor,” in 29th Symposium of the Combustion Institute, Sapporo, Japan, July 21-26, 2002.

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[21] Paschereit, C.O. and Gutmark, E., “Passive combustion control applied to premix burners. AIAA-2002-1007, 40th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, January 14-17, 2002. [22] Paschereit, C.O., Stuber, P., Knoepfel, H.-P. Steinbach, C., Bengtsson, K., Flohr, P., “Combustion control by extended EV burner fuel lance,” ASME Transactions, 2002. [23] Greenblatt, D., “Management of vortices tailing flapped wings via separation control,” AIAA Paper 2005-0061, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Jan. 2005 (Submitted to AIAA Journal). [24] Greenblatt, D., Melton, L., Yao, C., Harris, J., “Control of a wing tip vortex” AIAA Paper 2005-4851, 23rd AIAA Applied Aerodynamics Conference, Westin Harbour Castle, Toronto, Ontario, 6-9 June 2005. [25] Greenblatt, D. and Wygnanski, I, “Control of separation by periodic excitation,” Progress in Aerospace Sciences, Volume 37, Issue 7, pp. 487-545, 2000. [26] Greenblatt, D., Paschal, K., Yao, C., Harris, J., Schaeffler, N., and Washburn, A., “A Separation Control CFD Validation Test Case Part 1: Baseline & Steady Suction”, AIAA Paper 2004-2220, 2nd AIAA Flow Control Conference, Portland, Oregon, 28 June - 1 July 2004. [27] Pack L. G. and Seifert, A., “Periodic Excitation for Jet Vectoring and Enhanced Spreading,” AIAA Journal of Aircraft, Vol. 38, No. 3, May–June 2001.

Active Control to Improve the Aerodynamic Performance and Reduce the Tip Clearance Noise of Axial Turbomachines with Steady Air Injection into the Tip Clearance Gap L. Neuhaus and W. Neise Deutsches Zentrum für Luft- und Raumfahrt e.V., Institut für Antriebstechnik, Abteilung Triebwerksakustik, Müller-Breslau-Str. 8, 10623 Berlin, Germany

Summary The secondary flow over the impeller blade tips of axial fans is important for the aerodynamic and acoustic performance of the fans. Pressure coefficient and efficiency drop, and the usable range of the performance characteristic is diminished as the rotor flow stalls at higher flow rates. By applying active flow control in the tip region of the impeller, it is possible to reduce the negative effects of the tip clearance flow. In the present paper, steady air injection into the tip clearance gap is investigated. Three different air injection configurations are studied. Two configurations use up to 24 separate slit nozzles, which are evenly distributed over the circumference of the casing. In the third injection configuration, the individual nozzles are replaced by a continuous circumferential slit.

Nomenclature A a0 aĬ b c D f fLRD h Lp LW m min M MTip n ∆pt

cross sectional area speed of sound azimuthal phase angle speed variable blade chord length impeller diameter frequency rotating frequency of the rotor harmonic order of the BPF pressure level sound power level azimuthal mode order injected mass flow in percent of fan mass flow by ϕ = 0.3 = u/a0; flow Mach number Mach number at the impeller blade tip impeller speed total fan pressure (∆pt0 = 1 Pa)

Q volume flow (Q0 = 1 m3/s) s tip clearance u flow velocity uin jet exit flow velocity U impeller tip speed V number of stator vanes Z number of impeller blades Znoz number of injection nozzles

ε ζ ηel ηt

hub-to-tip ratio non-dimensional tip clearance efficiency of the drive motor = ∆pt Q/(ηelPel + Pin); approximate total fan efficiency θ blade stagger angle Ω =2πfLRD; angular velocity ωĬ phase angle velocity ρ0 air density

R. King (Ed.): Active Flow Control, NNFM 95, pp. 293–306, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

294 Pel Pin

L. Neuhaus and W. Neise electric power input to drive motor = Min⋅uin2 aerodynamic power of injected air flow

ϕ = 4Q/(πD2U); flow coefficient ψ = 2∆pt/(ρ0 U2); pressure coefficient

1 Introduction Axial turbomachines have a radial gap between the casing and the rotor blades. The static pressure difference between the suction and the pressure side of the impeller blades produces a secondary flow over the tip of the rotor blades (Fig. 1a). This tip clearance flow is important for the aerodynamic and acoustic performance of the machine. Pressure coefficient and efficiency drop, and the usable range of the performance characteristics is diminished as the rotor flow is stalled at higher flow rates. Previous work at DLR-Berlin [1]-[4] investigating the effects of varying tip clearance on noise and performance showed the existence of a broad-band noise source for large tip clearance. This source appeared in the rotor wall pressure spectrum at about half the blade passing frequency (BPF) and radiated a fluctuating tonal component into the far field, the tip clearance noise (TCN). Interpretation of the spectra and circumferential mode analysis led to the model of a rotating source mechanism, called rotating instability (RI), which moves relative to the blade row at a fraction of the shaft speed, similar to the cells of rotating stall (Kameier [1], Kameier and Neise [2]). The effect was also observed in the third stage of the low-speed research compressor at the TU Dresden when the tip clearance was enlarged (Mailach et al. [5]). Legros et al. [6] have found rotating instability to be a noise source of the axial fan in air condition systems of commercial aircraft.

(a)

(b)

Fig. 1. Schematic view of (a) the secondary flow through the tip clearance gap and (b) of a turbulence generator inserted in the gap

Kameier [1] was successful in reducing the tip clearance noise and increasing the aerodynamic performance by mounting a turbulence generator into the tip clearance gap (Kameier [1], Kameier and Neise [3]), compare Figure 1b. The aim of the present work is to reproduce and possibly improve the effect achieved with the turbulence

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generator without modifications of the tip clearance gap itself to make the method applicable also to flow machines where the tip clearance gap is variable, e.g. due to usage of different stagger angles of the impeller blades. One method for increasing aerodynamic performance and reducing the radiated noise level is to control the flow in the tip clearance gap by injecting air into the gap. A number of authors reported successful applications of air injection upstream of the fan rotor in order to inhibit rotating stall and compressor surge (D’Andrea et al. [7], Weigl et al. [8], Spakovszky et al. [9], [10]) or to increase the usable range of the performance characteristic (Nie et al. [11]). Efficiency and noise emission were not addressed in the studies mentioned above. Neuhaus and Neise [12], [13] investigated the influence of air injection on pressure rise, efficiency, increased stall margin, and noise emission.

2 Experimental Facility The test fan is a low-speed high-pressure axial fan with outlet guide vanes, the same as used for the experiments in references [1] - [4]. The principal impeller dimensions are as follows: impeller diameter D = 452.4 mm; hub-to-tip ratio ε = 0.62; NACA 65 blade profile; blade number Z = 24; blade chord length at the tip c = 43 mm; maximum blade thickness 3 mm; blade stagger angle at the tip θ = 27°. The design speed is n = 3000/min. The stator row comprises V = 17 unprofiled vanes. The axial distance between the rotor and the stator at the outer circumference is ∆x/c = 1.3. The tip clearance can be varied by exchanging casing segments while the impeller diameter remains constant. Four casing segments are available to give the following tip clearances: s = 0.3, 0.6, 1.2, and 2.4 mm (ζ = s/c = 0.7%, 1.4%, 2.8%, and 5.6%).

Fig. 2. Schematic view of the experimental setup (dimensions in mm)

Figure 2 shows the experimental setup with its major dimensions. The measurement facility is in accordance with the requirements of DIN 24136 [14] for measurement of aerodynamic fan performance. On the inlet side, there is a short duct section with a bellmouth nozzle; there are no flow straighteners or screens in the inlet duct.

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The anechoically terminated outlet duct is in accordance with the international standard ISO 5136 [15]. In the outlet duct a ½-inch microphone equipped with a turbulence screen is mounted in a rotatable duct section to measure the circumferentially averaged sound pressure level at a specified radial distance from the duct axis. In order to measure the unsteady blade pressures, a miniature pressure sensor is mounted on the suction side of one impeller blade at 36% of the chord length without changing the original outer blade contour. The radial distance from the blade tip is 7% of the chord length.

Fig. 3. Schematic view of the circumferential slit arrangement at the axial position ξ = 16.6%

To control the flow conditions in the tip clearance gap, air is injected into the gap through different injection configurations. In the first set up, 24 slit nozzles mounted flush with the inner casing wall are distributed uniformly over the circumference of the casing. In the second configuration, 17 nozzles are employed instead of 24, the arrangement of the nozzles is the same as in the first set up. In the third injection configuration, the individual nozzles are replaced by a continuous circumferential slit. The axial position of all three injection configurations is ξ = x/c = 16.6% downstream of the leading edge of the rotor blades, which is the point of the maximum profile thickness of the impeller blade. Figure 3 shows the circumferential slit arrangement at the axial position ξ = 16.6%. The angle between the jet axis and the interior casing wall is 15°. The jet axis is inclined towards to the main flow direction in the fan duct.

3 Steady Air Injection Through Different Injection Configurations In this paper only results for steady air injection are presented. Results for pulsed injection are reported in references [12] and [13]. The axial injection position is the optimal position with regard to the required injection rate, compare Neuhaus and Neise [16].

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3.1 Air Injection Through 24 Slit Nozzles The first experiments were conducted with steady air injection using Znoz = 24 nozzles which is equal to the number of impeller blades. With this arrangement, it was possible to control the tip vortex of all impeller blades simultaneously. Measurements were made at the design impeller speed of n = 3000/min. Experiments at reduced speed are shown in the references [12] and [13]. 0.5 0.8 0.7

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min = 0 min = 0.3 % min = 0.6 % min = 0.8 %

0.15

0.20 flow coefficient ϕ

0.25

0.30

Fig. 4. Pressure coefficient, efficiency, and sound pressure level in the outlet duct as functions of the flow coefficient for different steady air injection mass flow rates; n = 3000/min, Znoz = 24, ζ = 5.6%, ξ = 16.6%

Figure 4 shows the aerodynamic and acoustic performance curves at the design speed. For symbols and the definitions of the non-dimensional fan performance parameters used, see the nomenclature. The injected mass flow is given in percent of the maximum mass flow delivered by the fan (i.e., at ϕ = 0.3). With steady air injection, the pressure rise and the efficiency increase at low flow rates, and the stall point is shifted towards lower flow rates. For the case with an injection rate of min = 0.8%, the stall point is shifted to the operating point ϕ = 0.161, i.e. the useful range of the fan characteristic is extended by 36%. At this point the fan pressure is increased by about 28% and the efficiency by about 10%. With mass flow injection rates of min = 0.6% and 0.8%, the optimum efficiency is increased, and for even higher injection rates (not shown here), the maximum efficiency decreases slightly. Note that the aerodynamic power of the injected air flow is taken into account when calculating the fan efficiency, compare the equation given in the nomenclature. This is valid for all efficiency data presented in this paper. The sound pressure characteristic without air injection exhibits a peak at operating points around ϕ = 0.2 which is caused by the tip clearance noise. With air injection, the noise level is reduced in this range. At higher flow rates, the level in the outlet

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duct increases with the injected mass flow. At lower flow rates, the noise is nearly the same as without air injection. When the injected air flow is increased further (min > 0.8%, not shown here), the sound pressure level is larger than without air injection, except for the operating points where tip clearance noise exists. 90 110 Outlet duct

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min = 0.3%, ψ = 0.374, η = 0.743 min = 0.8%, ψ = 0.386, η = 0.743

Fig. 5. Sound power spectra in the fan outlet duct and wall pressure spectra on the rotor blade suction side for different steady air injection rates; n = 3000/min, Znoz = 24, ζ = 5.6%, ξ = 16.6%, ϕ = 0.2

The influence of the air injection on rotating instability and tip clearance noise is shown in Fig. 5 where sound power spectra in the fan outlet duct and wall pressure spectra on the suction side of a rotor blade are plotted for the operating point ϕ = 0.2, where tip clearance noise is strong. Without air injection and for min = 0.3%, rotating instability is visible in the blade wall pressure spectrum and tip clearance noise in the outlet duct. When the injected mass flow rate is min = 0.6%, both of these components disappear. The level of the blade passing frequency (BPF) is found to increase with the injected air flow which is due to the interaction between the jets from the Znoz nozzles and the Z impeller blades. The azimuthal mode order m of the BPF-component generated by this interaction can be determined following Tyler and Sofrin´s [17] rotorstator-interaction theory:

m = hZ + bZ noz

(1)

where h = 1,2,3,.. for the BPF fundamental and harmonics, and b = ..,-2,-1,0,1,2,.. With Znoz = Z = 24 nozzles in the present configuration, the interaction mode order is m = 0, i.e., the plane wave which is propagational in the outlet duct at all frequencies. Despite the increase in BPF-level, the overall sound pressure level is reduced, e.g., from 121 dB to 113 dB at min = 0.6% while the BPF level increases from 104 dB to 112 dB. For higher injection rates, the blade tone level rises even more which limits the usefulness of the method despite the improvement in aerodynamic performance and reduction in tip clearance noise.

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3.2 Air Injection Through 17 Slit Nozzles To avoid the increase in BFP-level, the number of nozzles is changed to obtain a cutoff rotor-jet-interaction, similar to the choice of the number of stator vanes in modern aero engine design. The angular phase velocity ȦĬ of the rotor-jet-interaction pressure fluctuations is given by

ωΘ =

hZΩ = hZ + bZ noz

Ω bZ 1 + noz hZ

,

(2)

and the corresponding azimuthal phase velocity aĬ at the casing wall is

aΘ = ωΘ

D . 2

(3)

The rotor tip speed Mach number is M Tip =

ΩD 2a 0

.

(4)

A cut-off rotor-jet-interaction mode is obtained when the azimuthal phase velocity at the casing wall becomes subsonic, i.e., when

ωΘ

D < a0 2

hZ + bZ noz > hZ ⋅ M Tip .

(5) (6)

For positive values of b, the above condition is always fulfilled. For negative values, two solutions exist: hZ + bZ noz > hZ ⋅ M Tip − bZ noz < hZ ⋅ (1 − M Tip )

(7)

−hZ − bZ noz > hZ ⋅ M Tip − bZ noz > hZ ⋅ (1 − M Tip )

(8)

For the present experiments, 17 injection nozzles are chosen to give a rotor-jetinteraction mode order of m = 7. This mode cannot propagate through the fan duct at the blade passing frequency, i.e., it decays exponentially with increasing axial distance from the source (i.e. the interaction region). Figure 6 shows the aerodynamic and acoustic performance curves for Znoz = 17 nozzles at the design speed with the air injection rate as a parameter. With air injection, the fan pressure is increased at low flow rates and the stall point is shifted towards lower flow rates. With small injection rates, the fan efficiency is improved at low mean flow rates (ϕ < 0.19). With the largest injection rate min = 1.5%, the fan pressure is increased over the entire characteristic curve. The usable range of the fan characteristic is enlarged by 48%, and at the operating point ĳ = 0.149 the fan pressure is raised from ȥ = 0.29 to 0.41. However, the fan efficiency is impaired over a large regime of the fan characteristic due to the high power which is necessary to sustain the high velocity of the air jets.

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0.5

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Fig. 6. Pressure coefficient, efficiency, and sound pressure level in the outlet duct as functions of the flow coefficient for different steady air injection mass flows; n = 3000/min, Znoz = 17, ζ = 5.6%, ξ = 16.6%

The range of operating conditions over which the sound pressure level is reduced increases with the injected mass flow. Only for the largest injection rate (min = 1.5%) and mean flow rates ĳ > 0.235, the radiated level is higher than without air injection. The optimum injection rate depends on the operating point: for ϕ < 0.185, the maximum level reduction is obtained with min = 1.5% and for ϕ > 0.185 with min = 0.6%. The sound power spectra in the outlet duct and the wall pressure spectra on the suction side of one rotor blade are shown in Figure 7, again for the operating point ϕ = 0.2 where tip clearance noise is strong. Rotating instability and tip clearance noise

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Fig. 7. Sound power spectra in the fan outlet duct and wall pressure spectra on the rotor blade suction side for different steady air injection rates; n = 3000/min, Znoz = 17, ζ = 5.6%, ξ = 16.6%, ϕ = 0.2

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are visible in the spectra for zero injected mass flow and for min = 0.5%. For higher mass flow rates min ≥ 0.6%, rotating instability and tip clearance noise are suppressed. The BPF-level in the outlet duct no longer increases when the injection rate goes up, but remains almost constant at about 96 dB. Incidentally, a strong peak appears at 850 Hz in the blade wall pressure spectra. This “jet-passing-frequency” is picked up by the sensor mounted on one impeller blade which crosses 17 jets during each revolution. In the fixed frame of reference, i.e., the fan outlet duct, this frequency component does not exist. With the cut-off injection design employing 17 nozzles, the overall sound pressure level in the outlet duct can be lowered from 119 dB at the operating point ϕ = 0.2 down to 106 dB with min = 0.6% and to 108 dB with min = 1.5%. 3.3 Air Injection Through a Circumferential Slit Arrangement

Another method to avoid the increase in fan tone level due to rotor-jet-interaction is to inject air through a continuous and uniform circumferential slit, see the schematic depicted in Figure 3. With this configuration, there are no individual jets and, therefore, no noise generating interactions with the rotor blades. The results obtained with the continuous slit injection are shown in Figure 8 and Figure 9. The usable range of the fan characteristic increases monotonously with the injected mass flow. With min = 1.5%, the improvement amounts to 62%, and at the operating point ĳ = 0.137, the fan pressure increases from ȥ = 0.29 to 0.41. 0,5 0,8 0,7

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0,30

Fig. 8. Pressure coefficient, efficiency, and sound pressure level in the outlet duct as functions of the flow coefficient for different steady air injection mass flows injected through the continuous slit arrangement; n = 3000/min, ζ = 5.6%, ξ = 16.6%

The efficiency increases only with air injection at low flow rates. With mass flow injection rates of min = 0.5% and 0.8%, the optimal efficiency is increased. With min = 1.5%, the efficiency decreases slightly to the right of the optimum fan efficiency. Again, the optimum air injection rate depends on the operating point.

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A comparison of the fan efficiency data obtained with the continuous slit arrangement and with the “cut-off design” (section 0), both with min = 1.5% mass flow rate, reveals the influence of the injection velocity on the aerodynamic power necessary to produce it: Lower jet injection velocities require a smaller amount of power than larger ones and, therefore, are aerodynamically more efficient. The radiated sound pressure level is reduced over a large range of flow rates. The higher the injection rate, the larger the regime of flow coefficients with reduced sound pressure level. At the operating point ϕ = 0.2, the overall sound pressure level in the outlet duct is reduced from 117 dB down to 106 dB with min = 0.8% injection rate and to 105 dB with min = 1.5%. When the injected air flow is min = 0.5%, rotating instability is still visible in the blade wall pressure spectrum and tip clearance noise in the sound power spectrum, compare Figure 9. Beginning at min = 0.8%, rotating instability and tip clearance noise disappear. The BPF-level decreases from 95 dB to 92 dB when the air injection rate is raised up to 1.5%. Hence, with the continuous slit arrangement, the improvements in aerodynamic performance are accompanied by lower noise levels. Outlet duct

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Fig. 9. Sound power spectra in the fan outlet duct and wall pressure spectra on the rotor blade suction side for different steady air injection rates through the continuous slit arrangement; n = 3000/min, ζ = 5.6%, ξ = 16.6%, ϕ = 0.2

4 Experiments with a Smaller Tip Clearance Gap The previous discussion has shown that air injection into the tip clearance gap can improve the aerodynamic and acoustic performance of axial-flow fans. The results that have been presented in this paper so far are for a large tip clearance gap of ζ = s/c = 5.6% of the blade chord at the tip. In this chapter, the potential of the flow control method for a small tip clearance gap of ζ = 0.7% is examined. Only data for the aerodynamic and acoustic fan performance are reported, see Figure 10. For these experiments, the continuous slit arrangement in the fan casing was used for air injection into the tip clearance gap. For comparison, the results for the large tip clearance gap (ζ = 5.6%) are presented in Figure 8.

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The stall point of the fan without air injection is ĳ = 0.2 for the large and ĳ = 0.153 for the small tip clearance gap. For both the small and the large tip clearance gap, the usable range of the performance curve is enlarged quite substantially, e.g., from 0,153 to 0,115 with min = 1.5% in case of the small tip clearance. This is the last measured point on the curve and no stall is observed in the characteristic. However, while there is a positive effect of the air injection over a wide range of operating conditions when the tip clearance is large, a beneficial effect can be observed for the small tip clearance gap only at operating conditions where the blade flow would stall without control. At these operating points, both fan pressure and efficiency increase, and the radiated noise is reduced with increasing injected mass flow. When air is injected at operating points with stable flow conditions, the characteristics either remain unchanged or become worse. 0,5

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min = 0 min = 0.5 % min = 0.8 % min = 1.5 % 0,15

0,20 flow coefficient ϕ

0,25

0,30

Fig. 10. Pressure coefficient, efficiency, and sound pressure in the outlet duct as functions of the flow coefficient for different steady air injection mass flows through the circumferential slit arrangement; n = 3000/min, ζ = 0.7%, ξ = 16.6%

Figure 11 shows the aerodynamic and acoustic performance curves for one air injection rate and different tip clearance gaps. The stall point for an injection rate of min = 1,5% is nearly the same for the gaps ζ = 2.8% and ζ = 5.6%. For small gaps (ζ = 0.7% and ζ = 1.4%) no stall point is visible in the pressure rise curve plotted. It is possible however to find the stall point by further closing the throttle. For tip clearance gaps ζ = 0.7%, 1.4% and 2.8%, the measured efficiencies are almost equal, except for operating points with stalled flow conditions for ζ = 2.8%. The efficiency curve of the fan with the largest gap is lower than the other three over the whole flow range. The acoustic performance curves show that the reduced noise levels are independent of the tip clearance gap.

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0,15

0,20 0,25 flow coefficient ϕ

0,30

Fig. 11. Pressure coefficient, efficiency, and sound pressure in the outlet duct as functions of the flow coefficient for different tip clearance gaps with air injection; n = 3000/min, min = 1.5%, ξ = 16.6%

5 Conclusion The secondary flow over the tip of the impeller blades of axial fans affects the aerodynamic and acoustic performance of the fans. Pressure coefficient and efficiency drop, and the usable range of the performance characteristic is reduced as the rotor flow stalls at higher flow rates. This is in particular true if the tip clearance gap is large. By applying active flow control in the tip regime of the impeller, it is possible to reduce the negative effects of the tip clearance flow. In the present paper, the effect of steady air injection into the tip clearance gap is investigated. An axial flow fan with D = 452.4 mm impeller diameter is used for the experiments. The largest tip clearance gap is ζ = s/c = 5.6% of the blade chord at the tip. Three different air injection configurations are studied. In the first configuration, air is injected through 24 slit nozzles distributed uniformly over the circumference of the fan casing. With steady air injection, it is possible to achieve either – with small injected mass flow rates – a significant reduction of the radiated noise level together with small improvements of the aerodynamic performance or – with high injected mass flow rates – significant improvements of the aerodynamic performance at the expense of a strong increase of the radiated noise level. The increase in noise level is caused by the interaction between the jets from the injection nozzles and the impeller blades, resulting in higher blade tone levels. The interaction of the 24 rotor blades with the 24 jets produces a blade passing frequency component propagating through the duct as a plane wave at all frequencies. The rotating instability component and the tip clearance noise are completely suppressed when the injected mass flow min ≥ 0.6%. By appropriate choice of the number of injection nozzles, the unsteady forces generated by the rotor-jet-interaction can be made cut-off, i.e. the pressure fluctuations produced in the source region can no longer propagate as sound waves through the duct but decay exponentially with axial distance from the source. With 17 injection

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nozzles in the second setup, the rotor-jet-interaction yields an azimuthal mode order m = 7 for the blade passing frequency which is not propagational in the outlet duct. Consequently, the BPF-level in the outlet duct does not increase when the injected flow rate is increased. Similar to the air injection through 24 nozzles, a significant increase of the fan pressure is observed with steady air injection. In the case min = 1.5%, the usable range of the fan characteristic is enlarged by 48%, and the fan pressure at the operating point ĳ = 0.149 is increased from ȥ = 0.29 to 0.41. The fan efficiency is improved for small injection rates and low flow rates. At very high injection rates, the fan efficiency is impaired over a large range of the flow coefficient because of the aerodynamic power necessary to produce the jet flow which is taken into account when calculating the efficiency. Another method to avoid the increase in blade tone level is to replace the individual nozzles by a continuous circumferential slit in the fan casing. In this case, no unsteady forces are generated on the fan blades, and hence, the blade passing frequency level is not altered. With this air injection configuration, significant improvements of the aerodynamic and acoustic performance are achieved. With an injected mass flow of min = 1.5%, the usable range of the fan characteristic is enlarged by 62%, and the fan pressure at the operating point ĳ = 0.137 is increased from ȥ = 0.29 to 0.41. The efficiency is increased at low flow rates. The radiated sound pressure level is reduced over a large range of flow rates. The higher the injection rate, the larger the range of flow coefficient with reduced sound pressure level. For mass flow rates min ≥ 0.8%, rotating instability and tip clearance noise are suppressed. Another successful method to control the flow in the tip clearance gap is to blow the pressurized air out of the tips of the rotor blades, see Neuhaus and Neise [18]. With this method, the air injection rate necessary to achieve the aerodynamic and acoustic improvements is lower than with air injecting through the casing wall. Air injection into the tip clearance gap through the continuous circumferential slit in the fan casing was also tested with a small tip clearance gap of ζ = 0.7%. The usable range of the performance curve is enlarged also with the small tip clearance gap, however, this is the only positive effect found, which is in contrast to the results obtained with the large tip clearance gap where the air injection resulted in improvements of the aerodynamic and acoustic fan performance over a broader rang of the characteristic curve.

Acknowledgement The investigation is supported by the German Research Foundation as part of the Collaborative Research Center 557 “Control of complex turbulent shear flows” conducted at the Berlin University of Technology.

References [1] F. Kameier. “Experimentelle Untersuchungen zur Entstehung und Minderung des Blattspitzen-Wirbellärms axialer Strömungsmaschinen”. PhD., Fortschr.-Ber. VDI Reihe 7 Nr. 243, VDI-Verlag, Düsseldorf, 1994.

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[2] F. Kameier and W. Neise. “Rotating blade flow instability as a source of noise in axial turbomachines”. Journal of Sound and Vibration, Vol. 203, 1997, pp. 833-853. [3] F. Kameier and W. Neise. “Experimental study of tip clearance losses and noise in axial turbomachinery and their reduction”. ASME Journal of Turbomachinery, Vol. 119, 1997, pp. 460 – 471. [4] J. März, Ch Hah and W. Neise. “An experimental and numerical investigation into the mechanisms of rotating instability”. ASME Journal of Turbomachinery, Vol. 124, 2002, pp. 367 – 374. [5] R. Mailach, I. Lehmann and K. Vogeler. “Rotating instability in an axial comprossor originating from the fluctuating blade tip vortex”. ASME Journal of Turbomachinery, Vol. 123, 2001, pp. 453 – 460. [6] J.-C. Legros, M. Lemasson and S. Pauzin. “Contribution to noise reduction of an air conditioning turbomachine”. 4th AIAA/CEAS Aeroacoustics Conference (19th AIAA Aeroacoustics Conference), AIAA-98-2254, Toulouse, France, 1998. [7] R. D’Andrea, R. L. Behnken and R. M. Murray. “Rotating stall control of an axial flow compressor using pulsed air injection”. ASME Journal of Turbomachinery, Vol. 119, 1997, pp. 742 – 752. [8] H. J. Weigl, J. D. Paduano, L. G. Frechette, A. H. Epstein, E. M. Greitzer, M. M. Bright and A. J. Strazisar. “Active stabilisation of rotating stall and surge in a transonic singlestage axial compressor”. ASME Journal of Turbomachinery, Vol. 120, 1998, pp. 625 – 636. [9] Z. S. Spakovszky, H. J. Weigl, J. D. Paduano, C. M. van Schalkwyk, K. L. Suder and M. M .Bright. “Rotating stall control in a high-speed stage with inlet distortion: Part I – radial distortion”. ASME Journal of Turbomachinery, Vol. 121, 1999, pp. 510 – 516. [10] Z. S. Spakovszky, C. M. van Schalkwyk, H. J. Weigl, J. D. Paduano, K. L. Suder and M. M. Bright. “Rotating stall control in a high-speed stage with inlet distortion: Part II – circumferential distortion”. ASME Journal of Turbomachinery, Vol. 121,, 1999 pp. 517 – 524. [11] C. Nie, G. Xu, X. Cheng, and J. Chen. “Micro air injection and its unsteady response in a low-speed axial compressor”. ASME Journal of Turbomachinery, Vol. 124, 2002, pp. 572 – 579. [12] L. Neuhaus, and W. Neise. “Active control of the aerodynamic and acoustic performance of axial turbomachines”. 8th AIAA/CEAS Aeroacoustics Conference, AIAA-2002-2499, 17. – 19. June, Breckenridge, Colorado, USA, 2002. [13] L. Neuhaus, and W. Neise. “Active control to improve the aerodynamic and acoustic performance of axial turbomachines”. 1st Flow Control Conference, AIAA-2002-2948, 24. – 27. June, St. Louis, Missouri, USA, 2002. [14] DIN 24163, Ventilatoren, Teil 1 - 3 Leistungsmessung. Deutsche Norm, Berlin, 1985, Deutsches Institut für Normung e.V. [15] ISO 5136, Acoustics – Determination of sound power radiated into a duct by fans and other air-moving devices – In-duct method, International Organization for Standardization Geneva, 2003. [16] L. Neuhaus, and W. Neise. “Active flow control to reduce the tip clearance noise and improve the aerodynamic performance of axial turbomachines”. Fan Noise 2003, 23.-25. September 2003, Senlis, France, 2003. [17] J. M. Tyler and T. G. Sofrin. “Axial Flow Compressor Noise Studies”. Transactions of the Society of Automotive Engineers, Vol. 70, 1962, pp. 309-332. [18] L. Neuhaus, and W. Neise. “Active Control to Improve the Aerodynamic Performance and Reduce the Tip Clearance Noise of Axial Turbomachines” 11th AIAA/CEAS Aeroacoustics Conference, AIAA-2005-3073, 23. – 25. May, Monterey, California, USA, 2005.

Drag Minimization of the Cylinder Wake by Trust-Region Proper Orthogonal Decomposition Michel Bergmann, Laurent Cordier, and Jean-Pierre Brancher LEMTA, UMR 7563 (CNRS - INPL - UHP) 2, avenue de la forˆet de Haye BP 160 - 54504 Vandoeuvre cedex, France

Summary In this paper we investigate the optimal control approach for the active control of the circular cylinder wake ﬂow considered in the laminar regime (Re = 200). The objective is the minimization of the mean total drag where the control function is the time harmonic angular velocity of the rotating cylinder. When the Navier-Stokes equations are used as state equations, the discretization of the optimality system leads to large scale discretized optimization problems that represent a tremendous computational task. In order to reduce the number of state variables during the optimization process, a Proper Orthogonal Decomposition (POD) Reduced-Order Model (ROM) is then derived to be used as state equation. Since the range of validity of the POD ROM is generally limited to the vicinity of the design parameters in the control parameter space, we propose to use the Trust-Region Proper Orthogonal Decomposition (TRPOD) approach, originally introduced by Fahl (2000), to update the reduced-order models during the optimization process. Beneﬁting from the trust-region philosophy, rigorous convergence results guarantee that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem deﬁned with the Navier-Stokes equations. A lot of computational work is indeed saved because the optimization process is now based only on low-ﬁdelity models. The key enablers to an accurate and robust POD ROM for the pressure and velocity ﬁelds are the extension of the POD basis functions to the pressure data, the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem, and the inclusion in the POD ROM of diﬀerent non-equilibrium modes. When the TRPOD algorithm is applied to the wake ﬂow conﬁguration, this approach converges to the minimum predicted by an open-loop control approach and leads to a relative mean drag reduction of 30% for reduced numerical costs (a cost reduction factor of 1600 is obtained for the memory and the optimization problem is solved approximately 4 times more quickly). R. King (Ed.): Active Flow Control, NNFM 95, pp. 309–324, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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1 1.1

Introduction

Managing the Use of Approximation Models in Optimization

During the last decade, the optimal control theory [1] has emerged as a new approach to solve active ﬂow control and aerodynamic shape design problems. Indeed, these problems can be reduced [2] to the minimization or maximization of an objective functional J (drag or lift coeﬃcients, concentration of pollutant, emitted noise, mixing . . . ) according to n control or design parameters c = (c1 , c2 , · · · , cn ) (unsteady blowing/suction velocities, heat ﬂows, . . . ) under some constraints (Navier-Stokes equations, geometric constraints . . . ). Roughly, these optimization problems can be solved by two diﬀerent class of numerical methods, on the one hand, the methods of descent type which at least require an approximation of the gradient of the objective functional, and, on the other hand, the stochastic methods whose principle consists in studying the evolution of a population of potential solutions during successive generations (genetic algorithms, simplex method . . . ). Whatever the speciﬁc class of numerical methods considered, the computational costs (CPU and memory) related to the resolution of optimization problems are so important that they become unsuited to the applications of ﬂow control. This situation is even worse in an optimization setting where the governing equations need to be solved repeatedly, or in closedloop control problems, for which the controller needs to determine his action in real time. Consequently, an alternative approach is necessary. In this communication, we propose to solve the aforementioned problem of optimization by an optimal control approach in which the Navier-Stokes equations - called high-ﬁdelity model in the multidisciplinary optimization literature - are replaced by a Proper Orthogonal Decomposition (POD) Reduced-Order Model (ROM) - low-ﬁdelity model - of the dynamics for the controlled ﬂow. The POD was originally introduced in Turbulence [3] as an unbiased method of extraction of the Coherent Structures widely known to exist in a turbulent ﬂow. Essentially, this technique leads to the evaluation of POD functions that deﬁne a ﬂow basis, optimal in an energetic sense. Thereafter, these POD modes can be used through a Galerkin projection on the Navier-Stokes equations to derive a POD ROM of the controlled ﬂow [4]. The POD basis is determined a posteriori using experimental or numerical data previously obtained for the conﬁguration under study. In ﬁrst approximation, the POD can be viewed as a method of information compression. Essentially, the POD algorithm try to remove ”redundant” information (if any) from the data base. As a consequence, the ability of POD modes to approximate any state of a complex system is totally dependent of the information originally contained in the snapshot set used to generate the POD functions. Then, despite the energetic optimality of the POD modes, it seems diﬃcult to build once for all, at the beginning of the optimization process, a POD ROM able to approximate correctly the diﬀerent controlled states encountered by the ﬂow along the optimal path (see the discussion in [5]). Some kind of reactualization of the POD basis during the optimization process seems essential, the main diﬃculty consisting in determining the moment when a new

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resolution of the Navier-Stokes equations is necessary to evaluate a new POD basis. Thereafter, we will use a speciﬁc adaptive method called Trust-Region POD (TRPOD) to update the reduced-order models during the optimization process. This approach, originally introduced in [6], couples a trust-region method of optimization and reduced-order models based on POD (see Sect. 2). The principal advantages of this approach are, on the one hand, that the radius of the trust-region corresponding to the POD ROM does not have to be ﬁxed by the user, but is evaluated automatically during the optimization process, and that on the other hand, there are results of convergence proving that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem deﬁned with the Navier-Stokes equations. 1.2

A Prototype of Massively Separated Flows: The Cylinder Wake Flow

In this study, we are interested to control the unsteady wake ﬂow downstream from a circular cylinder (Fig. 1). The objective is the mean drag minimization Γt

ey

Ω

Γi VT (t) U∞

θ 0

Γo ex

Γc D

Γb Fig. 1. Controlled ﬂow conﬁguration

of the wake ﬂow by rotary oscillation of the cylinder. The ﬂow is considered as incompressible and the ﬂuid is supposed to be viscous and Newtonian. Wake ﬂows dynamics are characterized [7] by the Reynolds number Re and by the natural Strouhal number Stn at which vortices are shed in the wake of the cylinder (Fig. 8(a)). Traditionally, the Reynolds number is deﬁned as Re = U∞ D/ν where D is the cylinder diameter (R is the corresponding radius), U∞ the uniform velocity of the incoming ﬂow and ν the kinematic viscosity of the ﬂuid.

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As for the natural Srouhal number, the common deﬁnition is Stn = f D/U∞ where f is the frequency characteristic of the periodic behavior of the ﬂow. The cylinder wake is considered in the laminar regime (Re = 200). The rotary control is characterized by the non dimensional velocity γ(t) deﬁned as the ratio of the tangential velocity VT to the upstream velocity U∞ i.e. γ(t) = VT (t)/U∞ . For γ = 0, the ﬂow is naturally said uncontrolled. Contrary to the case considered in [5] where no particular assumption was done on the variation of the control function, γ(t) is hereafter sought as an harmonic function of the form: γ(t) = A sin (2πStf t) where the amplitude A and the forcing Strouhal number Stf correspond to two degrees of freedom for the control. The optimal control theory is then used to determine the control vector c = (A, Stf )T which minimizes the mean time drag coeﬃcient of the wake. For a circular cylinder, this quantity estimated over a ﬁnite horizon T equal to a few vortex shedding periods writes: CD T = −

1 T 1 T

T

2π

0 T

0 2π

0

0

2 p nx R dθ dt 2 ∂u ∂u nx + ny R dθ dt , Re ∂x ∂y

(1)

where nx and ny are the projections of the external normal vector n onto the cartesian basis vectors ex and ey respectively, and θ is an angle deﬁning the curvilinear coordinate of a point on Γc (see Fig. 1). Furthermore, let us recall (see [8] for example), that in the supercritical regime of the wake ﬂow, every mean quantity consists of two terms, the basic ﬂow i.e., the unstable, steady, symmetric ﬂow, and the mean ﬂow correction which is due to the vortex shedding. Consequently, the mean time drag coeﬃcient CD T writes: basic 0 CD T = CD + CD ,

(2)

basic 0 where CD and CD represent the drag of the basic ﬂow and the mean ﬂow correction respectively (see Fig. 2). Of course, at a given Reynolds number, the contribution of the basic ﬂow to drag cannot be modiﬁed. Then controlling the wake ﬂow by rotary oscillations can only reduce the contribution of the mean ﬂow correction to drag. If we consider as in [8] that the drag of the mean ﬂow correction ﬁeld can be only positive, the minimal value of drag that can be obtained under periodic forcing conditions is that corresponding to the basic ﬂow. In conclusion, this ﬂow would be thus a natural ’desired’ ﬁeld in a ﬂow tracking procedure of optimization. In order to validate a posteriori the control law obtained with the TRPOD algorithm, an open-loop control study is ﬁrst performed numerically. The contours of the mean temporal drag coeﬃcient, estimated over approximately 6 periods of vortex shedding after the transients have died out, are visualized in Fig. 3 in the space spanned by the forcing parameters A and Stf .

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2.5

Unsteady ﬂow Basic ﬂow

2.25

CD T

2

1.75

CD T

1.5

0 CD

1.25

1

basic CD

0.75 25

50

75

100

125

150

175

200

Re Fig. 2. Variation with the Reynolds number of the mean drag coeﬃcient. Contributions and corresponding ﬂow patterns of the basic ﬂow and unsteady ﬂow.

2

Optimization by Trust-Region Methods and POD Reduced-Order Models

In this section, only the principle of the Trust-Region Proper Orthogonal Decomposition approach for ﬂow control is exposed. For all the details of the algorithms and in particular the proofs of convergence, the reader is referred to [6,11]. We consider that the ﬂow control problem discussed in Sect. 1.1 can be formulated as an unconstrained optimization problem min J (φN S (c), c) c∈Rn

(3)

where J : Rm × Rn → R represents the objective functional and where φN S and c respectively represent the state variables obtained by numerical resolution of the state equations and the control variables. The subscript N S means that the state equations which connect the control variables c to the state variables are the Navier-Stokes equations. Since an accurate computation of the state variables φ for given c is computationally expensive when the Navier-Stokes equations are used as the state equations, the evaluation of J during the solution of the optimization process (3) is computationally expensive. A reduction of numerical cost can be achieved by employing a POD ROM as the state equation. In such a way an approximate solution φP OD of the state variables φ is obtained and

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1.3919

6 3.5867

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0.9931 4 6.9876

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[9] 1.0450

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0

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1

Fig. 3. Variation of the mean drag coeﬃcient with A and Stf at Re = 200. Numerical minimum: (Amin , Stfmin ) = (4.25, 0.74) ; [9]: (A, Stf ) = (3., 0.75) ; [10] (not shown): (A, Stf ) = (3.25, 1.13) ; [5]: (A, Stf ) = (2.2, 0.53).

the optimization problem (3) is then replaced by a succession of subproblems of the form (4) min J (φP OD (c), c). c∈Rn Usually, a POD ROM is constructed for a speciﬁc ﬂow conﬁguration, e.g., for an uncontrolled ﬂow or for a ﬂow altered by a speciﬁed control. Therefore, the range of validity of a given POD ROM is generally restricted to a region located in the vicinity of the design parameters in the control parameter space, the so-called trust-region. It is then necessary to update the POD ROM during the iterative process, the crucial point being to determine when such a reactualization must take place. Let ∆k > 0 be the trust-region radius and ck be the control parameters obtained at an iterate k of the optimization process. To evaluate the function J (φN S (ck ), ck ), it is necessary to determine the variables φN S (ck ). These variables are obtained by resolution of the high-ﬁdelity model, the Navier-Stokes equations. Then, we compute snapshots that correspond to the ﬂow dynamics forced by ck . These snapshots form the input ensemble necessary [3] to generate a POD basis {Φki }i=1,...,NP OD (here, NP OD corresponds to the number of POD modes). This POD basis can then be used via a Galerkin projection of the Navier-Stokes equations onto the POD eigenvectors to derive a

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POD ROM for ck [11,5]. After integration in time of this POD ROM, the state variables φP OD (ck ) are estimated, and thus the function J (φP OD (ck ), ck ) is evaluated. Since this POD ROM can be employed for an optimization cycle, we deﬁne (5) mk (ck + sk ) = J (φP OD (ck + sk ), ck + sk ), as a model function for f (ck + sk ) = J (φN S (ck + sk ), ck + sk ),

(6)

on the trust-region sk ≤ ∆k around ck . One is then brought to solve approximately1 the corresponding trust-region subproblem deﬁned as min mk (ck + s), s∈Rn

s.t.

s ≤ ∆k .

(7)

In order to estimate the quality of the presumed next control parameters ck+1 = ck +sk where sk is an approximate solution of (7), we compare the actual reduction in the true objective, aredk = f (ck + sk ) − f (ck ), to the predicted reduction obtained with the model function predk = mk (ck + sk ) − mk (ck ). Essentially, it is this comparison that gives a measure for the current models prediction capability. If the trial step sk yields to a satisfactory decrease in the original objective functional in comparison to the one obtained by the model function, the iteration is successful, the trial step sk is accepted and the model mk is updated i.e. a new POD ROM is derived that incorporates the ﬂow dynamics as altered by the new control2 ck+1 . Furthermore, if the achieved decrease in f indicates a good behavior of the model mk , the trust-region radius ∆k can be increased. Now, if there is a limited predicted decrease compared to the actual decrease, we have the possibility to decrease slightly the value of the trust-region radius. For unsuccessful iterations, the trial step sk is not accepted, the trustregion radius ∆k is decreased and the trust-region subproblem (7) is solved again within a smaller trust-region. With the contraction of the trust-region it is more likely to have a good approximation to the true objective functional with the POD ROM. The corresponding TRPOD algorithm is schematically described in Fig. 4.

3

Drag Minimization of the Cylinder Wake Flow by the TRPOD Algorithm

The objective of this section is to implement the TRPOD approach presented in Sect. 2 for minimizing the mean drag coeﬃcient of the cylinder wake ﬂow. 1 2

Following the trust-region philosophy [12], it is suﬃcient to compute a trial step sk that achieves only a certain amount of decrease for the model function. Since a new snapshot ensemble is available, a new POD basis can then be determined, and ﬁnally, a new POD ROM can be derived.

316

M. Bergmann, L. Cordier, and J.-P. Brancher Initialization: c0 , Navier-Stokes resolution, J0 . k = 0.

c0

∆0

k = k+1

Construction of the POD ROM and evaluation of the model objective function mk

∆k+1 > ∆k ∆k+1 = ∆k ∆k+1 < ∆k Solve the optimality system based on the POD ROM under the constraints ∆k

ck+1 and mk+1

bad

Jk+1

Solve the Navier-Stokes equations and estimate a new POD basis

ck+1

ck

k = k+1

medium

ck+1

good

Evaluation of the performance (Jk+1 − Jk )/(mk+1 − mk )

Fig. 4. Schematic representation of the TRPOD algorithm

3.1

A Robust POD Surrogate for Drag Function

In order to simplify the future notations, one introduces the drag operator CD deﬁned as: CD : R3 → R

U → 2 0

2π

1 ∂u 1 ∂u nx − ny R dθ, pnx − Re ∂x Re ∂y

(8)

where U = (u, v, p)T denotes the vector corresponding to the velocity and pressure ﬁelds. By deﬁnition, CD (U ) = CD (t) where CD represents the instantaneous drag coeﬃcient. The velocity component u and pressure p present in the relation (8) can be obtained either by resolution of the Navier-Stokes equations, or by estimation using a POD ROM. In this study, a special care is taken to the development of the POD ROM. First, a POD basis Φi representative of the velocity ﬁelds u and v, as of the pressure ﬁeld p was determined [11]. Then, to improve the robustness of the POD ROM, the POD basis functions which represent the dynamics of the reference operating condition c, were increased by adding Nneq non-equilibrium modes corresponding to new operating conditions, following the procedure described in [13]. In addition, a calibration procedure

Drag Minimization of the Cylinder Wake by TRPOD

317

of the POD-Galerkin system similar to the methods recently introduced in [14] for the pressure model was used to take into account some modelling errors (Galerkin truncation, pressure model,. . . ). Here, this calibration procedure corresponds to the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem (see [15] for all the details). Finally, the control function method introduced in [16] is used to determine POD basis functions with homogeneous boundary conditions. The velocity and pressure ﬁelds can then be expanded into the POD basis functions Φi as:

Ngal

U (x, t) =

Ngal +Nneq

ai (t) Φi (x) +

i=0

Galerkin modes

i=Ngal +1

ai (t) Φi (x) + γ(c, t) U c (x) , control function

(9)

non-equilibrium modes

where Ngal is the number of Galerkin modes and where U c is called the control function. Mathematically, U c is determined as a solution of the NavierStokes equations satisfying speciﬁc boundary conditions (the procedure is fully described in [11,5]). The Galerkin projection of the Navier-Stokes equations on the space spanned by the ﬁrst Ngal + Nneq + 1 POD modes yields [11] to d ai (t) = dt

Ngal +Nneq

j=0

Ngal +Nneq

Bij aj (t) +

Cijk aj (t)ak (t)

j,k=0

⎛ ⎞ Ngal +Nneq dγ ⎝ + Di + Ei + Fij aj (t)⎠ γ(c, t) + Gi γ 2 (c, t), dt j=0

(10a)

with the following initial conditions: ai (0) = (u(x, 0), Φi (x)).

(10b)

The coeﬃcients Bij , Cijk , Di , Ei , Fij and Gi depend explicitly on Φi and U c . Their expression could be found in [5]. Let φN S (c) = (uN S , vN S , pN S )T represent the state variables obtained by resolution of the Navier-Stokes equations and φP OD (c) = (uP OD , vP OD , pP OD )T be the corresponding values estimated with the POD ROM (10), the objective functional is 1 T f (c) = J (φN S (c)) = CD (φN S (c)) dt, T 0 and the model function, introduced and justiﬁed in [11], is mk (c) = J (φP OD (c)) = where Ni = CD (Φi ).

1 T

0

+Nneq T Ngal i=0

ai (t)Ni dt,

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These two functions can then be used in a procedure of optimization coupling trust-region methods and POD reduced-order models, following the method presented in Sect. 2. 3.2

Solution of the Subproblem (7)

The convergence behavior of trust-region methods for general model functions with inexact gradient information is usually based on a suﬃcient decrease condition of the objective function (see [12] for example). In [6], these classical results were extended and it was demonstrated that the exact solution of the subproblem (7) is not necessary to prove global convergence of the TRPOD algorithm. Here, because of the low computational costs of solving the POD reduced-order model (10), an optimality system based on the POD ROM is derived (see [2] for example) and solved. By deﬁnition, this reduced-order optimality system is a system of three coupled partial diﬀerential equations [11] formed by: 1. the state equations (10) 2. the adjoint equations d ξi (t) =− dt

Ngal +Nneq

(Bji + γ(c, t) Fji ) ξj (t)

j=0

(11a)

Ngal +Nneq

−

(Cjik

j,k=0

1 + Cjki ) ak (t)ξj (t) − Ni , T

with terminal conditions : ξi (T ) = 0.

(11b)

3. the optimality conditions 1 ∇c J = T with

0

⎛ T

⎝

Ngal +Nneq

⎞ Li ⎠ ∇c γ dt,

(12)

i=0

⎛ ⎞ Ngal +Nneq dξi Li = − Di + ξi ⎝Ei + Fij aj + 2γ(c, t)Gi ⎠ . dt j=0

This system of coupled ordinary diﬀerential equations could be solved directly using a ”one-shot method”. However, due to large storage and CPU costs, an iterative process described in [11] is usually preferred. In this study, the directions of descent are estimated using the Fletcher-Reeves version of the Conjugate Gradient Method [17]. The linear search parameter is computed at each iteration by the backtracking Armijo method [17], in which the length of the step, along each direction of descent, checks the constraint imposed by the trust-region approach. The iterative method is stopped when two following values of the functional J are suﬃciently close i.e. when |∆J (a)| = |J (n+1) (a) − J (n) (a)| < 10−5 .

7 6 5 4 3 2 1 0

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Drag Minimization of the Cylinder Wake by TRPOD

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(a) c0 = (1.0 , 0.2)T .

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A

(b) c0 = (1.0 , 1.0)T .

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0

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15

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0

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1 0.8 Stf

A

(c) c0 = (6.0 , 0.2)T .

0.6 0.4 0.2

0

10 5 Iteration number

15

0

0

(d) c0 = (6.0 , 1.0)T .

Fig. 5. Variations of the forcing amplitude (left) and Strouhal number (right) with respect to the iteration number. Results obtained with the TRPOD algorithm.

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3.3

Numerical Results and Discussion

6

6

5

5

4

4

A

A

As it was discussed for example in [18], a possible drawback of solving a minimization problem with a gradient-based optimization approach is that the algorithm may converge to the global minimum or to some other local minimum of the cost function depending on the relative position of the starting point to the minima. To alleviate this diﬃculty and evaluate the robustness of the TRPOD algorithm, the optimization process will be initialized starting from several different control parameters c0 = (A ; St) chosen at random in the control parameter space retained for the open-loop control procedure (see Fig. 3). Hereafter, four diﬀerent initial values are employed: c0 = (1.0 , 0.2)T , c0 = (1.0 , 1.0)T , c0 = (6.0 , 0.2)T and c0 = (6.0 , 1.0)T . According to the TRPOD algorithm (see Fig. 4), the radius of the trustregion ∆ is automatically either increased, or decreased during the resolution of the optimization process. Figure 5 represents for the diﬀerent initial control parameters c0 , the variations of the values of the forcing amplitude and

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1

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St (d) c0 = (6.0 , 1.0)T

Fig. 6. Evolution of the control parameters during the optimization process for the diﬀerent initial values c0

Drag Minimization of the Cylinder Wake by TRPOD 1.5

1.5 basic ﬂow γ(t) = γopt (t) γ=0

1.4

1.2

1.4 1.3 CD

CD

1.3

1.1

0.9

0.9 25

75 50 t (a) Drag coeﬃcient.

100

γ = γopt

1.1 1

0

γ=0

1.2

1

0.8

321

0.81

-0.5

0 0.5 CL (b) Polar curves.

1

Fig. 7. Time evolutions of the aerodynamic coeﬃcients for the basic ﬂow, uncontrolled ﬂow (γ = 0) and optimally controlled ﬂow (γ(t) = γopt (t)). Control was started at time t = 0.

Strouhal number with respect to the iteration number.When the numerical convergence of the iterative procedure is achieved, the optimal control parameters are A = 4.25 and Stf = 0.738. These values of parameters, which entirely deﬁne the optimal control law γopt (t), are obtained in less than ten resolutions of the Navier-Stokes equations, whatever the initial condition considered (a more signiﬁcant number of iterations is however represented in Fig. 5 to highlight the convergence).As it was expected by the global convergence properties of the TRPOD algorithm [11,6], these optimal control parameters tend towards the values predicted by an open-loop control approach (§ 1.2), and this, whatever the initial values used for the control parameters (see Fig. 6 which represents the convergence in the control parameter space). This proves the performance and the robustness of the TRPOD algorithm. Figure 7(a) represents the time evolutions of the drag coeﬃcients, for an uncontrolled ﬂow and for the ﬂow forced by the optimal control law γopt (t). These results are compared to those obtained for the basic ﬂow. According to [8], the basic ﬂow generates a priori the lowest3 coeﬃcient of drag for the conﬁguration under study. The mean drag coeﬃcient varies from a value equal to CD unc T = 1.39 in the uncontrolled case to a value equal to CD opt T = 0.99 when the optimal control parameters are applied. The corresponding relative mean drag reduction, deﬁned as CD unc T − CD opt T /CD unc T , is equal to more than 30%. The value of the drag coeﬃcient for the optimally controlled ﬂow tends towards that obtained for basic = 0.94), but with a value always slightly higher. The polar the basic ﬂow (CD curves (time evolution of the drag coeﬃcient versus the lift coeﬃcient) are represented for the uncontrolled and optimally controlled ﬂow in Fig. 7(b). The shape 3

Recently, numerical evidence were brought [19] that, for the circular cylinder wake ﬂow at Re = 200, a partial control restricted to an upstream part of the cylinder surface could lead to a mean ﬂow correction ﬁeld with negative drag.

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(a) Uncontrolled ﬂow (γ = 0).

(b) Optimally controlled ﬂow (γ(t) = A sin(2πSt t) with A = 4.25 and St = 0.738).

(c) Basic ﬂow (γ = 0). Fig. 8. Vorticity contour plot of the wake for the uncontrolled (a), optimally controlled (b) and basic ﬂow (c). The dashed lines correspond to negative values.

of these curves indicates characteristically that, for the two ﬂow regimes, the drag coeﬃcient oscillates at a frequency equal to twice that of the lift coeﬃcient. Finally, in Figs. 8(a)-8(c) we represent the vorticity ﬁelds of the uncontrolled ﬂow, the optimally controlled ﬂow, and the basic ﬂow, respectively. The signiﬁcant vortex-shedding phenomenon observed in Fig. 8(a) has been substantially reduced when the control is applied and the ﬂow has been quasisymmetrized. The resulting ﬂow approaches the symmetric state characteristic of the corresponding basic ﬂow as can be awaited from the results presented in [8] and the discussion in [5]. Our results are qualitatively similar to the eﬀects observed in [20] and [9] and conﬁrm the arguments of [21] that the mean drag reduction is associated with control driving the mean ﬂow toward the unstable state.

4

Conclusions

The Trust-Region POD algorithm originally introduced in [6] was used to minimize the total mean drag coeﬃcient of a circular cylinder wake ﬂow in the laminar

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regime (Re = 200). The key enablers to an accurate and robust POD ROM for the pressure and velocity ﬁelds were the extension of the POD basis functions to the pressure data, the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem, and the inclusion in the POD ROM of diﬀerent non-equilibrium modes. Finally, the optimal control parameters obtained with the TRPOD algorithm are A = 4.25 and Stf = 0.738. The relative mean drag reduction is equal to 30%. Compared to previous similar studies where the Navier-Stokes equations are used as state equations in the optimal control problem, the main advantage of the TRPOD algorithm is that it leads to a signiﬁcant reduction of the numerical costs because the optimization process itself is completely based on reduced-order models only. Indeed, when the state equations of the optimality system are POD ROMs instead of the Navier-Stokes equations, a cost reduction factor of 1600 is obtained for the memory and the optimization problem is solved approximately 4 times more quickly. Now, if we compare to our preceding study [5], where a POD ROM was coupled to an optimal control approach without any strategy for updating the reduced-order model during the optimization process, the cost reduction factors, found here, are lower. However, in this study, the use of the TRPOD algorithm mathematically proves that the solutions converge at least to a local optimum for the original high-ﬁdelity problem, and less than ten resolutions of the Navier-Stokes equations are necessary. Due to the low computational costs involved in the optimization process and the mathematical proofs of global convergence, the TRPOD algorithm is a promising method of optimization in ﬂow control. This approach that can easily be adapted to other conﬁgurations, should ﬁnally lead to the current resolution of unsteady, three-dimensional optimization problems for turbulent ﬂows around complex geometries.

Acknowledgments The authors acknowledge M. Braza (Institut de M´ecanique des Fluides de Toulouse) and D. Ruiz (ENSEEIHT) to kindly provide us with an original version of their Matlab Navier-Stokes solver. Stimulating and fruitful discussions with the low-dimensional modelling and control team at the Technische Universit¨at Berlin, in particular Bernd R. Noack, are acknowledged.

References [1] Gunzburger, M.: Flow control. Springer, New York (1995) [2] Gunzburger, M.: Introduction into mathematical aspects of ﬂow control and optimization. In: Lecture series 1997-05 on inverse design and optimization methods. Von K´ arm´ an Institute for Fluid Dynamics (1997) [3] Cordier, L., Bergmann, M.: Proper Orthogonal Decomposition: an overview. In: Lecture series 2002-04 on post-processing of experimental and numerical data. Von K´ arm´ an Institute for Fluid Dynamics (2002)

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[4] Cordier, L., Bergmann, M.: Two typical applications of POD: coherent structures eduction and reduced order modelling. In: Lecture series 2002-04 on postprocessing of experimental and numerical data. Von K´arm´ an Institute for Fluid Dynamics (2002) [5] Bergmann, M., Cordier, L., Brancher, J.P.: Optimal rotary control of the cylinder wake using POD Reduced Order Model. Phys. Fluids 17 (2005) 097101:1–21 [6] Fahl, M.: Trust-Region methods for ﬂow control based on Reduced Order Modeling. PhD thesis, Trier university (2000) [7] Williamson, C.: Vortex dynamics in the cylinder wake. Ann. Rev. Fluid. Mech. 28 (1996) 477–539 [8] Protas, B., Wesfreid, J.: Drag force in the open-loop control of the cylinder wake in the laminar regime. Phys. Fluids 14 (2002) 810–826 [9] He, J.W., Glowinski, R., Metcalfe, R., Nordlander, A., P´eriaux, J.: Active control and drag optimization for ﬂow past a circular cylinder. Part 1. Oscillatory cylinder rotation. J. Comp. Phys. 163 (2000) 83–117 [10] Homescu, C., Navon, I., Li, Z.: Suppression of vortex shedding for ﬂow around a circular cylinder using optimal control. Int. J. Numer. Meth. Fluids 38 (2002) 43–69 [11] Bergmann, M., Cordier, L.: Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models. Soumis `a J. Fluid Mech. (2006) [12] Conn, A., Gould, N., Toint, P.: Trust-region methods. SIAM, Philadelphia (2000) [13] Noack, B., Afanasiev, K., Morzy´ nski, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497 (2003) 335–363 [14] Galletti, B., Bruneau, C.H., Zannetti, L., Iollo, A.: Low-order modelling of laminar ﬂow regimes past a conﬁned square cylinder. J. Fluid Mech. 503 (2004) 161–170 [15] Bergmann, M.: Optimisation a´erodynamique par r´eduction de mod`ele POD et contrˆ ole optimal. Application au sillage laminaire d’un cylindre circulaire. PhD thesis, Institut National Polytechnique de Lorraine, Nancy, France (2004) [16] Graham, W., Peraire, J., Tang, K.: Optimal Control of Vortex Shedding Using Low Order Models. Part 1. Open-Loop Model Development. Int. J. for Numer. Meth. in Engrg. 44 (1999) 945–972 [17] Nocedal, J., Wright, S.: Numerical Optimization. Springer series in operations research (1999) [18] Bewley, T.: Flow control: new challenges for a new Renaissance. Progress in Aerospace Sciences 37 (2001) 21–58 [19] Bergmann, M., Cordier, L., Brancher, J.P.: On the generation of a reverse Von K´ arm´ an street for the controlled cylinder wake in the laminar regime. Phys. Fluids 18 (2006) 028101:1–4 [20] Tokumaru, P., Dimotakis, P.: Rotary oscillatory control of a cylinder wake. J. Fluid Mech. 224 (1991) 77–90 [21] Protas, B., Styczek, A.: Optimal rotary control of the cylinder wake in the laminar regime. Phys. Fluids 14 (2002) 2073–2087

Flow Control on the Basis of a F EATFLOW-M ATLAB Coupling Lars Henning1, Dmitri Kuzmin3, Volker Mehrmann2, Michael Schmidt2 , Andriy Sokolov3, and Stefan Turek3 1

Technische Universit¨at Berlin, Fachgebiet Mess- und Regelungstechnik, Hardenbergstr. 36a, 10623 Berlin, Germany [email protected] 2 Technische Universit¨at Berlin, Fachgebiet Numerische Mathematik, Str. des 17. Juni 136, 10623 Berlin, Germany {mehrmann,mschmidt}@math.tu-berlin.de 3 Universit¨at Dortmund, Fachbereich Mathematik, Vogelpothsweg 87, 44227 Dortmund, Germany {kuzmin,asokolow,stefan.turek}@math.uni-dortmund.de

Summary For the model-based active control of three-dimensional ﬂows at high Reynolds numbers in real time, low-dimensional models of the ﬂow dynamics and efﬁcient actuator and sensor concepts are required. Numerous successful approaches to derive such models have been proposed in the literature. We propose a software environment for a comfortable and performant testing of control, actuator and sensor concepts which may be based on such models. It is realized by providing an easily manageable Matlab control interface for the k-ε-model from the Featﬂow CFD package. Potentials and limitations of this tool are discussed by considering exemplarily the control of the recirculation bubble behind a backward facing step.

1 Introduction Active control of ﬂuid ﬂow is one of the major challenges in many key technologies, e.g. in chemical process engineering or aeronautics. In this paper we will discuss a model problem for such applications, the active control of the length of a recirculation bubble behind a backward facing step which is very difﬁcult to solve with current mathematical methods. Active control methods require the interaction of the controller with either the physical system (via measurements) or with a numerical simulation (via an observer). Here we only discuss the latter interaction. A major problem in designing real time controllers is that the observations have to be provided fast enough and that the computational method needed in the controller design is efﬁcient enough. These two requirements call for fast numerical simulation methods and small state dimensions in the dynamical system that describes the ﬂow. On the other hand the observed data have R. King (Ed.): Active Flow Control, NNFM 95, pp. 325–338, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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to be good enough to allow the computation of a good control, which typically means that the simulation method has to produce sufﬁciently accurate approximations of the observed data. In order to design such controllers it is therefore very important to use the most suitable model assumptions for this interactive task and to coordinate the interaction of controller design and numerical simulation by creating an efﬁcient interface between simulation and controller design. Such an efﬁcient coupling of current commercial CFD packages and modern control software requires detailed knowledge about the CFD code and how it can be coupled with the control software. For this reason we discuss the coupling of the CFD code F EATFLOW[1] with classical control techniques as they are available from M ATLAB[2] or S LICOT[3] toolboxes. In Section 2 we present as new feature a M ATLAB-control interface, which has been developped for F EATFLOW’s RANS solver with k-ε turbulence model and which is planned for F EATFLOW’s DNS and LES solvers (see Fig. 1). We present in this paper as example just the interface for the RANS solver, though the discussion in Section 4 will show that DNS or LES may be a better choice for certain control conﬁgurations. The active control of the length of the recirculation bubble by insufﬂation and suction at the edge of the backward facing step is described in Section 3. We will demonstrate how control strategies can be easily developed and implemented, only requiring minimal insight in the operation of the CFD code F EATFLOW. Finally, in Section 4 we discuss the potentials and limitations of this tool.

CFD with Featflow DNS (2D, 3D)

Control with Matlab easy implementation of − actuator concepts − sensor concepts − control laws and online visualization

RANS (3D) LES (3D)

Fig. 1. M ATLAB control interface for F EATFLOW CFD codes, based on Direct Numerical Simulation (DNS), Reynolds-averaged Navier-Stokes equations (RANS) and Large Eddy Simulation (LES)

2

New Features of F EATFLOW

2.1 A Realization of the k − ε Turbulence Model As a ﬁnite element 3D solver for the incompressible Navier-Stokes equations we use the CFD code F EATFLOW. The underlying numerical algorithm is based on (nonconforming) FEM discretizations, adaptive implicit time-stepping and (geometric) multigrid solvers on general domains. F EATFLOW is an open-source software package built on the FEAT2D and FEAT3D libraries written in FORTRAN77 [1].

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Dealing with three-dimensional ﬂows at high Reynolds numbers, the numerical costs of DNS (Direct Numerical Simulation) are extremely high. For instance, a 3D simulation of the backward facing step with Reynolds number Re = 30 000 requires about 9 Re 4 ≥ 1010 nodes to resolve the smallest eddies [4]. Therefore, a k-ε turbulence model was added to F EATFLOW, in order to make the calculation of such ﬂows on meshes of moderate size possible. The corresponding CFD code PP3D-KE was developed by D. Kuzmin building on the laminar F EATFLOW version (http://www.featﬂow.de). The mathematical foundations of the program can be described as follows. We consider the following system of Reynolds-averaged Navier-Stokes (RANS) equations: ∂v + v · ∇v = −∇p + ∇ · ((ν0 + νT )D(v)) , ∂t (1) ∇ · v = 0. Here v = v(t; x) with v = (vx , vy , vz )T ∈ R3 is a time-averaged velocity and p = p(t; x) ∈ R is a time-averaged pressure, both deﬁned on a time-space domain (0, T ) × Ω with T > 0 and Ω ⊂ R3 . D(v) = 12 ∇v + (∇v)T is the strain tensor and 2 νT = Cµ kε is the turbulent viscosity. The turbulent kinetic energy k = k(t; x) and its dissipation rate ε = ε(t; x) are modeled by the following scalar transport equations: ∂k νT + ∇ · kv − ∇k = Pk − ε, ∂t σk (2) ∂ε νT ε + ∇ · εv − ∇ε = (C1 Pk − C2 ε), ∂t σε k where Pk = ν2T |∇v + ∇vT |2 . The default values of the involved empirical constants are Cµ = 0.09, C1 = 1.44, C2 = 1.92 , σk = 1.0, σε = 1.3. Additionally, appropriate boundary conditions for v, k and ε have to be prescribed on ∂Ω = Γin ∪ Γout ∪ Γwall ∪ Γsym . As usual, Dirichlet boundary conditions for v, k and ε are prescribed on the inﬂow boundary Γin , k 3/2 v = g, k = cab |v|2 , ε = Cµ , (3) l0 where cab is an empirical constant being appropriate for the step conﬁguration [5], l0 is a mixing length and g is a given inﬂow velocity proﬁle. Let (t, n) be the local orthogonal basis for a wall node, where t and n are the tangential and normal directions, respectively. At the outlet Γout the following ’do-nothing’ boundary conditions are prescribed: ∂v = 0, ∂n

∂k =0 ∂n

∂ε = 0, ∂n

n · [pI − νT D(v)] = 0.

(4)

Convective (Robin) boundary conditions would be a physically convincing alternative choice. An implementation of this type of boundary conditions is currently in process.

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In the k − ε model the behavior of a ﬂuid near an impervious solid wall is modeled by wall functions. The computational wall boundary Γwall is located at a distance δ from the real geometrical wall boundary. In our case we assume that the computational domain is already reduced by a layer of width δ, which is a user-deﬁned parameter. We set the following boundary conditions on Γwall : n · v = 0,

(D(v)n · t)t = −

u2τ v , νT |v|

u3 k = τ Cµ

,

ε=

uτf , κδ

(5)

where κ = 0.41 is the von K´arm´an constant and uτ is the friction velocity. Here uτ is the solution of the logarithmic wall law equation: 1 |v| = uτ ( log y + + 5.5), κ where y + =

uτ δ ν

(6)

is the local Reynolds number.

In our numerical simulations of the backward facing step we try to avoid the inﬂuences of the side walls and the upper wall on the recirculation zone, therefore we impose symmetry boundary conditions on Γsym [6]: n · v = 0,

D(v) · n = 0.

(7)

The discretization in space is performed by an unstructured grid ﬁnite element method [7]. A detailed description of the numerical algorithm for the k − ε model can be found in [5], [8]. 2.2 A F EATFLOW-M ATLAB Control Interface Dealing with ﬂow control problems, we assume that we are able to inﬂuence the ﬂow in Ω by manipulating the Dirichlet boundary conditions in a subset Γctrl ⊂ Γwall , i.e. v(t; x(c) ) = u(t; x(c) )

for (t; x(c) ) ∈ (0, T ) × Γctrl ,

with a control or input function u(t; x). We assume further that we can observe or measure the ﬂuid’s velocity ﬁeld and/or pressure ﬁeld in subsets Ωmeas ⊂ Ω and/or Γmeas ⊂ Γ , i.e. we know the observation or output function y = y(t, x(m) ),

y = (v, p),

where (t, x(m) ) ∈ (0, T ) × (Ωmeas ∪ Γmeas ).

A (feedback) controller u(t, x(c) ) = f (t, y(s ≤ t, x(m) )) can then be used for the calculation of appropriate controls u, possibly on the basis of current and former observations y, in order to achieve some control objective. Aiming to implement such controllers comfortably in M ATLAB, but carrying out the ﬂow calculations in a perfomant manner with F EATFLOW, a F EATFLOW-M ATLAB control interface has been developed. The interaction between the PP3D-KE program and M ATLAB was realized with the help of the M ATLAB Engine (function engOpen), which operates by running in the background as an independent process. This offers several advantages: under UNIX,

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the M ATLAB engine can run on the user’s machine, or on any other UNIX machine on the user’s network, including those of a different architecture. Thus, the user can implement a user interface on his/her workstation and perform the computations on a faster machine located elsewhere on a network (see M ATLAB Help). The M ATLAB-controller part is realized as an m-ﬁle MatlabController.m. During the simulation the PP3D-KE code calls this M ATLAB function at every time step. The transaction phase consists of three stages: receiving the required data (geometry, velocity, pressure) from the output domain Γmeas , execution of MatlabController.m and calculation of a control u = (ux , uy , uz ), setting v = u as a Dirichlet boundary condition for velocity in the input domain Γctrl . So MatlabController.m has, in principle, the following interface: (c)

(m,v)

˜ (xi ) = MatlabController(tn , xj v (m,v)

Here, tn is the current time, xj

(m,p)

, xk

(m,p)

, xk

(m,v)

˜ (xj ,v

(m,p)

), p˜(xk

)),

are the coordinates of the velocity and pres(m,v)

(m,p)

˜ (xj sure nodes lying in Γmeas ∪ Ωmeas , and v ) and p˜(xk ) are the corresponding discrete velocity ﬁeld and pressure ﬁeld, respectively, all communicated by F EATFLOW to M ATLAB. After the control is executed on the basis of this information (and possi(c) ˜ (xi ) bly of similar data computed at previous time steps), the discrete velocity ﬁeld v (c) with respect to xi ∈ R3 , the coordinates of the velocity nodes lying in Γctrl , is communicated by M ATLAB to F EATFLOW. A detailed description of the subroutines, their interfaces and communication can be found in [10].

3

Controlling the Flow over a Backward Facing Step

3.1 A Benchmark Conﬁguration We illustrate the facilities and limitations of the new F EATFLOW-M ATLAB control interface by considering as example the ﬂow over a backward facing step. One of the main features of such ﬂows is a recirculation region just downstream of the step, and we aim to control its length by means of insufﬂation and suction at the edge of the step, see Fig. 2.

control

recirculation zone

shear layer

reattachment length

Fig. 2. Flow over a backward facing step (see [9])

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The uncontrolled ﬂow over the step is a classical benchmark for the validation of CFD codes, since it is well-understood analytically and extensive experimental results are available (see e.g. [11,12,13,14,15]). F EATFLOW’s PP3D-KE has also been validated for it [8]. The control of the recirculation length is the subject of numerous experimental, numerical and analytical studies, and has, e.g., been intensively investigated in the DFGCollaborative Research Center ”Control of complex turbulent shear ﬂows” (see e.g. [9,16,17,18,19,20,21]). We mention some model-based active control concepts for the step ﬂow from the literature, a more extensive survey can be found in [22]. Applying the theory of optimal control to the NSE, open loop controls for the step ﬂow are investigated and numerically calculated in [23,24,25]. In order to reduce the numerical costs and to obtain a more robust feedback-like control, suboptimal control strategies like instantaneous control are applied in [25,26,27,28]. Based on the full NSE, these approaches will mostly be limited in the near future to low Reynolds number ﬂows in two-dimensional domains. For the real-time control of three-dimensional ﬂows at high Reynolds numbers lowdimensional models are required. In [29,30,31,32], low dimensional Galerkin models are derived from the full NSE by means of the method of proper orthogonal decomposition (POD) on the basis of snapshots from a DNS. In [19,20] low-dimensional Galerkin models are derived by combining classical POD modes and physically motivated transition modes in order to better capture the ﬂow dynamics. In [21] the ﬂow dynamics are described by low-dimensional vortex models without that the NSE have to come into play. Finally, in [9] black-box models are identiﬁed on the basis of experimental and numerical data and used for the design of robust controllers which then worked in real-time experiments and in LES simulations, respectively. Whereas the approaches in [23,24,25,26,27,28,29,30,31,32] use general mathematical reduction methods and control concepts, the approaches in [19,20,21,9] use ﬂowspeciﬁc physical insight for the modeling and for the control concept. For instance, they make use of the so-called Kelvin-Helmholz instability of the shear layer above the recirculation zone with a characteristic frequency fshear . A systematic excitation of these instabilities can lead to a shortening of the recirculation zone, and provides the basis for efﬁcient and well-realizable controls via periodic suction and insufﬂation. It is not the purpose of this paper to propose another control strategy for the backward facing step. We will show to what extent F EATFLOW in connection with the k-ε-model and the M ATLAB control interface can provide a comfortable and performant software environment for testing such control concepts for three-dimensional and high Reynolds number ﬂows. And we will see that it may even be used to design simple but efﬁcient controls. 3.2 Implementation in F EATFLOW and M ATLAB As a speciﬁc test case, we consider the ﬂow over a step of dimensionless height h = 1. The inlet section has the length 5h and the wake section the length 20h. The height as well as the width of the domain are 3h. We choose x, y and z as coordinates for the downstream, the vertical and the spanwise direction, respectively, see Fig. 3.

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Γsym 25

Γ

3

in

2 y

h=1 z

x

Γwall

3 20

Γctrl

Ω meas

Γout

Γwall

Fig. 3. The computational domain

We assume that we can blow out and suck in ﬂuid in an angle of 45 degree in positive x-y-direction at a slot at the edge of the step of width 0.05h in the x and y direction, i.e. 1 v(t; x, y, z) = √ (u(t), u(t), 0)T 2

on (0, T ) × Γctrl ,

where u(t) is a scalar control function that can be freely varied in time and Γctrl = {(x, y, z) ∈ Γ : 4.95 ≤ x ≤ 5, 0.95 ≤ y ≤ 1, 0 < z < 3}. Note that the implementation of distributed vector-valued controls v(t; x, y, z) = u(t; x, y, z) is also possible. The length of the recirculation zone is deﬁned via the reattachment position xr of the shear layer detaching at the edge of step. For each z, xr (t, z) is deﬁned by a zero wall shear stress τw (t; x, z) = 0, with τw (t; x, z) = η

∂vx ∂y

|(t;x,y=0,z) ,

(8)

where η denotes the viscosity [9]. We will determine τw (t; x, z) by measuring vx (t; x, y, z) in the domain Ωmeas = {(x, y, z) : 5 < x < 20,

0 < y < 0.125,

0 < z < 3}

and deﬁne a scalar reattachment length xr (t) by averaging τw (t; x, z) in spanwise direction. The ﬂow is governed by the dimensionless NSE (1). Prescribing a bulk ﬂow proﬁle v∞ = (1, 0, 0)T at the inﬂow boundary, we apply the boundary conditions v(t; x, y, z) = v∞ ∂v (t; x, y, z) = 0 ∂z ∂v (t; x, y, z) = 0 ∂n v(t; x, y, z) = 0

on Γin

(inhom. Dirichlet),

(9a)

on Γout

(hom. Neumann),

(9b)

on Γsym , (symmetry),

(9c)

on Γwall , (no slip).

(9d)

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Fig. 4. Mesh on multigrid level 3 (22848 elements)

We now discuss the numerical implementation of the mathematical model. Dealing with nonstationary high Reynolds number ﬂows in a three-dimensional domain, we use the RANS-solver PP3D-KE from the F EATFLOW. We have to provide a coarse mesh as basis for F EATFLOW’s multigrid solvers and use a mesh which is locally reﬁned near the edge of the step and near the ﬂoor of the expected recirculation zone, see Fig. 4. The boundary conditions (9), actuator positions Γctrl and sensor positions Ωmeas are easily speciﬁed in a F EATFLOW data ﬁle. In each time step of the simulation, the M ATLAB routine MatlabController.m receives from PP3D-KE the discrete velocity ﬁeld ˜ x (tn ; xi , yj , zk ) with respect to the mesh nodes (xi , yj , zk ) lying in Ωmeas . Correv sponding to (8) we approximate the wall shear stress in (xi , 0, zk ) by τ˜w (tn ; xi , zk ) = ˜ x (tn , xi , yj , zk )/yj and average for each xi over all corresponding zk . We then deﬁne v xr (tn ) as a reasonable zero of the polynomial ﬁtting of τ˜w (xi ), see Fig. 5. The implementation of open loop controls u(t) = f (t) or closed loop controls u(t) = f (t, xr (s ≤ t)) for given control laws f in the M ATLAB function is straight ˜ (tn , xi , yj , zk ) in all mesh velocforward by prescribing the corresponding values of v ity nodes (xi , yj , zk ) lying on Γctrl .

0.15 0.1

τw

0.05 0 −0.05 −0.1 0

5

10

15

x/h

Fig. 5. Approximated wall shear stress distribution τ˜w (x) of the steady state and the ﬁtting polynomial, with x denoting the distance to the step. The oscillations are mainly due to the numerical approximation of τw .

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3.3 Numerical Simulation and Control Results As an example, we aim to design a feedback controller providing controls u(t) such that the resulting reattachment length xr (t) tracks a given reference length xref (t). First the uncontrolled ﬂow is simulated with Reh = 30 000, which leads to a steady state solution. Here Reh is the Reynolds number with respect to the height h = 1 of the step and the inﬂow velocity |v∞ | ≡ 1. As initial data, corresponding steady state solutions from calculations with lower Reynolds numbers are taken. Figures 6 and 7 show the vx (x, z) and the k(x, z) distributions in the x − y−cut-plane at z = 1.5. The typical recirculation zone can be observed, as well as the typical second vortex in the bottom corner of the step. The steady τ˜w (x) distribution and the resulting reattachment length xr /h = 6.21 are shown in Figure 5.

Fig. 6. Downstream velocity component vx (with isolines) of the steady state solution in x-ycutplane at z = 1.5

Fig. 7. Turbulent kinetic energy k of the steady state solution in x-y-cutplane at z = 1.5

However the typical shear layer instabilities with the characteristic frequency fshear and thus a time-oscillation of the reattachment length cannot be observed. This is explained by the time-averaged character of the k − ε turbulence model and by the fact, that the near wall zone at the step (where shear layer instabilities originate) is not calculated directly, but is modeled by logarithmic wall functions, which are derived from the boundary layer theory. The numerical results for the backward facing step and its comparison with others can be found in [8], [6] and [33]. We only observe an excessive turbulent kinetic energy in the shear layer zone. Next the system’s response to a number of open loop controls is calculated with F EATFLOW’s PP3D-KE and via the F EATFLOW-M ATLAB interface, which provides

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sufﬁcient information to identify black-box models of the system’s input/output behavior and to design a robust closed-loop control for the reattachment length. In fact, classical step experiments are performed. Thereby the actuation amplitude u(t) is switched from zero to different levels a0 to obtain different operating points of the system, see Fig. 8. A family of linear time-continuous models of 4th order is ﬁtted to the measured data by application of subspace methods [34]. The corresponding bode responses of the identiﬁed transfer functions Gi (s) are given in Fig. 9. To synthesize a robust controller C(s), a H∞ -controller design scheme is chosen. In H∞ -control, stability and/or performance of the ’worst’ plant used to describe the process can be guaranteed. The goal is to ﬁnd a good trade-off between the closedloop sensitivity function S(s), giving the performance, the restriction of the magnitude of the plant input signals, given by the transfer function C(s)S(s), and robustness, given by the complementary sensitivity T (s) = 1 − S(s). Here, the sensitivity transfer

6.3 xr(t)

6.2 6.1 6 5.9 0

0.5

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time t

u(t)

0.4 0.3 0.2 0.1 0

0

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Fig. 8. Step responses of the plant with damped oscillations of the control output

Magnitude [dB]

0

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−60 0 10

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180

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Fig. 9. Bode responses of all identiﬁed models

4

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d r= -xcref-6

C(s)

u

-

? Plant

y= xr

Fig. 10. Control-loop with H∞ -controller C(s) (xref - reference, u - manipulated variable, d disturbances, y = xr - output)

function is given by S(s) = 1/(1+C(s)GN (s)) with GN being a nominal model showing minimal distance to all identiﬁed models over a certain frequency range. The tradeoff is achieved by solving the mixed sensitivity problem, i.e. the closed-loop transfer functions are weighted with WT (s), WCS (s) and WS (s) depending on the frequency, and then the optimal controller C(s) is obtained by minimizing the combined cost functional ⎤ ⎡ WT (s) T (s) min N (C(s)) ∞ , N = ⎣ WCS (s) C(s)S(s) ⎦ . C WS (s) S(s) For more details on H∞ -control the reader is referred to standard textbooks, e.g. [35]. The synthesized H∞ -controller is added in a classical control-loop as shown in Fig. 10. For the implementation into the F EATFLOW-M ATLAB-interface the controller transfer function C(s) is converted into a discrete state-space form. The successful operation of the H∞ -controller is illustrated in Fig. 11, where a tracking of a reference trajectory xref (t) by the reattachment length xr (t) is performed.

4 Discussion By using the newly developed PP3D-KE module from the CFD code F EATFLOW for the ﬂow calculations, the simulation of nonstationary ﬂows at high Reynolds numbers and in three-dimensional domains becomes feasible. The implementation of the ﬂow and control conﬁguration about easily manageable F EATFLOW data ﬁles and M ATLAB m-ﬁles allows to easily test different actuator, sensor and control concepts and may even be a tool for the development of controllers, and only minimal insight into the CFD-code is required. In this sense, the presented F EATFLOW-M ATLAB coupling can be considered as a general purpose tool for ﬂow control purposes. However, the performance of this tool is also subject to some substantial restrictions. Using the RANS approximation of the NSE with k − ε turbulence model, some physically important ﬂow phenomena may be no longer resolved. This is for instance often the case for ﬂow conﬁgurations with detaching and separating ﬂows, since the wall-layer models in the turbulence models are no longer valid. In our example of the backward facing step, the Kelvin-Helmholz shear layer instabilities cannot be observed in the numerical results. However, these instabilities may be the basis for very efﬁcient control concepts, i.e. the control of the re-attachment length by enhancing the shear

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9 xref(t) xr(t)

8

7

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0.1

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0

−2

−4

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Fig. 11. Tracking response of the closed-loop system. Stepwise changings of the reference variable show oscillations of the controlled output due to the fast controller performance. This effect does not occur after rampwise changing of the reference variable in the time-intervall [0.4s, 0.7s].

layer instabilities via a harmonic actuation with about the characteristic frequency of the instabilities, see [9]. Furthermore, the simulated ﬂow is basically two-dimensional and the simulated transfer of inputs (insufﬂation amplitude) to outputs (reattachment length) is linear, though experiments and LES-simulations show distinct three-dimensional structures and a nonlinear input/output behavior. Here Large Eddy Simulations (LES) prove to be the better choice in order to numerically observe and simulate these physical phenomena [9]. The presented ﬂow control environment based on F EATFLOW’s k-ε model aims to present one tool to tackle nonstationary three-dimensional ﬂow control problems. The example of the backward facing step clearly shows that the simultaneous use of different models and experiments for simulation, control design, test and validation purposes is recommended. Therefore the development of M ATLAB-interfaces for the DNS and LES solvers in the F EATFLOW package is an important future task.

References [1] S. Turek et al.: ”FEATFLOW. Finite element software for the incompressible Navier-Stokes equations: User Manual, Release 1.2”, 1999. (www.featﬂow.de) [2] The MathWorks, Inc., Cochituate Place, 24 Prime Park Way, Natick, Mass, 01760. M ATLAB Version 7.0.4.352 (R14) , 2005.

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[3] P. Benner, V. Mehrmann, V. Sima, S. Van Huffel und A. Varga, ”SLICOT-A Subroutine Library in Systems and Control Theory”, Applied and Computational Control, Signals and Circuits. 1, 1999, pp. 499-532. [4] M. Griebel, T. Dornseifer, T. Neunhoeffer: ”Numerical Simulation in Fluid Dynamics: A Practical Introduction”. SIAM Monographs on Mathematical Modeling and Computation, 1998. [5] D. Kuzmin, S. Turek: ”Numerical simulation of turbulent bubbly ﬂows”. 3rd International Symposium on Two-Phase Flow Modelling and Experimentation 2004. Editors: G.P. Celata, P.Di Marco, A. Mariani, R.K. Shah. Pisa, 2004. [6] G. Medi´c, B. Mohammadi: ”NSIKE - an incompressible Navier-Stokes solver for unstructured meshes”. INRIA Research Report No 3644, 1999. [7] S. Turek: ”Efﬁcient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach”. LNCSE 6, Springer, 1999. [8] D. Kuzmin, R. L¨ohner and S. Turek (Eds.), ”Flux-Corrected Transport”. Springer, 2005. [9] R. Becker, M. Garwon, C. Gutknecht, G. B¨arwolff, R. King: ”Robust control of separated shear ﬂows in simulation and experiment”. Journal of Process Control 15, 2005, pp. 691700. [10] M. Schmidt, A. Sokolov: ”A F EATFLOW-M ATLAB-Coupling for Flow Control: User Manual”, 2006. (www.featﬂow.de) [11] B.F. Armaly, F. Durst, J.C.F. Pereira, B. Schonung: ”Experimental and theoretical investigation of backward-facing step ﬂow”. J. Fluid Mech. 127, 1983, pp. 473-496. [12] K. Gartling: ”A Test Problem For Outﬂow Boundary Conditions - Flow Over a Backward Facing Step”. International Journal for Numerical Methods in Fluids 11, 1990, pp. 953-967. [13] J. Kim, P. Moin: ”Application of a Fractional-Step Method to Incompressible Navier-Stokes Equation”. Journal of Computational Physics, 59, 1985 pp. 308-314. [14] H. Le, P. Moin, J. Kim: ”Direct numerical simulation of turbulent ﬂow over a backwardfacing step”. J. Fluid Mech. 330, 1997, pp. 349-374. [15] L. Kaiktsis, G. E. Karniadakis, S. A. Orszag: ”Onset of Three-Dimensionality, Equilibria, and Early Transition in Flow over a Backward facing Step”. Journal of Fluid Mechanics, 231, 1991, pp. 501-528. [16] T. Weinkauf, H.-C. Hege, B.R. Noack, M. Schlegel, A. Dillmann: ”Coherent Structures in a Transitional Flow around a Backward-Facing Step”. Physics of Fluids, 15(9), 2003. [17] M. Garwon, R. King: ”A multivariable adaptive control strategy to regulate the separated ﬂow behind a backward-facing step”. 16th IFAC World Congress, Prague, Czech Republic, July 2005. [18] L. Henning, R. King: ”Multivariable closed-loop control of the reattachement length downstream of a backward-facing step”. 16th IFAC World Congress, Praha, Czech Republic, July 2005. [19] J. Gerhard, M. Pastoor, R. King, B.R. Noack, A. Dillmann, M. Morzynski and G. Tadmor: ”Model-based control of vortex shedding using low-dimensional Galerkin models”. AIAAPaper 2003-4262, 2003. [20] O. Lehmann, D.M. Luchtenburg, B.R. Noack, R. King, M. Morzynski, G. Tadmor: ”Wake stabilization using POD Galerkin models with interpolated modes ”, Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005, Invited Paper 1618, 2005. [21] M. Pastoor, R. King, B.R. Noack, A. Dillmann and G. Tadmor: ”Model-based coherentstructure control of turbulent shear ows using low-dimensional vortex models”. AIAAPaper 2003-4261, 2003. [22] M. Hinze: ”Optimal and instantaneous control of the instationary Navier-Stokes equations”. Habilitation thesis, Fachbereich Mathematik, Technische Universit¨at Berlin, 2000.

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[23] R. Glowinski: ”Finite element methods for the numerical simulation of incompressible viscous ﬂow; Introduction to the Control of the Navier-Stokes Equations”. Lectures in Applied Mathematics, 28, 1991. [24] M. D. Gunzburger, S.Manservisi: ”The velocity tracking problem for Navier-Stokes ﬂows with boundary controls”. Siam J. Control and Optimization, 39, 2000, pp. 594-634. [25] M. Hinze, K. Kunisch: ”Control strategies for ﬂuid ﬂows - optimal versus suboptimal control”. In H.G.Bock et al., editor, ENUMATH 97, 1997, pp. 351-358. [26] M. Hinze, K. Kunisch: ”Suboptimal Control Strategies for Backward Facing Step Flows”. Proceedings of the 15th IMACS World Congress on Scientiﬁc Computation, Modelling and Applied Mathematics, Editor A. Sydow, Vol. 3, 1997, pp. 53-58. [27] H. Choi, M. Hinze, K. Kunisch: ”Instantaneous control of backward-facing step ﬂows”. Appl. Numer. Math. 31, 1999, pp. 133-158. [28] T. R. Bewley, P. Moin, R. Temam: ”DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms”. J. Fluid Mech., 447, 2001, pp. 179-225. [29] K. Afanasiev, M. Hinze: ”Adaptive control of a wake ﬂow using proper orthogonal decomposition”. In Shape Optimization & Optimal Design, Lecture Notes in Pure and Applied Mathematics 216. Marcel Dekker, 2001. [30] M. Hinze, K. Kunisch: ”On suboptimal Control Strategies for the Navier-Stokes Equations”. ESAIM: Proceedings 4, 1998, pp. 181-198. [31] K. Ito, S. S. Ravindran: ”Reduced basis method for optimal control of unsteady viscous ﬂow”. Int. J. Comput. Fluid Dyn., 15, 2001, pp. 97-113. [32] S. S. Ravindran: ”Control of ﬂow separation over a forward-facing step by model reduction.” Comput. Methods Appl. Mech. Engrg. 191, 2002, pp. 4599-4617. [33] B. Mohammadi, O. Pironneau: ”Analysis of the k-epsilon turbulence model”. Wiley, 1994. [34] L. Ljung: ”System identiﬁcation - Theory for the user.” Prentice Hall PTR, 1999. [35] S. Skogestad, I. Postlethwaite: ”Multivariable feedback control - Analysis and design”. 2 edn. John Wiley & Sons, Ltd, 2005.

On the Choice of the Cost Functional for Optimal Vortex Reduction for Instationary Flows Karl Kunisch1 and Boris Vexler2 1

2

University of Graz, Institute for Mathematics, Heinrichstraße 36, A-8010 Graz, Austria [email protected] Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria [email protected]

Summary In this paper we discuss an appropriate choice of a cost functional for vortex reduction for unsteady ﬂows described by the Navier-Stokes equations. We discuss different possibilities for a deﬁnition of a vortex and for a corresponding cost functionals. The resulting optimal control problem is analyzed and numerical experiments are provided.

1 Introduction This work focuses on the choice of proper cost-functionals in optimal control formulations for vortex reduction in incompressible ﬂuids. The formalization of vorticity is still a major challenge and subject of intense research within ﬂuid mechanics research itself. In the context of optimal control the quantiﬁcation must satisfy the additional requirement that it allows the description of vorticity as a scalar-valued functional in terms of observables of the ﬂuid. Moreover the mathematical properties of the functional have signiﬁcant consequences for mathematical programming considerations and for the numerical realization of the resulting optimization problems. Let us at ﬁrst summarize some of the cost functionals that were already used in the optimal control literature to formulate vortex reduction problems. We denote by y(t, x) the velocity vector and by p(t, x) the pressure of an incompressible ﬂuid which extends over the time horizon [0, T ] and the spatial domain Ω ⊂ Rd , d = 2, 3. A prototypical optimal control problem with (Dirichlet) boundary control is formulated as follows: Minimize J(y, p) + G(u) R. King (Ed.): Active Flow Control, NNFM 95, pp. 339–352, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

(1)

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subject to the state equation ⎧ ⎪ ⎪ yt − ν∆y + y · ∇y + ∇p = f in (0, T ] × Ω, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −div y = 0 in (0, T ] × Ω, ⎪ ⎪ y(0, ·) = y0 on Ω, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˆ on (0, T ] × ∂Ω. y = Bu

(2)

ˆ is a control operator Here, u denotes a control variable from a control space U and B mapping the control variable to the space of admissible Dirichlet boundary condition, see e.g. [HK] and the references given there. The data ν > 0, f, y0 are given. The two parts of the cost functional J and G are real-valued functionals vorticity and control-action respectively. A typical choice of the functional G is G(u) = α2 |u|2 , where | · | denotes an appropriate norm on the control space with α ≥ 0. For rigorous frameworks for boundary control of the Navier-Stokes equations, we refer to [FGH] and [HK], for example. Frequently used choices for the functional J are: T J(y, p) = J1 (y) =

|y(t, x) − ydes (t, x)|2 dx dt,

(3)

|curl y(t, x)|2 dx dt.

(4)

0 Ω

T J(y, p) = J2 (y) = 0 Ω

In the deﬁnition of J1 (y), ydes stands for a given desired ﬂow ﬁeld which contains some of the expected features of the controlled ﬂow ﬁeld without the undesired vortices. Typically ydes is chosen as the solution to the Stokes problem on the same ﬂow geometry. This functional is referred to as tracking-type functional. For the second functional J2 (y), we deﬁne curl y(t, x) for the dimension d = 2, 3 in a standard way. This function is used e.g. in [AT,G]. One of the objections against both these functionals is that they are not Galilean invariant, i.e. they are not invariant under frame transformations of the form Qx + d t of the ﬂow ﬁeld y, where Q is a time-independent matrix and d is a constant vector. Another objection against the functional J2 (y) is the fact, that the value of this functional can be large in the parts of the domain with shear ﬂow. For instance in Ω ⊂ R2 with ﬂow given by y1 = ax2 (1 − x2 ), y2 = 0, we have that J2 (y) = 0 and J2 (y) is proportional to a. We will illustrate this fact in our numerical example. There is a number of publications discussing the choice of a Galilean invariant deﬁnition of a vortex, e.g, [CPC,BMC,JH,O,W,HWM] and the references cited therein. In case the spatial domain Ω is two-dimensional, several formally different criteria are equivalent to “det”-criterion, which deﬁnes a vortex region as a region

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where det ∇y(t, x) > 0. Based on this criterion, one may deﬁne a cost functional J as T J(y, p) = J3 (y) = g3 (det ∇y) dx dt, (5) 0 Ω

where g3 ∈ C (R) is a smoothing of the function max(0, x), chosen for example as: ) 0, t≤0 t3 . g3 (t) = , l(t) = 2 t +1 l(t), t > 0 2

A similar cost-functional was used for optimal vortex-reduction in a driven-cavity problem in [HKSV]. Galilean invariant vortex criteria allow a classiﬁcation which is invariant under frame changes that move at constant speed relative to each other. In a variety of different theoretical and example driven approaches, see e.g. [H1], [H2], [LHK], [LKH], [TK], it was established that Galilean invariance is not sufﬁcient for reliable vortex identiﬁcation. Rather criteria must be invariant also under coordinate transformations of the form Q(t)x + d(t), where Q is a time-dependent orthogonal matrix and d a time-dependent velocity vector. Such transformations are called objective in continuum mechanics and in particular they allow time-dependent rotations. In [LKH] an objective criterion is obtained in 2-D which deﬁnes a rotation dominated region by means of |r(y, p)| > 1 (6) where r(y, p) =

ω σs (px1 x1 − px2 x2 ) − 2σn px1 x2 − , 3 σ σ2

(7)

and ω = (y2 )x1 − (y1 )x2 , σs = (y2 )x1 + (y1 )x2 , σn = (y1 )x1 − (y2 )x2 , σ = (σs2 + σn2 )1/2 . It can readily be used for vortex reduction by introducing the cost-functional T J(y, p) = J4 (y, p) =

g4 (r(y, p)) dx dt,

(8)

0 Ω

with g4 ∈ C 2 (R) given e.g. by: ⎧ ⎪ ⎨l(−t − 1), t < −1, g4 (t) = 0, −1 ≤ t ≤ 1 , ⎪ ⎩ l(t − 1), t>1

l(t) =

t3 . t2 + 1

In [TK] a 2d objective criterion is introduced, which is shown to be equivalent to (6). Another 2d and 3d objective criteria based on Lyapunov functionals are introduced in [H1,H2]. In this paper we shall show the practical efﬁciency of the objective functional (8) for vortex reduction. We shall further conduct a comparison among the four functionals (3), (4), (5) and (8). Optimal control problems based on these four functionals can give

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surprisingly different results. A comparison among different cost functionals is, at ﬁrst, impeded by the following difﬁculty: As indicated in the prototype problem (1,2) usually a term G(u) representing control costs is utilized. Mathematically it guarantees a-priori bounds on minimizing sequences for (1,2) and subsequently existence of a minimizer to (1,2). The optimal solution depends on G and therefore, if J is taken as one of the four functionals (3), (4), (5) or (8), the question must be addressed, how to eliminate the effect of the control-cost term on the solutions of these optimal control problems. Here we take the approach that we eliminate G altogether. As a consequence, we have to consider a-typical existence problems for optimal control problems with the Navier-Stokes equations as constraints. In fact, there are no obvious a-priori bounds for the control. In general, one can not guarantee the existence of a solution for such problems. Therefore we restrict ourselves to the consideration of ﬁnite dimensional control spaces only. This will allow us to show the existence of solutions to (1,2) with G(u) ≡ 0. Another reason for consideration of ﬁnite dimensional control spaces is that in the practice the possibilities for the changes in the control variable are often restricted to a ﬁnite dimensional space. The outline of the paper is as follows: In the next section we discuss the formulation of vortex reduction as an optimal control problem in more details, provide existence results and discuss the optimality conditions. In section 3 we describe some algorithmic aspects on the continuous level and present the space-time ﬁnite element discretization in Section 4. Numerical examples for a channel ﬂow with an obstacle are given in Section 5.

2

Optimal Control Problem

In this section we formulate the optimal control problem for vortex reduction more detailed, discuss the existence of solutions and the optimality conditions. For given spacial domain Ω ⊂ R2 and the time interval I = (0, T ), the space-time cylinder is denoted by Q = (0, T )×Ω. Moreover, we make use of the following spaces: V = H 1 (Ω)2 ,

V0 = H01 (Ω)2 ,

L = L2 (Ω)/R,

for an arbitrary space Y we abbreviate L2 (Y ) = L2 (0, T ; Y ) and set * + * + W = w ∈ L2 (V ) : wt ∈ L2 (V ∗ ) , W0 = w ∈ L2 (V0 ) : wt ∈ L2 (V0∗ ) , (9) where V ∗ and V0∗ denote the dual spaces of V and V0 respectively. The pressure space P is deﬁned as follows: +∞ , P = r ∈ D (Q) : R(t) = r(s) χ[0,t] (s) ds,

R ∈ C(0, T ; L) ,

−∞

where χ[0,t] is the characteristic function of [0, t]. The linear space P is a subset of H −1 (L). Moreover we introduce the space of admissible Dirichlet boundary conditions WΣ , see [HK]: WΣ = {g = τ gˆ : gˆ ∈ W } ,

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where τ : W → L2 (H 1/2 (Ω)2 ) is the trace operator onto the lateral boundary Σ = (0, T ) × ∂Ω of the cylinder Q. The scalar product in L2 (Ω) is denoted by (·, ·). ˆ : U → WΣ is a Let U ∼ = Rn (n ∈ N) be a ﬁnite dimensional control space, and B linear control operator satisfying: ˆ · n = 0 a.e. in (0, T ) for all u ∈ U. Bu ∂Ω

ˆ has the following form: Due to linearity the operator B ˆ = Bu

n

ui ψˆi ,

ψˆi ∈ WΣ .

i=1

ˆ to be injective, i.e. the functions {ψˆi } are linearly Throughout we assume the operator B independent. To set up a weak formulation of the state equation we introduce the state space X = W × P and X0 = W0 × P . For x = (y, p) ∈ X and ζ = (ψ, ξ) ∈ X0 we deﬁne the semi-linear form a : X × X0 → R as follows:

T

{(yt , φ) + ν(∇y, ∇ψ) + (y∇y, ψ) − (p, div ψ) − (f, ψ)} dt

a(x)(ζ) = 0

T

(div y, ξ)} dt + (y(0) − y0 , ψ(0)).

+ 0

ˆ Moreover, we need a continuous prolongation B : U → X of the control operator B with the property: ˆ for all u ∈ U, Bu = (BW u, 0) with τ (BW u) = Bu where τ is the trace operator. The operator BW can be deﬁned, for example by prolongation of {ψˆi }, i = 1, 2, . . . n using time-dependent Stokes equations. Then the corresponding weak formulation of (2) is given by: Find x ∈ Bu + X0 such that: a(x)(ζ) = 0 for all ζ ∈ X0 . (10) Let J : X → R be one of the cost functionals J1 , J2 , J3 and J4 from the previous section, and consider the optimization problem in the form: Minimize J(x) subject to (10),

x ∈ Bu + X0 , u ∈ U.

(11)

In [KV] we discuss the existence of optimal solutions for this atypical optimal control problem without control costs. We prove the following theorem: Theorem 1. There exists a solution for the optimal control problem (11) for both choices of the cost functional J = J1 and J = J2 . Moreover, in [KV] we discuss the extendibility of this result to other cost functionals.

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It is well known, see e.g. [HK] or [FGH], that the ﬁrst order necessary optimality conditions for problem (11) are: x ∈ Bu + X0 : a(x)(ζ) = 0 for all ζ ∈ X0 , (12) (13) z = (λ, π) ∈ X0 : a (x)(ζ, z) = J (x)(ζ) for all ζ ∈ X0 , T u∈U : (ν∇λ · n − π · n)BW v ds dt = 0 for all v ∈ U. (14) 0 ∂Ω

The direct discretization of this optimality system does not lead to a discrete scheme of optimal order of convergence with respect to the maximal cell-size h, see [V]. To avoid this, we use similar techniques as in [V] and base our discretization on the following (equivalent) formulation of the optimality system: x ∈ Bu + X0 : a(x)(ζ) = 0 for all ζ ∈ X0 , z = (λ, π) ∈ X0 : a (x)(ζ, z) = J (x)(ζ) for all ζ ∈ X0 , u ∈ U : J (x)(Bv) − a (x)(Bv, z) = 0 for all v ∈ U.

(15) (16) (17)

This system can be obtained from (12) – (14) by integration by parts. We emphasize, that these two systems are equivalent only on the continuous level. The discretized versions are not equivalent. Note that the integration over ∂Ω in (14) is replaced by the integration over Ω in (17), which leads to better approximation properties of the ﬁnite element discretization.

3 Optimization Algorithms In this section we brieﬂy discuss the optimization algorithm based on Newton’s method on the continuous level. Since a ﬁnite element discretization is used, the continuous algorithm can then be simply translated to a discrete one. This is well known, see e.g. [T], that there exists a continuous solution operator S : U → X for the state equation (10) such that: S(u) ∈ Bu + X0 : a(S(u))(ζ) = 0 for all ζ ∈ X0

for all u ∈ U.

This gives rise to the introduction of a reduced cost functional j : U → R by: j(u) = J(S(u)),

(18)

and allows us to reformulate the optimization problem (11) as the unconstraint problem: Minimize j(u),

u ∈ U.

(19)

For the application of Newton’s method to this optimization problem, we have to compute the derivatives of the reduced cost functional j. This is done in the following proposition: Proposition 1. The reduced cost functional j deﬁned in (18) is continuously differentiable and its directional derivatives can be expressed as follows:

On the Choice of the Cost Functional for Optimal Vortex Reduction

345

(a) For an arbitrary direction δu ∈ U we have: j (u)(δu) = J (x)(Bδu) − a (x)(Bδu, z) where x = S(u) is the solution of the state equation (10) and z ∈ X0 is the solution of the adjoint equation (16). (b) For arbitrary directions δu, τ u ∈ U we have: j (u)(δu, τ u) = J (x)(δx, Bτ u) − a (x)(δx, Bτ u, z) − a (x)(Bτ u, δz), where z ∈ X0 is the solution of the adjoint equation (16), δx ∈ X is determined by the tangent equation: δx ∈ Bδu + X0 : a (x)(δx, ζ) = 0

for all ζ ∈ X0 ,

(20)

and δz ∈ X0 is the solution of the dual Hessian equation: δz ∈ X0 : a (x)(ζ, δz) = J (x)(δx, ζ) − a (x)(δx, ζ, z)

for all ζ ∈ X0 . (21)

Proof. The proof is similar to [HK] or [BMV]. In the following we describe the solution of the optimization problem (19) by Newton’s method on the continuous level. Starting with an initial guess u0 ∈ U , the next iterate un+1 is computed by an update step un+1 = un + δun , where δun solves: j (un )(δun , v) = −j (un )(v)

for all v ∈ U.

(22)

To solve (22) we use the conjugate gradient method (cg), which requires only the evaluation of the right-hand side and of matrix-vector products. Thus we have to evaluate j (un )(v) and j (un )(δun , v) for ﬁxed v. This can be done efﬁciently based on Proposition 1. Remark 1. For one step of the cg-method, we have to solve one tangent equation (20) and one dual-Hessian equation (21). In some cases, if the dimension of U is small, it can be more efﬁcient to build up the Hessian ∇2 j(un ), see [BMV] for a detailed discussion and a comparison.

4

Finite Element Discretization

In order to apply Newton’s method described before, we consider a space-time ﬁnite element discretization of the optimal control problem. For the time discretization we use the dG (discontinuous Galerkin) method, see e.g. [EJT]. For the time grid 0 = t0 < . . . < tl < . . . tM = T,

kl = tl − tl−1 ,

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and a space mesh Th consisting of quadrilaterals, we use a space of spatially continuous and cell-wise biquadratic and discontinuous in time piecewise polynomial functions r of order r, Xhk . In order to set up a Galerkin method using this space as trial and test spaces, one modiﬁes the semi-linear form a by adding jump terms ensuring the convergence to a continuous solution for the step size k = max kl tending to zero. For the detailed description of discrete equations to be solved within an application of the Newton method we refer to [BMV]. We emphasize that the space-time ﬁnite element discretization leads to a direct translation of the formulas for the derivatives of the reduced cost functional (see Proposition 1) from the continuous to the discrete level. This allows for the exact representation of the ﬁrst and second derivatives of the discrete reduced cost functional and is important for the convergence of the optimization algorithm. Remark 2. The solution of the underlying state equation is required in the whole time interval for the computation of the dual, tangent and dual Hessian equations. If all data are stored, the storage grows linearly with respect to the number of time intervals in the time grid and also linearly with respect to the number of degrees of freedom in the space discretization. This makes the optimization procedure prohibitive for ﬁne discretizations. This difﬁculty can be overcome by using storage reduction techniques known as “check-pointing” or “windowing”, see e.g. [Gr], [BGL] and [BMV] for the application to the optimization by parabolic equations. Remark 3. For the examples that will be presented in the following section we use isoparametric biquadratic ﬁnite elements for the space discretization of both pressure and velocities. We add further terms to the semi-linear form a in order to obtain a stable formulation with respect to both the pressure-velocity coupling and convection dominated ﬂows. This type of stabilization techniques is based on local projections of the pressure gradients (LPS-method) ﬁrst introduced in [BB]. This type of stabilization for optimal control problems is analyzed in [RV,BV].

5 Numerical Results In this section we discuss a numerical example illustrating the effect of different choices of the cost functional in the context of optimal vortex reduction. For this example we chose the dG(0) method for time and biquadratic elements for space discretization, as described in the previous section. For globalization we use trust region techniques, see e.g. [CGT,NW]. The use of such techniques in the example described below is necessary particularly for the optimization of the cost functional J4 . For the numerical test we consider the following conﬁguration: The computational domain Ω ⊂ R2 is sketched in Figure 1. We start with the following uncontrolled situation: We have constant parabolic inﬂow on Γin , “no-slip” boundary conditions on ∂Ω \ (Γin ∪ Γout ) and “do nothing” boundary conditions on Γout , see [HRT], i.e. ν∇y · n − p · n = 0 on Γout .

On the Choice of the Cost Functional for Optimal Vortex Reduction

Γ1

~ Ω Γout

Γin

Fig. 1. Computational domain

Fig. 2. Uncontrolled ﬂow, t = 2.4

Fig. 3. Values of curl y(t, x) for t = 2.4 for the uncontrolled ﬂow

347

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Fig. 4. Stokes ﬂow, used as the desired state for the “tracking type” functional 22 uncontrolled tracking curl det lkh

20 18 16

g(u)(t)

14 12 10 8 6 4 2 0

0.5

1

1.5 t

2

2.5

3

Fig. 5. Optimal controls g(u)(t) for four different cost functionals

The ﬂow with Reynolds number Re ≈ 103 is considered on the time horizon (0, T ) with T = 3. The initial velocity ﬁeld y0 is chosen as the solution of the non-stationary Stokes equation on the same conﬁguration after several time steps. The solution of the uncontrolled state equation for t = 2.4 is shown in Figure 2. In this ﬁgure we observe two primary “vortex regions”. First, we emphasize shortcomings of the cost functional J2 based on curl y(t, x) for vortex identiﬁcation and reduction. In Figure 3 the values of curl y(t, x) for t = 2.4 are plotted. We observe high (positive and negative) values of curl y(t, x) , e.g., in the inﬂow part of the domain, where no recirculation zones (vortices) are given.

On the Choice of the Cost Functional for Optimal Vortex Reduction

349

Fig. 6. Optimal ﬂow w.r.t. four different cost functionals (from top to bottom): J1 (tracking), J2 (curl), J3 (det), J4 (LKH)

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In order to compare different cost functionals for vortex reduction, we consider the ﬂow in the conﬁguration described above with the following (controlled) Dirichlet boundary conditions: y=0 on ∂Ω \ (Γin ∪ Γout ), y = g(u) yˆin on Γin , where yˆin is a ﬁxed parabolic proﬁle and g(u)(t) =

n c0 + u2k−1 sin(2πkt/T ) + u2k cos(2πkt/T ) . T k=1

The control variable u is searched for in the space U = R2n . For this setting we have for all u ∈ U : T y · n ds dt = c0 yˆin · n ds 0 Γin

Γin

independently of u. This condition has the following physical interpretation: The total ﬂux through the inﬂow boundary in the time horizon (0, T ) does not depend on the control action. Thus we aim for the vortex reduction under the constraint that the total ﬂux remains unchanged. In Figures 5, 6 we collect the results for the four cost functionals in the speciﬁed conﬁguration. For the “tracking type” functional we use the solution of the Stokes equation, see Figure 4, as the desired state ydes . In Figure 5 we show the optimal trajectories g(u)(t) of the controls for the four cost functionals under consideration. In Figure 6 we collect the solutions of the state equation for the optimal control u with respect to the four different cost functionals J1 , J2 , J3 , J4 . It can be noted that from the point of view of graphical vortex representation there is a signiﬁcant reduction of “vorticity” as we move from J1 to J4 .

Acknowledgments The authors would like to thank Prof. Haller for a very helpful exchange of emails.

References [AT] [BB] [BMV] [BV]

F. Abergel and R. Temam: On some control problems in ﬂuid mechanics, Theoret. Comput. Fluid Dynamics, 1, 1990, pp. 303-325. R. Becker and M. Braack: A ﬁnite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo 38, 2001, pp. 137-199. R. Becker, D. Meidner and B. Vexler: Efﬁcient numerical solution of parabolic optimization problems by ﬁnite element methods, submitted, 2005 R. Becker and B. Vexler: Optimal control of the convection-diffusion equation using stabilized ﬁnite element methods, submitted, 2005.

On the Choice of the Cost Functional for Optimal Vortex Reduction [BGL]

351

M. Berggren, R. Glowinski and J.-L. Lions: A computational approach to controllability issues for ﬂow-related Models. (I): Pointwise control of the viscous burgers equation, Int. J. Comput. Fluid Dyn. 7, 1996, pp. 237-253. [BMC] H.M. Blackburn, N.N. Mansour and B.J. Cantwell: Topology of ﬁne-scale motions in turbulent channel ﬂow, J. Fluid Mech. 310, 1996, pp. 293-324. [CGT] A.R. Conn, N. Gould and Ph.L. Toint. Trust-region methods, SIAM, MPS, Philadelphia, 2000. [CPC] M.S. Chong, A.E. Perry and B.J. Cantwell: A general classiﬁcation of threedimensional ﬂow ﬁelds, Phys. Fluids A 2, 1990, pp. 765-777. [EJT] K. Eriksson, Cl. Johnson and V. Thom´ee: Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modelisation Math. Anal. Numer. 19, 1985, pp. 611-643. [JH] J. Jeong and F. Hussain: On the identiﬁcation of vortex, J. Fluid Mech. 285, 1995, pp. 69-94. [FGH] A.V. Fursikov, M.D. Gunzburger and L.S. Hou: Boundary value problems and optimal boundary control for the Navier-Stokes system: The two-dimensional case, SIAM J. Control Optim. 36, 1998, pp. 852-894. [Gr] A. Griewank: Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optim. Methods Softw., 1, 1992, pp. 35-54. [HRT] J.G. Heywood, R. Rannacher and S. Turek. Artiﬁcial boundaries and ﬂux and pressure conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Math. Fluids, 22, 1992, pp. 325-352. [H1] G. Haller: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Phys. Fluids, 13, 2001, pp. 3365-3385. [H2] G. Haller: An objective deﬁnition of a vortex, J. Fluid Mech. 525, 2005,pp. 1-26. [HKSV] M. Hinterm¨uller, K. Kunisch, Y. Spasov and S. Volkwein: Dynamical systems based optimal control of incompressible ﬂuids, Int. J. Numer. Meth. Fluids 46, 2004, pp. 345-359. [HK] M. Hinze and K. Kunisch: Second order methods for boundary control of the instationary Navier-Stokes system, ZAMM Z. Angew. Math. Mech. 84, 2004, pp. 171-187. [HWM] J.C.R. Hunt, A.A. Wray and P. Moin: Eddies, stream and convergence zones in turbulent ﬂows, Center for Turbulence Research Report CTR-S88, 1988, 193ff. [G] M. Gunzburger: Flow Control, IMA 68, Editor M.D. Gunzburger, Springer, Berlin, 1995. [KV] K. Kunisch and B. Vexler: Optimal vortex reduction for instationary ﬂows based on translation invariant cost functionals, submitted, 2005. [LHK] G. Lapeyre, B.L. Hua and P. Klein: Dynamics of the orientation of active and passive scalars in two-dimensional turbulence, Phys. Fluids 13, 2001, pp. 251-264. [LKH] G. Lapeyre, P. Klein and B.L. Hua: Does the tracer gradient vector align with the strain eigenvectors in 2D turbulence? Phys. Fluids 11, 1999, pp. 3729-3737. [NW] J. Nocedal and S.J. Wright: Numerical Optimization, Springer Series in Operations Research, Springer New York, 1999. [O] A. Okubo: Horizontal dispersion of ﬂoutable trajectories in the vicinity of velocity singularities such as convergencies, Deep-Sea Res. 17, 1970, pp. 445-454. [RV] A. R¨osch and B. Vexler: Superconvergence in ﬁnite element methods for the optimal control problem of the stokes equations, submitted, 2005. [T] R. Temam: Navier-Stokes equations, Theory and Numerical Analysis, North Holland, Amsterdam, 1984.

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[TK]

M. Tabor and I. Klapper: Stretching and alignment in chaotic and turbulent ﬂows, Chaos, Solitons & Fractals 4, 1994, pp. 1031-1055. B. Vexler: Finite element approximation of elliptic dirichlet optimal control problems, submitted, 2005. J. Weiss: The dynamics of enstrophy transfer in 2-dimensional hydrodynamics, Physica D 48, 1991, pp. 273-294.

[V] [W]

Flow Control with Regularized State Constraints J.C. de los Reyes1,2 and F. Tr¨oltzsch1 1

Technical University of Berlin Str. des 17. Juni 136 10623 Berlin, Germany 2 Escuela Polit´ecnica Nacional Quito Ladr´on de Guevara E11-253 Quito, Ecuador [email protected] , [email protected]

Summary We consider the distributed optimal control of the Navier-Stokes equations in presence of pointwise state constraints. A Lavrentiev regularization of the constraints is proposed and a ﬁrst order optimality system is derived. The regularity of the mixed constraint multiplier is investigated and second order sufﬁcient optimality conditions are studied. In the last part of the paper, a semi-smooth Newton method is applied for the numerical solution of the control problem and numerical experiments are carried out.

1 Introduction In the recent past, optimal control of ﬂuid ﬂow has become an attractive multidisciplinary research ﬁeld with a wide range of ongoing and promising applications. The optimization problems in this context consist in minimizing or maximizing an objective functional (e.g. drag, lift, etc.) subject to the constitutive ﬂuid ﬂow equations and additional control and/or state constraints. The controls involved are usually considered in distributed form on a sub-domain or as boundary condition acting on some wall sectors. While the design of boundary controls is technically posible, the implementation of a distributed control action presents important difﬁculties. Lately, an increasing attention has been paid to this kind of controls, mainly within the ﬁeld of magneto-hydro-dynamics (MHD). Weakly conductive ﬂuids are controlled through the action of Lorentz forces, induced by magnetic ﬁelds (see [17,25]). Let us brieﬂy comment on the literature. The distributed optimal control problem of the Navier-Stokes equations has been mathematically analyzed and numerically studied in many research papers, see for example [1,3,10,11,16,23]. In these articles optimality conditions and/or numerical methods for the solution of the control problem were discussed. The same topics were considered, for the boundary optimal control problem, in [6,10,12,13]. In [6,12,13] Dirichlet controls were studied, while in [10] the action of Neumann boundary conditions was investigated. In presence of pointwise R. King (Ed.): Active Flow Control, NNFM 95, pp. 353–366, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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control constraints, optimality conditions and numerical methods have been treated in [4,15,24]. In particular semi-smooth Newton methods have been applied in this context (see [6,15,24]). In presence of pointwise state constraints the problem has received much less attention. The mathematical analysis of the optimal control problem has been considered in [5] and [9] for the stationary and time dependent problems, respectively. In [7] the numerical solution utilizing a penalized problem together with a semi-smooth Newton method has been studied. In this paper we consider a bounded two-dimensional domain Ω ⊂ R2 and pointwise state constraints of box type a(x) ≤ y(x) ≤ b(x),

(1.1)

where y = (y1 , y2 ) stands for the velocity vector ﬁeld. These constraints are imposed in order to reduce backward ﬂow and, consequently, diminish recirculations. Among other applications, such restrictions can have an important effect in avoiding ﬂow separation or reducing the drag of a body. For the numerical solution of the control problem we propose a Lavrentiev regularization of the pointwise state constraints, i.e. we consider the modiﬁed box constraints a(x) ≤ y(x) + εu(x) ≤ b(x),

ε > 0.

(1.2)

Due to the mixed nature of the pointwise constraints (1.2), the corresponding Lagrange multiplier is expected to be more regular than in the state constrained case (cf. [5]). It is also expected that, as ε tends to zero, the solutions converge to the optimal solution of the state constrained problem (see [18]). Based on the methodology developed in [19] for semilinear elliptic equations, we locally reformulate the mixed problem as a control constrained control problem in a new variable. After that, necessary and sufﬁcient conditions for optimality are studied. Also, thanks to the efﬁciency of semi-smooth Newton methods for nonlinear control constrained optimal control problems (cf. [6,14]), we apply a method of this type for the numerical solution of the control problem. The outline of the paper is as follows. In Section 2, the optimal control problem is stated and existence of a global optimal solution is veriﬁed. In Section 3, the problem is reformulated as a control constrained optimal control problem and ﬁrst order necessary optimality conditions are obtained. Sufﬁcient conditions of second order type are the topic of Section 4. In Section 5, a semi-smooth Newton algorithm for the solution of the problem is stated. Reports on numerical experiments are summarized in Section 6.

2

Problem Statement and Existence of Solution

Consider a bounded regular domain Ω ⊂ R2 . Our objective is to ﬁnd the optimal control u∗ and its associated state y ∗ , solution of the following problem:

Flow Control with Regularized State Constraints

⎧ min J(y, u) = 12 |y − zd |2 dx + ⎪ ⎪ ⎪ ⎪ Ω ⎪ ⎪ ⎪ subject to ⎪ ⎪ ⎨ −ν∆y + (y · ∇)y + ∇p = u ⎪ ⎪ div y = 0 ⎪ ⎪ ⎪ ⎪ ⎪ y| Γ =g ⎪ ⎪ ⎩ a ≤ εu + y ≤ b a.e.,

α 2

355

|u|2 dx

Ω

(2.1)

where α > 0, ε > 0 is the Lavrentiev regularization parameter, zd is the desired state, g is a non-homogeneous Dirichlet boundary condition and a(·), b(·), with a(x) ≤ b(x), are the lower and upper constraint functions, respectively. The constant ν > 0 stands for the viscosity coefﬁcient of the ﬂuid and Re := 1/ν for its Reynolds number. It is well known that there exists a solution for the stationary two-dimensional Navier-Stokes system (cf. [21]). Moreover, if ν is sufﬁciently large or u sufﬁciently small, an appropriate estimate and uniqueness of the solution are obtained. Next, we verify the existence of an optimal solution for our control problem. For that purpose let us deﬁne the set of admissible solutions Tad = {(y, u) which satisfy the restrictions in (2.1) }. Theorem 1. If Tad is non-empty, then there exists an optimal solution for (2.1). Proof. We refer to [8, p. 3]. In the previous result the existence of a feasible solution was assumed. This hypothesis makes sense, since no pure control constraints are involved. In presence of control constraints the admissible set could possibly be empty.

3 First-Order Necessary Optimality Conditions Once the existence of an optimal solution is veriﬁed, it is important to derive conditions that characterize any local solution of the optimization problem. To this aim a necessary condition involving ﬁrst order derivatives is obtained. This condition takes the form of a system of partial differential equations (Navier-Stokes and adjoint equations) coupled with a nonlinear complementarity problem. Let us consider the interior of the set of controls for which a unique associated Navier-Stokes solution exists and let us denote this set by U . Introducing the controlto-state operator G : u → y(u) that assigns to each u ∈ U the corresponding NavierStokes solution y(u), problem (2.1) can equivalently be expressed in reduced form as ⎧ ⎨minu∈U J(u) = 1 |G(u) − zd |2 dx + α |u|2 dx 2 2 Ω Ω (P) ⎩subject to: a ≤ εu + G(u) ≤ b a.e. in Ω.

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Let us introduce the constant M(y) := sup

|

Ω

(v·∇)y·v dx| , |∇v|2 dx

Ω

v∈V

where V is the space

of divergence free square integrable functions, with square integrable weak derivatives, that vanish at the boundary Γ . If ν > M(y(u)), it can be veriﬁed that the control to state operator G is twice Fr´echet differentiable at u and its derivatives w := G (u)h and z := G (u)[h]2 are given by the unique solutions of the systems: −ν∆w + (w · ∇)y + (y · ∇)w + ∇π = h div w = 0 w|Γ = 0

(3.1)

and −ν∆z + (z · ∇)y + (y · ∇)z + ∇ = −2(w · ∇)w div z = 0

(3.2)

z|Γ = 0. The idea now consists in reformulating problem (P) in a new variable v := εu+G(u) and treat it as a control-constrained optimal control problem. In order to express u as a function of v we consider the operator F : (v, u) → εu + G(u) − v and the solvability of the equation F (v, u) = 0. It can be veriﬁed (see [8]), that there are constants r, r0 > 0 such that for each v with 1/2 1/2 ∗ 2 ≤ r0 , there exists a unique u := K(v) with Ω |u − u∗ |2 dx Ω |v − v | dx ≤ r such that εK(v) + G(K(v)) = v. (3.3) Moreover, since F is twice continuously Fr´echet differentiable, the implicit function theorem also implies that K is twice continuously Fr´echet differentiable. Let us denote by K (v)[ξ, η] the second derivative of K in directions ξ and η and introduce K (v)[ξ]2 := K (v)[ξ, ξ]. Taking the ﬁrst and second derivatives on both sides of (3.3) in direction ξ yields (ε + G (K(v)))K (v)ξ = ξ,

(3.4)

(ε + G (K(v)))K (v)[ξ] = −G (K(v))[K (v)ξ] , which implies that and

2

2

(3.5)

K (v) = (ε + G (K(v)))−1

K (v)[ξ]2 = −(ε + G (K(v)))−1 G (K(v))[K (v)ξ]2 .

Locally around u∗ , our control problem can therefore be formulated as: ⎧ ⎪ ⎨min J (v) =: J(y(K(v)), K(v)) subject to a ≤ v ≤ b a.e. ⎪ ⎩ v ∈ Br0 (v ∗ ).

(Pr )

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357

Next an optimality system which characterizes the solutions to (P) is stated. The proof is given in the Appendix. Theorem 2. Let u∗ be a local optimal solution of (P) with ν > M(y(u∗ )). Then there exist adjoint variables λ, q and Lagrange multipliers µa , µb such that −ν∆y ∗ + (y ∗ · ∇)y ∗ + ∇p = u∗ div y ∗ = 0

(3.6)

∗

y |Γ = g, −ν∆λ − (y ∗ · ∇)λ + (∇y ∗ )T λ + ∇q = zd − y ∗ + µa − µb div λ∗ = 0 λ∗ |Γ = 0,

(3.7)

λ − αu∗ = ε(µb − µa ),

(3.8)

a ≤ εu + y ∗ ≤ b, µa , µb ≥ 0, ∗ ∗ µai (ai − εui − yi ) dx = µbi (bi − εu∗i − yi∗ ) dx = 0, for i = 1, 2. Ω

(3.9)

Ω

Optimality systems are important to understand the regularity of the control, state and adjoint variables and to apply a wide variety of numerical methods for the solution of the optimization problem. If no inequality constraints are present, the system can be solved as a system of partial differential equations. In general, however, it is constituted by the state equations, the adjoint equations and a nonlinear complementarity system and, in this case, additional methods for the solution of complementarity problems have to be considered.

4 Second Order Sufﬁcient Condition Next, we turn to second order sufﬁcient optimality conditions for problem (P). This type of conditions allows the identiﬁcation of a stationary point (a solution to optimality system (3.6)-(3.9)) as a minimum for the optimal control problem. Additionally, they are of importance in the convergence analysis of Newton type methods applied to the optimization problem. Following [19], the idea consists in utilizing the second order sufﬁcient optimality properties of the pure control constrained problem (Pr ) and translate them to the original setting. By introducing the Lagrangian L(y, u, λ) = J(y, u) + ν

∇λ : ∇y dx +

Ω

(y · ∇)y · λ dx −

Ω

λ · u dx, Ω

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the equivalence of its second derivative with the one of the reduced functional J can be veriﬁed. The second derivative of the reduced cost functional in direction ξ therefore satisﬁes J (v ∗ )[ξ]2 = L (y ∗ , u∗ , λ)(w, h)2 , where h = K (v ∗ )ξ and w is the solution to (3.1) with h on the right hand side. Let us now introduce the set of strongly active constraints Aτ,i := {x ∈ Ω : |µi (x)| ≥ τ } and the critical cone ⎧ ⎫ vi (x) = 0 if x ∈ Aτ,i ⎪ ⎪ ⎨ ⎬ ∗ 2 ˜ Cτ = v ∈ L (Ω) : vi (x) ≥ 0 if vi (x) = ai , x ∈ Aτ,i . ⎪ ⎪ ⎩ ⎭ vi (x) ≤ 0 if vi∗ (x) = bi , x ∈ Aτ,i For the investigation of optimality for a given stationary pair (y ∗ , u∗ ) let us hereafter assume that for some δ > 0 the following second order condition holds: there exist τ > 0, δ > 0 such that ∗ ∗ 2 L (y , u , λ)(w, h) ≥ δ |h|2 dx, (SSC) Ω

for all (w, h) ∈ Cτ , where Cτ consists of all pairs (w, h) such that system (3.1) is satisﬁed and εh + w ∈ C˜τ . Theorem 3. If u∗ is a stationary point of (P) and (SSC) holds for some δ > 0, τ > 0, then there exist constants ρ > 0 and σ > 0 such that J(y, u) ≥ J(y ∗ , u∗ ) + σ |u − u∗ |2 dx (4.1) Ω

for all (y, u) such that y = G(u), a ≤ εu + y ≤ b and

Ω

1/2 |u − u∗ |2 dx ≤ ρ.

Proof. See [8, p. 10] Remark 4. For the analysis of second order numerical methods, a stronger condition is needed: there exist constants τ > 0, δ > 0 such that |h|2 dx (SSC) L (y ∗ , u∗ , λ)(w, h)2 ≥ δ Ω

for all pairs (w, h) that solve (3.1) and satisfy εhi + wi = 0 on Aτ,i , for i = 1, 2.

5

Semi-smooth Newton Method

In this section we propose a semi-smooth Newton method for the numerical solution of (P). These generalized Newton methods for nonsmooth equations are based on the notion of Newton differentiability, which, differently from othe differentiability concepts, allows to prove local superlinear convergence of the method (cf. [14]). For the application of the method to the optimality system (3.6)-(3.9) we introduce the variable x = (y, u, λ, q, µ), with µ := µb − µa . The system can then be reformulated as an operator equation T (x) = 0 and a semi-smooth Newton step is given by G(xk )δx = −T (xk ), where G is the Newton derivative of T .

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359

In our case the difﬁculty is given by the complementarity system (3.9). Using the max and min functions, this problem can be reformulated as the operator equation µ = max(0, µ + c(v − b)) + min(0, µ + c(v − a))

(5.1)

for all c > 0. The Newton differentiability of the max and min function then imply the Newton differentiability of the whole system. The derivative candidates ) ) 1 if y(x) ≥ 0 1 if y(x) ≤ 0 Gmax (y)(x) = Gmin (y)(x) = (5.2) 0 if y(x) < 0; 0 if y(x) > 0, constitute Newton derivatives of max(0, y) and min(0, y), respectively (cf. [14]). By choosing c := α/ε2 in (5.1) and considering the derivatives (5.2), the complete algorithm can be formulated as an active set strategy through the following steps. Algorithm 5 1. Initialize the variables u0 , y0 , µ0 = 0 and set k = 1. 2. Until a stopping criterion is satisfied, set for i = 1, 2 α Anbi = {x : µn−1 + 2 εun−1 + yin−1 − bi > 0}, i i ε α n−1 n Aai = {x : µi + 2 (εun−1 + yin−1 − ai ) < 0}, i ε Iin = Ω\(Anbi ∪ Anai ). and find the solution (y, p, λ, q) of: −ν∆yi + y1n−1 ∂1 yi + y2n−1 ∂2 yi + y1 ∂1 yin−1 + y2 ∂2 yin−1 ⎧ 1 ⎪ ⎨ ε (bi − yi ) n−1 n−1 n−1 n−1 +∂i p = y1 ∂1 yi + y2 ∂2 yi + λαi ⎪ ⎩1 ε (ai − yi )

on Anbi on Iin on Anai

div yi = 0 yi |Γ = g 1 − y2 ∂2 λn−1 −y1n−1 ∂1 λi − y2n−1 ∂2 λi + λ1 ∂i y1n−1 −ν∆λi + λi − y1 ∂1 λn−1 i i ε +λ2 ∂i y2n−1 + λn−1 ∂i y1 + λn−1 ∂i y2 +∂i q = zd,i − yi − y1n−1 ∂1 λn−1 1 2 i ⎧ α ⎪ (b − y ) on Anbi i ⎨ ε2 i −y2n−1 ∂2 λn−1 + λn−1 ∂i y1n−1 + λn−1 ∂i y2n−1 + λεi on Iin 1 2 i ⎪ ⎩α on Anai ε2 (ai − yi ) div λi = 0 λi |Γ = 0. ⎧ 1 n ⎪ ⎨ ε n(bi − yi ) λ i Set (y n , pn , λn , q n ) = (y, p, λ, q), uni = α ⎪ ⎩1 n ε (ai − yi ) 1 n n ε (λ − αu ), and goto step 2.

on Anbi on Iin , µn = on Anai .

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Note that the system to be solved in step (2) corresponds to the optimality system of a quadratic control problem with afﬁne constraints. Under the satisfac 1/2 tion of the second order condition (SSC) and if Ω |∇(y n−1 − y ∗ )|2 dx and 1/2 n−1 ∗ 2 |∇(λ − λ )| dx are sufﬁciently small, convexity of the optimization probΩ lem can be argued. Therefore, under these conditions, there exists a unique solution for the system in step (2). Sufﬁcient conditions for local superlinear convergence of the semi-smooth Newton method applied to (P) are investigated in [8].

6 Numerical Results For the numerical tests, a ”forward facing step channel” was utilized (see Figure 1). The ﬂuid ﬂows from left to right with parabolic inﬂow condition and ”do nothing” output condition. In the remaining boundary parts an homogeneous Dirichlet condition was imposed. The geometry was discretized using a staggered grid and an upwinding ﬁnite differences scheme was applied. The behavior of the uncontrolled ﬂuid ﬂow with Reynolds number Re = 1000 is depicted in Figure 2. Two main recirculation zones, which increase their size together with the Reynolds number, can be clearly identiﬁed from the graphic. These results can be veriﬁed experimentally (see [2,20]). The target of our control problem is to properly diminish the recirculations of interest by considering, together with the tracking type cost functional, adequate pointwise control-state constraints.

0.5

Ω !b

0

0.5

Fig. 1. Forward facing step channel

!b Fig. 2. Streamlines of the uncontrolled state

1

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For the solution of the optimality system, Algorithm 5 was utilized. The semi-smooth Newton algorithm stops when the state increment norm is lower than 10−4 . Unless otherwise speciﬁed, the mesh step h = 1/240 was considered. For the solution of the linear systems, Matlab’s exact solver was utilized. 6.1 Example 1 In this ﬁrst experiment we consider the elimination of bubbles in the channel by imposing the constraint y1 + εu1 ≥ −10−7 . For ε sufﬁciently small, this constraint avoids backward ﬂow in the channel and thus possible recirculations. Additionally, the tracking type component of the cost functional is responsible for a more linear behavior of the ﬂow ﬁeld. The remaining parameter data utilized are h = 1/240, Re = 1000, ε = 10−4 and α = 0.1. The semi-smooth Newton method (SSN) stops after 9 iterations, with the ﬁnal active set containing 28 grid points. The cost functional takes the ﬁnal value J(y ∗ , u∗ ) = 0.00445224 and the NCP function residuum the value 2.2737 × 10−9 . The optimal control ﬁeld is depicted in Figure 3, where the concentration of the control action on the recirculation zones can be observed. The desired recirculation diminishing effect of the control can be visualized from the plot of the reached controlled state streamlines in Figure 4. In Table 1 the number of SSN iterations, the ﬁnal cost functional value and the size of the active set are registered for different ε values. It can be observed that as ε tends to 0, the problem becomes harder to solve and more SSN iterations are required.

Fig. 3. Example 1: control vector ﬁeld with tracking component

Fig. 4. Example 1: streamlines of the controlled state with tracking component

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J.C. de los Reyes and F. Tr¨oltzsch Table 1. Example 1: h=1/240, tol = 10−4 ε SSN Iter. 10−1 5 10−2 6 10−3 8 10−4 9 10−5 9

J(y ∗ , u∗ ) |Aa ∪ Ab | 0.00399972 33 0.00410360 42 0.00438273 29 0.00445224 28 0.00445989 32

Fig. 5. Example 1: streamlines of the controlled state without tracking component

Fig. 6. Example 1: control vector ﬁeld without tracking component

Subsequently we consider the limit case where the tracking type part of the cost functional is dismissed. We aim to ﬁnd the control of minimum norm that allows the satisfaction of the state constraint y1 + εu1 ≥ 10−7 over the domain of interest. As before, the constraint takes care that no important backward ﬂow arises. By considering the constraint on the whole domain, i.e. ΩS = Ω, both recirculations before and after the step are diminished (see Figure 5). From Figure 5 it can also be observed that the behavior of the ﬂuid ﬂow, mainly before the step, is not as closer to a Stokes ﬂow as in the case where the tracking type component is present (see Figure 4). From the control vector plot (see Figure 6) it can be observed that the control action in this case is even more concentrated on the recirculations zones. The parameter values for this case are Re = 1000, ε = 10−4 and α = 0.1. The number of SSN iterations needed is 29 and the cost functional takes the ﬁnal value 8.99816 × 10−4 .

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ΩS

Fig. 7. Example 1: streamlines of the controlled state without tracking component; state constraint subdomain

In many practical cases, the recirculations reduction or elimination on the whole domain is not necessary, if not undesirable. In such cases the state constraint may be imposed in the sectors where the bubble to be diminished is localized. In the case of our geometry the essential recirculation to be diminished is the one after the step. By considering the state constraint on the subdomain ΩS := [0.5, 0.75] × [0.25, 0.5], this elimination is attached with the cost functional value 8.98898 × 10−4 in 6 SSN iterations. The ﬁnal controlled state is shown in Figure 7, where it can be observed that the recirculation after the step is numerically eliminated, although the one before the step becomes bigger than in the uncontrolled case. 6.2 Example 2 As an alternative strategy for the reduction of the recirculation after the step, we consider in this example a state constraint that guarantees an homogeneous outgoing velocity. The constraint imposed is y1 + εu1 ≤ 1.7 and the remaining parameter values are Re = 1000, ε = 10−3 and α = 0.01. In this case, the SSN algorithm stops after 15 iterations and the resulting active set contains 2283 grid points. The cost functional takes the ﬁnal value J(y ∗ , u∗ ) = 0.003470768. The controlled state is depicted in Figure 8, where an important reduction of the recirculations can be visualized. Since the outgoing velocity is the quantity of interest, it is natural to consider the case where the constraint is imposed only in the last part of the channel. By considering the domain ΩS := [0.5, 0.75] × [0.25, 0.5], the recirculation diminishing

Fig. 8. Example 2: streamlines of the controlled state

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ΩS

Fig. 9. Example 2: streamlines of the controlled state; state constraint subdomain

effect does also take place (see Figure 9), but with a lower ﬁnal cost functional value J(y ∗ , u∗ ) = 0.0031112131. The SSN algorithm stops after 10 iterations with a ﬁnal active set containing 906 active points. The remaining parameter values are the same as in the case ΩS = Ω.

7 Conclusion In this paper the optimal control problem of the Navier-Stokes equations with regularized pointwise state constraints of box type was considered. The problem was mathematically analyzed, yielding optimality conditions of ﬁrst and second order. A semismooth Newton method for the solution of the problem was proposed. For the numerical realization a forward facing step channel was considered and a ﬁnite differences scheme was utilized for the partial differential equations involved. The results show that the state constrained approach succeeded in reducing the recirculations of interest. This happened in the case where the constraint held all over the domain and also in the more realistic case, when it was restricted to a subdomain. Both limiting backward ﬂow and imposing a more homogeneous outgoing velocity proﬁle showed a positive effect with respect to recirculation reduction Distributed controls are currently applied in magneto-hydro-dynamic problems, where the results obtained here can be used. It seems also possible to extend the analysis to the case of boundary optimal control problems with state constraints.

References [1] F. Abergel and R. Temam: On some control problems in ﬂuid mechanics, Theoretical and Computational Fluid Mechanics, 303-325, 1990. [2] T. Ando and T. Shakouchi: Flow characteristics over forward facing step and through abrupt contraction pipe and drag reduction, Res. Rep. Fac. Eng. Mie Univ., Vol. 29, 1-8, 2004. [3] E. Casas: Optimality conditions for some control problems of turbulent ﬂows. Flow control (Minneapolis, MN, 1992), IMA Vol. Math. Appl., 68, 127–147, Springer Verlag, New York, 1995. [4] J. C. de los Reyes: A primal-dual active set method for bilaterally control constrained optimal control of the Navier-Stokes equations, Numerical Functional Analysis and Optimization, Vol. 25, 657-683, 2005.

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[5] J. C. de los Reyes and R. Griesse: State constrained optimal control of the stationary NavierStokes equations. Preprint 22-2005, Institute of Mathematics, TU-Berlin, 2005. [6] J. C. de los Reyes and K. Kunisch: A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations, Nonlinear Analysis: Theory, Methods and Applications, Vol. 62, 1289-1316, 2005. [7] J. C. de los Reyes and K. Kunisch: A semi-smooth Newton method for regularized state constrained optimal control of the Navier-Stokes equations, Computing, Vol. 78, 287-309, 2006. [8] J. C. de los Reyes and F. Tr¨oltzsch: Optimal control of the stationary Navier-Stokes equations with mixed control-state constraints. Preprint 32-2005, Institute of Mathematics, TUBerlin, 2005. [9] H. O. Fattorini and S. S. Sritharan: Optimal control problems with state constraints in ﬂuid mechanics and combustion, Applied Math. and Optim., Vol. 38, 159-192, 1998. [10] M. D. Gunzburger, L. Hou, and T. P. Svobodny: Analysis and ﬁnite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls, Mathematics of Computation, Vol. 57, 195, 123-151, 1991. [11] M. D. Gunzburger and S. Manservisi: Analysis and approximation of the velocity tracking problem for Navier-Stokes ﬂows with distributed control, SIAM Journal on Numerical Analysis, Vol. 37, 1481-1512, 2000. [12] M. D. Gunzburger and S. Manservisi: Analysis and approximation of the velocity tracking problem for Navier-Stokes ﬂows with boundary control, SIAM Journal on Control and Optimization, Vol. 39, 594-634, 2000. [13] M. Heinkenschloss: Formulation and analysis of sequential quadratic programming method for the optimal Dirichlet boundary control of Navier-Stokes ﬂow, Optimal Control (Gainesville, FL, 1997), Kleuver Acad. Publ., 178-203, Dordrecht, 1998. [14] M. Hinterm¨uller, K. Ito, and K. Kunisch: The primal dual active set strategy as a semismooth Newton method, SIAM Journal on Optimization, Vol. 13, pp. 865-888, 2003. [15] M. Hinterm¨uller and M. Hinze: A SQP-semi-smooth Newton-type algorithm applied to control of the instationary Navier-Stokes system subject to control constraints, submitted. [16] M. Hinze and K. Kunisch: Second order methods for optimal control of time dependent ﬂuid ﬂow, SIAM Journal on Control and Optimization, Vol. 40, 925-946, 2002. [17] M. Hinze: Control of weakly conductive ﬂuids by near wall Lorentz forces, SFB609Preprint-19-2004, Sonderforschungsbereich 609, Technische Universitt Dresden, 2004. [18] C. Meyer, A. R¨osch and F. Tr¨oltzsch: Optimal control of PDEs with regularized pointwise state constraints. Preprint 14-2003, Institute of Mathematics, TU-Berlin, 2003. [19] C. Meyer and F. Tr¨oltzsch: On an elliptic optimal control problem with pointwise mixed control-state constraints, Recent Advances in Optimization. Proceedings of the 12th FrenchGerman-Spanish Conference on Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 563, pp. 187-204, Springer-Verlag, 2006. [20] H. St¨uer: Investigation of Separation on a Forward Facing Step, Ph. D. Thesis, ETH Z¨urich, 1999. [21] R. Temam: Navier Stokes Equations: Theory and Numerical Analysis, North Holland, 1979. [22] F. Tr¨oltzsch: Optimalsteuerung bei partiellen Differentialgleichungen, Vieweg Verlag, 2005. [23] F. Tr¨oltzsch and D. Wachsmuth: Second order sufﬁcient optimality conditions for the optimal control of Navier-Stokes equations, to appear in ESAIM: Control, Optimisation and Calculus of Variations. [24] M. Ulbrich: Constrained optimal control of Navier-Stokes ﬂow by semismooth Newton Methods, Systems and Control Letters, Vol. 48, 297-311, 2003.

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[25] T. Weier, G. Gerbeth, G. Mutschke, O. Lielausis, G. Lammers: Separation control by stationary and time periodic Lorentz forces, SFB-Preprint SFB609-03-2004, Sonderforschungsbereich 609, Technische Universitt Dresden, 2004.

Appendix: Proof of Theorem 2 We consider the space of square integrable functions on Ω, denoted by L2 (Ω), and introduce the bold notation for the product of spaces. We denote by (·, ·) the inner product in L2 (Ω) and by · the associated norm. Since u∗ is a locally optimal solution of (P), we get for some r > 0 that J(y ∗ , u∗ ) ≤ J(y(u), u), for all u ∈ Br (u∗ ) with a ≤ εu + y(u) ≤ b. Equivalently, since u = K(v) holds locally, J (v ∗ ) ≤ J (v), for all v ∈ Br0 (v ∗ ) with a ≤ v ≤ b, and for an appropriate constant r0 > 0. Therefore, the following ﬁrst order necessary condition follows J (v ∗ )(v − v ∗ ) ≥ 0, ∀a ≤ v ≤ b

(7.1)

Applying the chain rule, the derivative of J (v ∗ ) in direction ξ ∈ L2 (Ω) is given by (J (v ∗ ), ξ) = (y ∗ − zd , G (u∗ )K (v ∗ )ξ) + α(u∗ , K (v ∗ )ξ),

(7.2)

which, by h := K (v ∗ )ξ, yields (J (v ∗ ), ξ) = (y ∗ −zd , G (u∗ )h)+α(u∗ , h). Denoting by µ ∈ L2 (Ω) the Riesz representative of −J (v ∗ ) and using explicitly the derivative of K we obtain (µ, ξ) = (µ, (ε + G (u∗ ))h) = ε(µ, h) + (µ, G (u∗ )h). Therefore, equation (7.2) is equivalent to (y ∗ − zd + µ, G (u∗ )h) + (αu∗ + εµ, h) = 0.

(7.3)

We now consider the adjoint equations (3.7). Since, by hypothesis ν > M(y ∗ ), the ellipticity of the adjoint operator can be easily veriﬁed and, therefore for zd − y ∗ − µ ∈ L2 (Ω), there exists a unique solution λ ∈ V for the adjoint system. Consequently, equation (7.3) can be rewritten as λ − αu∗ = εµ. Utilizing the decomposition µ = µb − µa , with µb := µ+ = 12 (µ + |µ|) and µa := µ− = 12 (−µ + |µ|), where |µ| = (|µ1 |, |µ2 |)T , the optimality condition (7.1) can be rewritten as (J (v ∗ ), v ∗ ) = mina≤v≤b {(µa,1 , v1 )−(µb,1 , v1 )+(µa,2 , v2 )−(µb,2 , v2 )}. By ﬁxing the second component of the new control variable v2 = v2∗ and considering the mutual disjoint sets {x : µa,1 (x) > 0} and {x : µb,1 (x) > 0}, we obtain that (J (v ∗ ), v ∗ ) = (µa,1 , a1 ) − (µb,1 , b1 ) + (µa,2 , v2∗ ) − (µb,2 , v2∗ ) and, consequently, (µa,1 , a1 − εu∗1 − y1∗ ) − (µb,1 , b1 − εu∗1 − y1∗ ) = 0. Fixing now the ﬁrst component of v and proceeding in a similar manner we get that (µa,2 , a2 − εu∗2 − y2∗ ) − (µb,2 , b2 − εu∗2 − y2∗ ) = 0. Taking into account that, by deﬁnition, µa , µb ≥ 0 componentwise, the complementarity system (3.9) follows.

Feedback Control Applied to the Bluﬀ Body Wake Lars Henning1 , Mark Pastoor1, Rudibert King1 , Bernd R. Noack2 , and Gilead Tadmor3 1

3

Measurement and Control Group, Berlin University of Technology, Hardenbergstr. 36a, 10623 Berlin, Germany [email protected] http://mrt.tu-berlin.de 2 Institute of Fluid Dynamics and Technical Acoustics, Berlin University of Technology, M¨ uller-Breslau-Str. 8, 10623 Berlin, Germany [email protected] Department of Electrical and Computer Engineering, Northeastern University, 360 Huntington Avenue, Boston MA 02115-5000, USA [email protected]

Summary In the present study the ﬂow around a 2D bluﬀ body with blunt stern is investigated experimentally and theoretically. The goal is to decrease and stabilize drag by active control. Low-dimensional vortex models are used to describe actuation eﬀects on the coherent structures and the pressure ﬁeld. Open-loop actuation as well as feedback control is applied using robust H∞ -controllers and slope-seeking feedback for a range of Reynolds numbers based on the height from 20 000 to 60 000. As expected, a decreased drag is observed to be related to delayed vortex shedding, i.e. an extended recirculation zone. Intriguingly, a control which mitigates the natural coupling between separating upper and lower shear-layer and the vortex street serves that purpose.

1

Introduction

The ﬂow around an elongated bluﬀ body with blunt stern (Fig. 1) is investigated in the present study. The framework aims at understanding the mechanism of active ﬂow control with respect to reduce aerodynamic drag, which is induced by coherent structures in the wake. The suppression or a desired control of separation phenomena has been addressed in the ﬂuid dynamics community for many decades. Various passive means for bluﬀ body ﬂow control are well-investigated and already applied in experiments. For example, in Bearman [1] the eﬀect of a splitter-plate on the wake ﬂow of a bluﬀ body is described. Tanner [2] examined the drag reduction for various kinds of a spanwise modulation of the trailing edge, such as segmented, curved, and M-shaped trailing edges. Tombazis & Bearman [3] investigated modiﬁcations of vortex shedding by adding a set of wavy trailing R. King (Ed.): Active Flow Control, NNFM 95, pp. 369–390, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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Fig. 1. Sketch of the bluﬀ body conﬁguration with the Cartesian coordinate system

edges to a blunt-based model. These studies show that modiﬁcations of the geometry lead to a signiﬁcant drag reduction. However, when shaping of the geometry has reached an optimum or when passive means, such as vortex generators, have positive and negative eﬀects, active devices in open- or closed-loop control can further improve the performance. Furthermore, active control is able to adapt the actuation to a wide range of operating conditions, even in an optimal sense. Most of the work published so far is dedicated to open-loop control. Literature surveys on feedforward ﬂow control, including actuation mechanisms and sensor applications, are given in Fiedler & Fernholz [4], or Gad-el-Hak et al. [5]. The eﬀect of the active base bleed for drag reduction is proved by Bearman [6]. An active control application is investigated in Kim et al. [7] by spanwise distributed forcing at the trailing edges of a two-dimensional bluﬀ body. Here, in large eddy simulations and wind tunnel experiments a signiﬁcant drag reduction together with a suppression of vortex shedding in the wake was reached by openloop forcing. The ﬁrst real active open-loop ﬂow control demonstration for an airplane was done 2003 with a XV-15 tilt-rotor aircraft [8]. Closed-loop ﬂow control of bluﬀ bodies mainly concentrates on a well established benchmark problem, the control of the ﬂow over a circular cylinder. In Gerhard et al. [9] a model-based ﬂow control strategy is proposed for the suppression of vortex shedding downstream a circular cylinder. A closed-loop ﬂow control combined with a model-based sensor using a Galerkin model is presented in Tadmor et al. [10]. Experimental validations of closed-loop ﬂow control are still rare. In Siegel et al. [11] a simple low-dimensional model is used to control the ﬂow over a cylinder ﬂow. Allan et al. [12] use tuning rules for the control of a generic model of an airfoil. In Henning & King [13] a robust multivariable H∞ controller is used to control the spanwise reattachment length downstream of a backward-facing step. In King et al. [14] robust and adaptive controllers are compared and used to control the spanwise and the spanwise averaged

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recirculation length behind a backward-facing step, the lift of a generic highlift conﬁguration, and the pressure recovery in a diﬀusor ﬂow. In our experimental set-up the maximal actuation frequency is limited to the range of the shear-layer instability frequency. To hamper the development of coherent structures that can interact with the wake instability by forcing significantly higher frequencies are not possible. The break-up of coherent structures by forcing 3D structures and thus mimicking passive control, for instance using spanwise distributed actuation, is beyond the scope of this study. Instead eﬃcient means to synchronize 2D coherent structures for achieving the control goal are proposed. The paper is organized as follows: The ﬂow conﬁguration is outlined in Sec. 2 with a characterization of the physical processes. Experimental results of openloop actuation are explained and exploited for the application of active ﬂow control in Sec. 3. Sec. 4 describes the synthesis of various feedback control schemes (robust H∞ -control, slope seeking feedback, phasor control) and their application in experiments. The paper closes with a conclusion and an outlook.

2

Flow Conﬁguration

In this section the experimental set-up and the natural ﬂow are outlined. 2.1

Experimental Set-Up

The experiments are conducted in an Eiﬀel-type wind tunnel. The maximum free stream velocity is approximately 20 m s−1 with a turbulence level less than 0.5%. The closed measuring section is 2500 mm × 555 mm × 550 mm in the streamwise (x), transverse (y) and spanwise (z) direction, respectively. The bluﬀ body with chord length L = 262 mm, body height H = 72 mm and spanwise width W = 550 mm has a rounded nose to prevent the ﬂow from separating. Trip tapes were placed 30 mm downstream of the nose in order to trigger transition. The model is mounted on two aluminium rods and is vertically centred in the wind tunnel. Reynolds and Strouhal numbers are given with respect to body height and free stream velocity. All experiments are conducted with Reynolds number 40 000 unless otherwise stated. Sinusoidal zero net ﬂux actuation through spanwise slots (slot width 1 mm, spanwise length 250 mm) located at the upper and lower edge of the stern is eﬀected by individually controllable loudspeakers. Harmonic actuation a(t) = A sin(ωA t), with actuation amplitude A(t) and actuation frequency ωA (t) is applied to each slot, thus generating periodic sucking and blowing. The cost of actuation is characterized by the non-dimensional excitation momentum coeﬃcient cµ =

2 2 s qA , 2 H u∞

where qA is the eﬀective velocity of the actuation.

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Base pressure is monitored by 3 × 9 diﬀerence pressure gages mounted in three parallel rows on the stern at z/W = {−0.15, 0.0, 0.15}. Pressure gages are calibrated and temperature compensated. Their operating pressure range is ±2.5 mbar. The reference pressure is taken above the body. Four strain gages are applied to the aluminium rods for drag measurements. Pressure recovery and drag are described by non-dimensional pressure and drag coeﬃcients ∆p

cP =

, 1 2 2 ρ u∞

cD =

Fx , 1 2 A ρ u ∞ cs 2

respectively. Here, ∆p is the pressure diﬀerence between the stern-mounted pressure gage and the reference, ρ denotes the density, u∞ is the free stream velocity. Additional variables and constants are Fx , the drag force in streamwise direction, and Acs , the cross section of the bluﬀ body. Data acquisition and the implementation of the controllers is realized by rapid prototyping hardware (dSPACE controller). The sampling frequency is 5000 Hz. 2.2

Vortex Model

Vortex models provide an approximation of the vorticity distribution by adding point-vortices to a potential ﬂow ﬁeld. Superposition of Biot-Savart’s law and potential theory yields a solution of the Euler equation, the inviscid equation of motion. A vortex model operating with Oseen-vortices as proposed in [15] is used here to describe the coherent structures of the ﬂow. 2.3

Description of the Natural Flow

The ﬂow around a bluﬀ body is governed by an absolute wake instability. This mechanism generates the typical von-K´ arm´an vortex street with an alternating sequence of vortices at characteristic frequencies. The ﬂow is predominantly twodimensional. Smoke visualizations of the ﬂow with bent shear-layers at the upper and lower edge are displayed in Fig. 2a (left). Two large vortical structures appear almost at the centre line, whereas the lower vortex is closer to the stern. Pressure readings indicate a pressure minimum at the bottommost pressure gages. The timeaveraged base pressure coeﬃcient is cP,0 = −0.5. This corresponds to an average drag coeﬃcient of cD,0 = 1.2. On the right hand of Fig. 2a the pressure ﬁeld obtained from a vortex model is shown. It indicates low pressure at the vortex locations and high pressure on the convex side of the shear-layers. Apparently, the lower vortex pulls the upper shear-layer down and induces the roll-up of new vortex. As the lower vortex convects downstream the new upper vortex grows and moves towards the centre line. Here, it triggers the creation of a new vortex in the lower shear-layer. As a result, both shear-layers do not evolve independently and are strongly bent already close to the stern.

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Fig. 2. Smoke visualization at ReH = 40 000 and pressure ﬁeld obtained from a vortex model. (a) Natural ﬂow. (b) Open-loop actuation with StA = 0.17 and cµ = 0.015.

The pressure ﬁeld can be obtained by solving the (normalized) Poisson-equation. ∆cP = −2 ∇u · ·∇uT = 2 Q

(1)

The source term Q is the local diﬀerence of the double contraction of the rotation and strain tensors, respectively. Inside a vortex, rotation dominates and Q >> 0, which according to Eq. (1) indicates falling pressure. In a straight shear-layer rotation and strain are balanced and Q = 0. On the convex side of a bent shearlayer and between vortices strain dominates. With Q << 0 this marks a region with raising pressure. Velocities and velocity gradients are rather small in the dead water. With Q = 0 the pressure gradients remain constant, thus, the base pressure along the stern is determined by the near wake pressure ﬁeld. Hence, vortices in the near wake left their footprints in the pressure readings. Dominant Strouhal numbers of the wake are usually within a range of 0.2 for circular cylinders and 0.26 for D-shape bodies. In Fig. 3 frequency spectra of hotwire measurements taken at x/H = 1, y/H = 0.7, z/H = 0 are plotted for the natural ﬂow at various Reynolds numbers. The spectra indicate a maximum ﬂuctuation level at Strouhal numbers of StW = 0.28. How can control be applied in order to reduce drag? Strong vortices in alternating sequence yield a short dead water and strongly bent shear-layers. As explained, this yields low pressure recovery and high drag. To elongate the dead water the alternating character of the wake has to be hampered or delayed.

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Fig. 3. Experimental frequency spectra of the natural ﬂow. Power spectra are based on velocity ﬂuctuations from the natural ﬂow measured at x/H = 1, y/H = 0.7, z/H = 0 for Reynolds numbers 20 000 (−), 30 000 (−−), 40 000 ( ), 50 000 (−·), 60 000 (· · · ).

From the control perspective the suppression of the wake instability is the key to increase pressure recovery and to decrease drag. Before the control scheme is outlined the mechanism of the wake instability is elucidated. Since the dominant coherent structures are two-dimensional a vortex model is used to explain the transient development of the wake (Fig. 4). State a) - Initial vortices: The ﬂuid starts to move and separates at both trailing edges. The shear-layers roll up into almost symmetrical vortices. These initial vortices convect downstream and eventually dissipate due to mutual circulation. State b) - Shear-layer vortices: Shear-layers are sensitive to disturbances in the natural ﬂow. Due to convective Kelvin-Helmholtz-type instabilities these disturbances are ampliﬁed in streamwise direction, hence, shear-layer vortices evolve. Since the characteristic frequency of this process scales with the shearlayer thickness, it is one magnitude above the characteristic frequency of the fully developed wake. Thus, shear-layer vortices in this phase are rather weak, because only small portions of vorticity are lumped in each vortex. Further downstream the small vortical structures may roll up to stronger vortices (pairing). State c - Wake instability: Some two or three body heights downstream interaction between upper and lower shear-layer vortices starts. Any perturbation of the symmetry between upper and lower vortex conﬁguration is ampliﬁed. Vortices in the far wake are aligned in an alternating order with stretched spatial wave length. The wake instability locks on large vortical structures. Thus, the alternating sequence of vortices in the far wake has an impact even at some distance upstream, hence, the term absolute instability. Natural noise is outpaced by almost monofrequent perturbations. The far wake vortices trigger alternating shedding of larger vortices in the near wake.

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Fig. 4. Point-vortices and isobares from the vortex model illustrating transient behavior of coherent-structures in the wake: (a) initial vortices, (b) almost independent shear-layers, (c) interaction (wake instability) and (d) periodical wake

State d - Vortex street: Mutual interaction of both shear-layers intensiﬁes alternating shedding. The dead water becomes shorter and shorter. After a few convective time units a fully developed vortex street appears rather close to the stern.

3

Open-Loop Control

A reduction of drag is achieved by a suppression of the wake instability, as outlined in the previous section. The eﬀect of elongating the dead water by forcing is illustrated by smoke visualization in Fig. 2b. Base pressure recovery is increased as the shear-layers evolve straight and symmetrical. In Sec. 3.1 experimental results of the actuated ﬂow without feedback are presented. The eﬀects and processes of open-loop actuation are explained in Sec. 3.2. 3.1

Parameter Sweep

The sensitivity of the ﬂow for open-loop actuation was investigated for a wide range of Reynolds numbers, actuation frequencies and amplitudes. Upper and lower actuator were operated in two modes: in-phase and anti-phase. In Fig. 5 pressure and drag coeﬃcients are plotted versus the Strouhal number of the actuation. Actuators operated with a constant amplitude (cµ = 0.015) and

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Fig. 5. Normalized and time-averaged pressure (left) and drag coeﬃcient (right) as a function of the Strouhal number for cµ = 0.015 (in-phase forcing) at Reynolds numbers 20 000 (− · −), 30 000 (− ◦ −), 40 000 ( × ), 50 000 (− · + − ·), 60 000 (· · · ∗ · · · )

in-phase. Both coeﬃcients are normalized by the pressure and drag coeﬃcient of the natural ﬂow, respectively. There is a maximum pressure recovery at StA ≈ 0.17 for all checked Reynolds number, which yields a drag reduction of almost 15%. For Strouhal numbers below StA = 0.1 and above StA = 0.4 there is no beneﬁcial modiﬁcation to pressure recovery (experiments where performed to Strouhal numbers up to StA = 4). There is a striking collapse of pressure recovery in a small band centred at the Strouhal number of the wake instability at StA = 0.28, which is slightly above the drag of the natural ﬂow. Fig. 6 illustrates the eﬀect of the actuation amplitude on pressure recovery and drag for the most eﬃcient Strouhal number StA = 0.17. Again, both actuators operated in-phase. Obviously, there are Reynolds number dependent thresholds of cµ before forcing signiﬁcantly increases pressure recovery. Above a certain amplitude pressure recovery runs into a saturation level of cP /|cP,0 | = −0.6. ≈ 0.003 and cmax ≈ 0.007. The For ReH = 40 000 these threshold are cmin µ µ intermediate behaviour almost linearly connects natural and saturation level. Similar behaviour is cognizable for the drag coeﬃcient. Operating the actuators with a phase shift of 180◦ (anti-phase) still increases pressure recovery and reduces drag in the same range of Strouhal numbers as in-phase forcing (Fig. 7). However, it is less eﬃcient and for StA = 0.28, the frequency of the wake instability, pressure recovery drops to cP /|cP,0 | = −1.4 and drag increases by almost 15%. Obviously, the wake instability is ampliﬁed. 3.2

Description of the Actuated Flow

Zero net ﬂux actuation increases the magnitude of perturbations in the initial shear-layer and rectiﬁes the roll-up process. The convective instability starts with a higher initial condition. In free mixing-layers actuators also command the phasing of vortices. Transferring this behaviour to the bluﬀ body, synchronization of upper and lower shear-layer should be achieved by forcing in-phase.

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Fig. 6. Normalized and time-averaged pressure (left) and drag coeﬃcient (right) as a function of the momentum coeﬃcient with StA = 0.17 (in-phase forcing) at Reynolds numbers 20 000 (−·−), 30 000 (−◦−), 40 000 ( × ), 50 000 (−·+−·), 60 000 (· · ·∗· · · )

Perturbations in the symmetry would be reduced and therefore evolution of the wake instability could be delayed. However, in the far wake the alternating character imposed by the wake instability still asserts. In Fig. 8 the ability of the actuator to command phasing is illustrated. Only pressure readings of the topmost and the bottommost pressure gages are used here. Plot (8a) displays phase angles between pressure readings and upper and lower actuation variables, respectively, as a function of the Strouhal number. Plot (8c) shows the rms-value of pressure ﬂuctuations. In plot (8b) the amplitudes of Fourier projections of the pressure readings are illustrated. Fourier projections are conducted with the wake instability frequency and the forced frequency. Amplitudes are normalized by the rms-value of the pressure ﬂuctuations. In the following three cases of actuations with varying frequencies and modes are closer examined at Reynolds number 40 000. Amplitude is maintained constant at cµ = 0.015. Case 1 (StA < StW ): Forcing with a frequency below the wake instability frequency with both actuators in-phase yields high pressure recovery (Fig. 5). This can be explained by triggering stronger and more synchronized structures in both shear-layers, which allows for a more independent evolution of both shear-layers. By forcing 0.15 ≤ StA ≤ 0.22 vortex creation in the upper and lower shearlayer occur in the same phasing (Fig. 8a). Pressure readings at the stern and actuator output show a phase lag of approximately 180◦, vortices appear when actuators switch from blowing to sucking. Fig. 8b indicates that most of the ﬂuctuations are imposed by vortices shed at the actuation frequency. The overall noise level in the pressure readings is low (Fig. 8c). This is an indicator for proper synchronization. Obviously, there is a certain threshold amplitude required to rectify phasing (Fig. 6).

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Fig. 7. Normalized and time-averaged pressure (left) and drag coeﬃcient (right) as a function of the Strouhal number for cµ = 0.015 (anti-phase forcing) at Reynolds numbers 20 000 (− · −), 30 000 (− ◦ −), 40 000 ( × ), 50 000 (− · + − ·), 60 000 (· · · ∗ · · · )

There is a lower limit to actuation frequencies, where forcing has no beneﬁcial eﬀect. Below StA < 0.1 phasing between upper and lower vortices diﬀer, hence, synchronization is poor (Fig. 8a). Moreover, sensitivity of the shear-layer for lowfrequency perturbations is small compared to the noise level and perturbations imposed by the wake instability (Fig. 8b–c) dominate. Case 2 (StA > StW ): Beneﬁcial eﬀects to pressure recovery decline for actuation frequencies above the wake instability frequency using in-phase forcing (Fig. 5). In this frequency range there is still some authority to command phasing, vortices are released synchronously (Fig. 8a). But ﬂuctuations imposed by these vortices and the wake instability are almost of equal size (Fig. 8b–c). Higher frequencies create smaller vortices. Further roll-up and vortex pairing is triggered by the wake instability in the far wake. Thus, large alternating vortices still appear close to the stern, which reduces pressure recovery. Case 3 (StA = StW ): The experiments presented above indicate a collapse of pressure recovery when forcing the frequency of the wake instability. The explanation is poor authority of the actuators to command phasing of vortex shedding (Fig. 8a), as the phase relation between actuation and pressure readings diﬀers. Fig. 8b) indicates a failure of the lower actuator to keep in pace with the upper shear-layer. Apparently, the perturbations imposed by the wake instability outpace the attempt to rectify the lower shear-layer evolution. In fact, the actuator increases the perturbation level of the shear-layer in the matching frequency, thus amplifying the wake instability. As a result, pressure recovery is decreased by 10% whilst drag is increased by some 5% (Fig. 5). Operating with anti-phase forcing cause a signiﬁcant increase of drag by merely 15%. As with in-phase forcing larger vortical structures are created. But now phasing enforced by the actuators and that imposed by the wake instability

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Fig. 8. Open-loop control with in-phase forcing at ReH = 40 000 with cµ = 0.015 and various Strouhal numbers. (a) Phase angles between actuator and pressure for upper (∗) and lower (◦) actuators and pressure gages. (b) Normalized amplitudes of Fourier projections of the pressure readings on the wake instability frequency (upper − · −, lower −−·−−) and the actuation frequency (upper −∗−, lower −◦−). (c) Rms-values of upper (− · −) and lower (− − · − −) pressure readings.

are identical. Alternating shedding is ampliﬁed and thus the pressure recovery is decreased (Fig. 7). 3.3

Deduced Recommendations for Feedback Control

The ﬁndings from open-loop control can be used to give some recommendations for the design and application of feedback control. In this study we distinguish closed-loop control and phasor control. The former designates a control scheme where a control variable, e.g. the base pressure, is monitored and fed back in order to match a reference variable. A black-box-model obtained from experiments gives the relation between actuating and control variable. The control goal is to stabilize a certain base pressure by a modulation of the amplitude of the actuator. For such a controller inphase forcing at Strouhal number 0.15 ≤ StA ≤ 0.22 with a amplitude range of 0.003 ≤ cµ ≤ 0.007 seems appropriate for Reynolds number 40 000. Phasor control refers to a control scheme, where dynamic states of the ﬂow model are monitored and fed back in order to achieve synchronization of

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coherent structures, here the vortex shedding in the upper and lower shearlayer. The control goal is to maximize the base pressure for a ﬁxed amplitude of the actuator. It also provides the possibility to prove our understanding of the instability mechanisms. A single actuator at the upper trailing edge is operated with a constant amplitude, while the phase is obtained from pressure readings at the lower trailing edge. A pressure minimum is a phase mark of a vortex in the lower shear-layer. The actuator at the upper edge has to force the roll-up of a vortex with a certain phase-relation. Open-loop actuation with Strouhal numbers in the range 0.15 ≤ StA ≤ 0.22 features a phase lag of φ = 180◦. Thus, we expect a maximum pressure recovery when the phase of the pressure signal is fed back with φ ≈ 180◦ and a corresponding frequency of approximately 0.17.

4

Feedback Control

In this section, diﬀerent feedback control schemes are proposed for drag reduction of the bluﬀ body in experiments based on the results of Sec. 3. One focus is to underline the superiority of feedback control in contrast to open-loop control. Especially when the well deﬁned conditions of wind tunnels are left, all ﬂow control problems will have to tackle situations in which disturbances act on the system. Hence, only by applying feedback control concepts, the beneﬁts of active ﬂow control can be fully exploited. First, a model-based robust closed-loop control is proposed which proﬁts from the well-known advantages of tracking performance and disturbance rejection. Second, a model-free control method, which automatically receive the maximum pressure recovery with respect to the optimal actuation energy, is proposed by the application of slope seeking feedback. In both approaches the recommendations for feedback-control given in Sec. 3.3 are considered. In-phase forcing using both actuators at a constant Strouhal number StA = 0.17 is applied. In a third approach, a model-based sensor is used for synchronization the vortex shedding in the near wake ﬂow. Here, only one actuator is necessary. This approach is called phasor control. 4.1

Robust Control

Model Identiﬁcation and Controller Synthesis. Many if not all real systems can only be described in an approximate manner by mathematical models. For robust control, the process is described by black-box models, i.e. only the input/output behaviour is examined. The control input or manipulated variable u corresponds to the excitation momentum coeﬃcient cµ . The spatial averaged and low-pass ﬁltered pressure coeﬃcient cP is taken as the surrogate control variable and the output y of the control problem. For the low-pass ﬁltering a second order low-pass ﬁlter with a cut-oﬀ frequency of 1 Hz is applied. To identify linear black-box models, classical step response experiments are performed. Thereby the excitation momentum coeﬃcient cµ is switched from zero to diﬀerent levels to obtain diﬀerent operating points of the system at

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Reynolds number 40 000. A family of linear time-continuous models of 2nd order with a time delay is ﬁtted to the measured data by application of subspace methods, see [16] for details. From all identiﬁed models, a frequency-based nominal model GN (s) is generated, with a minimum deviation lM (ω) to all identiﬁed models over a certain frequency range. The plant G(s), on which the robust controller synthesis is based, is described by a multiplicative uncertainty ∆M (jω) [17]: G(s) = GN (s)(1 + ∆M (s)), ∆M (jω) ≤ lM (ω). Here, ∆M (s) comprises all models, which, at each frequency ω, are less than or equal to lM (ω) in magnitude. The uncertainty is the result of the system behaviour by the application a simple linear, 2nd order model for this nonlinear system. The nonlinearity of the investigated bluﬀ body ﬂow causes a wide spreading of the model parameters in Gi (s) and, hence, large values of lM (ω), i.e. a conservative controller design is required for robust closed-loop stability. Analysing the identiﬁed linear model family, a correlation between the excitation momentum coeﬃcient used for the step response experiments and the values of the static gain KS = GN (0) for diﬀerent excitation momentum coeﬃcients can be found. This dependency can be approximately described by a static map f (u). With the inverse of this map, the uncertainty of the models can be compensated, partially. f (u)−1 is implemented in the closed-loop as a pre-compensator, as shown in Fig. 9. A model-based dynamic feedforward controller is used to obtain a better tracking performance of the control loop, as proposed in [18]. For such uncertain systems, many mature controller synthesis methods do exist. Here, a H∞ -synthesis scheme is chosen. In H∞ -control, stability and/or performance of the worst-ﬁtted plant used to describe the process can be guaranteed. To ﬁnd a trade oﬀ between the closed-loop sensitivity function S(s), giving the performance, the restriction of the magnitude of the plant input signals, given by the transfer function C(s)S(s), and robustness, given by the complementary sensitivity T (s) = 1 − S(s), the mixed sensitivity problem is solved. Here, the - Cf (s) uf d ∗

r - cuc u C(s) - c? - f (u∗ )−1 -6

u-

? Plant

y-

Fig. 9. Control-loop with pre-compensator f (u∗ )−1 and dynamic feedforward controller Cf (s) (C(s) - H∞ -controller, r - reference base pressure, u = cµ - manipulated variable, d - disturbances, y = cP - output)

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sensitivity transfer function is given by S(s) = 1/[1 + C(s)GN (s)]. Closed-loop transfer functions are weighted with WT (s), WCS (s), and WS (s) depending on the frequency, and then combined to a cost functional ⎤ ⎡ WT (s) T (s) min N (C(s)) ∞ , N = ⎣ WCS (s) C(s)S(s) ⎦ , C WS (s) S(s) which has to be minimized. Here, C(s) is the optimal controller. For more details the reader is referred to standard textbooks, e.g. [17].

Fig. 10. Behavior of the closed-loop system in experiment. Tracking response of the base pressure coeﬃcient (top left), corresponding time series of the drag coeﬃcient (bottom left) and time series of manipulated variables (uf : feedforward controller, uc : H∞ -controller) (top right). Perturbations are imposed by a variation of the Reynolds number (bottom right).

Experimental Results. The behaviour of the controlled system is tested thoroughly in wind tunnel experiments with respect to tracking response and disturbance rejection by using the robust controller synthesized with H∞ -design. Fig. 10 shows the tracking response of the closed-loop system after stepwise changing of the reference command, which corresponds to an increase of the base pressures coeﬃcient cP of approximately 40%. To demonstrate the robustness of the closed-loop control the Reynolds number is varied from approximately 60000 to 35000, as shown in the bottom right plot of Fig. 10. Although the operating

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point is changed continuously, a good tracking performance can be observed. High-frequency disturbances are not rejected because of both the system’s inherent limited tracking dynamics and the requirement of robustness giving a limitation of the closed-loop performance. The bottom left plot of Fig. 10 shows the time series of the aerodynamic drag coeﬃcient cD . Here, a reduction of approximately 15% can be observed. The corresponding time series of the manipulated variables of both the feedforward controller and the H∞ -controller are given in the top right plot. 4.2

Slope Seeking Feedback

Slope Seeking Feedback Scheme. Slope seeking feedback is an appropriate, non-model-based method for the control of non-linear plants characterized by a plateau in the steady state. It is an extension of the extremum seeking scheme, for details see [19] and [20]. The plant is considered as a block with the input u, i.e. the manipulated variable, and the output y to be controlled, with a static input-output map y = f (u). The idea is to achieve a commanded or reference = ∂f (u)/∂u of the steady-state map. slope fref

Fig. 11. Block diagram of the basic slope seeking feedback scheme (LP: low-pass ﬁlter, HP: high-pass ﬁlter)

A block diagram of the slope seeking controller and a plateau-type static map y = f (u) with the commanded slope fref at y ∗ = f (u∗ ) is shown in Fig. 11. Side constraints for this kind of feedback are fast process dynamics in comparison of variations of the input and a plateau-type static map as displayed in Fig. 11 (right). As the manipulated variable u tends towards the optimal value, the output variable y increases. The output passes through a high-pass ﬁlter (HP) which removes the mean value, but not the harmonic part with frequency ω. The product of the ﬁltered output and the zero-mean sine leads to a non zero-mean signal (actually detected slope) at the output of a low-pass ﬁlter (LP) as long as the plateau is not obtained.

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The negative value of the commanded slope fref is added to the actually detected slope. By an integration and multiplication by KP , u ˆ is increased until the commanded slope is reached. The slope seeking scheme is an adaptive closed-loop type of control and guarantees closed-loop stability if designed properly, see [19] and [20]. The choice of certain design parameters, as the gain KP , the cut-oﬀ frequencies of the highpass ﬁlter and the low-pass ﬁlter, the amplitude a and the frequency ω of the sine signal, determine the speed of convergence. In the present contribution the momentum coeﬃcient cµ is chosen as the input variable, and the output is given by the spatially averaged base pressure coeﬃcient cP . For the experiments the parameters of the slope seeking controller have be chosen appropriately. The cut-oﬀ frequencies of both the high-pass ﬁlter and the low-pass ﬁlter are set to π rad s−1. The frequency of the sine signal ω is set equal to the cut-oﬀ frequencies of the ﬁlters and for the amplitude a = 1.5 × 10−3 is chosen. The gain is set to KP = 0.05. For the commanded slope a small value of fref = 5 is suitable for the application in the experiments.

Experimental Results. The right hand of Fig. 12 illustrates the static map with the base pressure coeﬃcient as a function of the momentum coeﬃcient at a constant Reynolds number 40 000 obtained from the open-loop experiments. This static map cP = f (cµ ) is characterized by a plateau for cµ ≥ 0.007. The idea for the slope seeking feedback is to ﬁnd the maximum pressure recovery where the least control input is necessary. This condition is considered satisﬁed is below the threshold. where the slope fref Experimental data is illustrated in Fig. 12. The extremum seeking controller starts at cµ = 4 × 10−3 . Sinusoidal modiﬁcations of cµ are applied to obtain the slope. According to the local slope cµ is raised until the plateau is reached. This leads to a signiﬁcant increase of the base pressure coeﬃcient as shown in the middle plot, corresponding to the static map cP = f (cµ ). In this experiment a reduction of the drag coeﬃcient by 15% can be observed as expected from the results of open-loop actuation (Sec. 3). The static maps for various Reynolds numbers are given in the right plot of Fig. 13. It can be observed that the optimal point shifts by changing the Reynolds number. Thereby the optimal momentum coeﬃcient decreases with increasing Reynolds number. The left plots of Fig. 13 show an experimental example when the Reynolds number is increased continuously from 35 000 up to 60 000 (bottom plot). The slope seeking controller maintains the desired high base pressure at cP = −0.3 (middle plot) with a minimal control input u = cµ . In the upper plot a decrease of the input variable cµ can be observed corresponding to the static maps. This experiment shows an advantage of the slope seeking feedback with respect to changing operation points, as e.g. the Reynolds number.

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Fig. 12. Slope seeking feedback at ReH = 40 000 in experiment (left), and static map cP = f (cµ ) obtained from open-loop experiments (right). In-phase forcing with Strouhal number StA = 0.17 is applied.

Fig. 13. Slope seeking feedback in experiment for an increasing Reynolds number (left). Static maps cP = f (cµ ) at various Reynold numbers 20 000 (− · −), 30 000 (− ◦ −), 40 000 ( × ), 50 000 (− + −), 60 000 (− ∗ −). In-phase forcing with Strouhal number StA = 0.17 is applied.

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Fig. 14. Principle scheme of the phasor control

4.3

Phasor Control

The applications of both the robust controller and the slope seeking feedback suffer from high cost due to rather large values of the manipulated variables. Hence, a second approach is used here with focus on energy saving actuation, using only one actuator. The idea is to maximize the base pressure by synchronizing the upper shear-layer to the unforced evolution of the lower shear-layer. Phasor Control Scheme. The principle of the phasor control is shown in Fig. 14. The wall-pressure ﬂuctuations induced by the vortex shedding at the lower edge of the bluﬀ body stern are measured at x/L = 0, y/H = −0.4, z/W = 0. This pressure ﬂuctuations can be approximated by a simple sine function y(t) = a0 sin(θ) + y0 , with the phase θ(t) = 2πf t. To use this signal for a realtime state estimation, these parameters are interpreted as states of a dynamical system. The four states are given by the frequency f , the amplitude a0 , the phase θ and the oﬀset y0 . A simple stochastic state-space model for the description of the pressure ﬂuctuations in discrete time is given by ⎡ ⎤ 1 000 ⎢ 0 1 0 0⎥ ⎥ x(k + 1) = ⎢ ⎣ 2π∆t 0 1 0 ⎦ x(k) + w(k) 0 001 y(k) = x2 (k) sin(x3 (k)) + x4 (k) where ∆t denotes the sampling interval and the elements of vector w are Gaussian white noise. The model-based sensor algorithm, namely an extended Kalman ﬁlter, is used for the real-time estimation of the states. For more details about the Kalman ﬁlter algorithm, see e.g. [21]. Based on the estimated phase θ obtained by the extended Kalman ﬁlter, a harmonic actuation signal a(t) = A sin(θ + φ) is only applied the upper actuator

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Fig. 15. Results of phasor control at ReH = 40 000 and cµ = 0.005. Normalized and time-averaged base pressure (− ◦ −) and normalized and time-averaged drag coeﬃcient (− −) as a functions of the phase φ (left). Forcing frequency StA as a function of the phase φ (right).

slot. φ represents the actuation phase with respect to the phase of the lower edge vortex shedding. The actuation amplitude A is set to a constant momentum coeﬃcient cµ = 0.005. Experimental Results. Experimental results using phasor control are shown in Fig. 15 at Reynolds number 40 000. The left ﬁgure shows both the timeaveraged base pressure coeﬃcient and the time-averaged drag coeﬃcient as functions of the phase φ in the range of 0◦ ≤ φ ≤ 360◦ . Both coeﬃcients are normalized by the values of the unforced case. An increased base pressure can be observed in a range 40◦ ≤ φ ≤ 260◦ . There is a maximum pressure recovery at φ ≈ 200◦ , which yields a drag reduction of 14%. The corresponding actuation frequencies are plotted versus the phase φ in the right plot of Fig. 15. Increased pressure recovery is obtained in the frequency range 0.14 ≤ StA ≤ 0.29 as expected. The maximum pressure recovery is yield at a forcing frequency of approximately 0.2. It should be mentioned, that a drawback of the phasor control is the restricted applicability to a smaller Reynolds number range. Experiments with ReH ≥ 60 000 have for example shown that the estimation of the state by the extended Kalman ﬁlter fails. An extension of the phasor control approach is provided in [22]. Here, an extremum seeking feedback is used to determine the optimal phase for a maximum pressure recovery.

5

Conclusions and Outlook

As mentioned in the introduction, suppression of the vortex street is a suitable method for drag reduction of a bluﬀ body. This can be achieved by

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Open-loop control

Robust closed-loop control Slope seeking feedback

Phasor control

Advantage • data base for low-order modelling • simple control scheme

Challenges • no consideration of disturbances • parameter study required • no tracking capability • robust • model identiﬁcation re• tracking capability quired • sensors required • non-model approach • slow adaptation • optimal operating condi- • restricted to a small class tions of non-linear systems • sensors required • energy saving actuation • poor robustness • intuitive control scheme • no tracking capability regarding the dynamics of • sensors required the ﬂow

high-frequency excitation or 3D perturbation of the shear-layers, in order to mitigate coherent 2D structures. In this study, another possibility to suppress the vortex street is proposed. The formation of vortex street is delayed by synchronisation the upper and lower shear-layer evolution. Thus, the natural coupling between the shear-layer and vortex street is mitigated and early formation of von K´arm´an vortex is suppressed. Low-dimensional 2D vortex models predict the description of this mechanism. In particular, the eﬀects and processes of open-loop actuation can be explained and recommendations for feedback control be provided. Based on these recommendations diﬀerent feedback control schemes are successfully applied in experiment. A comparison of the methods is provided in Tab. 1. A combination of the elucidated control methods in order to combine the advantages and eliminate the drawbacks is possible. In the future, the applied low-dimensional models and the feedback control methods will be extended to more complex three dimensional bluﬀ body conﬁgurations, e.g. the grounded reference car model with a base slant angle, such as the so-called ’Ahmed Body’.

Acknowledgements We acknowledge funding of the Deutsche Forschungsgemeinschaft (DFG) and the U.S. National Science Foundation (NSF). DFG support was via the Collaborative Research Centre (Sfb 557) “Control of complex turbulent shear ﬂows” at the Berlin University of Technology and by the grant NO 258/1-1. NSF support was

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via grants ECS-0136404, ECS-0524070 and CNS-0410246. We appreciate stimulating discussions with Andr´e Brunn, Mark Luchtenberg, Marek Morzy´ nski, Erik Wassen, and Tino Weinkauf.

References [1] Bearman, P.W.: Investigation of the ﬂow behind a two-dimensional model with a blunt trailing edge and ﬁtted with splitter plates. Journal of Fluid Mechanics 21 (1965) 241–255 [2] Tanner, M.: A method of reducing the base drag of wings with blunt trailing edges. Aeronautical Quarterly 23 (1972) 15–23 [3] Tombazis, N., Bearman, P.W.: A study of three-dimensional aspects of vortex shedding from a bluﬀ body with a mild geometric disturbance. Journal of Fluid Mechanics 330 (1997) 85–112 [4] Fiedler, H.E., Fernholz, H.: On management and control of turbulent shear ﬂows. Progress in Aeronautical Science 27 (1990) 305–387 [5] Gad-el-Hak, M., Pollard, A., Bonnet, J.P.: Flow control - Fundamentals and practices. Springer, Berlin, Heidelberg (1998) [6] Bearman, P.W.: The eﬀect of base bleed on the ﬂow behind a two-dimensional model with a blunt trailing edge. Aeronautical Quarterly 18 (1967) 207–224 [7] Kim, J., Hahn, S., Kim, J., Lee, D., Choi, J., Jeon, W., Choi, H.: Active control of turbulent ﬂow over a model vehicle for drag reduction. Journal of Turbulence 5 (2004) [8] Wygnanski, I.: The variables aﬀecting the control of separation by periodic excitation. AIAA Paper 2004-2505 (2004) [9] Gerhard, J., Pastoor, M., King, R., Noack, B., Dillmann, A., Morzynski, M., Tadmor, G.: Model-based control of vortex shedding using low-dimensional Galerkin models. AIAA Paper 2003-4262 (2003) [10] Tadmor, G., Noack, B., Dillmann, A., Gerhard, J., Pastoor, M., King, R., Morzynski, M.: Control, observation and energy regulation of wake ﬂow instabilites. In: Proceedings of the 42nd IEEE Conference on Decision and Control 2003. Number WeM10-4, Maui, HI, U.S.A. (2003) 2334–2339 [11] Siegel, S., Cohen, K., McLaughlin, T.: Experimental variable gain feedback control of a cylinder wake. AIAA Paper 2004-2611 (2004) [12] Allan, B.G., Juang, J.N., Raney, D.L., Seifert, A., Pack, L.G., Brown, D.E.: Closed-loop separation control using oscillatory ﬂow excitation. In: ICASE Report 2000-32. (2000) [13] Henning, L., King, R.: Multivariable closed-loop control of the reattachement length downstream of a backward-facing step. In: Proceedings of the 16th IFAC World Congress, Praha, Czech Republic (2005) [14] King, R., Becker, R., Garwon, M., Henning, L.: Robust and adaptive closed-loop control of separated shear ﬂows. AIAA Paper 2004-2519 (2004) [15] Pastoor, M., King, R., Noack, B., Dillmann, A., Tadmor, G.: Model-based coherent-stucture control of turbulent shear ﬂows using low-dimensional vortex models. AIAA Paper 2003-4261 (2003) [16] Ljung, L.: System identiﬁcation - Theory for the user. Prentice Hall PTR (1999) [17] Skogestad, S., Postlethwaite, I.: Multivariable feedback control - Analysis and design. John Wiley & Sons, Chichester, England (1996)

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[18] Hagenmeyer, V., Zeitz, M.: Flatness-based design of linear and nonlinear feedforward controls. at-Automatisierungstechnik 52 (2004) 3–12 [19] Krstic, M., Wang, H.H.: Stability of extremum seeking feedback for general nonlinear dynamic systems. Automatica 36 (2000) 595–601 [20] Ariyur, K., Krstic, M.: Real-time optimization by extremum-seeking control. John Wiley & Sons, Hoboken (2003) [21] Gelb, A.: Applied optimal estimation. The M.I.T. Press, Cambridge (1986) [22] Henning, L., King, R.: Drag reduction by closed-loop control of a separated ﬂow over a bluﬀ body with a blunt trailing edge. In: Proceedings of the 44th IEEE Conference on Decision and Control, and European Control Conference ECC’05, Seville, Spain (2005) 494–499

Active Blade Tone Control in Axial Turbomachines by Flow Induced Secondary Sources in the Blade Tip Regime O. Lemke1, R. Becker2, G. Feuerbach2, W. Neise3, R. King2, and M. Möser4 1

Technische Universität Berlin, Sonderforschungsbereich 557, Müller-Breslau-Straße 8, 10623 Berlin, Germany 2 Technische Universität Berlin, Institut für Prozess- und Anlagentechnik, Hardenbergstraße 36a, 10623 Berlin, Germany 3 Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR), Institut für Antriebstechnik, Abteilung Triebwerksakustik, Müller-Breslau-Straße 8, 10623 Berlin, Germany 4 Technische Universität Berlin, Hermann-Föttinger Institut, Fachgebiet Technische Akustik, Einsteinufer 25, 10587 Berlin, Germany

Summary In conventional active noise control experiments, loudspeakers are used to generate the secondary anti-phase sound field to be superimposed destructively with the sound waves radiated from the primary source. In the present study, aerodynamic sound sources are used to actively control the tonal noise of an axial fan. This is achieved by disturbing the flow field around the blade tips in such a way that additional periodic forces are set up which in turn form the secondary sound sources. To disturb the flow, air is blown into the blade tip region through the casing wall. The resulting aerodynamic sound sources are adjustable in both amplitude and phase. Results for automatic control of the blade passing frequency are presented along with flow measurements in the interstage regime between rotor and stator.

1 Introduction The tonal noise of axial turbomachines is due to the periodic forces that are exerted by the flow on the rotor blades, stator vanes, and the casing. As shown by Tyler and Sofrin [1], the interaction between the rotor and inlet flow distortions and the interaction of the wake flow of the impeller blades with the downstream stator vanes are the main causes for the blade tone spectrum of axial turbomachines. In conventional active noise control experiments, loudspeakers are used to generate the secondary antiphase sound field which is superimposed destructively on the sound waves radiated from the primary source. In this study, the required anti-phase sound is produced by additional aerodynamic sound sources. The advantage of the aeroacoustic secondary sources is that their intensity follows the same flow speed (or rotor speed) dependence as that of the primary ones because the generating mechanism is aerodynamic in nature in both cases. R. King (Ed.): Active Flow Control, NNFM 95, pp. 391–407, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007

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The tonal noise reduction can be achieved by disturbing the flow field around the blade tips by using appropriate wall flush mounted actuators in the fan casing (see Schulz et al. [2], [3] and Neuhaus et al. [4] for details). Hereby, additional periodic forces are set up on the rotor blade tips which in turn form the secondary aerodynamic sound sources which are adjustable in both amplitude and phase. A principal sketch of this arrangement is depicted in Figure 1a. The most effective method so far is to blow air jets through small wall-flush mounted cylindrical nozzles into the blade tip flow regime, see Figure 1b. The interaction between the air jets and the wakes of the rotor blades leads to additional periodic unsteady forces generating a secondary sound field. Amplitude and phase of the secondary sound field are governed by the injected mass flow and the circumferential position of the jets.

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Fig. 1. a) Principle of the experimental set-up with actuators mounted in the fan casing. b) Air injection through cylindrical nozzles.

Amplitude and phase of the primary sound field are governed by the rotor-statorinteraction which in turn depends on the operating condition of the fan and other external parameters like temperature and pressure. Changes of these parameters require readjusting amplitude and phase of the secondary sound field by altering the injected mass flow rate and/or the circumferential nozzle position. When using flow induced secondary sources for active noise control, typical feed-forward algorithms or adaptive filters cannot be employed because the aeroacoustic secondary field cannot be regarded independent of the primary one: The sound fields of both the rotor-statorinteraction and the rotor-distortion-interaction are generated simultaneously. Hence, other means of closed-loop control strategies have to be found for this application. As a solution to the problem, an extremum-seeking control technique is used in a multiple-input-single-output (MISO) configuration to automatically adjust amplitude and phase of the secondary sound field via the injected mass flow rate and the circumferential nozzle position.

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Detailed flow measurements were made to obtain a better understanding of the physical mechanism of the active noise control technique using injected air jets. Schulz et al. [5] performed experiments on a small two-dimensional stationary cascade placed in a wind tunnel flow showing additional flow structures produced by the injected air jets. In chapter 5 of this paper, results of hot wire measurements between rotor and stator are presented to illustrate the three-dimensional fan flow at this interstage location.

2 Experimental Facility The experiments were performed with a low speed high pressure fan with outlet guide vanes in a ducted inlet/ducted outlet configuration, see the schematic presentation in Figure 2. The impeller diameter is D = 357.4 mm, the casing diameter DC = 358 mm, and the hub-to-tip ratio ε = 0.62. The rotor has Z = 18 NACA 5-63-(10) profiled blades with a chord length of c = 53.6 mm and a blade stagger angle of ν = 27° at the tip. The tip clearance gap width is s = 0.3 mm (ζ = s/c = 0.6%). 16 unprofiled stator vanes are mounted downstream of the rotor. A throttle is placed at the end of the outlet duct to control the fan operating condition. The sound fields in the anechoic ducts are monitored by using 16 wall-flush mounted 1/4-inch microphones equally spaced circumferentially. The circumferential sound pressure distribution is resolved into azimuthal duct modes. The azimuthal mode structure of the tonal sound field generated by the rotor-stator-interaction is described by the theory of Tyler and Sofrin [1]. At the blade passage frequency (BPF), the only azimuthal mode that can propagate through the fan duct is the mode m = 2.

Fig. 2. Experimental set-up (dimensions in mm)

The injection nozzles are mounted in a rotatable casing segment driven by a DCmotor via a timing belt (Figure 3a) to continuously vary the circumferential nozzle position i relative to the stator vanes. A DC-voltage supplied by an electronic controller is used to automatically set the circumferential nozzle positions. The injected mass flow is controlled by electric proportional valves which are also driven by the electronic controller.

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Compressed air is injected through cylindrical nozzles with a diameter of dnozzle = 1.5 mm at an angle of α = 45° relative to the casing wall (Figure 3c). The axial nozzle position is ∆x/c = 0.22 downstream of the rotor trailing edge. The circumferential injection angle β = 117° follows the rotor blade chord direction, see Figure 3b. In previous experiments the nozzle parameters α, β, dnozzle, ∆x/c were varied. The presented configuration turned out to be the most effective so far for a maximal BPF-level reduction (see Schulz et al. [2]).

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Fig. 3. a) Rotatable casing, b) Positioning of the injection nozzles in the rotor blade tip regime. c) Cross sectional view of an injection nozzle.

3 Results Obtained with Steady Air Injection Figure 4a shows sound pressure spectra measured in the fan outlet duct at a fan operating point ϕ = 0.216 and a rotor speed of n = 3000/min with and without active control. The flow coefficient is defined as ϕ = 4QFan/(πD2U) with QFan = volume flow, D = impeller diameter, and U = tip speed of the fan. The maximum reduction of the blade passage frequency level of 21.1 dB was achieved at a circumferential nozzle position of i = 14° and a mass flow rate of MJets/MFan = 0.61% where MFan is the main mass flow rate delivered by the fan without air injection. This best adaptation of amplitude and phase of the secondary sound field in the outlet duct was found by rotating the casing segment carrying the injection nozzles in small incremental steps relative to the stator vane leading edges, and by changing the injected mass flow rate through online observing the sound pressure level of the dominant mode. The axial position of the injection nozzles and the angle of injection were kept constant (Figure 4c). The azimuthal mode spectra at the blade passage frequency component of 900 Hz with and without steady air injection are plotted in Figure 4b. As expected, the dominant azimuthal mode without control is m = 2. When steady air injection is applied, this mode is reduced remarkably, and consequently the BPF-level decreases. The reduction in BPF-level is accompanied by small increases of the broadband noise level at higher frequencies, which is due to the noise of the jets themselves, and level increases of the higher blade tone harmonics. The additional interaction of the air jets with the rotor blades generates a secondary sound field with many higher harmonics, which are generally not in anti-phase with the higher harmonics of the

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120 Control off Lp,BPF = 114.4dB Control on Lp,BPF = 93.3dB ∆Lp,BPF = 21.1dB

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primary source. As a consequence the higher blade tone harmonics are increased in level. This limits the overall effectiveness of the noise reduction method. Still, the reduction of the overall sound pressure level is 9.8 dB. The results shown in Figure 3 are consistent with those presented earlier by Schulz et al. [2]. Flow visualisation and PIV experiments with a simplified two-dimensional blade cascade in a wind tunnel flow carried out by Schulz et al. [5] gave insights into the physical mechanisms dominating the interaction of the air jets with the rotor blades: The injected air jets roll up to form longitudinal vortex pairs which exist as additional flow structures in the tip blade flow regime and interact with the rotor blade wakes. Further, numerical investigations by Ashcroft et al. [6] indicate that the potential flow field of a high energy air jet is similar to the potential field around a cylinder. The interaction of the potential fields of the air jets and the rotor blades lead to unsteady periodic forces set up on the rotor blades which are interpreted as secondary aeroacoustic sound sources. Figure 5a shows the maximum BPF-level reduction ∆Lp,BPF measured in the outlet duct as function of the circumferential nozzle position i over one stator passage Θ = 22.5°. At each position i, the injected mass flow MJets was adjusted to give the maximum BPF-level reduction ∆Lp,BPF in the outlet duct. A small range of optimal anti-phase adjustment of the secondary sound field can be identified. Figure 5b shows the BPF-level Lp,BPF as a function of both the circumferential nozzle position i and the injected mass flow rate MJets/MFan. The transparent horizontal plane specifies the baseline configuration. Clearly, BPF-level reductions as well as increases are observed in

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Figure 5b when varying circumferential position and injected mass flow. This characteristic diagram is used for the controller design described in the next chapter.

4 Control Strategy Since the secondary sound sources employed in the present study are generated by injecting air jets into the blade tip flow regime, they cannot be operated without the fan running, i.e., without the primary sources being present at the same time. Thus it is not possible to describe the secondary source system by transfer functions between sensor and actuator signals as is done in conventional active noise control experiments, and other closed-loop control strategies have to be used for this application. Gawron et al. [7] and Lemke et al. [8] developed and tested a closed-loop controller based on an extremum seeking control technique. 4.1 Extremum Seeking Control Figure 6 shows the control circuit in a multiple input/single output (MISO) setup. The plant is considered as a block with two inputs, the injected mass flow MJets and the

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circumferential nozzle position i, and the output to be controlled with a static inputoutput-map. The idea is to perform a gradient based online optimization in order to adjust the control inputs such that the minimal steady-state system output is achieved without knowing the steady-state input-output-map and especially its minimum. The structure consists of two decoupled single input/single output control loops for the adjustment of each input with a common output. Each control loop works as follows: The initial control input, marked by the index 0, calculated by some higher control hierarchy or set to zero, is superimposed by a sine signal a⋅sinωt with a small amplitude a. If the period of the harmonic perturbation is larger than the largest time constant of the dynamic plant, an approximate sinusoidal output perturbation will be obtained, initially around Mjets0 and i0, respectively. This output perturbation is analyzed in order to detect the slope (gradient) of the input-output-map and then used for gradient based optimization. To do so, the system output is passed through a band-pass filter (BP) which extracts the sinusoidal perturbation with frequency ω. The product of this filtered output, i.e. the output perturbation, and the zero-mean sine signal sinωt gives a measure of the slope. It leads to a non-zero mean signal as long as the maximum is not reached. If the plant is initially to the left of the minimum, the input and output perturbations are in anti-phase, i.e. the product will be negative. On the other side, an in-phase relation giving a positive product will be an indication of being to the right of the minimum. This product is then passed through a low-pass filter (LP) to extract a mean value. Additional terms, i.e. ∆Mjets(t) and ∆i(t), respectively, added to the input signal, are then calculated by time integration of this mean value and multiplication by a negative gain g. As long as the output of the LP is negative, i.e. the system is on the left side of the minimum, an increasing input value, i.e. Mjets and i, respectively, is obtained. For a positive output of the LP, the opposite is true. A maximum is obtained accordingly with this method. The extremum seeking scheme is an adaptive closed-loop type of control and guarantees closed-loop stability if designed properly, see Kristic and Wang [9] for details. The choice of the perturbation frequencies, and the filter and integrator design parameters determine the speed of convergence. Two different perturbation frequencies ω1 and ω2 are chosen in order to decouple both control loops. Here, the faster control

Fig. 6. MISO extremum seeking controller in a decoupled arrangement to automatically adjust the optimal amount of injected mass flow MJets (inner circuit) and optimal circumferential nozzle position i (outer circuit)

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adjusts the injected mass flow Mjets. If the plant behaviour varies due to uncertainties, the time scales of the perturbation signal and the filters have to be larger with respect to the slowest possible plant dynamics. The main advantage of this extremum seeking control is that no plant model is needed for controller synthesis. One price paid are the permanent harmonic input and output perturbations. 4.2 Experimental Results at Constant Operating Condition

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Programming and implementation of the controller were performed with MATLABSimulink by using an 8-channel DSP-system from the company dSpace. The plant output variable was a short time (10 ms) RMS value of the measured band pass filtered microphone signals. A band pass was used to separate the BPF component. To suppress uncorrelated noise signals, the average RMS value of 8 of the 16 wall-flush mounted microphones was used. It was assumed that a reduction of the BPF-level accompanies a reduction of the dominate azimuthal mode of order m = 2 at this frequency. First tests of the controller were performed at constant fan operating condition ϕ = 0.216 and a rotor shaft speed of n = 3000/min. The controller is able to find the minimum BPF-level for the same parameters as found by manual control as shown in Figure 4. Figure 7 shows the temporal development of BPF-level (top) and both input values, i.e. injected mass flow rate (middle) and circumferential nozzle position (bottom). The actuation starts with the initial values i = 10° and MJets/MFan = 0.35% inside the characteristic diagram shown in Figure 5b. Both input values are superimposed with their current sine perturbation. The plant follows the inputs. At t = 14 s, feedback control starts by closing the loop. The plant reaction to the given inputs is clearly shown in the way the BPF-level develops. The slowly oscillating signal part is the result of the circumferential input perturbation with an amplitude of iR = 0.6° at a perturbation frequency of ω2/2π = 0.2 Hz . This corresponds to a phase perturbation of 115 110 105 100 95 90 2.5 2.0 1.5 1.0 0.5 0.0

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the secondary sound field of ∆Φ = 10°. At t = 80 s, the steady state is achieved. The steady state is characterised by a noisy behaviour of the BPF-level signal. This is due to the fact that minimal BPF-levels occur only over a small range of circumferential nozzle positions and injection rates, compare Figure 5b, while the circumferential input perturbations are continued. The mechanical properties of the rotatable casing segment prevent smaller perturbation amplitudes to be realised: the inertial mass of the rotatable casing segment together with sudden transition from static to dynamic friction in its friction type bearing set a lower limit to the perturbation amplitude. Such non-linear effects lead to distortion of the harmonic sine of the input pertubation. The small temporal discontinuities in the minima and maxima of the circumferential sine input perturbation, seen in the bottom diagram of Figure 7, are needed to correct for the play of the servo motor gear. The slowly oscillating BPF-level signal is superimposed by a fast oscillating signal, which is given by the mass flow perturbation at a frequency of ω1/2π = 2 Hz . Due to the two different perturbation frequencies and the band pass filters used, the controller is able to separate their reactions for both amplitude and phase of the secondary sound field. The controller can shift the injected mass flow to a local optimum for every circumferential nozzle position i. In the steady-state condition the mean value of the injected mass flow rate is oscillating in anti-phase but at the same perturbation frequency as the oscillation of circumferential nozzle position. Small phase errors of the secondary sound field as a result of continued circumferential oscillation during the optimum condition are corrected by the injected mass flow. Here, a changed injected mass flow rate leads to small deflection of the air jets and, consequently, the phase of the secondary sound field is altered, due to a shifted position of the secondary sound sources. Figure 8 shows the sound pressure spectra and the azimuthal mode spectra for the optimum achieved by using closed loop control. For comparison with the results obtained with manual adjustment, see Figure 4. The controller is able to adjust the amplitude and phase of the secondary sound field by simultaneous variation of both injected mass flow and circumferential nozzles position to give a maximum reduction of the dominant azimuthal mode m = 2 by controlling the BPF component. However, there are small differences in the control parameters between the cases of manual and closed-loop control. For example, the injected mass flow rate MJets/MFan = 0.56% in the latter case, is 0.05 percent points lower than in the former. The average circumferential nozzle position is i = 13.5° with closed-loop control compared to i = 14° with manual control. For the feedback controlled experiment, the BPF-level reduction achieved with closed-loop control is only 18.4 dB compared to 21.1 dB with manual control, but the level reduction of the dominant azimuthal mode m = 2 is 1.5 dB better. These differences in the results are due to the fact, that the optimum injected mass flow rate and the circumferential position are kept fixed in the manual-control case while they are superimposed by perturbations in the closed-loop control case even after the optimum has been reached.

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4.3 Feedback Control Under Changing Fan Operating Conditions The main purpose of feedback control is to be able and respond to changes of the external parameter like fan operating condition, temperature, or pressure in the present case. For example, changing the fan operating condition alters the entire flow field in the fan interstage regime and, in turn, the rotor-stator-interaction process responsible for the generation of the fan blade tones. In other words, amplitude and phase of the primary sound field are affected by changes in the fan operating conditions and automatic readjustments of amplitude and phase of the secondary sound field are required to maintain efficient active noise control. Figure 9 shows traces of the plant output signal (BPF-level, top diagram), the two input signals (injected mass flow in the middle and circumferential nozzle position at the bottom), and the fan operating condition in terms of the flow coefficient in the middle diagram. The fan operating condition is changed from the best efficiency point ϕ = 0.260 to a point next to the stall line ϕ = 0.200. The fan speed is n = 4000/min. After closed-loop control has been initiated, the controller has first to find the optimum for the flow coefficient ϕ = 0.260. Feedback control starts at t = 20 s and the steady state is achieved after 65 s at a circumferential nozzle position of i = 10.5°. At t = 133 s the flow coefficient ϕ is changed by moving the throttle at the end of the fan outlet duct. This is like a step change of the input signal to the slow controller. The

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BPF-level increases because of the now less-than-optimum setting of amplitude and phase of the secondary sound. The controller responds by moving the input signals to optimum values for the new flow coefficient ϕ = 0.200. After 140 s, the new steady state condition is reached with MJets = 33 g/s injected mass flow and i = 13.2° circumferential nozzle position.

5 Hot Wire Measurements As mentioned before, Schulz et al. [5] performed flow visualisation and PIV experiments with a simplified two-dimensional blade cascade in a wind tunnel flow to gain a better understanding of the interaction of the air jets with the rotor blades. The injected air jets form longitudinal vortex pairs in the tip blade flow regime that interact with the rotor blade wakes and thereby set up unsteady periodic forces on the rotor blades which act as the required secondary sound sources. The aim of the present hot wire measurements is to show how the flow field between rotor and stator is affected by steady air injection. The measurements were made at the same fan operating conditions as in the active noise control experiments described in Chapter 3. 5.1 Measurement Technique The standard X-wire data analysis algorithm is based on the assumption of twodimensional flow conditions and uses a cosine relation to relate the two bridge voltages to the two velocity components to be measured. This approach leads to errors in the calculated velocity components because the influence of the third velocity component is neglected in the analysis, i.e., the radial component in the case of the uv-probe (u,v being the axial and circumferential components, respectively) and the circumferential component in the case of the uw-probe (w = radial component). A solution to the problem is the parallel analysis of all four hot wire signals in the iterative

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four-wire-probe algorithm described by Rosemann [10]. The relation between the time series of the two probes, measured sequentially, is given by a reference trigger signal from the fan. It is often attempted, to determine the velocity components of a three-dimensional flow field by two subsequent X-wire measurements with different probe orientations. Two different X-wire probes are used for the measurements, see Figure 10a. In the first step, the uv-probe is used to measure the axial and the circumferential velocity components cx and cu of the fan exit flow. In the second step the uw-probe is placed at the same location and circumferential incident angle η as the uv-probe before to measure the axial and radial velocity components cx and cr.

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Fig. 10. a) Sketch of X-wire probes with coordinate system (denoted by index s). b) Top sketch: notation of flow velocity components at the impeller exit.Bottom: Transformation of the velocity components from the probe coordinate system (denoted by index s) to the fan coordinate system (xz,Θz).

Fig. 11. Positions of the X-wire probes

G The X-wire probes are positioned in the rotor wake flow ( c2 = mean flow velocity vector in the fixed frame of reference) at a certain incident probe angle η, see Figure 10b, to assure that the unsteady velocity vector lies within the range of the probe´s

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G flow resolution cone. The flow components vs = (us , vs , ws ) calculated by the iterative algorithm are defined in the probe coordinate system s also depicted in Figure 10b. To G obtain the exit flow velocity vector c2 in the fan coordinate system (xz,Θz) from the probe-related velocity components, the probe incident angle η has to be taken into account as described in the following relation:

§ c2 x · § cosη ¨ ¸ ¨ ¨ c2u ¸ = ¨ sinη ¨c ¸ ¨ 0 © 2r ¹ ©

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Between rotor and stator, the X-wire probes are placed ∆x/c = 0.47 downstream of the rotor tip trailing edge. Measurements are taken at seven circumferential positions with an incremental distance of ∆i = 2° to cover the angle between two injection nozzles (Figure 11). Due to the casing design, the nearest possible position to a nozzle is 5.25°. At each circumferential position, measurements are made at 41 radial points. On the fan intake side, only one circumferential probe position at ∆x/c = 0.52 axial distance from the rotor blades leading edges is used, because the incoming flow field is fairly uniform. 5.2 Flow Profiles With and Without Steady Air Injection

Figure 12a shows the radial profile of the axial flow velocity component c1x at the rotor intake measurement position. The radial coordinate is normalized by the radial width of the fan annulus, hk = r/(DC – Dhub)/2, with hk = 0 at the hub and hk = 1 at the inner casing wall. The local flow velocity is divided by the mass flow averaged axial velocity component c1x,mean at the same axial location (with no air injection) which amounts to 20m/s in this particular case. This average flow velocity can be determined either from the hot wire flow profile data or from the fan volume flow measured directly via the static flow pressure in the inlet nozzle of the test duct. The difference between the results of the two calculations is less than 1%. G The radial and circumferential components of the incoming flow vector c1 are negligibly small. The axial flow velocity profile at the casing wall is characterised by a boundary layer thickness of about 30% of the blade span. Due to the 6.5 m long inlet duct and the perforated wall inside the anechoic duct termination, the flow upstream of the impeller plane is turbulent as is shown in Figure 12b. The effect of the hub boundary layer is visible in the radial profile of the turbulence level (Figure 12b). The hub boundary layer is much thinner than the casing boundary because of its shorter axial length beginning at the stagnation point of the fan spinner to the measurement plane. Injecting air (the mass flow rate necessary to give maximum BPF-level reduction in the outlet duct) has no influence on the turbulence level, while the axial velocity component c1x is reduced by about 1.5%. As a consequence the fan flow coefficient decreases to ϕ = 0.213. The axial velocity component c2x and the swirl component ∆cu⋅r of the flow field between rotor and stator are plotted in Figure 13, both are averaged circumferentially. The axial velocity profile upstream of the rotor blades is also given for comparison.

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Beginning at the hub, the axial flow is decelerated while passing through the blade channels over almost half of the blade span. Very likely, this is due partly to the narrower blade channels in this region, which increases frictional losses, and partly to flow separation occurring in the corner between the hub and the rotor blade suction sides. The corner separation itself is caused by the interaction of the boundary layers along the hub and the blade suction sides and results in a blockage of the mean flow in the hub region. As a result, the outer flow is deflected radially outwards and accelerated by up to 12% compared to the incoming flow between 40% and 80% of the

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blade span. The casing wall flow region is characterised by the interaction of casing boundary layer with the blade tip vortex structure. When air is injected through the casing wall, the near wall flow is accelerated due to the axial momentum of the air jets. Investigations of Schulz et al. [5] and Ashcroft et al. [6] have shown that the primary effect of air injection occurs at the casing wall. A secondary effect is to decelerate the mean flow over the entire blade span as is observed in the inlet velocity profiles plotted in Figure 12. Figure 13b shows the swirl component ∆cu ⋅ r = r ⋅ (c2u − c1u ) , which is proportional to the energy supplied to the fan. The swirl component is calculated from the difference between the local circumferential velocity components at the rotor exit and

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inlet multiplied by the radial coordinate. In the present experiments, the air is injected in the direction of the blade chord, compare Figure 11. Thus, the air jets have a circumferential velocity component against the impeller rotation and, hence, diminish the swirl component ∆cu⋅r in the range hk = 0.7 to 1.0 as compared to the case without injection. As a consequence, the fan pressure is reduced by about 1.5% with air injection. The above results show that the injection configuration which gives maximum tone level reductions is not necessarily optimal also for the aerodynamic performance of the axial flow machine. In future experiments, other orientation of the air jets will be investigated to minimize the negative effect on the swirl component. 1.0 0.9 0.8 Incoming flow ϕ = 0.216 Control off ϕ = 0.216 Control on ϕ = 0.213

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6 Conclusions The tonal noise components of an axial fan can be reduced by using aeroacoustic sound sources for active noise control. The secondary sound field is generated by actively controlling the flow around the impeller blade tips. Both amplitude and phase of the secondary sound field can be controlled in such a way, that a destructive superimposition with the primary sound field is possible. In this paper, results are presented for the reduction achieved by injecting air through the casing wall via cylindrical nozzles to disturb the flow around the blade tips. With steady air injection, the sound pressure level at the blade passage frequency is reduced by up to 21.1 dB. Azimuthal

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mode analyses show that the dominant azimuthal mode of the order m = 2 is lowered by as much as 26.5 dB. For noise control with flow induced secondary sound sources conventional control techniques used in active noise control with loudspeakers, like feed forward controllers or adaptive filters, do not work because the secondary sources can not be operated independently of the primary ones and, therefore, measurements of transfer functions between sensors and actuators is not possible. Extremum seeking control techniques are shown to be appropriate tools for controlling the BPF component of an axial fan by using flow induced secondary sound sources. In this study a multiple input/single output controller in a decoupled arrangement was developed to control amplitude and phase of the secondary sound field by automatically changing the injected mass flow rate and the circumferential nozzle position. In the first step the controller was tested to adjust the control parameters for a minimal BPF-level which was founded earlier by manual adjustment at a fixed operating condition of the fan. The main purpose of automatic control is to be able and respond to changes of the system to be controlled, e.g., to varying fan operating conditions in the present study. The extremum seeking controller used turned out to be effective in readjusting amplitude and phase of the secondary sound field by finding new optimal values for the injected mass flow rate and circumferential nozzles position. For active noise control in flow ducts it is important that the secondary sound field is not only of the same amplitude and frequency as the primary field, but also has the same modal structure. This is achieved in the present study by making the number of injected air jets equal to the number of stator vanes. Hence, the remaining control variables are amplitude and phase which are set via the injected mass flow rate and the circumferential nozzles position. In this way the degrees of freedom for the extremum seeking controller are greatly reduced: only two input signals and one output signal are necessary to control the secondary sound field. For a better understanding of the complex interaction of the injected air jets with the three-dimensional fan flow in the interstage regime, hot wire measurements were carried out upstream and downstream of the rotor. Two different X-wire probes were used sequentially to measure the axial, radial and circumferential velocity components. The time series of all four hot wire signals were recorded together with a rotor trigger signal. An iterative four-wire algorithm was employed to determine the three dimensional velocity vector of the fan flow. The experimental results show a strong influence of the blade tip flow. For the given nozzles configuration, the axial velocity component in this region is accelerated and the swirl component decreased due to the injection of air against the rotational direction of the rotor. Consequently, the fan pressure is also decreased. A second effect is that the mean flow velocity in the fan annulus, and hence the fan volume flow, is diminished by about 1.5%. In summary, with the present nozzle configuration, optimal BPF-level reductions achieved by steady air injection are accompanied by small losses in the aerodynamic fan performance. Future investigations will seek air injection configurations to give maximum tone level reductions while avoiding the aerodynamic losses observed in this study.

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Acknowledgement This investigation is supported by the Deutsche Forschungsgemeinschaft as part of the Sonderforschungsbereich 557 “Beeinflussung komplexer turbulenter Scherströmungen” (Control of complex turbulent shear flows) carried out at the Technische Universität Berlin in cooperation with Deutsches Zentrum für Luft- und Raumfahrt, Institut für Antriebstechnik, Abteilung Triebwerksakustik Berlin.

References [1] Tyler, J.M., Sofrin, T.G.: Axial flow compressor noise studies. Transactions of the Society of Automotive Engineers 70, 309-332, 1962. [2] Schulz, J., Neise, W., Möser, M.: Active noise control in axial turbomachines by flow induced secondary sources. 8th AIAA/CEAS Aeroacoustics Conference, paper AIAA2002-2493. Breckenridge, Colorado, USA, June 17-19, 2002. [3] Schulz, J., Neise, W., Möser, M.: Active control of the blade passage frequency noise level of an axial fan with aeroacoustic sound sources. Fan Noise 2003, Senlis, France, 23.-25. September 2003. [4] Neuhaus, L., Schulz, J., Neise, W., Möser M.: Active control of aerodynamic performance and tonal noise of axial turbomachines. 5th European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics, Paper No. 195. Czech Rep., Prague, March 17-22, 2003. [5] Schulz, J., Fuchs, M., Neise, W., Möser, M.: Active flow control to reduce the tonal noise components of axial turbomachinery. 1st Flow Control Conference, paper AIAA2002-2949. St. Louis, Missouri, USA, June 24-26, 2002. [6] Ashcroft, G., Schulz, J.:Numerical modelling of wake-jet interaction with application to active noise control in turbomachinery. 10th AIAA/CEAS Aeroacoustics Conference, paper AIAA-2004-2853. Manchester, UK,10-12 May 2004. [7] Garwon, M., Schulz, J., Satriadarma, B., King, R., Möser, M., Neise, W.: Adaptive and robust control for the reduction of tonal noise components of axial turbomachinery with flow control, 30. Deutsche Jahrestagung für Akustik, 7. Congrès Français d'Acoustique, CFA/DAGA 04, Straßburg, Frankreich. 22.-25. März 2004. [8] Lemke, O., Becker, R., Neise, W., Möser, M., King, R.: Aktive Minderung des Drehklangs axialer Turbomaschinen durch Strömungsbeeinflussung unter Einsatz einer Mehrgrößenregelung., 5. VDI Fachtagung Ventilatoren, Braunschweig, 7./8. März 2006. [9] Kristic, M.; Wang H.-H.,: Stability of extremum seeking feedback for general non-linear dynamic systems. Automatica 36, pp. 595-601, 2000. [10] Rosemann, H.: Einfluß der Geometrie von Mehrfach-Hitzdrahtsonden auf die Messergebnisse in turbulenten Strömungen., Forschungsbericht DLR, Institut für Experimentelle Strömungsmechanik, ISSN 0171-1342, Göttingen 1989.

Phase-Shift Control of Combustion Instability Using (Combined) Secondary Fuel Injection and Acoustic Forcing Jonas P. Moeck, Mirko R. Bothien, Daniel Guyot, and Christian Oliver Paschereit Institute for Fluid Dynamics and Technical Acoustics Technical University Berlin, M¨uller-Breslau-Str. 8, 10623 Berlin, Germany

Summary Phase-shift control was applied to an atmospheric swirl-stabilized premixed combustor. Two different actuators were tested in the control scheme. An on-off valve was used to modulate secondary pilot fuel and a loudspeaker provided excitation of the air mass ﬂow upstream of the burner. In individual mode, both actuators were able to successfully control a low-frequency combustion instability with similar levels of suppression. The pilot valve could also be triggered at subharmonics of the dominant oscillation frequency without loosing control performance. Using the two actuators simultaneously gave an even better suppression compared to individual operation. It was further shown that the pulsed pilot fuel could be used to assist purely acoustic control in the case of limited actuator authority.

1 Introduction Modern gas turbine technology relies on lean premixed combustion to satisfy stringent governmental emission restrictions. Premixing the fuel with large quantities of air before injecting both into the combustor signiﬁcantly reduces the peak temperatures in the combustion zone and thereby leads to lower NOx emissions. However, combustion systems operating in the lean premixed mode are highly susceptible to the excitation of high amplitude pressure ﬂuctuations called thermoacoustic instability [1,2]. These self-excited oscillations are a result of the interaction between unsteady heat release in the ﬂame and the combustion chamber’s acoustic ﬁeld. The main consequences of thermoacoustic instabilities are increased noise, reduced system performance and reduced system durability. As described by Rayleigh’s Criterion [3], self-excitation of a combustion system occurs if the ﬂuctuations in heat release from the combustion process are in phase with the pressure ﬂuctuations. Although the real instability processes are somewhat more complex (due to combustor dynamics, ﬂuid dynamics, chemical kinetics, transport processes, ﬂame kinematics, heat transfer, etc.), Rayleigh’s Criterion pinpoints how control of thermoacoustic instability can be achieved. A common approach is to induce heat release ﬂuctuations, which are out of phase with the pressure ﬂuctuations. Active and passive strategies have been realized for control of combustion instabilities. The ﬁrst can be subdivided into open-loop and closed-loop control [4,5,6,7]. Of all R. King (Ed.): Active Flow Control, NNFM 95, pp. 408–421, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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these control strategies, closed-loop control was proven to be most effective due to its higher sensitivity to the dynamics of the system being controlled. In contrast, passive or open-loop control techniques cannot react to the excitation of unforeseen instabilities that result from changing operating conditions or different fuel composition and are thus generally less effective [8,9]. Two means of inducing forced excitations into a combustion system will be addressed here: secondary fuel injection and acoustic actuation. Modulated secondary fuel injection was recognized to be an efﬁcient active control method for attenuation of combustion instabilities [10,11]. This is due to the high control authority that is provided by the chemical energy of the fuel. In certain circumstances, adding small amounts of pilot fuel to premixed combustion strongly affects the combustion process. Paschereit et al. [11] studied the inﬂuence of secondary pilot fuel on premixed combustion with constant and modulated secondary pilot fuel injection. McManus et al. [12] also investigated the effect of main premix fuel modulation on instability suppression. This type of control is also applicable to full-scale engines [13]. Besides secondary fuel injection, acoustic actuators also have proven their ability to suppress combustion instabilities effectively [14], while simultaneously reducing NOx emissions [8]. The advantage of acoustic actuators over a modulation of the fuel ﬂow, is their ability to induce high bandwidth proportional pressure signals into the combustion system. Dowling and Morgans [5] pointed out that one of the main issues in active control of combustion instabilities is to ﬁnd suitable actuators. The implementation of loudspeakers, for example, may not be feasible for practical applications containing high energy densities due to the relatively large amount of power needed to stabilize an unstable combustion system [15]. For this reason, a combination of secondary fuel injection and acoustic actuation is proposed here. The idea is to ﬁrst dampen the main instability frequency using modulated secondary fuel injection with its high control authority. After stabilization of the combustion system has been achieved, acoustic actuation is employed to maintain stability. The combination of different actuators could therefore result in a higher variety of applicable actuators. According to Auer et. al [16], the wide variety of actuators’ control effectiveness reported in literature is partly due to the fact that the speciﬁc actuators do not supply the required modulation. This underscores the difﬁculties in providing appropriate actuators. This paper presents experimental results obtained with a premixed swirl-stabilized atmospheric combustor. A digital closed-loop control system was employed to suppress combustion instabilities. The pressure and the OH chemiluminescence in the combustor were recorded and processed. The processed pressure signal was used to drive the controllers. Control was achieved by means of two different mechanisms: secondary pilot fuel was injected through an on-off valve and a loudspeaker was used as an acoustic actuator. These two actuation methods were tested separately and simultaneously. The main control parameters investigated were (1) the delay time between the recorded pressure signal and the transmitted controller signal; (2) the controller gain in the case of acoustic actuation; and (3) the secondary fuel mass ﬂow when using the pilot valve. In case of secondary fuel ﬂow modulation, actuation at subharmonics of the main instability frequency was also tested. Subharmonic excitation reduces the need for high

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frequency actuation and extends actuator lifetime. Furthermore, subharmonic excitation has been shown to be effective. Jones et al. [7] reported attenuation of the sound level by 22 dB when modulating secondary fuel with the fourth subharmonic of the dominant frequency of oscillation.

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2.1 Combustion Facility The combustion test rig has a 500 mm long, water-cooled combustion chamber, which is equipped with an inner ceramic layer. To generate a thermoacoustic instability with a frequency close to those typically occuring in full-scale engines, a resonance tube (900 mm long) was attached to the combustion chamber. The combustor incorporates an environmental burner (EV-10) designed by ABB with a cross-sectional area expansion ratio of 4 for ﬂame stabilization. Figure 1 shows a detailed sketch of the burner. It is composed of two half cones shifted in such a way that the air is forced to enter the cone circumferentially through two slots. The resulting swirling airﬂow generates a recirculation zone along the centerline at the burner outlet. The main fuel is injected through holes, which are distributed equidistantly along the two air slots. Mixing of swirling air and main fuel results in a nearly premixed combustion. Pilot fuel can be injected at the EV-10 cone apex using a pilot lance.

Fig. 1. EV-10 burner

To show the capabilities of the actuators an operating point with a high instability level was chosen. Before entering the combustion chamber, the air was heated to a temperature of 523 K, thereby simulating a temperature that is closer to typical conditions in gas turbines. All tests described in the following were conducted with an air mass ﬂow of 200 kg/h and a premix fuel ﬂow (natural gas) of 6.6 kg/h. This corresponded to an equivalence ratio of 0.57 and a thermal power of approximately 100 kW. 2.2 Actuators and Sensors Two actuation devices allowed for control of combustion instabilities. The secondary pilot fuel was modulated using a Bosch injection valve, thereby generating ﬂuctuations in heat release. This type of valve is commonly used in the automobile industry for the injection of gaseous fuel. It features an on-off characteristic with a duty cycle period of 4 ms. The valve was installed 2 m upstream of the burner inlet.

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In addition to the modulated secondary fuel, a loudspeaker provided controlled excitation of the air mass ﬂow entering the burner. This woofer had a nominal power of 300 W. It was mounted at the plenum 720 mm upstream of the burner. Because the combustion air was pre-heated, the loudspeaker was purged with cold air (approx. 293 K) to protect the speaker’s membrane. Pressure ﬂuctuations in the combustion chamber were measured using two Br¨uel and Kjær 1/4-in. condenser microphones. Because of the harsh temperature conditions, the microphones were assembled in probe holders developed at DLR Berlin, which operated like semi-inﬁnitive tubes to suppress reﬂections of the acoustic waves. The probe holders were continuously purged with Nitrogen to avoid water condensation and to provide for cooling. One probe was installed at the ﬂame position, the other one further downstream. The heat release ﬂuctuations were measured using a photomultiplier equipped with a narrow 308 nm band-pass ﬁlter. At this wavelength, the photomultiplier captured light from OH chemiluminescence which is proportional to the heat release rate [17]. Both microphone and photomultiplier signals were ampliﬁed and low-pass ﬁltered at 1 kHz to avoid aliasing. The pressure ﬂuctuations recorded by the microphone at the ﬂame served as input to the controller. The description of the control circuit, which was implemented in Matlab/Simulink, follows in Section 4. The Simulink model ran on a DS1103 PPC Controller Board (dSPACE) and generated command signals for both actuators. To suppress Larsen effects, the input signal of the woofer was low-pass ﬁltered with a cut-off frequency of 4 kHz.

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The operating point chosen for the investigations corresponded to a very strong instability with high pulsation amplitudes. Although control efﬁciency at one operating point cannot be considered representative for all possible sets of parameter combinations, this operating point was the most demanding in terms of actuator authority. Sound pressure levels recorded during unstable operation could reach up to 5600 Pa. 1

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Figures 2 and 3 show spectra of heat release and pressure signals acquired at stable and unstable conditions. In Fig. 2, a probability distribution (pdf) of the combustor pressure ﬂuctuations for the unstable case is also shown. Stable operation at an initially unstable operating point was made possible by reducing the combustor length by approximately two thirds. The strongest peaks in both graphs correspond to the 100 Hz quarter wave mode in the downstream combustor tube. Clearly, the pressure signal primarily consists of a single harmonic component, whereas the heat release exhibits several harmonics, some having almost the same magnitude as the fundamental. Obviously, this supports the idea that the acoustics can be considered linear even on the limit-cycle and the pulsation amplitude is determined by nonlinear saturation in the ﬂame response [18,19]. Figure 4 shows ensemble-averaged perturbation cycles of pressure and heat release computed from 3200 realizations. The pressure is basically a sinusoid, whereas the signal from the OH emission, representing the heat release, is strongly distorted by higher harmonic components. Note here, that the symbols in the graph correspond to actual sampling points. Two cycles of the ensemble-averaged Rayleigh-Index (product of unsteady pressure and heat release), computed from the pressure perturbation at the ﬂame and the OH signal, are plotted in Fig. 5. Clearly, the Rayleigh-Index is positive over the dominant part of the cycle, especially where the heat release peaks. This indicates a distinct net production of acoustic energy [20]. It is important to note here that the system is indeed in a limit-cycling state. In combustion systems operating closer to the stoichiometric case, high oscillation amplitudes may also be produced by a stable noise-driven system [21]. Yet, the obvious nonlinearity in the heat release and the double hump pdf of the pressure signal are well-deﬁned features of limit-cycling behavior [22,23].

4 Phase-Shift Control with Individually and Combined Working Actuators Figure 6 shows a schematic of the control set-up used for this investigation. The unsteady pressure data acquired by the lower microphone was fed back to generate the control signal for the actuators. In principal, the OH signal could also be used for

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Fig. 6. Schematic of the control set-up

controller input but it usually contains more noise. Also, employing a pressure signal in the feedback loop rendered band-pass ﬁltering unnecessary. The reason for choosing the signal of the lower microphone was that it detected a higher amplitude due to the quarter-wave mode shape. In the ﬁrst feedback loop, the microphone signal was phase-shifted and ampliﬁed. This signal was used to drive the loudspeaker, which modulated the air mass ﬂow into the burner and directly generated sound waves. In the second loop, the dSPACE board generated a trigger pulse of the phase-shifted input signal. The control board ran at a sampling frequency of 10 kHz. The triggered valve modulated the secondary pilot fuel resulting in a heat release ﬂuctuation. Additionally, the Simulink model allowed for setting up a threshold value. If the pressure oscillations remained below this threshold value, the pilot valve would not open. This can save pilot fuel without adversely affecting the suppression of heavy oscillations [7]. The input signal to the controller was not band-pass ﬁltered. Using a narrow bandpass ﬁlter on the input signal can help to track the dominant mode. However, such a ﬁlter generally induces a rapid phase change along the passband and adversely affects the control scheme if the frequency of oscillation is shifted by the controller. This mechanism can cause an intermittent loss of control [24]. As will be shown in Section 4.2, the dominant frequency was indeed affected by the phase-shift controller due to the so-called peak-splitting phenomenon [21,25]. 4.1 Pulsed Pilot Fuel The inﬂuence of time-delay and secondary fuel mass ﬂow on the performance of the pilot fuel injector to suppress thermoacoustic instabilities was studied. The pilot actuator was operated in three modes: generation of secondary fuel pulses at the fundamental frequency of the dominant mode, at its 2nd subharmonic and at its 4th subharmonic. In Figure 7 the normalized peak amplitude (with respect to baseline conditions) is plotted as a function of time-delay. The pilot to premix fuel mass ﬂow ratio was held ﬁxed at 3.5%. This corresponded to the maximum suppression obtained from a pilot mass ﬂow variation (see Fig. 8). Maintaining constant mean pilot mass ﬂow had the effect of doubling and quadrupling the fuel pulse per stroke in the case of control at the 2nd and 4th subharmonics, respectively. For all tests (modulation at fundamental and

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subharmonics), the time-delay was ﬁrst set to zero and then increased in steps of 0.4 ms to a maximum time-delay of 10 ms. This maximum time-delay corresponded to a phaseshift of 360◦ with respect to the dominant oscillation frequency at baseline conditions. For fuel injection at the fundamental, the suppression performance was also investigated for decreasing time-delay (from 10 ms to 0 ms) to check for hysteresis behavior. The series denoted as ’fundamental A’ in Fig. 7 corresponds to increasing time-delay, the series denoted as ’fundamental B’ corresponds to decreasing time-delay. Regarding the inﬂuence of time-delay (Fig. 7), suppression performance of the injector was best when operating at the fundamental instability frequency with a time-delay of 0–1.2 ms or 8.8–10 ms. At 0.8 ms time-delay an attenuation of more than 20 dB was achieved. The lowest suppression occurred at 4.4 ms time-delay. However, the peak amplitude was still well below baseline conditions (6 dB). For fuel modulation at the 2nd subharmonic, the change of peak amplitude with time-delay was similar to modulation at the fundamental. The suppression was slightly lower, achieving a maximum attenuation of 20 dB at a time-delay of 9.2 ms. In the case of control at the 4th subharmonic, maximum attenuation was again lower than for the 2nd subharmonic. However, the corresponding curve appears to be time-shifted by about -4 ms from the result one would expect from looking at fundamental and 2nd subharmonic excitation. To rule out a measurement error, the corresponding experiments were repeated. The same results were obtained. Note that only for fuel modulation at the 4th subharmonic noise ampliﬁcation was obtained in a narrow time-delay interval between 1.6 and 2.8 ms corresponding to a phase interval of 43◦ . Comparison of series ’fundamental A’ and ’fundamental B’ shows no indication of hysteresis behavior for the variation of phase-shift. Figure 8 shows the normalized peak amplitude (with respect to baseline conditions) as a function of the pilot to premix fuel mass ﬂow ratio at zero time-delay. Generally, the mean fuel mass ﬂow was increased during the measurements. Only for the series ’2nd subharmonic B’ the mass ﬂow was ﬁrst set to its maximum and then reduced to check again for hysteresis behavior. The data presented clearly shows that a threshold of the secondary fuel ﬂow must be exceeded (here at least 3% of the premix fuel mass ﬂow) to achieve signiﬁcant sound

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suppression. For all excitation frequencies an optimum pilot to premix fuel ratio was found within the interval investigated. For excitation at the fundamental and at the 2nd subharmonic, the optimum pilot to premix mass ﬂow ratio was 3.8%; for the 4th subharmonic it was 6%. As for the variation of time-delay, no hysteresis behavior was found for the variation of pilot fuel ﬂow. From Fig. 7 it is noted that (excluding control at the 4th subharmonic) for all phaseshifts, the controlled oscillations are smaller than in the baseline case. Typically one would expect that generating additional heat release ﬂuctuations in phase with the instability should amplify the pressure oscillations. However, a pilot ﬂame generally has a stabilizing effect on the premixed ﬂame even when burning in steady state [11]. The fact that the optimal phase-shift for modulation at the 4th subharmonic differs considerably from those found for modulation at the fundamental frequency and the second subharmonic is not fully understood. Detailed investigations on the actuator response where performed and exhibited a rather complex behavior. From heat release response measurements in the combustion chamber, it could be seen that the initially square fuel pulses strongly smeared out and partially overlapped. One reason for that was the relatively large distance between the valve and the fuel injection point (approx. 2 m). In addition to that, the fuel was not injected directly into the premixed ﬂame but at the cone apex of the burner (see Fig. 1). Before reaching the ﬂame, it was subject to extensive mixing due to the swirling ﬂow. Usually it is undesirable to use a large amount of pilot fuel. Since the pilot ﬂame burns in diffusion mode, it will generally increase NOx emissions due to locally higher temperatures. In an earlier investigation conducted on the same test rig, it was shown that the pulsed pilot fuel indeed increased NOx emissions [26]. The advantage of high control authority associated with pilot fuel injection is accompanied by the drawback of higher NOx emissions. 4.2 Acoustic Excitation Phase-shift control with proportional actuators has essentially two parameters: gain and time-delay. These parameters can be tuned manually to yield an optimal suppression of pressure oscillations or can be adaptively adjusted by an extremum seeking controller [27,28]. The latter has the obvious advantage that the controller can compensate for a change in operating conditions. Since in this study the operating parameters were ﬁxed, controller gain and delay were set manually. When searching for the best combination of parameters, it was assumed that the optimal phase-shift (the one that results in minimal peak amplitudes at the instability frequency) was independent of the gain and vice versa. Hence, not the complete 2D parameter ﬁeld was investigated but only the variation of the time-delay at one ﬁxed gain and the variation of the gain for one ﬁxed time-delay. Figure 9 shows the peak amplitude of ﬂuctuating pressure and heat release in the combustor as a function of the controller time-delay. The gain was ﬁxed to its optimal value. All values have been normalized with their associated values at unstable conditions. The time-delay interval shown corresponded to 360◦ in phase with respect to the frequency of dominant oscillation. As can be seen, the range of time-delays where the controller is effective is rather narrow. The highest sound attenuation is obtained for a time-delay of 4.6 ms, which corresponded to a phase-shift of approximately 165◦ .

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Input and output lines and the DSP board introduce an additional time-delay of less than one millisecond. Hence, the loudspeaker actuated in antiphase, as expected. Using the optimal phase-shift, the amplitude of the dominant mode was reduced by about 20 dB. The overall sound pressure level decreased by 14 dB (not shown). For the larger part of possible phase-shifts, the instability is, in fact, ampliﬁed. In the worst case, the heat release oscillations at the unstable frequency grew almost twice as large as in the uncontrolled case (Fig. 9). The inﬂuence of the controller gain on instability suppression is shown in Fig. 10. Here, values of the control gain have been normalized with the optimal gain. The timedelay was ﬁxed to the optimum, i.e. 4.6 ms. Starting from small values, raising the gain has the effect to rapidly diminish the peak amplitude. After a minimum is attained at the optimal value, a further increase in the gain causes the amplitude of the dominant mode to grow again. However, for all values of the gain tested the pulsations were attenuated compared to the uncontrolled case. A typical phenomenon encountered when using phase-shift controllers to suppress combustion instabilities is the so-called peak-splitting [21,22,25,29]. Under the inﬂuence of the phase-shift controller, the frequency of the dominant mode slightly changes. At the optimal phase-shift two peaks are present in the frequency spectrum of the combustor pressure, one slightly shifted to the left and one slightly shifted to the right. Figure 11 shows sections of the pressure spectrum for different controller phase shifts. Increasing the phase-shift from 0◦ has the effect of shifting the frequency of the dominant mode to lower values in addition to attenuating the amplitude. For phase-shifts larger than the optimal value, the frequency of the dominant peak was shifted to the right with respect to the uncontrolled peak. The variation of the peak frequency over the full phase-shift interval is presented in Fig. 12. At the optimal phase-shift (lowest peak amplitude) the peak frequency jumps from 96 Hz to 107 Hz. In fact, two peaks are present here: One that is decreasing and one that is increasing. At the optimal phaseshift both peaks are of the same amplitude. The presence of two peaks in the combustor pressure spectrum can also be seen in Fig. 13 in the next section. Peak-splitting also appeared in the case of control with modulated pilot fuel (not shown).

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It should be noted here, that the peak-splitting phenomenon is not inherent to the controlled combustion process but rather a property of the phase-shift controller. Rowley et.al. [30] e.g. observed this phenomenon when using a phase-shift controller to suppress cavity ﬂow oscillations. 4.3 Simultaneous Pulsed Pilot and Acoustic Control As a ﬁrst test for combined control, both actuators were used simultaneously. The control parameters were set to the optimal values determined in the experiments with only one actuator (see Sections 4.1 and 4.2). Figure 13 shows combustor pressure spectra for the uncontrolled case, for control with pilot fuel modulation, for control with acoustic forcing and for simultaneous control. Compared to the cases in which only one actuator was used to suppress the instability a simultaneous operation of both actuators resulted in an additional attenuation of 3 dB; hence a total reduction of 24 dB was achieved. It should be noted here that these results were obtained without varying the control parameters in simultaneous mode. Since there are more parameters to tune when using both actuators at the same time, it can be expected that even better results could be obtained. Also note the peak-splitting evident in Fig. 13. As mentioned earlier, purely acoustic forcing might not have the control authority necessary to suppress thermoacoustic instabilities in practical applications due to the high energy densities present. However, to maintain stability far less power is required. In other words, an acoustic actuator might not have enough authority to suppress an instability initially but might succeed in keeping it in a controlled state. To investigate this case, artiﬁcial saturation was added to the acoustic control command (see Fig. 6). This essentially simulated a power limitation of the loudspeaker. The pressure sensor signal was not ampliﬁed proportionally but saturated at a predeﬁned limit. Figure 14 shows time traces of the unsteady combustor pressure, the pilot control command and the acoustic control command. After 6 s the acoustic control was switched on but due to its limited power it could not attenuate the pressure oscillations signiﬁcantly. After 18 s pilot control was switched on and the pressure amplitude decreased almost immediately. Acoustic control was added again at 23 s but now, after

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taking away the pilot control, the loudspeaker was able to keep the pressure oscillations at a controlled level for 20 more seconds. However, once heavily perturbed from the controlled state, the acoustic actuator was not able to re-attain control due to its limited power. This can be seen in Fig. 15. Initially, the combustor pressure was maintained in a controlled state using both actuators. After 3 s, the pilot control was switched off. The acoustic actuator was able to maintain a low level of pressure ﬂuctuations for a period of 9 s but then suddenly lost control. Note that Figs. 14 and 15 have different time scales. This drawback could be overcome by enabling the pilot actuator only to inject fuel if a predeﬁned threshold value for the pressure oscillations was reached. If the pressure oscillations fell again below this threshold no additional pilot fuel was added and the loudspeaker suppressed the instability on its own. Figure 16 shows results from this conﬁguration. As can be seen, an existing instability coud not be suppressed by purely acoustic control (8 s to 15 s). Therefore, pilot fuel was injected. After eight injection pulses, the pressure oscillation decreased and pilot fuel injection was terminated. Every

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Fig. 15. Time traces of unsteady combustor pressure (top), pilot control command (middle) and acoustic control command (bottom). Acoustic actuator is not able to maintain control.

Fig. 16. Instability control with combined acoustic control and pilot fuel injection above a certain threshold for the pressure amplitude

time the acoustic actuator started to loose control and the pressure amplitude grew to the threshold value secondary fuel injection was activated again. To show that the loudspeaker was not able to keep the instability suppressed, the threshold was set to inﬁnity at 42 s, i.e. no pilot fuel was added irrespective of how high the pressure level grew. After 3 additional seconds, the loudspeaker lost control and only after resetting the previous threshold (at 48 s) control was re-attained. In comparison to the cases depicted in Figs. 14 and 15 a lower saturation value for acoustic actuation was set. As can be seen in Fig. 16 (lowest curve) this had the effect that the acoustic control was working at its limit most of the time.

5 Conclusions Different ways of suppressing a thermoacoustic instability using phase-shift controllers were examined. The pressure ﬂuctuations in the combustion chamber were fed back to control the actuation mechanisms. Both secondary pilot fuel modulation by an on-off

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valve and acoustic control with a loudspeaker resulted in an attenuation of the pressure oscillation of 20 dB. The control effectiveness exhibited a strong dependence on phase-shift and gain. While the injection of secondary pilot fuel exhibited a wide phase range, in which the maximum attenuation of the instability was achieved, the best loudspeaker performance occurred only in a relatively narrow phase interval. Combining both actuators resulted in a total reduction of 24 dB. The approach of both actuators mutually supporting each other was examined. An artiﬁcial saturation function was used to to limit the loudspeaker’s power. In case of limited loudspeaker control authority an instability could not be suppressed initially by pure acoustic forcing but could be kept in a controlled state. Heavy perturbations exceeding a critical threshold value were additionally suppressed with secondary fuel injection. This strategy resulted in sufﬁcient suppression of the instability, while consuming a minimal amount of pilot fuel.

Acknowledgements We kindly acknowledge dedicated support from our undergraduate student co-workers Sebastian Schimek and Henning Kroll. Furthermore, we would like to thank the combustion team of DLR Berlin, for making part of their measurement hardware available to us. This work was supported by the German Research Foundation as part of the Collaborative Research Center 557 ”Control of Complex Turbulent Shear Flows” conducted at the Technical University Berlin.

References [1] Poinsot, T.J., Trouve, A.C., Veynante, D.P., Candel, S.M., Esposito, E.J.: Vortex-driven acoustically coupled combustion instabilities. J. Fluid Mech. 177 (1987) 265–292 [2] Candel, S.M.: Combustion instabilities coupled by pressure waves and their active control. 24th Symposium (International) on Combustion, The Combustion Institute (1992) 1277–1296 [3] Rayleigh, J.W.S.: The Theory of Sound. Volume 2. Dover Publications, New York (1945) [4] Paschereit, C.O., Gutmark, E.: The Effectiveness in Passive Combustion Control Methods. (2004) ASME Paper 2004-GT-53587. [5] Dowling, A.P., Morgans, A.S.: Feedback Control of Combustion Oscillations. Ann. Rev. Fluid Mech. 37 (2005) 151–182 [6] McManus, K.R., Poinsot, T., Candel, S.: A review of active control of combustion instabilities. Prog. Energy Combust. Sci. 19 (1993) 1–29 [7] Jones, C.M., Lee, J.G., Santavicca, D.A.: Closed-Loop Active Control of Combustion Instabilities Using Subharmonic Secondary Fuel Injection. J. Propulsion Power 15 (1999) 584–590 [8] Paschereit, C.O., Gutmark, E.: Proportional Control of Combustion Instabilities in a Simulated Gas-Turbine Combustor. J. Propulsion Power 18 (2002) 1298–1304 [9] Docquier, N., Candel, S.: Combustion control and sensors: a review. Prog. Energy Combust. Sci. 28 (2002) 107–150 [10] Hathout, J.P., Fleiﬂ, M., Annaswamy, A.M., Ghoniem, A.F.: Combustion Instability Active Control Using Periodic Fuel Injection. J. Propulsion Power 18 (2002) 390–399

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[11] Paschereit, C.O., Gutmark, E., Weisenstein, W.: Control of Thermoacoustic Instabilities in a Premixed Combustor by Fuel Modulation. (1999) AIAA Paper 99-0711. [12] McManus, K., Han, F., Dunstan, W., Barbu, C., Shah, M.: Modeling and Control of Combustion Dynamics in Industrial Gas Turbines. (2004) ASME paper GT2004-53872. [13] Seume, J.R., Vortmeyer, N., Krause, W., Hermann, J., Hantschuk, C.C., Zangl, P., Gleis, S., Vortmeyer, D., Orthmann, A.: Application of Active Combustion Instability Control to a Heavy Duty Gas Turbine. J. Eng. Gas Turbines and Power 120 (1998) 721–726 [14] Lang, W., Poinsot, T., Candel, S.: Active Control of Combustion Instability. Combustion and Flame 70 (1987) 281–289 [15] Fung, Y.T., Yang, V., Sinha, A.: Active Control of Combustion Instabilities with Distributed Actuators. Combust. Sci. Tech. 78 (1991) 217–245 [16] Auer, M.P., Gebauer, C., M¨osl, K.G., Hirsch, C., Sattelmayer, T.: Active Instability Control: Feedback of Combustion Instabilities on the Injection of Gaseous Fuel. J. Eng. Gas Turbines and Power 127 (2005) 748–754 [17] Haber, L.C., Vandsburger, U., Saunders, W.R., Khanna, V.K.: An Examination of the Relationship between Chemiluminescent Light Emission and Heat Release Rate under Nonadiabatic Conditions. (2000) ASME Paper 2000-GT-0121. [18] Stow, S., Dowling, A.P.: Low-Order Modelling of Thermoacoustic Limit Cycles. (2004) ASME paper GT2004-54245. [19] Dowling, A.: Nonlinear self-excited oscillations of a ducted ﬂame. J. Fluid Mech. 346 (1997) 271–290 [20] Culick, F.E.C.: A Note on Rayleigh’s Criterion. Combust. Sci. Tech. 56 (1987) 159–166 [21] Banaszuk, A., Jacobson, C.A., Khibnik, A.I., Mehta, P.G.: Linear and Nonlinear Analysis of Controlled Combustion Processes. Part I: Linear Analysis. (1999) Proceedings of the 1999 IEEE International Conference on Control Applications. [22] Banaszuk, A., Jacobson, C.A., Khibnik, A.I., Mehta, P.G.: Linear and Nonlinear Analysis of Controlled Combustion Processes. Part II: Nonlinear Analysis. (1999) Proceedings of the 1999 IEEE International Conference on Control Applications. [23] Lieuwen, T.C., Zinn, B.T.: Experimental Investigation of Limit Cycle Oscillations in an Unstable Gas Turbine Combustor. (2000) AIAA paper 2000-0707. [24] Yu, K.H., Wilson, K.J., Schadow, K.C.: Liquid-Fueled Active Instability Suppression. 27th Symposium (International) on Combustion, The Combustion Insitute (1998) 2039–2046 [25] Hibshman, J.R., Cohen, J.M., Banaszuk, A., Anderson, T.J., Alholm, H.A.: Active Control of Pressure Oscillations in a Liquid-Fueled Sector Combustor. (1999) ASME paper 99-GT-215. [26] Albrecht, P., Bauermeister, F., Bothien, M.R., Lacarelle, A., Moeck, J.P., Paschereit, C.O., Gutmark, E.: Caracterization and Control of Lean Blowout Using Periodically Generated Flame Balls. (2006) ASME paper GT2006-90340. [27] Schneider, G., Ariyur, K.B., Krstic, M.: Tuning of a Combustion Controller by Extremum Seeking: A Simulation Study. Volume 5 of Proceedings of the 39th IEEE Conference on Decision and Control. (2000) 5219–5223 [28] Murugappan, S., Gutmark, E.J., Acharya, S.: Application of Extremum Seeking Controller for Suppression of Combustion Instabilities in Spray Combustion. (2000) AIAA paper 2000-1025. [29] Cohen, J.M., Banaszuk, A.: Factors Affecting the Control of Unstable Combustors. J. Propulsion Power 19 (2003) 811–821 [30] Rowley, C.W., Williams, D.R., Colonius, T., Murray, R.M., Macmynoski, D.G.: Linear models for control of cavity ﬂow oscillations. J. Fluid Mech. 547 (2006) 317–330

Vortex Models for Feedback Stabilization of Wake Flows Bartosz Protas Department of Mathematics & Statistics, McMaster University, 1280 Main Street West, Hamilton L8S 4K1, Ontario, Canada [email protected] http://www.math.mcmaster.ca/bprotas

Summary This paper reviews recent progress concerning development of point–vortex reduced– order models for feedback stabilization of the cylinder wake ﬂow. First, we recall brieﬂy some earlier results related to the design of linear feedback stabilization strategies based on the F¨oppl system. Then we present derivation of a higher–order F¨oppl system based on solutions of the Euler equations which desingularize the original F¨oppl vortices. We argue that such higher–order F¨oppl systems possess important advantages over the classical F¨oppl system which are relevant from the control–theoretic point of view. In particular, we present computational results indicating that a higher–order F¨oppl system can be stabilized completely in contrast to the classical F¨oppl system for which this is not possible owing to the presence of a stable center manifold spanned by uncontrollable modes.

1 Introduction The Navier–Stokes equations and most of their simpliﬁed versions such as, for instance, the Euler equations, are inﬁnite–dimensional dynamical systems and therefore their solutions are characterized by an inﬁnite number of parameters. Despite the fact that the inﬁnite–dimensional Control Theory is well developed, most of the readily available computational control algorithms have ﬁnite–dimensional systems as their point of departure. This justiﬁes the need for deriving ﬁnite–dimensional representations of the steady solutions that one intends to stabilize and deriving ﬁnite–dimensional descriptions of the system dynamics in the neighborhood of such unstable solutions, the so–called “reduced–order models”. As regards derivation of such ﬁnite–dimensional representations, there are two main approaches which can be roughly classiﬁed as “data–based” and “equation–based”. The approaches belonging to the ﬁrst family rely on ﬁnding empirical basis functions which optimally span, in some suitable sense, the data characterizing the system and collected during its evolution. The best known approach in this category is the Proper Orthogonal Decomposition (POD) whose application for ﬂow control purposes was reviewed by [1]. Application of such data–based approaches to development of reduced–order models is however limited to regimes well–represented by the data available. Such models may therefore provide rather poor representation of the system response to arbitrary forcing, as it may push the system trajectories towards R. King (Ed.): Active Flow Control, NNFM 95, pp. 422–436, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

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regimes not described by the available empirical data. On the other hand, equation– based approaches seek to construct ﬁnite–dimensional representations as solutions of truncated ﬁnite–dimensional forms of the governing equations. The most common approaches belonging to this category use standard space discretizations of the original partial differential equations to generate ﬁnite–dimensional models of the system. Examples of ﬂow control techniques developed based on such models are described, for instance, by [2] and [3]. In the present investigation we will pursue an alternative approach where we will consider simpliﬁed forms of the governing equations which, while remaining inﬁnite–dimensional, are easier to solve and analyze. Solutions of such simpliﬁed equations can be treated using analytical techniques, so that suitable truncation of the obtained expressions will lead to the reduced–order models. The present paper is concerned with development of a systematic procedure for generation of such reduced–order models of certain hydrodynamic systems. We are interested here in constructing simple reduced–order models for vortex– dominated ﬂows, hence we will assume that the ﬂow is incompressible and inviscid. Therefore, instead of solutions of the Navier–Stokes equations, we will consider solutions of the Euler equations. It is well–known [4] that 2D steady–state Euler equations can be equivalently represented as ⎧ ⎪ ⎨ ∆ψ = f (ψ) in Ω, ψ=0 on ∂Ω, (1) ⎪ ⎩ for |(x, y)| → ∞, ψ → U∞ y where ψ is the streamfunction, which allows the velocity components to be expressed ∂ψ as u = ∂ψ ∂y and v = − ∂x , and f is an arbitrary function representing the relationship between the streamfunction and the vorticity ω as ω = f (ψ). In this investigation we are interested only in solutions symmetric with respect to the ﬂow centerline, so without loss of generality we can restrict Ω in (1) to the upper half–plane (i.e., points with y > 0). We note that the indeterminacy of the function f in (1) reﬂects the nonuniqueness of solutions of the Euler equations in a given domain Ω. For instance, expressing the function f (ψ) on the RHS as a linear combination of 2K Dirac delta distributions 2K k=1 Γk δ(x − xk )δ(y − yk ) with weights Γk , where K is the total number of singularities and their images, leads to systems known from the classical potential ﬂow theory corresponding to 2K point vortices located at the points {xk , yk }2K k=1 . An example of such a solution was found in closed form by F¨oppl in [5], where the potential ﬂow was obtained by placing behind the obstacle two counter–rotating point vortices located symmetrically with respect to the centerline (Figure 1a). With points √ of the plane characterized by their complex coordinates z = x + iy, where i = −1, the complex potential of this ﬂow W0 (z) = (ϕ + iψ)(z), where ϕ and ψ are, respectively, the potential and the streamfunction, can be expressed as W0 (z) = WC (z) + WF,0 (z),

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where R2 WC (z) = U∞ z + , (2a) z Γ0 R2 R2 − ln(z − z 0 ) + ln z − (2b) WF,0 (z) = ln(z − z0 ) − ln z − 2πi z0 z0 and Γ0 and z0 = x0 + iy0 represent, respectively, the circulation and position of the F¨oppl vortices. In Equation (2) WC (z) represents the base ﬂow symmetric with respect to the OY axis and due to the cylinder only, whereas WF,0 (z) corresponds to the two F¨oppl vortices located at z0 and z 0 and their two images located inside the obstacle. Two one–parameter families of steady vortex conﬁgurations given by (2) were found by F¨oppl in [5]: the conﬁguration characterized by the condition ⎧ 2 2 2 2 2 ⎪ ⎨ (|z0 | − R ) = 4|z0 | y0 , (3) (|z0 |2 − R2 )2 (|z0 |2 + R2 ) ⎪ , ⎩ Γ0 = −2π 5 |z0 | hereafter referred to as the “classical F¨oppl system” (Figure 1a), and the conﬁguration characterized by the condition (z0 ) = 0, i.e., corresponding to the vortices located on the OY axis. The latter solution, however, does not correspond to any physical situation and will not be discussed further in this investigation. The classical F¨oppl system has recently been used as a reduced–order model in the development of a simple feedback stabilization strategy for the cylinder wake ﬂow in [6]. The cylinder rotation ΓC = ΓC (t) was used the ﬂow actuation (i.e., the control variable) and observations of the centerline velocity downstream of the obstacle as the system output. The advantage of this approach is that, due to simplicity of the F¨oppl model, the synthesis of the stabilization algorithm becomes a simple task with a signiﬁcant part of the calculations carried out analytically. The performance of the

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stabilization strategy, while quite encouraging, also showed some limitations of such very simple point–vortex systems employed as reduced–order models. The purpose of this investigation is to identify sources of these limitations and propose possible improvements. The structure of the paper is as follows: ﬁrst in Section 2 we review the formulation of the Linear–Quadratic–Gaussian (LQG) compensator designed based on the F¨oppl system as a reduced–order model, in Section 3 we introduce a family of higher–order F¨oppl systems characterized by more desirable properties as candidates for reduced–order models, then some computational results are presented in Section 4 and conclusions are deferred to Section 5.

2

Control Design Based on the F¨oppl System as a Reduced–Order Model

We begin this Section by analyzing the stability properties of the classical F¨oppl system linearized around the equilibrium solution. This analysis will motivate the design of an LQG compensator for feedback stabilization of the linearized F¨oppl system. Careful analysis of the linear stability of the F¨oppl system and its relevance to modeling the onset of vortex shedding in 2D wake ﬂows was presented by Tang and Aubry in [7]. Our discussion of the control–theoretic aspects will be here necessarily concise and the reader is referred to the publication [6] for further details. Different ﬂow control problems also based on the F¨oppl system as the reduced–order model were studied in [8,9]. We will assume that the cylinder has unit radius R = 1 and the free stream at inﬁnity has unit magnitude U∞ = 1. In addition, we will also assume that all quantities are nondimensionalized using these values. The F¨oppl model can be regarded as a nonlinear dynamical system with evolution described by ⎡ ⎤ [V1 (z1 , z2 , Γ1 , Γ2 )] ⎢−[V1 (z1 , z2 , Γ1 , Γ2 )]⎥ d ⎥ X = F(X) + b(X)ΓC ⎢ (4) ⎣ [V2 (z1 , z2 , Γ1 , Γ2 )]⎦ + b(X)ΓC , dt −[V2 (z1 , z2 , Γ1 , Γ2 )] where X [x1 y1 x2 y2 ]T is the state vector and the control matrix b(X) is expressed as ⎡ ⎤ −y1 /|z1 |2 2⎥ 1 ⎢ ⎢ x1 /|z1 |2 ⎥ . (5) b(X) ⎣ y /|z | 2π 2 2 ⎦ x2 /|z2 |2 The expressions for V1 and V2 in (4) are given by the velocity ﬁeld 1 Γ1 1 1 V (z) = 1 − 2 − − z 2πi z − z1 z − 1/z1 1 Γ2 1 ΓC , + − + 2πi z − z2 z − 1/z2 2πiz

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1 1 evaluated at z1 and z2 with the singular “self–induction” terms ( z−z and z−z , re1 2 spectively) omitted. For the moment we will ﬁx attention on the properties of the F¨oppl system without control, hence we will assume that ΓC ≡ 0, which renders (4) autonomous. The linear stability analysis of the F¨oppl system is performed by adding the perturbations (x1 , y1 ) and (x2 , y2 ) to the coordinates of the upper and lower vortex of the stationary solution and then linearizing the system (4) around X0 [x0 y0 x0 − y0 ]T assuming small perturbations. Thus, evolution of the perturbations is governed by the system d X = AX , (7) dt

where X [x1 y1 x2 y2 ]T is the perturbation vector and the system matrix is given ∂F by the Jacobian of the nonlinear system at the equilibrium A = ∂X (X0 ). We remark that (7) is a linear time–invariant (LTI) system. Eigenvalue analysis of the matrix A reveals (see [7] for details) the presence of the following modes of motion (Figure 1b): – unstable (growing) mode α corresponding to a positive real eigenvalue λ1 = λr > 0, – stable (decaying) mode β corresponding to a negative real eigenvalue λ2 = −λr < 0, – neutrally stable oscillatory mode γ corresponding to a conjugate pair of purely imaginary eigenvalues λ3/4 = ±iλi . These qualitative properties are independent of the downstream coordinate x0 characterizing the equilibrium solution. The analysis of the orientation of the unstable eigenvectors carried out in [7] revealed that the initial stages of instability of the F¨oppl system closely resemble the onset of vortex shedding in an actual cylinder wake undergoing Hopf bifurcation. The free parameter characterizing the equilibrium solution (3) of the F¨oppl system (i.e., the downstream location of the singularities x0 ) is chosen here, so that the length of the recirculation zone in the F¨oppl system is the same as the recirculation length in the steady unstable solution of the Navier–Stokes system at a prescribed Reynolds number. Further justiﬁcation as well as details of calculations are described in [6]. In the examples presented hereafter the downstream position of the vortices was chosen, so that the recirculation length is the same as in the steady unstable solution of the Navier–Stokes system at the Reynolds number Re = 75. After including the control term representing the cylinder rotation the linearized F¨oppl system becomes d X = AX + BΓC , (8) dt where ⎡ ⎤ −y0 ⎥ 1 ⎢ ⎢ x0 ⎥ . B b(X0 ) = (9) 2 ⎣ y 2πr0 0⎦ x0 As mentioned in Section 1 our control objective is attenuation of vortex shedding which can be quantiﬁed by measuring the velocity at a point on the ﬂow centerline with the

Vortex Models for Feedback Stabilization of Wake Flows

427

streamwise coordinate xm (note that in the stationary symmetric solution the transverse velocity component vanishes on the centerline). Choosing this quantity as an output of system (4) we obtain the following output equation [V (xm )] h(z1 , z2 ) (10) + DΓC , −[V (xm )] 1 T where the matrix D 2πx represents the control–to–measurements map. 2 [0 xm ] m Linearization of equation (10) yields

h(z0 + z1 , z 0 + z2 ) ∼ = h(z0 , z 0 ) + CX ,

(11)

where zk = xk + iyk , k = 1, 2, and the linearized observation operator C is given by ∂u(xm ) ∂u(xm ) ∂u(xm ) ∂u(xm ) ∂x1 ∂y1 ∂x2 ∂y2 (x ,y ) (x ,y ) (x ,y ) (x ,y ) 0 0 0 0 0 0 0 0 C = ∂v(xm ) . (12) ∂v(xm ) ∂v(xm ) ∂v(xm ) ∂x1

(x0 ,y0 )

∂y1

(x0 ,y0 )

∂x2

(x0 ,y0 )

∂y2

(x0 ,y0 )

Uncertainty of the reduced–order model is represented by the presence of noise w which affects the linearized system dynamics via a [4 × 1] matrix G and the linearized system output via a [2 × 1] matrix H. Moreover, we assume that the velocity measurements may be additionally contaminated with noise m [m1 m2 ]T , where m1 and m2 are stochastic processes. With these deﬁnitions we can now put the linearized reduced– order model in the standard state–space form (see [10]) d X = AX +BΓC +Gw, dt Y = CX +DΓC +Hw + m.

(13a) (13b)

Prior to designing a controller for system (13) we have to verify whether the system is controllable and observable which is done by studying the ranks Nc and No of the controllability and observability Grammians (14) Nc rank B AB A2 B A3 B = 2, T T T T 2 T T 3 T (15) No rank C A C (A ) C (A ) C = 4. We conclude that the matrix pair {A, B} is not controllable and only two out of four modes present in the system can be controlled. On the other hand, the matrix pair {A, C} is completely observable. Converting system (13) to the minimal representation which consists of those modes only which are both controllable and observable will allow us to determine which modes are in fact controllable. We accomplish this by introducing an orthogonal transformation matrix ⎡ ⎤ 1/2 0 −1/2 0 √ ⎢ 0 1/2 0 1/2⎥ ⎥ (16) Tc 2 ⎢ ⎣1/2 0 1/2 0 ⎦ 0 1/2 0 −1/2

428

B. Protas

and making the following change of variables Xab

Xa = Tc X . The correspondXb

ing form of system (13) is now d Xa Ga Aa 0 Xa Ba ΓC + w, = + X X 0 Gb 0 A dt b b b H1 0 Cb Xa D1 Yb = + Γ + w + m. Ya Ca 0 Xb D2 C H2

(17a) (17b)

Our minimal representation is thus given by the upper row in equation (17a) and the lower row in (17b), i.e., d X = Aa Xa +Ba ΓC +Ga w, dt a Ya = Ca Xa +D2 ΓC +H2 w + m2 .

(18a) (18b)

Eigenvalue analysis of the matrices Aa and Ab reveals that Aa has two real eigenvalues (positive and negative) corresponding to the growing and decaying modes α and β, whereas the matrix Ab has a conjugate pair of purely imaginary eigenvalues which correspond to the neurally stable mode γ (Figure 1b). Hence, the uncontrollable part of the model system dynamics is associated with the neutrally stable oscillatory mode γ and the original system (13) is thus stabilizable, but not controllable. Practical effectiveness of the proposed algorithm depends on the location of the velocity sensor xm . As argued in [6], the distance xm is chosen so as to maximize the observability residual of the unstable mode α. Our objective here is to ﬁnd a feedback control law ΓC = −KX , where K is a [4 × 1] feedback matrix, that will 1. stabilize system (13) and 2. minimizing a performance criterion represented by the following cost functional ∞ T (Y QY + ΓC RΓC )dt , (19) J (ΓC ) E 0

where E denotes the expectation, Q is a symmetric positive semi–deﬁnite matrix and R > 0. Note that the cost functional (19) represents a sum of the linearized system output Y [i.e., the velocity at the sensor location (xm , 0)] and the control effort. The feedback control law determines the actuation (i.e., the circulation of the control vortex ΓC representing the cylinder rotation) based on the state of the reduced–order model (i.e., the perturbation X of the stationary solution). In practice, however, the state X of the ˜ = [Y˜b Y˜a ]T of the actual model (13) is not known. Instead, noisy measurements Y system [i.e., the nonlinear F¨oppl model (4) or an actual wake ﬂow] are available and can be used in an estimation procedure to construct an estimate Xe of the model state X . The evolution of the state estimate Xe is governed by the estimator system d ˜ − Ye ), X = AXe +BΓC + L(Y dt e Ye = CXe +DΓC ,

(20a) (20b)

Vortex Models for Feedback Stabilization of Wake Flows

w (system noise)

429

~ Y (measurements)

PLANT Γc (control) COMPENSATOR CONTROLLER

X’e

ESTIMATOR

v (measurement noise)

Γc (control) Fig. 2. Schematic of a compensator composed of an estimator and a controller

where L is a feedback matrix that can be chosen in a manner ensuring that the estimation error vanishes in the inﬁnite time horizon, i.e., that Xe → X as t → ∞. Thus, the estimator assimilates available observations into the system model, so as to produce an evolving estimate of the system state. Finally, the controller and the estimator can be combined to form a compensator in which the feedback control is determined based on the state estimate Xe as (21) ΓC = −KXe . The ﬂow of information in a compensator is shown schematically in Figure 2. The design of a Linear–Quadratic–Gaussian (LQG) compensator can be accomplished using standard methods of Linear Control Theory (see, e.g., [10]). Here we only remark that, since system (8) is stabilizable, but not controllable, the controller can in fact be designed based on the minimal representation. On the other hand, since system (8) is observable, the estimator is designed based on the original representation. Given small dimensions of systems (7) and (18), solution of the Riccati equations at the heart of these problems does not pose any difﬁculties. The reader is referred to [6] for further details.

3 Higher–Order F¨oppl Systems In this Section we describe potential ﬂow solutions generalizing F¨oppl’s classical point– vortex system. They can approximate with desired accuracy the velocity ﬁeld of the steady–state solutions of the Euler equations characterized by arbitrary compact vorticity support. Such system have the same dimension as the original F¨oppl model, however, are characterized by an arbitrary number of adjustable parameters, hence are more ﬂexible as reduced–order models. We consider again solutions of system (1), however, instead of a collection of Dirac delta functions now we allow for more general forms of the RHS function f (ψ). A family of interesting solutions was computed by Elcrat et al. in [11] by choosing f (ψ) as follows −ω, ψ ≤ σ, f (ψ) = (22) 0, ψ > σ,

430

B. Protas

y

z

U

8

P

δΩ

z’

zs

ω<0

1 0 0 1

Ω

x ω>0

Q Fig. 3. Schematic showing EFHM solutions with two opposite–sign vortex patches P (solid line) and Q (dashed line) located symmetrically with respect to the ﬂow centerline

where σ is an adjustable parameter. Such a vorticity distribution is sometimes referred to as the Rankine core. These solutions will play an important role in our development and we will hereafter refer to them as the “EFHM ﬂows”. When σ < 0, such ﬂows are characterized by compact regions of constant vorticity embedded in an otherwise irrotational ﬂow and are therefore related to the so–called Sadovskii ﬂows [12]. In addition to this solution, shown schematically in Figure 3, it was found in [11] that regions of opposite–sign vorticity, hereafter denoted P and Q, may also exist above and below the obstacle, as well as in front of it. Since these solutions do not correspond to any physical situation, they will not be considered hereafter. Consider a compact region P of vorticity embedded in an irrotational ﬂow past a circular cylinder (Figure 3) and characterized by the vorticity distribution ω = ω(z). When ω = Const, the corresponding steady–state solutions of the Euler equations deﬁned by (1) and (22) are given by families of the EFHM ﬂows computed in [11]. Below we construct algebraic systems that are approximations of such solutions. Our approach is conceptually related to the method devised by [13] in which moments of vorticity distribution are used to characterize the evolution of a system of vortex patches. Using complex Green’s function for the Laplace equation in a 2D unbounded domain 1 ln(z − z ), the complex potential induced by a vortex patch in such a G(z, z ) = 2πi domain can be expressed for points outside the patch z ∈ / P as ˜ P (z) = (ϕ + iψ)(z) = 1 ln(z − z )ω(z ) dA(z ), (23) W 2πi P where dA(z ) = dx dy . Tilde (˜) indicates that this potential represents a ﬂow in an unbounded domain (i.e., without the obstacle), whereas the subscript indicates that the potential is due to the patch P . We now choose a point zs ∈ P as the origin of the local coordinate system associated with the patch P and set ζ = z − zs (see Figure 3). The complex potential (23) can now be expressed as

Vortex Models for Feedback Stabilization of Wake Flows

ln 1 −

˜ P (z) = Γ0 ln(z − zs ) + 1 W 2πi 2πi

P

ζ z − zs

431

ω(zs + ζ) dA(ζ).

(24)

The second term in (24) can, for |z − zs | > |z − zs |, be expanded in a Taylor series which yields ∞ cn ˜ P (z) = Γ0 ln(z − zs ) − 1 (z − zs )−n , W 2πi 2πi n=1 n

where

|z − zs | > ζm ,

(25)

ω(zs + ζ)ζ n dA(ζ)

cn (zs ) =

(26)

P

and ζm = max(zs +ζ)∈P |ζ|. Thus, the point zs represents also the location of a singularity which, for the moment, remains unspeciﬁed. The quantities cn (zs ), n = 1, . . . , N are the moments of the vorticity distribution in the patch P with respect to the point zs and therefore are related to the eccentricity of the patch (c1 ), its ellipticity (c2 ), etc. (unless required for clarity, hereafter we will skip the argument of cn ). The zeroth moment c0 is equal to the total circulation Γ0 of the patch. The complex potential due to a ﬁnite– area vortex patch P can be approximated for points of the plane lying outside this patch by truncating expression (25), i.e., replacing it with a ﬁnite sum of singularities located at the point zs N cn ˜ P (z) ∼ ˜ P,N (z) = Γ0 ln(z −zs )− 1 W (z −zs )−n , |z −zs | > ζm . (27) =W 2πi 2πi n=1 n

˜ . The complex The order of truncation is represented by the second subscript on W ˜ Q,N (z) due to the patch Q with the opposite–sign vorticity and located potential W symmetrically below the ﬂow centerline (Figure 3) can be represented using an analogous expression in which zs is replaced with z s and cn with −cn for n = 1, . . . , N . Below we use these expressions to construct potential ﬂows approximating solutions of the steady–state Euler equations (1) in the sense that the velocity ﬁeld of the potential ﬂow will converge, for z ∈ / P and z ∈ / Q, to the velocity ﬁeld of the Euler ˜ P,N (z) ﬂow as N → ∞. These potential ﬂows are constructed using the potentials W ˜ Q,N (z), and adding suitable “image singularities” located inside the obstacle in and W a way ensuring that the boundary conditions for the wall–normal velocity component are satisﬁed. In general, such ﬂows can be constructed using the “Circle Theorem” [4] which states that if w(z) ˜ is the complex potential of a ﬂow in an unbounded domain and with singularities at some points zk , such that ∀k, |zk | > R, then the complex potential of the corresponding ﬂow past the cylinder with radius R is given by the expression 2 ˜ Rz ). Thus, using this construction to enforce the boundary condiw(z) = w(z) ˜ + w( tions and including also the base ﬂow with the potential WC (z) [cf. Eq. (2)], we obtain the following expression for the complex potential

432

B. Protas

WN (z) = WC (z) + WF,N (z)

2 2 R R ˜ ˜ ˜ ˜ = WC (z) + WP,N (z) + WQ,N (z) + W P,N + W Q,N z z 2 2 R R Γ0 − = U∞ z + − ln(z − zs ) − ln z − z 2πi zs N cn R2 1 1 − ln(z − z s ) + ln z − zs 2πi n=1 n (z − zs )n n n z z cn cn cn n n n n , − (−1) − + (−1) n 2 2 zs (z − z s ) zs R z−R z − zs zs (28)

where WF,N (z) represents the truncated potential due to the ﬁnite–area vortex patches and their images. We notice that setting N = 0 in (28) we recover the complex potential (2) of the classical F¨oppl system discussed in Section 1. Therefore, the family of the complex potentials given in (28) represents N –th order corrections to the F¨oppl system regarded as approximations of the corresponding solution of the steady–state Euler equations and hereafter we will refer to them as the “higher–order (N –th order) F¨oppl systems”. By taking N large enough we can obtain an arbitrarily accurate representation of the velocity ﬁeld in the Euler ﬂow valid for points in the ﬂow domain outside the vortex patches, thereby improving applicability of the F¨oppl system as a model for steady wake ﬂows. In general, in an inviscid and incompressible ﬂuid singularities (e.g., a point vortex s ˆ located at zs ) move according to the velocity ﬁeld dz dt = VN (zs ), where complex conjugation is required to account for the fact that the complex velocity ﬁeld is given by VˆN = u ˆ − iˆ v . We note that the advection velocity VˆN of a singularity is not affected by its self–induction which can be seen by regarding the singularity as a limit of a sequence of ﬁnite–area circular distributions of the corresponding quantity. The velocity induced by such distributions can be shown to vanish at their center. Thus, the advection velocity of a singularity is obtained as N c Γ 1 0 n VˆN (z) = VN (z) − − + , (29) 2πi z − zs n=1 (z − zs )n+1 N (z) where VN (z) = dWdz , i.e., the terms which become singular as z → zs are removed from the velocity ﬁeld. Hereafter, hats (ˆ) will distinguish quantities with these self–inductions terms subtracted off. We are interested in steady–state solutions of the higher–order F¨oppl systems, therefore, for a given truncation order N , we need to ﬁnd the equilibrium points of system (29), i.e., the points zN such that setting zs = zN the following condition is satisﬁed for given Γ0 and {cn }N n=1

VˆN (zN ) = 0.

(30)

Vortex Models for Feedback Stabilization of Wake Flows

433

This condition can be expanded to ⎡ 2 1 1 R Γ 1 0 ⎣− − VˆN (zN ) = U∞ 1 − 2 − + R2 zN 2πi (z − z ) N N zN − z N zN − ⎡ N n−1 R 2 cn zN 1 ⎢ n+1 + ⎣(−1) n+1 n+1 2 2πi n=1 zN zN − zRN ⎤ −

cn − (−1)n+1 (zN − z N )n+1

R 2 cn zN −

R2 zN

n+1

⎤ R2 zN

⎦

1 ⎥ 2 ⎦ =0 zN

(31) which is a complex–valued equation characterizing one complex unknown zN . In the case when N = 0, one solution of (31) is given by (3). When N ≥ 1, solutions must be found numerically, e.g. using Newton’s method. Furthermore, since the order of the equation increases with N , it can be anticipated that so does the number of roots. In fact, it can be proved that there is always one root of (31) that is in a neighborhood of the solution characterized by (3) and the size of this neighborhood can be bounded by the magnitudes of the coefﬁcients cn . Thorough analysis of this and other analytical properties of the higher–order F¨oppl system is deferred to a forthcoming paper [14].

4 Computational Results In this Section we present some preliminary computational results concerning construction of a higher–order F¨oppl system for a given EFHM ﬂow and application of such a system as a reduced–order model to the design of an LQG–based stabilization strategy. Because of space limitations, our discussion here is necessarily short and the reader is referred to the forthcoming papers [14,15] for further particulars regarding the computational procedure and detailed results. To ﬁx attention, we will focus on the F¨oppl system with the singularities located at [x0 , y0 ] = [4.32, ±2.3596] and with the circulation of the vortices given by Γ0 = −29.6015 (this is the conﬁguration investigated in [6]). We will also consider an EFHM ﬂow with the area of the vortex patch A = 20.43 as a desingularization of the classical F¨oppl system and will take N = 10 as the truncation order in the construction of the higher–order system. As analyzed in detail in [14], such higher–order systems are characterized by multiple solutions, however, in our analysis below we will focus on the equilibrium [xN , yN ] in the neighborhood of [x0 , y0 ], both of which are shown in Figure 4a. As can be easily veriﬁed, the higher–order F¨oppl system linearized about the new equilibrium [xN , yN ] has the same properties in terms of controllability and observability as the classical F¨oppl system [cf. Eqns. (14) and (15)]. Assuming measurements of two velocity components on the ﬂow centerline as the observations and thecylinder rotation as the actuation, all four modes are observable, but only two of them are controllable. Stability analysis of this higher–order F¨oppl system indicates that, in

434

B. Protas

4

y/R

3

2

1

0

1

2

3

4

5

6

7

8

x/R (a)

2.6 2.5 2.4

y/R

2.3 2.2 2.1 2.0 1.9 1.8 3.5

4.0

4.5

5.0

5.5

6.0

x/R (b) Fig. 4. (a) Location of the equilibria of (circle) the classical F¨oppl system (3) and (square) the higher–order F¨oppl system (31) with N = 10. The boundary of the vortex patch in the EFHM ﬂows used to construct the higher–order F¨oppl system is represented by the dotted line and the cylinder boundary is represented by a thick solid line. (b) Trajectories of the state of (solid line) the classical and (dotted line) higher–order F¨oppl system stabilized with an LQG compensator in the neighborhood of the corresponding equilibrium solutions.

Vortex Models for Feedback Stabilization of Wake Flows

435

addition to a growing and decaying mode (corresponding to, respectively, the modes α and β, cf. Figure 1b) characterized by purely real eigenvalues, there exists also a mode characterized by pair of complex–conjugate eigenvalues (corresponding to the mode γ, cf. Figure 1b). However, in contrast to the classical F¨oppl system, these complex eigenvalues have negative real parts, hence the oscillatory mode in the higher–order F¨oppl system is in fact exponentially stable. This difference has important consequences when a linear stabilization strategy, such as LQG, is applied to the original nonlinear system. As illustrated in Figure 4b, when the LQG compensator is applied to the classical F¨oppl system, the state of the system does not return to the equilibrium, but lands instead on a closed orbit. One can prove rigorously using methods of dynamical systems that this orbit has in fact the structure of a center manifold and the trajectory of the system on this manifold is stable (see [15] for precise statements and proofs of these theorems). On the other hand, when the LQG compensator is applied to the higher–order F¨oppl system, the system trajectory returns to the equilibrium owing to the exponential stability of the uncontrollable modes.

5 Conclusions The dynamics of both the classical and higher–order F¨oppl systems in the neighborhood of an equilibrium point is characterized by four degrees of freedom. However, in contrast to the classical system which has just one parameter, the higher—order systems are characterized by an arbitrary number of adjustable parameters represented by the expansion coefﬁcients in (28). The number of these parameters is determined by the truncation order N . Therefore, by introducing a larger number of adjustable parameters, one can incorporate much more ﬂexibility into F¨oppl–type models, so that, while remaining four–dimensional, they can reproduce more accurately certain properties of realistic ﬂows. Advantages of having this additional ﬂexibility were illustrated by the computational results presented in Section 4. We showed that the state of the classical F¨oppl system with an LQG stabilization converges to a center manifold, whose persistence prevents this state from reaching the equilibrium and, as a result, the amplitude of the state oscillations does not decrease. We conjecture that this is a possible reason for the oscillations of the velocity ﬁeld in the near wake region occurring when this strategy was applied to stabilize an actual cylinder wake ﬂow at Re = 75 (see [6]). On the other hand, the ﬂexibility of the higher–order F¨oppl system investigated here made it possible to alter the stability properties of the new equilibrium in such way that the uncontrollable mode became stable. As a result, the same LQG compensation strategy was now able to stabilize completely the equilibrium. We anticipate that this additional ﬂexibility of higher–order F¨oppl systems will play a role when employing these systems as reduced–order models to stabilization of actual cylinder wake ﬂows. Veriﬁcation of performance of such approaches is underway.

Acknowledgments The author wishes to express his thanks to Profs. Alan Elcrat and Ken Miller for many interesting and helpful discussions regarding the EFHM ﬂows and for providing him

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with a code to reproduce the results of the paper [11]. This research was supported by an NSERC Discovery Grant (Canada) and CNRS (France).

References [1] J. Lumley and P. Blossey: “Control of Turbulence”, Ann. Rev. Fluid Mech. 30, 311-327, (1998). [2] T. R. Bewley and S. Liu: “Optimal and robust control and estimation of linear paths to transition”, J. Fluid Mech. 365, 305-349, (1998). [3] J. Kim: “Control of turbulent boundary layers”, Phys. Fluids 15, 1093-1105, (2003). [4] L. M. Milne–Thompson: “Theoretical Hydrodynamics”, MacMillan, (1955). [5] L. F¨oppl: “Wirbelbewegung hinter einem Kreiscylinder”, Sitzb. d. k. Bayr. Akad. d. Wiss. 1-17, (1913). [6] B. Protas: “Linear Feedback Stabilization of Laminar Vortex Shedding Based on a Point Vortex Model”, Phys. Fluids 16, 4473-4488, 2004. [7] S. Tang and N. Aubry: “On the symmetry breaking instability leading to vortex shedding”, Phys. Fluids 9, 2550-2561, (1997). [8] S. Tang and N. Aubry: “Suppression of vortex shedding inspired by a low–dimensional model”, J. Fluids and Struct. 14, 443-468, (2000). [9] F. Li and N. Aubry: “Feedback control of a ﬂow past a cylinder via transverse motion”, Phys. Fluids 15, 2163-2176, (2003). [10] R. F. Stengel: “Optimal Control and Estimation”, Dover Publications, (1994). [11] A. Elcrat, B. Fornberg, M. Horn and K. Miller: “Some steady vortex ﬂows past a circular cylinder”, J. Fluid Mech. 409, 13-27, (2000). [12] V. S. Sadovskii: “Vortex regions in a potential stream with a jump of Bernoulli’s constant at the boundary”, Appl. Math. Mech. 35, 729, (1971). [13] M. .V. Melander, N. J. Zabusky and A. .M. Styczek: “A moment model for vortex interactions of two–dimensional Euler equations. Part 1 — Computational validation of a Hamiltonian elliptical representation”, J. Fluid Mech. 167, 95-115, (1986). [14] B. Protas: “Higher–order F¨oppl models of steady wake ﬂows”, Phys. Fluids 18, 117109, (2006). [15] B. Protas: “Center Manifold Analysis of a Point–Vortex Model of Vortex Shedding with Control”, (submitted), 2006.

Keyword Index

acoustic forcing Phase-Shift Control … Closed-loop Active… Designing Actuators for … aerodynamic improvement Active Control to … air injection Active Control to… Active Blade Tone … Airfoil Steady and Oscillatory … alternating suction and blowing Designing Actuators for … artificial neural network State estimation of … axial turbomachine Active Control to … backward facing step Flow Control on … Control of Wing Vortices bluff body flow Feedback control applied… boundary layer Experimental and Numerical… boundary layer separation Closed-loop Active … cavity flow control Reduced-order Model-based … cavity resonance Supersonic Cavity Response… CFD Computational Investigation… Experimental and Numerical Investigations… Flow Control on… chaos control Turbulence: The Taming … circulation flow Pulsed Plasma Actuators …

408 85 69 293 293 391 190 69 105 293 325 137 369 56

coherent structures Turbulence: The Taming … combustion control Phase-Shift Control … combustion instabilities Phase-Shift Control of … computational fluid dynamics Flow Control on … control design Pulsed Plasma Actuators … Flow Control on … control of entrainment Active Management of … control of turbulence Turbulence: The Taming … critical flow regimes Turbulence: The Taming … cylinder wake State estimation of … delta wing Control of Wing Vortices… drag control Active Drag control … drag reduction Feedback control …

1 408 408 325 42 325 281 1 1 105 137 247 369

85 211

230 173 56 325 1 42

Electromagnetic flow control Electromagnetic control of … excitation wave form Electromagnetic control of … experimental-based model Reduced-order Model-based … extremum-seeking control Active Blade Tone … fan noise Active Control to Improve … FEATFLOW Flow Control on… feature extraction A Unified Feature …

27 27 211 391 293 325 119

438

Keyword Index

feedback control Closed-loop Active … Turbulence: The Taming … Feedback control applied … State estimation of … figure of merit Closed-loop Active … flat plate Experimental and Numerical... flow analysis A Unified Feature… flow separation Electromagnetic control … flow stability Continuous mode … flow topology A Unified Feature … flow visualization A Unified Feature … forward facing step Flow control with … Galerkin model Continuous mode interpolation… Galerkin projection Reduced-order Model-based Feedback … generic car model Active Drag control … high-lift Designing Actuators for … Computational Investigation … Hover Steady and Oscillatory …

85 1 369 105 85 56 119 27 260 119 119 353

260

211 247 69 173 190

interstage hotwire measurements Active Blade Tone …

391

jet control Active Management of …

281

Lavrentiev regularization Flow control with … leading edge stall Towards Active control …

353 152

LES Active Drag control … linear control theory Vortex Models for … linear-quadratic optimal control Reduced-order Model-based … longitudinal vortex Active Drag control … low dimensional modeling State estimation of … Continuous mode interpolation … low-dimensional vortex models Feedback control … Matlab-interface Flow Control on … micro aerial Pulsed Plasma Actuators … microactuators Turbulence: The Taming .. microfabrication Turbulence: The Taming... microsensors Turbulence: The Taming mode interpolation Continuous mode interpolation … model-based feedback control Reduced-order Modelbased … Navier-Stokes equations On the choice of … Newton method Flow control with … noise control Active Blade Tone … noise reduction Active Control to I … nonlinear dynamical systems theory Turbulence: The Taming … nonlinear system identification State estimation of …

247 422

211 247 105 260 369 325 42 1 1 1 260

211 339 353 391 293

1 105

Keyword Index

open-loop control Supersonic Cavity Response … optimal flow control On the choice … Drag Minimization … Flow control with … periodic excitation Electromagnetic control of .. Computational Investigation ... phase-shift control Phase-Shift Control … PIV Active Drag control … Towards Active control of… Electromagnetic control of… plasma actuator Experimental and Numerical … pneumatic actuators Towards Active control … POD State estimation of … point vortex models Vortex Models for … pointwise state constraints Flow control with … proper orthogonal decomposition State estimation … Drag Minimization of … pulsed blowing Designing Actuators for … pulsed plasma actuators Pulsed Plasma Actuators … pulsed-blowing actuator Supersonic Cavity Response… RANS Computational Investigation… Flow Control on … reactive control Turbulence: The Taming … reattachment Control of Wing Vortices reduced order model Drag Minimization of … Reduced-order Model-based…

230 339 309 353 27 173 408 247 152 27 56 152 105 422 353 105 309 69 42 230 173 325 1 137 309 211

Reynolds-averaged Navier-Stokes equations Flow Control on … rotating instability Active Control to … secondary fuel injection Phase-Shift Control … semi-smooth Flow control with … separation control Designing Actuators for .. Steady and Oscillatory … Active Drag control … Computational Investigation … Pulsed Plasma Actuators … shear flows Flow Control on … snapshot POD Reduced-order Model-based … soft computing Turbulence: The Taming … spanwise vortex Active Drag control … stochastic estimation Reduced-order Model-based … streamwise vortices Active Management of … supersonic cavity Supersonic Cavity Response … tip clearance noise Active Control to … tonal fan noise Active Blade Tone … transient flow estimation State estimation of … transition Experimental and Numerical … trust-region optimization Drag Minimization of …

439

325 293 408 353 69 190 247 173 42 325

211 1 247

211 281

230 293 391 105

56 309

440

Keyword Index

turbulent boundary layers Turbulence: The Taming … unsteady aerodynamics Pulsed Plasma Actuators … V-22 model Steady and Oscillatory … Vehicles Pulsed Plasma Actuators … vortex breakdown Control of Wing Vortices…

Printing: Mercedes-Druck, Berlin Binding: Stein + Lehmann, Berlin

1 42 190 42

137

vortex core lines A Unified Feature … vortex reduction On the choice … Vortices Control of Wing Vortices… wake flow Drag Minimization of … Vortex Models for …

119 339

137 309 422

Active Flow Control II: Papers Contributed to the Conference Active Flow Control II 2010, Berlin, Germany, May 26 to 28, 2010 (Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 108)

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Flow control

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100 Volumes of 'Notes on Numerical Fluid Mechanics': 40 Years of Numerical Fluid Mechanics and Aerodynamics in Retrospect (Notes on Numerical Fluid Mechanics and Multidisciplinary Design)

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Control of Fluid Flow (Lecture Notes in Control and Information Sciences, 330)

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Optimal Control of Viscous Flow

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Fluid flow measurement: a practical guide to accurate flow measurement

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Vibration Control of Active Structures (Solid Mechanics and Its Applications)

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Modern Numerical Methods for Fluid Flow (draft)

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Perspectives in flow control and optimization

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Viscous Fluid Flow

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Introduction to Modern Traffic Flow Theory and Control

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Perspectives in Flow Control and Optimization

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Compressible Fluid Flow

Compressible Fluid Flow

Viscous Fluid Flow

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Active Fault Tolerant Control Systems: Stochastic Analysis and Synthesis (Lecture Notes in Control and Information Sciences)

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