96
Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM)
Editors E.H. Hirschel/München W. Schröder/Aachen K. Fujii/Kanagawa W. Haase/München B. van Leer/Ann Arbor M.A. Leschziner/London M. Pandolfi/Torino J. Periaux/Paris A. Rizzi/Stockholm B. Roux/Marseille Yu. Shokin/Novosibirsk
New Results in Numerical and Experimental Fluid Mechanics VI Contributions to the 15th STAB/DGLR Symposium Darmstadt, Germany, 2006 Cameron Tropea, Suad Jakirlic, Hans-Joachim Heinemann, Rolf Henke, Heinz Hönlinger (Editors)
ABC
Prof. Dr. Cameron Tropea TU Darmstadt Petersenstraße 30 64287 Darmstadt Germany
Prof. Rolf Henke RWTH Aachen Wüllnerstraße 7 52062 Aachen Germany
Dr. Suad Jakirlic TU Darmstadt Petersenstraße 30 64287 Darmstadt Germany
Prof. Dr. Heinz Hönlinger DLR - AE Bunsenstraße 10 37073 Göttingen Germany
Dr. Hans-Joachim Heinemann DLR Bunsenstraße 10 37073 Göttingen Germany
Library of Congress Control Number: 2007934013 ISBN-10 3-540-74458-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-74458-0 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:12112951 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 E-mail: 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]
Foreword
This volume contains the paper presented at the 15th DGLR/STAB-Symposium held at the Technical University Darmstadt, Germany, November, 29 to December 01, 2006. STAB is the German Aerospace Aerodynamics Association, founded towards the end of the 1970's, whereas DGLR is the German Society for Aeronautics and Astronautics (Deutsche Gesellschaft für Luft- und Raumfahrt - Lilienthal Oberth e.V.). The mission of STAB is to foster development and acceptance of the discipline “Aerodynamics” in Germany. One of its general guidelines is to concentrate resources and know-how in the involved institutions and to avoid duplication in research work as much as possible. Nowadays, this is more necessary than ever. The experience made in the past makes it easier now, to obtain new knowledge for solving today's and tomorrow's problems. STAB unites German scientists and engineers from universities, research-establishments and industry doing research and project work in numerical and experimental fluid mechanics and aerodynamics for aerospace and other applications. This has always been the basis of numerous common research activities sponsored by different funding agencies. Since 1986 the symposium has taken place at different locations in Germany every two years. In between STAB workshops regularly take place at the DLR in Göttingen. The changing meeting places were established as focal points in Germany's Aerospace Fluid Mechanics Community for a continuous exchange of scientific results and their discussion. Moreover, they are a forum where new research activities can be presented, often resulting in new commonly organised research and technology projects. It is the sixths time now that the contributions to the Symposium are published after being subjected to a peer review. The material highlights the key items of integrated research and development based on fruitful collaboration of industry, research establishments and universities. Some of the contributions still present results from the "Luftfahrtforschungsprogramm der Bundesregierung (German Aeronautical Research Programme)". Some of the papers report on work sponsored by the Deutsche Forschungsgemeinschaft (DFG, German Research Council) in some of their Priority Programs (Verbundschwerpunkt-Programm) as well as in their Collaborative Research Centres (Sonderforschungsbereiche). Other articles are sponsored by the European Community and are therefore results of cooperation among different organisations. The main areas include numerical simulation and mathematics, aeroelasticity, small and large aspect ratio wings in the context of leading-edge vortices, wake vortices, high lift systems and propulsion integration, and new developments in wind tunnel facilities and measurement techniques. Therefore, this volume gives an almost complete review of the ongoing aerodynamics research work in Germany. The order of the papers in this book corresponds closely to that of the sessions of the Symposium. From 70 lectures presented at the Symposium 57 are included in this book. The Review-Board, partly identical with the Program-Committee, consisted of A. Altminus (München), J. Ballmann (Aachen), R. Behr (München), Chr. Breitsamter
VIII
Foreword
(München), A. D’Alascio (München), J. Delfs (Braunschweig), R. Eggels (Dahlewitz), K. Ehrenfried (Göttingen), R. Ewert (Braunschweig), M. Fischer (Bremen), R. Friedrich (München), R. Grundmann (Dresden), P. Hakenesch (München), H. Hansen (Bremen), R. Hartmann (Braunschweig), St. Hein (Göttingen), K. D. Hennecke (Darmstadt), P. Hennig (Unterschleißheim), R. Höld (München), F. Holzäpfel (Weßling), J. Kompenhans (Göttingen), G. Koppenwallner (KatlenburgLindau), W. Kordulla (Noordwijk), E. Krämer (Stuttgart), H.-P. Kreplin (Göttingen), N. Kroll (Braunschweig), D. Kröner (Freiburg), J. Longo (Braunschweig), A. Meister (Kassel), F. Menter (Otterfing), V. Mikulla (München), C.-D. Munz (Stuttgart), K.-P. Neitzke (Bremen), W. Nitsche (Berlin), H. Olivier (Aachen), R. Radespiel (Braunschweig), H.-J. Rath (Bremen), K. Richter (Göttingen), U. Rist (Stuttgart), H. Rosemann (Göttingen), G. Schrauf (Bremen), W. Schröder (Aachen), F. Seiler (Saint-Louis), J. Sesterhenn (München), J. Seume (Hannover), P. Thiede (Bremen), J. Thorbeck (Berlin), C. Tropea (Darmstadt), R. Voß (Göttingen), C. Wagner (Göttingen), C. Weiland (München), C. Weishäupl (München) and W. Würz (Stuttgart). Nevertheless, the authors sign responsible for the contents of their contributions. The editors are grateful to Prof. Dr. E.H. Hirschel as the General Editor of the "Notes on Numerical Fluid Mechanics and Multidisciplinary Design" and to the Springer-Verlag for the opportunity to publish the results of the Symposium.
May 2007
C. Tropea, Darmstadt S. Jakirlic, Darmstadt H.-J. Heinemann, Göttingen R. Henke, Bremen H. Hönlinger, Göttingen
Table of Contents
Airplane Aerodynamics An Airbrake Design Methodology for Steep Approaches B. Anil Mertol ….……………………………………………………........... ............
1
High-Reynolds-Number Design of a Wing Section Including Control of Boundary Layer Properties K. Richter, S. Koch, H. Rosemann ….…………………………………….…….......
9
.
Numerical Analysis of Wing Vortices F. Zurheide, W. Schröder ….…………………………………………………..….
.
17
Studies on Tailplane Stall for a Generic Transport Aircraft Wind Tunnel Model A. Grote, R. Radespiel ….………………………………………………………... .
26
URANS and DES Simulation of Flow Around a Rectangular Cylinder C. Mannini, A. Soda, R. Voß, G. Schewe ….……………………………………....
36
Active Manipulation of a Rectangular Wing Vortex Wake with Oscillating Ailerons and Winglet-Integrated Rudders R. Hörnschemeyer, C. Rixen, S. Kauertz, G. Neuwerth, R. Henke ….…………………………………………………………………..........
44
.
A New Actuator Disk Model for the TAU Code and Application to a Sailplane with a Folding Engine A. Raichle, S. Melber-Wilkending, J. Himisch ….…………….……………..…. ... 52 .
Design of a Retrofit Winglet for a Transport Aircraft with Assessment of Cruise and Ultimate Structural Loads Th. Streit, J. Himisch, R. Heinrich, B. Nagel, K.H. Horstmann, C. Liersch …… ... .
62
Flow Control Numerical Simulation of a Wing with a Gapless High-Lift System Using Circulation Control K.-C. Pfingsten, R. Radespiel ….………………………………………….…......... 71
X
Table of Contents
Shock Control Bumps on Flexible and Trimmed Transport Aircraft in Transonic Flow B. König, M. Pätzhold, T. Lutz, E. Krämer ….…………..................................... ...
80
A Flat Plate Experiment for Investigations of Vortex Generator Jets at High Reynolds Number J. Ortmanns, C.J. Kähler, R. Radespiel ….………………….……………….........
88
.
Experimental Simulation and Test Techniques Measurement of Unsteady Surface Forces by Means of Piezoelectrical Copolymer Coatings J. Domhardt, J. Leuckert, I. Peltzer, W. Nitsche ….……………………………....
96
Studies of Reynolds Number Effects on Wing Tip Vortex Positions by Means of Laser Light Sheet (LLS) and Background Oriented Schlieren (BOS) Technique Under Cryogenic Conditions D. Pallek, F. Klinge ….………………………………………………………...... 104 Project ForMEx – A New CFD Approach for Transposition of Wind Tunnel Data Towards Flight Conditions S. Melber-Wilkending, G. Wichmann ….………………………………………... 113 Pressure and Heat Flux Measurements on the Surface of a Low-Aspect-Ratio Circular Cylinder Mounted on a Ground Plate T. Rödiger, H. Knauss, U. Gaisbauer, E. Krämer ….…………………………... 121 An Investigation into Internal and External Force Balance Configurations for Short Duration Wind Tunnels M. Robinson, J. Martinez Schramm, K. Hannemann …..…...…………………...
129
Numerics Multiple Discipline Take-Off Weight Minimization for a Supersonic Transport Aircraft U. Herrmann …..…………………………………………………………….…...
137
Three-Dimensional Discontinous Galerkin Codes to Simulate Viscous Flow by Spatial Discretization of High Order and Curved Elements on Unstructured Grids C. Lübon, S. Wagner ….…………………………………………………….…....
145
The Space-Time Expansion DG Method F. Lörcher, G. Gassner, C.-D. Munz ….…………...……….................................
154
The Parallel Mesh Deformation of the DLR TAU-Code T. Gerhold, J. Neumann ….……………………………………………………...
162
Table of Contents
XI
Simulation of Oscillating Airfoils and Moving Flaps Employing the DLR-TAU Unsteady Grid Adaptation A.D. Gardner, K. Richter, H. Rosemann ….……………………………..……....
170
RANS Simulations with One and Six Degrees of Freedom Rigid Body Motions T. Schwarz ….……………………………………………………….…………...
178
Improvement of the Automatic Grid Adaptation for Vortex Dominated Flows Using Advanced Vortex Indicators with the DLR-Tau Code M. Widhalm, A. Schütte, T. Alrutz, M. Orlt ….………..........................................
186
Adjoint Algorithms for the Optimization of 3d Turbulent Configurations R.P. Dwight, J. Brezillon ….……………………………………………………..
194
Enhanced Runge-Kutta/Implicit Methods for Solving the Navier-Stokes Equations C.-C. Rossow …..……………..……………………………………………….....
202
Laminar Flow Control and Transition Navier-Stokes High-Lift Airfoil Computations with Automatic Transition Prediction Using the DLR TAU Code A. Krumbein, N. Krimmelbein …..……...……………………………………......
210
Manipulation of Attachment Line Transition by Geometry Modification at the Slat of a Multi-element Airfoil J. Wild, H. Dettmar …..…...…………………………………………………......
219
Using CryoTSP as a Tool for Transition Detection and Instability Examination at High Reynolds Numbers U. Fey, Y. Egami, C. Klein …..……...…………………………………………...
227
Direct Numerical Simulation of a Short Laminar Separation Bubble and Early Stages of the Bursting Process O. Marxen, D. Henningson ….…………………………………………….…......
235
Experimental Investigations of Controlled Transition in a Laminar Separation Bubble at an Axisymmetric Diffuser L. Hoefener, W. Nitsche ….………………………………………………….…...
244
Development of a Sensor-Actuator-System for Active Control of Boundary Layer Instabilities in Compressible Flows M. Engert, A. Pätzold, W. Nitsche ….……………………………… ………........
252
.
Evaluation of Initial Amplitudes of Free-Stream Excited Tollmien-Schlichting Waves from Flight-Test Data A. Seitz ….………………………………………………………………….…..... 260
XII
Table of Contents
The Role of Turbulent Dissipation for Flow Control of Near-Wall Turbulence B. Frohnapfel, P. Lammers, J. Jovanoviü ….…………………………………....
268
Hypersonic Flows Radical Farming in Scramjets J. Odam, A. Paull ….…………………………………………………………......
276
Aerothermodynamic Investigation of the Pre-X Configuration in HEG J. Martinez Schramm, B. Reimann ….………………………………..……….....
284
SHEFEX – A First Aerodynamic Post-flight Analysis T. Barth, T. Eggers ….…………………………………………………… ...........
291
Simulation of Magnetohydrodynamic Effects on an Ionised Hypersonic Flow by Using the TAU Code C. Böttcher, V. Hannemann, H. Lüdeke ….……………………………………...
300
Heat Fluxes Inside a Cavity Placed at the Nose of a Projectile Measured in a Shock Tunnel at Mach 4.5 F. Seiler, J. Srulijes, M. Gimenez Pastor, P. Mangold ….…………………….....
309
.
Experimental Investigation and Numerical Simulation on a Missile Radome at Mach 6 C. Dankert, H. Otto ….……………………………………………………..….... 317
Turbomachines Control of Flow Separation in Adverse Pressure Gradients by Means of Crosswise Grooved Surfaces for Turbo Machine Applications W. Hage, R. Meyer, C.O. Paschereit ….………………………….……………...
324
Wave Drag Reduction Approach for Lattice Wings at High Speeds E. Schülein, D. Guyot …….………………………………………………...........
332
Forces and Velocity Measurements in Ship Propulsion Systems J. Pêgo, H. Lienhart, F. Durst, K. Tigges ….………………………………..…...
340
Flows Past Bodies Numerical Simulation of the Flow Around a Finite Cylinder with Ground Plate in Comparison to Experimental Measurements O. Frederich, E. Wassen, F. Thiele, M. Jensch, M. Brede, F. Hüttmann, A. Leder ….……………………………………….…....
348
Table of Contents
XIII
Numerical Simulation of Aerodynamic Problems with SSG/LRR-Ȧ Reynolds Stress Turbulence Model Using the Unstructured TAU Code B. Eisfeld …….………………………………………………............................... 356 Numerical Simulation of the Flow Field Around the Stratospheric Observatory for Infrared Astronomy S. Schmid, T. Lutz, E. Krämer ………....….……………………………… …….. .
364
Large Eddy Simulation / Direct Numerical Simulation DNS of Compressible Inert and Infinitely Fast Reacting Mixing Layers I. Mahle, J. Sesterhenn, R. Friedrich ….…………………………........................
372
Development of a Numerical Procedure for Direct Simulations of Turbulent Convection in a Closed Rectangular Cell M. Karczorowski, A. Shishkin, O. Shishkina, C. Wagner ….................................. 381 Computation of Supersonic Base Flow Using Detached Eddy Simulation V.K. Togiti, H. Lüdeke ….………………………...…………….……………......
389
Large-Eddy Simulation of Tundish Flows Using Preconditioning N.A. Alkishriwi, M. Meinke, W. Schröder ….…...………………………...….......
397
Numerical Investigation of Effusion Cooling in Hypersonic Boundary-Layer Flow J. Linn, M.J. Kloker ….………………………………………………………......
405
Aeroacoustics Numerical Investigation of Flow-Induced Noise Generation at the Nozzle End of Jet Engines A. Babucke, M.J. Kloker, U. Rist ….……………………………………….….....
413
Numerical Analysis of Sound Generating Mechanisms of a High-Lift Device D. König, W. Schröder, M. Meinke …….………………..………………….….. .
421
On the Design of Silent Trailing-Edges M. Herr…..…………………………………………………………………….....
430
Investigation of Flow Induced Sound Radiated by a Forward Facing Step C. Hahn, S. Becker, I. Ali, M. Escobar, M. Kaltenbacher …..…...………….… ..
438
.
..
Miscellaneous Application of PIV Techniques for Rotor Blade Tip Vortex Characterization H. Richard, B.G. Van der Wall, M. Raffel, M. Thimm ….…….………………....
446
XIV
Table of Contents
Automatic Differentiation of an Entire Design Chain for Aerodynamic Shape Optimization N.R. Gauger, A. Walther, C. Moldenhauer, M. Widhalm .........................................
454
Computational Study of Mean Flow and Turbulence Structure in Inflow System of a Swirl Combustor S. Šariü, S. Jakirliü, D. ýavar, B. Kniesner, P. Altenhöfer, C. Tropea …..……………………………………...…………..............................
462
Author Index .....……………………...………………………………………....
471
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High-Reynolds-Number Design of a Wing Section Including Control of Boundary Layer Properties K. Richter, S. Koch, and H. Rosemann DLR AS-HK, Bunsenstr. 10, D-37073 Göttingen, Germany
Summary A novel high Reynolds number design method was investigated and conducted for transonic airfoils with an application in the outer part of a modern transport aircraft wing. In contrast to conventional low Reynolds number design methods, this method exploits full aerodynamic performance and controls the boundary layer properties. Three airfoils were created for transonic free flight conditions with very similar aerodynamic performance, but different boundary layer developments on the upper surface. The performance and boundary layer properties of the airfoils were discussed and validated by RANS computations.
1 Introduction In the aerodynamic design of wings for modern transport aircraft, the design of the outer wing part is especially important. The flow quality here can be crucial for the flight stability of the entire aircraft since the stall characteristics of the outer wing part affects the efficiency of the aileron. Conventional aerodynamic wing designs were conducted for low Reynolds numbers as an experimental verification at flight Reynolds numbers was not feasible. However, the stall characteristic is known to be highly dependent on the Reynolds number and to typically enhance performance at high Reynolds number. Thus the conventional design did not push the outer wing part to its free flight performance limit and did not exploit the full aerodynamic potential. Since high Reynolds number test facilities have now been established for the performance assessment at flight conditions, an aerodynamic design should be conducted also at flight Reynolds numbers. A first step towards such a design was undertaken in the frame of the German technology research programme LuFoIII High Lift Configurations and is presented in this paper. A high Reynolds number design of two-dimensional wing sections was carried out with respect to requirements for their application in the outer wing part. A newly developed design concept allowed for the control of boundary layer properties in the design phase. Three different twodimensional airfoils were created providing the same aerodynamic performance at the selected design points but having different boundary layer characteristics at the rear of the upper side. The design was validated by Navier-Stokes computations with respect to performance and boundary-layer properties. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 9–16, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007
10
K. Richter, S. Koch, and H. Rosemann
2 Design Environment For the aerodynamic design of airfoils for the outer wing part an existing optimisation environment was used and extended. The environment was based on the commercial optimisation software Pointer™ which performs nonlinear multi-dimensional constrained optimisations [1]. Pointer™ provides a graphical user interface for displaying, controlling and interacting with the optimisation. It offers different optimisation strategies for minimizing an arbitrary objective function and allows for single and multi point optimisations. Special interfaces for data input and output enable linkage with a user-specific optimisation problem and the communication with analysis tools. In our case Pointer™ was linked with the software module AerOpt developed by Airbus-D for the purpose of aerodynamic shape optimisation. AerOpt contains a two-dimensional geometry generation algorithm based on 25 design variables. Furthermore, AerOpt handles the pre-processing, the call and the postprocessing of aerodynamic analysis tools as well as the calculation of objective functions. For the means of the aerodynamic analysis the coupled potentialtheory/boundary-layer method EPPLER [2] was used for low speed problems and the coupled Euler/boundary-layer method MSES [3] was used for transonic problems. In the present activity for a high Reynolds number design the optimisation environment was extended with respect to the control of boundary layer properties by incorporating the evaluation of the shape parameter H12 into AerOpt. The shape parameter H12 is given by the ratio of the displacement thickness δ1 to the momentum loss thickness δ2 [4]. H12 gives information about the shape of the boundary layer velocity profile and allows conclusions about the separation tendency of the boundary layer. For a turbulent boundary layer without pressure gradient H12 is constant while it decreases for a negative pressure gradient and increases for a positive pressure gradient. It is known from experiments that turbulent boundary layers typically separate above H12 ≈ 2.6. The shape parameter at the airfoil top trailing edge H12,TE was included in the set of optimisation variables as well as the H12 distribution at different stations on the airfoil upper side. Objective functions were developed to optimise not only the conventional aerodynamic objectives but also to include desired boundary layer characteristics at the trailing edge. More advanced objective functions were also developed for controlling large regions of the upper surface boundary layer by including the shape parameter distribution.
3 Airfoil Shape Optimisation The aerodynamic shape optimisation was conducted at flow conditions typical for outboard wing applications in free-flight. Three new airfoil contours were to be developed for a transonic Mach number M = MDesign, a Reynolds number of Re = 20 x 106, and fixed transition at xtr/c = 6% on both sides. Unlike conventional low Reynolds number airfoil designs for outer wing application, the new high Reynolds number design should push the airfoil to a higher design lift coefficient.
High-Reynolds-Number Design of a Wing Section
11
Table 1. List of design points used for the high-Reynolds-number airfoil design including control of boundary layer properties Design point
Boundary Conditions for Aerodynamic Performance
Boundary Conditions for Boundary Layer Properties
D1
M = MDesign, Re = 20e6, xtr/c = 6%, cl,Design
H12,TE, H12-distribution
D2
M = MDesign, Re = 20e6, xtr/c = 6%, ∆cl =-0.1
—
D3
M = MDesign, Re = 20e6, xtr/c = 6%, ∆cl =+0.1 M = MLowSpeed, Re = 10e6, xtr/c = 6%, cl ≥ clmax Constraints: tmax/c = const., tTE/c = const.
—
D4
—
The first airfoil “AIRFOIL_REF” was designed to have a good performance, but to reach a desired shape parameter at the trailing edge H12,TE. Based on the AIRFOIL_REF, two other airfoils were designed to hold the aerodynamic performance and also H12,TE of the reference airfoil but exhibit different shape parameter distributions. In this way the airfoil shape design converts into a design of the boundary layer properties on the airfoil upper side. Airfoil “AIRFOIL_SLOW” should have a slower increase of the shape parameter towards H12,TE than AIRFOIL_REF while “AIRFOIL_STEEP” should have a steeper increase. The designs of all three airfoils were performed by four-point optimisations considering three cruise flight relevant design points D1, D2, and D3, and one design point D4 accounting for maximum lift requirements. Table 1 shows a list of the design points and their boundary conditions. Design point D1 was the main design point and considered the aerodynamic performance, in form of lift and drag, as well as the boundary layer properties, in form of the shape parameter distribution and the shape parameter at the trailing edge, at M = MDesign, Re = 20 x 106, cl = cl,Design and xtr/c = 6%. Design points D2 and D3 were additional cruise design points accounting for aerodynamic performance only but at lower (∆cl = -0.1) and higher lift coefficients (∆cl = +0.1) than D1. Design point D4 was a low speed design point to ensure sufficient maximum lift. The objective function used for the optimisation consisted of a rather complex weighting of performance (cl, cd) and boundary layer properties (H12 at different stations, H12,TE) at these design points, giving design point D1 the highest weight and introducing a penalty when the required cl,max was not reached for D4. All optimisations were constrained by the maximum airfoil thickness tmax and a fixed trailing edge thickness tTE.
4 Design Results With the novel high Reynolds number design three airfoils were generated. Three airfoils were chosen for further investigations. Figures 1a and 1b depict the rear part of the surface pressure distributions cp and the distributions of the upper surface shape
12
K. Richter, S. Koch, and H. Rosemann
AIRFOIL_REF AIRFOIL_SLOW AIRFOIL_STEEP
-cp [-]
H12 [-]
AIRFOIL_REF AIRFOIL_SLOW AIRFOIL_STEEP
'cp = 0.2
0.4
0.5
0.6
0.7
x/c [-]
(a)
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0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/c [-]
(b)
Fig. 1a and 1b. Distributions of surface pressure coefficient (left) and upper surface shape parameter (right) of designed airfoils computed with MSES at D1 (see table 1)
parameter H12 for these airfoils at the main design point D1. As intended, the pressure distributions of the designed airfoils are very similar and the airfoils themselves can be seen to constitute an airfoil family. The cp-distributions are identical in the leading edge region and also on almost the entire lower surface. Major differences can be observed only in the mid and the rear part of the upper surface, accounting for the very different development of the boundary layer in these three cases. The shape parameter distributions of the three airfoils are very similar on the upper surface until approx. x/c = 50%. After that, the H12-distributions exhibit the intended slow (AIRFOIL_SLOW), moderate (AIRFOIL_REF), and steep (AIRFOIL_STEEP) increase of H12 towards the trailing edge. As the shape parameter at the trailing edge H12,TE is constant for all airfoils, the different H12-distributions represent a variation in the streamwise boundary layer loading resulting in the same boundary layer characteristic at the trailing edge of the airfoil. This variation in loading can be observed also in the cp-distributions. The reference airfoil AIRFOIL_REF has an almost constant pressure gradient towards the trailing edge. The slow rise airfoil AIRFOIL_SLOW exhibits first a high gradient followed by a low gradient causing H12 to rise early and then to reduce the increase to hold the common H12,TE. The steep rise airfoil AIRFOIL_STEEP shows a low gradient followed by a high one causing H12 to stay almost constant at first and then to rush up to H12,TE. These characteristics are directly associated with a variation in the lift generation in the rear part of the airfoil and hence with a variation of the airfoil circulation. Lift curves and drag polars are shown in Figures 2a and 2b for M = MDesign, Re = 20 x 106 and xtr/c = 6%. Here, the three airfoils also appear as an airfoil family since their lift curves are virtually identical. However, due to different circulations they exhibit offsets in the angle of attack for AIRFOIL_STEEP and AIRFOIL_SLOW, reaching cl,Design (D1) at higher incidences than the reference airfoil. Correspondingly, AIRFOIL_REF exhibits the lowest pitching moment followed by AIRFOIL_STEEP and AIRFOIL_SLOW. As intended during the airfoil optimisation, the drag polars show only small differences. At low lift all polars exhibit the same drag while the polars diverge with increasing lift coefficient. At main design point D1, AIRFOIL_REF and AIRFOIL_SLOW exhibit the same drag, while
High-Reynolds-Number Design of a Wing Section
13
AIRFOIL_REF AIRFOIL_SLOW AIRFOIL_STEEP
cl,Design AIRFOIL_REF AIRFOIL_SLOW AIRFOIL_STEEP
cl [-]
cl [-]
cl,Design
'cl = 0.05
'cl = 0.05 'cd = 0.0002
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D [°]
(a)
cd [-]
(b)
Fig. 2a and 2b. Lift curves (left) and drag polars (right) of designed airfoils computed with MSES at M = MDesign, Re = 20 x 106 and xtr/c = 6%
AIRFOIL_STEEP shows reduced drag by ∆cd = 3 counts. At higher lift the polars merge again. The polar of the reference airfoil AIRFOIL_REF exhibits a kink at the design point D1 which is caused by a special formation of the supersonic flow field introducing increased wave drag. This characteristic was already present for the airfoil contour used as the starting point for the optimisation of AIRFOIL_REF. Since the changes between both contours were only small, the kink could not be suppressed although the additional transonic design points D2 and D3 were used. For AIRFOIL_SLOW and AIRFOIL_STEEP this characteristic could be avoided by the use of D2 and D3 since the changes in contours were larger.
5 Design Validation A detailed investigation of the designed airfoils was also undertaken using the NavierStokes solver DLR-TAU [5]. Two-dimensional RANS computations were performed with the Spalart-Allmaras turbulence model with Edwards modification [6] on hybrid grids generated with the commercial grid generator CENTAURTM [7]. The DLR-TAU grid adaptation method [8] was employed to achieve grid convergence. DLR-TAU computations were performed for different Reynolds numbers and compared to the results obtained with MSES for a detailed validation of the airfoil design done. Figure 3 shows drag polars of all airfoils computed with both numerical methods at M = MDesign, Re = 20 x 106 and xtr/c = 6%. The results of both codes are in good agreement although DLR-TAU generally predicts slightly increased drag. The DLRTAU result for airfoil AIRFOIL_SLOW matches best since it exhibits an offset not exceeding ∆cd = 1 count, while AIRFOIL_REF shows a constant offset of approx. ∆cd = 2 counts for the entire polar. Here, the RANS computation exactly reproduces the kink in the AIRFOIL_REF drag polar. The agreement for airfoil AIRFOIL_STEEP is still good but exhibits an offset of approx. ∆cd = 3 counts diminishing at higher lift. The reason for the airfoil-dependent deviations between MSES and DLR-TAU is unclear. Nevertheless, the DLR-TAU computations generally confirm the predicted aerodynamic performance of the designed airfoils.
14
K. Richter, S. Koch, and H. Rosemann
Fig. 3. Drag polars of designed airfoils computed with MSES and DLR-TAU at M = MDesign, Re = 20 x 106 and xtr/c = 6%
Comparisons of the surface pressure distribution show also good agreements for each airfoil. Figure 4a depicts the rear part of the cp-distribution for airfoil AIRFOIL_STEEP at the main design point D1, and shows only minor deviations in the region of the supersonic flow field (x/c<0.53) and in the shock position. The suction peak, the rear half of the upper surface and the entire lower side match exactly.
(a)
(b)
Fig. 4a and 4b. Distributions of pressure coefficient (left) and skin friction coefficient (right) of AIRFOIL_STEEP computed with MSES and DLR-TAU at D1 (see table 1)
A good overall agreement between both numerical methods was found also for the prediction of the skin friction distribution cf, Figure 4b and Figure 5. Minor differences can be seen in the region of the shock caused by different predictions of the shock location. However, deviations for all airfoils occur also in the prediction of the trailing edge flow indicating an important difference in the modelling of the boundary layer flow. MSES underestimates, or DLR-TAU overestimates respectively,
High-Reynolds-Number Design of a Wing Section
Fig. 5. Distributions of the skin friction coefficient of AIRFOIL_SLOW computed with MSES and DLR-TAU at D1
15
Fig. 6. Distributions of the shape parameter of AIRFOIL_STEEP computed with MSES and DLR-TAU at D1
the effect of the pressure loading on the boundary layer properties. Using the DLRTAU code, the upper surface skin friction is predicted to be lower, i.e. nearer to separation, for the high loaded airfoil AIRFOIL_STEEP, Figure 4b, and vice versa for the low loaded airfoil AIRFOIL_SLOW, Figure 5. This behaviour was not significantly dependent on the choice of DLR-TAU turbulence model. Using the boundary-layer analysis tool TAUBL [9] provided by Airbus-D, the shape parameter distribution was extracted from the DLR-TAU RANS flow solution and compared to the MSES result. The H12-distributions exhibit some quantitative discrepancies probably caused by the different boundary layer modelling. However, the distributions themselves match qualitatively, i.e. the H12-distributions provided by DLR-TAU also clearly distinguish between slow, moderate and steep cases. Figure 6 shows a comparison between MSES and DLR-TAU of the H12-distribution for airfoil AIRFOIL_STEEP confirming the designed boundary layer properties.
6 Conclusions A novel high Reynolds number design for airfoils with an application in the outer part of a modern transport aircraft wing was conducted within the frame of the German technology research programme LuFo III “High Lift Configurations”. Three airfoils were designed for transonic free flight conditions by four-point optimisations accounting for aerodynamic performance and boundary-layer properties using the optimisation software PointerTM and the MSES code for the aerodynamic analysis. An existing optimisation environment was extended for controlling the boundary layer shape parameter, and specific objective functions were developed. For similar aerodynamic performance, the designed airfoils realised different boundary layer developments by different shape parameter distributions on the upper surface with a slow, a moderate, and a steep rise towards the trailing edge. The shape parameter at the trailing edge was kept constant for all airfoils. The design was validated by DLRTAU RANS computations showing that lift, drag, pressure distribution, and skin
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K. Richter, S. Koch, and H. Rosemann
friction distribution were in good agreement with the MSES prediction. However, the precise prediction of the skin friction at the trailing edge exhibited differences probably caused by the differing modelling of the boundary layer flow. Comparisons of the shape parameter distributions confirmed the designed boundary layer properties.
References [1] A. Van der Velden: “Pointer Design Automation Software”, Synaps Inc., 1996 [2] R. Eppler: “Airfoil Program System, User’s Guide”, Institut für Mechanik, Universität Stuttgart, 1988 [3] M. Drela: “Two-dimensional aerodynamic design and analysis using the Euler equations”, Dissertation, Massachusetts Institute of Techn., 1985 [4] H. Schlichting: “Grenzschicht-Theorie”, Verlag Braun, Karlsruhe, 1982 [5] T. Gerhold, O. Friedrichs, J. Evans and M. Galle: “Calculation of Complex ThreeDimensional Configurations Employing the DLR-τ-Code”, AIAA 97-0167, 35th Aerospace Sciences Meeting & Exhibit, Reno, Nevada, January 6-10, 1997 [6] J.R. Edwards and S. Chandra: “Comparison of Eddy Viscosity-Transport Turbulence Models for Three-Dimensional, Shock-Separated Flowfields“, AIAA Journal, Vol. 34, No. 4, April 1996 [7] http://www.centaursoft.com [8] T. Alrutz: “Chapter 7: Hybrid Grid Adaptation in TAU”, in MEGAFLOW: Numerical Flow Simulation for Aircraft Design Results of the second phase of the German CFD initiative MEGAFLOW, pp. 109-116, 2002 [9] D. John: “Ein Verfahren zur Berechnung der Grenzschichtströmung aus Navier-Stokes Ergebnissen in unstrukturierten Netzen”, Airbus-D, interner Bericht, 2003
Numerical Analysis of Wing Vortices Frank Zurheide and Wolfgang Schr¨oder Aerodynamisches Institut, RWTH Aachen, W¨ullnerstr. zw. 5 u. 7, 52062 Aachen, Germany
[email protected] http://aia.rwth-aachen.de
Summary The temporal development of generic vortex systems based on Lamb-Oseen vortices is computed by large-eddy simulations. For a two-vortex system the growth of shortwave co-operative elliptical and long-wave Crow instabilities leads to a pronounced decay of its vorticity distribution. The four-vortex system possesses a smaller decay rate than the two-vortex system.
1 Introduction Due to the predicted increase in passenger numbers and aircraft traffic and the limitations on airport capacity the problem of aircraft wake vortex hazard is of growing importance. Heavy transport aircraft generate vortex wakes that endanger the following aircraft traffic, because they cannot be accurately detected. Since the decay of vortex wakes by diffusion and dissipation mechanisms is weak, scientific research focuses on the evolution of instabilities inherent to or artificially introduced into the wake structure [16]. These instabilities can lead to a rapid decay of vortex strength. The aim is to reduce the time needed for the onset of rapid vortex decay by promoting mechanisms that lead to an amplification of different types of instabilities. In counter-rotating vortex systems several three-dimensional instabilities are known. Crow [3] and Waleffe [18] performed stability analyses of a vortex pair with the vortex filament method resulting in basic findings for long and short-wave instabilities. Following these investigations we define instabilities having a wave length larger/smaller than 5/1 times the vortex distance are considered long/short waves, respectively. In this paper results based on large-eddy simulations are discussed to show the development of short wave and long wave instabilities for a counter-rotating vortex pair and the evolution of a four-vortex system.
2 Numerical Method 2.1 Governing Equations The three-dimensional compressible Navier-Stokes equations are solved using the MILES approach (monotone-integrated large-eddy simulation). The discretization C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 17–25, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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of the governing equations is based on a mixed central-upwind AUSM (advective upstream splitting method) scheme with low numerical dissipation, see Meinke et al. [12]. The temporal appoximation is based on an implicit dual time-stepping scheme. A convergence acceleration of the low Mach number problem is achieved using multigrid and preconditioning methods [1]. With this method convincing results could be reached for a variety of internal [13] and external turbulent flows [5].
2.2 Initialization Leweke and Williamson [11] showed that a generic flow field for the far wake generated by an airplane can be simulated by superposing Lamb-Oseen vortices. In this case the vortices are rolled up and no shear layer is considered. The circumferential velocity vθ and pressure (p) fields for a Lamb-Oseen vortex are given by −r2 Γ0 1 − exp 2 vθ (r) = 2πr r0 2 dp ρv (r) = θ dr r
(1) (2)
where the density is denoted by ρ and the circulation is defined as Γ = 2πrvθ (r) such that the root circulation Γ0 reads Γ0 = 2πr0 vθ (r0 ). The quantity rc represents the core radius and r0 depicts a scaled core radius r0 = rc /1.121. The distance between the vortices is b0 . The Reynolds number ReΓ = Γ0 /ν is based on the root circulation and the viscosity ν. The time scale of the system is based on the vortex separation b0 and root circulation Γ0 , that leads to the non-dimensional time t∗ = 2πb20 /Γ0 = b0 /w0 where w0 = Γ0 /2πb0 is the initial sink rate of the vortex system. The pressure is a non-linear function of the velocity field. That is, the initial pressure distribution in the computational domain caused by superposing two or four vortices is determined by solving the 3D Poisson pressure equation. The two-vortex system consists of a pair of counter-rotating vortices. The values of the system are shown in fig. 1 (left) and given in table 1. In the case of the fourvortex system two counter-rotating vortex pairs are prescribed (fig. 1 right). Table 1. Simulation parameters Case I II III
Type rc /b0 ReΓ Lx × Ly × Lz Short-wave 0.2 2400 4.5 × 5.5 × 4.5 b30 Crow 0.085 7400 8.68 × 4 × 4 b30 Four vortex – 4200 9.4285 × 4 × 4 b31
grid points 2.4 million 2.4 million 4.1 million
Without any perturbations in the flow field no intabilities develop and the vortex interaction results in a constant sink rate of w0 . For this reason, perturbations are
Numerical Analysis of Wing Vortices
19
introduced to the vortex system. Disturbances introduced into the velocity field lead to a displacement of the vortex cores. Due to this displacement instabilities in the system start to grow [3, 14]. Periodic boundary conditions were used in each direction. As discussed in [6, 7] the influence of the boundaries is negligible small when the distance of the vortices is small compared to the distance between a vortex core and the boundary. Details of the meshes are given in the section 3.
3
Results
Three different simulations were performed to evidence the impact of long and short-wave instabilities on the decay of vorticity in vortex systems. In the first analysis the short-wave co-operative elliptical instability (Case I) is considered. Then, the Crow instability (Case II) is computed and finally, a four-vortex system (Case III) is analyzed. Note, the main difference in the initial conditions between the short-wave and the Crow instability is the ratio of vortex core to vortex distance rc /b0 . The origin of the coordinate system is located in the center of the vortex system. The x-axis is oriented parallel to the vortex axis. The rotation direction of the vortices is also given, fig. 1. b1
b0
b2
z
Γ rc
y
−Γ rc
Γ1
−Γ2
r1
r2
Γ2
r2
−Γ1 r1
Fig. 1. Description of vortex systems. Left: two-vortex, right: four-vortex system.
3.1 Co-operative Elliptical Instability (Case I) The parameters (fig. 1) were chosen similar to the values of Laporte and Corjon and Leweke and Williamson [10, 11]. For the perturbations uniform white noise was used with a maximum value of 10−3 · vθ,max . In figure 3 the temporal development of the two-vortex system is shown. Contours of the vortex criterion λ2 [9] are displayed to identify the vortices. This criterion is based on the negative value of the second eigenvalue λ2 (S 2 + T 2 ) < 0 of the strain-rate tensor S and the vorticity tensor T to lead to a pressure minimum. The parallel vortex cores (t∗ = 0.1) are deformed (t∗ = 6.7). The deformation of the vortices has a phase shift of 180 degrees. The curvature of the vortices results from a self-induction of each vortex and the interaction with the other vortex. The interaction of the two vortices leads to secondary vortices (t∗ > 7.8). The vorticity components of ω = (ωx , ωy , ωz )T are integrated and averaged for each space direction separately. Following [10] the resulting quantities Ωx , Ωy , Ωz are used to measure the strength of the instabilities in the three space dimensions.
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F. Zurheide and W. Schr¨oder
While Ωy and Ωz are integrated over the complete domain, Ωx is computed only for one half of the flow field, i. e., for just one single vortex, since the two vortices are symmetric 1 |ωx (x, y, z)|dz dy dx (3) Ωx = Lx Lx Ly /2 Lz Ωy =
1 Ly
1 Ωz = Lz
Ly
Lz
|ωy (x, y, z)|dx dz dy
|ωz (x, y, z)|dx dy
Lz
Ly
(4)
Lx
dz.
(5)
Lx
Note, the integrated quantity ω equals the circulation Γ = ω · ndA. In fig. 6(a) the temporal development of Ωx , Ωy , Ωz is shown. The vorticity component Ωx remains constant up to t∗ ≈ 7. The vorticity components perpendicular to the vortex axis show some kind of dissipation at t∗ < 2, since the prescribed perturbations cause some rearrangement of the flow field. At t∗ > 4 a strong growth of three orders of magnitude of these components can be observed that lead to strongly curved vortex cores. At t∗ ≈ 7 the interaction of the two vortices results in a slight decrease of Ωx and the components Ωy and Ωz reach their peak values. After the vortex interaction (t∗ > 8) dissipation causes a loss of vorticiy, i. e., a slight decline of Ωy and Ωz occurs. To determine the growth of the modes of the instabilities the kinetic energy Ec = 1/2(u2 + v 2+ w2 ) is used to compute the discrete Fourier coefficients nx ˆ (y , z ) = 1 −2Iπki/nx in the yz-plane for each grid Ec k j m i=1 Ec(xi , yj , zm )e nx point xi , yj , zm using nx grid points in the x direction. √ The Fourier mode number is denoted by k = 1, 2, . . . , nx /2 and I represents −1. The discrete Fourier coefficients are calculated using the Fast-Fourier Transformation (FFT). The mean kinetic ˆ k (yj , zm ) energy spectrum Ec∗k in the axial direction is determined by averaging Ec Ec∗k
ny nz 1 ˆ k (yj , zm )∆y,j ∆z,m = Ec Ly Lz j=1 m=1
(6)
with the grid spacing ∆y,j , ∆z,m , the length of the integration domain Ly , Lz and the number of points ny , nz in the y and z direction. A dimensionless wave number kz∗ = kz rc = 2kπrc /Lx is defined for the wave number kz and the vortex-core radius rc . In fig. 6(b) the mean kinetic energy spectrum Ec∗k is shown as a function of time ∗ t and of the dimensionless wave number kz∗ . The mode having the strongest growth is the mode of the elliptical instability kz∗ ≈ 1.5. The related non-dimensional wave length is λk = Lx /k. The wave length of the elliptical instability λe related to the vortex separation b0 is λe /b0 = 0.9. The mean kinetic energy spectrum also shows a growth in all modes, not only in the mode of the elliptical instability.
Γ/Γ0
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21
1.2 0.8 0.4 0
2
4
6
8 10 12 14 16 18 20 t*
) Fig. 2. Temporal development of circulation for short-wave instability (—) LES, (∗, ×) measurements from [11], (· · ·) simulations from [10]
Fig. 3. Case I: Temporal development of short-wave instability of a two-vortex system. λ2 contours for non-dimensional times t ∗ = 0.1, 6.7, 7.2, 7.8 .
In fig. 2 the decay of vorticity is compared with numerical results from Laporte and Corjon [10] and experimental results from Leweke and Williamson [11] that were based on a slightly higher Reynolds number ReΓ = 2660. The present LES findings show a good agreement with the experimental results. 3.2 Crow Instability (Case II ) To simulate the Crow instability parameters given in table 1 are used. The white noise perturbations had an rms-value of 0.1 and were linearly dependent on the tangential velocity of the vortices [7]. The perturbations for the Crow instabilities were two orders of magnitude stronger than for the short-wave instabilities.
Fig. 4. Case II: Temporal development of Crow instability of a two-vortex system. λ2 contours for non-dimensional times t∗ = 0.1, 6.4, 7.6, 8.6 .
Figure 4 shows the temporal development of the Crow instability for four different timesteps. The Crow instability leads to a deformation of the two vortices and to an interaction, which in turn generates vortex rings. Note that short-wave instabili. 8..6 ). ties exist although they saturate at a very low amplitude (e. g. t∗ = 7.6,
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The development of the vorticity components as a function of time t∗ is evidenced in fig. 6(c). The component Ωx remains nearly constant until the interaction of the two vortices causes a decrease of circulation. The other components show a behavior comparable to that of the short-wave instability. That is, the oscillations dissipate and then the instabilities grow, however, at a much smaller rate. The mean kinetic energy spectrum Ec∗k in fig. 6(d) shows mainly the growth of the lowest mode, while the increase of the instabilities at higher modes is saturated at a low level. Sipp showed in [15] that for a vortex core rc /b0 < 0.1 short-wave instabilities saturate at a low level. This means that the energy spectrum of the short-wave modes reaches an low-level plateau such that the growing short-wave instabilities do not lead to the breakdown of the vortex system. Instead, the weak growing Crow instability causes an interaction of the vortices. Thus, the saturation of the short-wave instability yields a slower decay of vorticity since it is the breakdown phenomenon which enhaces the destruction of the vortices. 3.3 Four-Vortex System (Case III) The generic wake of an airplane in high-lift configuration is represented through a four-vortex system [2, 4, 8, 10]. Between two counter-rotating vortices exist two smaller vortices that rotate in the opposite direction, fig. 1 (right). The parameters for the presented calculation were similar to case 2 of Stumpf [17] Γ2 /Γ1 = −0.35, b2 /b1 = 0.35, r1 /b1 = 0.05 , r2 /b1 = 0.025. Other values can be found in tab. 1. The perturbations were based on uniform white noise with a maximum value of 10−3 · vθ,max .
Fig. 5. Case III: Temporal development of a four-vortex system. λ2 contours for nondimensional times t∗ = 0.03, 0.56, 0.76, 2.49.
Figure 5 shows the interaction of the vortices. After one rotation of the small vortices around the larger vortices the small structures have deformed to Ω-shaped loops and lead to a strong deformation of the main vortices. In figs. 6(e) and 6(f) the circulation components and the mean kinetic energy spectrum Ec∗k are displayed. The vorticity component Ωx shows only a small decrease after the fast interaction of the vortices. The other components Ωy and Ωz possess no uniform behavior. Unlike in the two-vortex system the growth of the instabilities cannot be verified. After the interaction the disturbances dissipate.
Ωx/Ωx(0), Ωy/Ωx(0), Ωz/Ωx(0)
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0.001 1e-04 1e-05 1e-06 9 1e-07 1e-08 8
0.1
Ec*k 0.01
0.01
Ωx/Ωx(0)
1e-08
Ωy/Ωx(0)
0.001
2
Ωz/Ωx(0)
4
6
1e-04
0
2
12
10
8 t*
6
4
14
t*
16
8 10 12 14
1
2
3
7 6 5 k * z 4
(b)
(a) Ωx/Ωx(0), Ωy/Ωx(0), Ωz/Ωx(0)
23
1
0.1
Ωx/Ωx(0)
Ωy/Ωx(0)
0
Ωz/Ωx(0)
2
4
0.01
t*
10
8
6
4
2
0.01 0.001 1e-04 1e-05 2 1e-06 1.6 1.2 0.8 kz* 0.4
Ec*k 0.01 0.001 1e-04 1e-05
6
8
10
t*
(d)
Ωx/Ωx(0), Ωy/Ωx(0), Ωz/Ωx(0)
(c) 1 0.1
1e-04
0.01 0.001 1e-04
Ωx/Ωx(0)
1e-05
Ωy/Ωx(0)
0
Ωz/Ωx(0)
1e-06
0
1
4
3
2
5
1e-05
Ec*k 1e-04 1e-05 1e-06
6
7
1e-06
1
2
t*
3
4
1 0.8 0.6 0.4 kz* 0.2
1e-07
5
t*
(f)
(e)
Fig. 6. Case I (a), (b); Case II (c), (d) and Case III (e), (f). Left column: Dimensionless vorticity components for dimensionless time t∗ . Right column: Mean kinetic energy spectrum Ec∗k for dimensionless time t∗ and wave number kz∗ .
4
Conclusion
In this article the temporal development of three different vortex systems were presented. The flow field of the generic vortex systems were created by superposing Lamb-Oseen vortices. Perturbations were added to the vortices to displace the vortex cores and to allow the growth of instabilities. Simulations for the development of a short-wave co-operative elliptical instability and the Crow instability were performed for two-vortex systems. The type of instabilities that led to an interaction of the vortices depended on the ratio of core
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F. Zurheide and W. Schr¨oder
radius to vortex distance rc /b0 . The short-wave instabilities saturated at a low level for the case rc /b0 = 0.085 < 0.1 such that the weak growing Crow instability caused an interaction of the vortices. In the case of rc /b0 = 0.2 the strong growing co-operative elliptical instability led to an interaction of the vortices. Aircraft with larger wing span have a ratio of rc /b0 0.1 such that fast growing small instabilities saturate at a lower level. The simulation of the four-vortex system showed a fast interaction of the vortices. However, the decrease rate of vortex flux component Ωx was lower than that for the two-vortex systems.
Acknowledgments The support of this research by the Deutsche Forschungsgemeinschaft (DFG) in the frame of SFB 401 is gratefully acknowledged.
References [1]
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Alkishriwi, N., Meinke, M., Schr¨oder, W.: A large-eddy simulation method for low Mach number flows using preconditioning and multigrid. Comput. & Fluids 35 (2006) 1126–1136 Crouch, J.D.: Instability and transient growth for two trailing-vortex pairs. J. Fluid Mech. 350 (1997) 311–330 Crow, S.C.: Stability theory for a pair of trailing vortices. AIAA J. 8 (1970) 2172–2179 Fabre, D., Jacquin, L.: Stability of a four-vortex aircraft wake model. Phys. Fluids 12 (2000) 2438–2443 Guo, X., Schr¨oder, W., Meinke, M.: Large-eddy simulations of film cooling flows. Comput. & Fluids 35 (2006) 587–606 Holz¨apfel, F., Gerz, T.: Two-dimensional wake vortex physics in the stably stratified atmosphere. Aerosp. Sci. Technol. 5 (1999) 261–270 Holz¨apfel, F., Gerz, T., Baumann, R.: The turbulent decay of trailing vortex pairs in stably stratified environments. Aerosp. Sci. Technol. 5 (2001) 95–108 Jacquin, L., Fabre, D., Sipp, D., Theofilis, V., Vollmers, H.: Instability and unsteadiness of aircraft wake vortices. Aerosp. Sci. Technol. 7 (2003) 577–593 Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285 (1995) 69–94 Laporte, F., Corjon, A.: Direct numerical simulations of the elliptic instability of a vortex pair. Phys. Fluids 12 (2000) 1016–1031 Leweke, T., Williamson, C.H.K.: Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360 (1998) Meinke, M., Schr¨oder, W., Krause, E., Rister, T.: A comparison of second- and sixthorder methods for large-eddy simulations. Comput. & Fluids 31 (2002) 695–718 R¨utten, F., Schr¨oder, W., Meinke, M.: Large-eddy simulation of low frequency oscillations of the Dean vortices in turbulent pipe bend flows. Phys. Fluids 17 (2005) Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge; New York (1992)
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[15] Sipp, D.: Weakly nonlinear saturation of short-wave instabilities in a strained lamboseen vortex. Phys. Fluids 12 (2000) 1715–1729 [16] Spalart, P.R.: Airplane trailing vortices. Annual Review of Fluid Mechanics 30 (1998) 107–138 [17] Stumpf, E.: Untersuchung von 4-Wirbelsystemen zur Minimierung von Wirbelschleppen und ihre Realisierung an Transportflugzeugen. Diss., Aerodyn. Inst. RWTH Aachen (2003) [18] Waleffe, F.: On the three-dimensional instability of strained vortices. Phys. Fluids 2 (1970) 76–80
Studies on Tailplane Stall for a Generic Transport Aircraft Wind Tunnel Model Arne Grote1 and Rolf Radespiel2 1
EDAG SIGMA Concurrent Engineering GmbH Papenreye 23, 22453 Hamburg, Germany
[email protected] 2 Institute of Fluid Mechanics, Braunschweig Technical University Bienroder Weg 3, 38106 Braunschweig, Germany
[email protected]
Summary A generic wind tunnel model for tailplane stall research was designed and experimentally investigated to establish a database for code validation. The configuration is numerically optimised to obtain large Reynolds numbers at the horizontal tailplane in a wind tunnel of limited size. It consists of a fuselage, a detachable horizontal tailplane and a tip-truncated wing, that mounts the model to the turntables of the closed test section. The wing was designed to reproduce a representative downwash in the tailplane region. The tests were conducted at a freestream Mach number of 0.16 and at a Reynolds number of 0.72×106. Tailplane stall in case of natural and fixed transition was visualised by the oil-flow technique and quantified by pressure and force measurements and by the Particle Image Velocimetry of the turbulent trailing-edge separation. Numerical simulations, using the unstructured Reynoldsaveraged Navier Stokes Code TAU, are in good agreement with the experiments. They show a separation of the boundary layer starting at the trailing edge with high crossflow velocities at the outer tailplane. Depending on the boundary-layer transition, the stall occurs abruptly for natural transition, initialised by a burst of a laminar separation bubble, or gradually for fixed transition.
1 Introduction The aerodynamic design of horizontal tailplanes for transport aircrafts is mainly driven by demands for longitudinal stability and reduced drag at cruise condition, for manoeuvrability at low speed, and for prevention of tail stall. Securing low speed manoeuvrability becomes more important, if future aircraft tailplane area will be reduced for improved aircraft performance. Therefore meaningful predictions of the complex viscous flow over the tail and the efficiency of stabiliser and elevator at low speed manoeuvre conditions are necessary already during early stages of the industrial design process. Advanced three-dimensional, numerical simulation tools cope with the demand to forecast the nonlinear characteristic of stabiliser and elevator efficiency for increasing aerodynamic load, but experimental data for code validation relating to tailplane stall are not freely available. To overcome this lack of high quality experimental tail flow data, the model of a generic transport aircraft was designed and experimentally investigated for flow C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 26–35, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
Studies on Tailplane Stall for a Generic Transport Aircraft Wind Tunnel Model
27
conditions at the horizontal tailplane from beginning trailing-edge separation until the onset of stall. The experiments took place in the subsonic wind tunnel MUB of Braunschweig Technical University. Aside three-dimensional Reynolds-averaged Navier Stokes computations were undertaken to allow comparisons between measured and simulated data of the flowfield around the tailplane. For low Reynolds numbers and high angles of incidence, transitional phenomena influence the flow by the occurence of a laminar separation bubble. In this case the increasing pressure gradient at the tailplane’s leading edge causes a bubble burst initiating the stall. A more characteristic stall behaviour for higher Reynolds numbers without laminar separation bubble can be achieved by a transition tripping close to the leading edge. Due to the thin boundary layer the flow is very sensitive to this specific flow control at the leading edge.
2
Design of the Research Configuration
The design of the wind tunnel model is driven by the demand of attaining maximum Reynolds number at the horizontal tailplane. For a typical tailplane geometry with an aspect ratio of 4.5 and a taper ratio of 0.42 Reynolds numbers about 1.0×106 can be realised in the test section of the wind tunnel MUB with a cross section of 1.3 m squared. In our concept the span of the taiplane is 880 mm, which corresponds to 68% of the test section width, leading to a mean aerodynamic chord of 206 mm. The main wing is clipped and serves as the model support. A typical single-aisle, 150 seat aircraft forms the geometric base of the generic model. The dimensions of the closed wind tunnel section determine wind tunnel interferences as mentioned by Krynytzky [5]. These influences were taken into account during the aerodynamic design of the model as described by Grote et al. [2],[3]. For the purpose of comparison between experimental and numerical results, this procedure is valid as long as the numerical simulations include the test section.
3 Experiments The structural layout in Figure 1 shows the resulting wind tunnel model. It can be divided into a front part generating the necessary flow conditions at the horizontal tailplane, including the main wing and the fuselage up to the disjunction ahead of the horizontal tailplane, and a rear part, consisting of tail and tailplanes with 30% chord flaps. Rear fuselage and horizontal tailplanes are manufactured of carbon fibre reinforced epoxy. The complete fuselage is 2050 mm long and has an axially symmetric section of 280 mm in diameter. The rear part of the fuselage is connected to the front via a 6-component strain-gauge balance, in order to measure the resulting air loads on the tail. The model’s portside horizontal tailplane is equipped with 100 pressure tabs in sections at y/sh = 0.55 and 0.75 semispan with a high resolution close to the leading edge to resolve expected separation bubbles. To evaluate pressure forces on the inside of the rear fuselage as a result of the gap between both parts, pressure tabs were located in this cut-plane as well.
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strain−gauge balance inner chassis
balance support tailplane shaft
step motor connection to main wing
Fig. 1. Structural layout of the model
Fig. 2. Model mounted in the test section
The subsonic wind tunnel MUB is a conventional closed-circuit atmospheric tunnel with exchangable test sections. For the current experiments the 1.3 m squared closed test section for velocities up to 60 m/s was used. The model is mounted in the test section using turntables at the side walls as illustrated in Figure 2. By this way sting-induced interferences on the flow in the rear fuselage and tailplane region could be avoided. The tests were carried out at a freestream Mach number of 0.16 and a Reynolds number at the horizontal tailplane of 0.72×106. Natural transition was allowed at all components of the model. Later, transition at the horizontal tailplane was controlled using a jagged adhesive tape of 25 µm height and 3 mm tooth width, located with its tips along the leading-edge line. The parameters angle of attack α, stabiliser incidence angle it , and deflection of the elevator (which is not treated here) were varied to maximum negative lift at the tailplane and beyond stall. The surface flow at the tailplanes and in the junction of the rear fuselage was visualised using the oil-flow technique. It delivered a detailed spatial resolution of the surface streamlines and provided a view of the structures within areas of separated flow at the trailing edge and the interaction with the rear fuselage. In case of natural transition the position and the size of the laminar separation bubble and its movement with increasing angle of incidence of the tailplane could be detected. To investigate the characteristics in the area of turbulent trailing edge separation the conventional Particle Image Velocimetry was applied. Planes parallel to the xz-plane at y/sh = 0.55 and 0.75 semispan and aligned in flow direction were illuminated with a frequency doubled Nd:YAG double-pulse laser system (Quantel Brilliant) with 150mJ pulse energy at a wavelength of 532 nm. The laser beam was formed into a light sheet of 1 mm in thickness to illuminate tracer particles of 1 µm, which were generated with a standard seeding atomizer from vegetable oil. The light scattered by the particles was recorded by means of a Peltier cooled LaVision ImagerPro CCD camera (resolution of 4008 × 2672 pixels) equipped with a TAMRON 180/3.5 Macro lens. The evaluation of the recordings was performed using an iterative multi-grid method of 2nd order accuracy with window-shifting und window-deformation algorithms (Davis 7.0.09 by LaVision). The spatial resolution was 0.29 mm and the number of vectors per recording was 501 × 334 with 50% oversampling. To attain reliable estimations of the near-wall velocity distribution, regions without particle-information like the models contour, regions shadowed by the twist of the tailplane or outshone by reflections of the surface were masked before the calculation of the velocity vectors.
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4 Numerical Flow Simulation The numerical investigations using the DLR hybrid unstructured flow solver TAU as described by Gerhold [1] simulate the flow around the model inside the wind tunnel test section at a mean velocity of 60 m/s. For reason of complexity only those boundary layers were resolved, which were expected to have an influence on the tailplane flow. Therefore, the boundary layers of the horizontal tailplane and the fuselage were resolved with prisms. The flow over the main wing and the wind tunnel walls was assumed to be inviscid. The viscous flow at the horizontal tailplane was resolved by at least 308 points in circumferential direction and 48 prism layers in normal direction. In regions of high grid resolution spanwise anisotropic cells were used. The fuselage was resolved by 32 prismatic layers. Grid adaptation locally improved the grid quality within the prismatic and tetrahedral regions in particular at the horizontal tailplane and increased the total number of nodes to about 10×106. Natural transition is not yet considered. Instead transition is prescribed at the horizontal tailplane for each upper and lower side separatly. On the suction side it was located 0.5 mm downstream of the leading edge to prevent the appearance of the laminar separation bubble. Due to the startup of the turbulence model this was even ahead of the position detected during the experiments. On the pressure side it was set to about 80% of the local chord shortly ahead of the beginning pressure rise at the trailing edge. By using the Spalart-Allmaras turbulence model, investigations were carried out to permit a better understanding of the model’s behaviour in the wind tunnel and to compare them with the results obtained experimentally. Details of the numerical simulation are given in Grote et al. [3].
5 Results and Discussion The occurence of negative tail stall was visualised using the oil-flow technique for different tailplane settings at constant angle of attack. As the gap between the tailplane and the rear fuselage was sealed, the model was rotated around its angle of attack from attached flow conditions to the required setting. Therefore the remaining oilfilm always includes flow field information of the current setting and the preceding, which was passed through. To assess the impact of the local downwash component in the region of the horizontal tailplane, numerical simulations of the model without tailplane were performed. For an angle of attack of −2◦ the local downwash angle ε, which is a relation of the downward component of the velocity w and the local velocity component in freestream direction u, evaluated at the quarter-chord line of the tailplane varies from ε = −4◦ at the inner tailplane to ε = −1.5◦ at the outer tailplane. By a variation of the angle of attack, a spanwise nonlinear change in the effective angle of incidence at the tailplane of up to 1.5◦ is induced leading to a change in spanwise lift distribution and in angle of attack for maximum lift. Assuming natural transition of the boundary layer, the flow over the lower tailplane’s surface, the suction side, is fully attached for small negative stabiliser incidence angles. Caused by high suction peaks with steep pressure gradients close
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to the leading edge, a laminar separation bubble appears near the leading edge at medium stabiliser incidence angles. It can be localised as paint is captured within the bubble during the experiment. For increased angles the separation bubble moves upstream towards the leading edge, while its extension is reduced. Besides a boundarylayer separation is observed at the trailing edge of the outer tailplane expanding upstream as shown in Figures 3 (a) and (b). The stall occurs abruptly initiated by a local burst, that expands quickly upon the whole spanwise separation bubble. In the result fluid from the outer trailing edge flows into the separated region up to the leading edge and further on in the direction of minimum pressure along the separation line near the leading edge. At the inner wing, the fluid is redirected forming a
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Fig. 3. Oil-flow visualisation on the suction Fig. 4. Oil-flow visualisation on the suction side, natural transition, α = −2◦ side, fixed transition, α = −2◦
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vortex with its centre in the junction between fuselage and tailplane. The footprint of this vortex is visualised in Figure 3 (c). -7
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The process of flow separation changes, if transition is controlled. As shown by the pressure distribution at the leading edge in the enlarged detail of Figure 5, the appearence of the separation bubble as indicated by the small region of nearly constant pressure could be prevented. Figure 6 shows velocity profiles of the boundary layer measured by the Particle Image Velocimetry and confirms the modest perturbation by the strip of tape. In case of controlled transition the process of trailing-edge flow separation is comparable (Figures 4 (a) and (b)), but expands more continuously up to the outer tailplane’s leading edge, where the reversed flow is split. The flow directed inboard discharges into a vortex located at the centre of the inner tailplane, the outboard directed flow discharges into a vortex near the leading edge at the junction to the tailplane tip. The inner vortex is well visible in Figure 4 (c) and moves
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further inboard with increased flow separation. The loss of lift by this constantly developing separation is moderate and shows no effects of hysteresis. To prevent the appearance of laminar separation bubbles in the numerical simulations, the location of completed transition was prescribed slightly downstream the leading edge. Figure 7 shows the resulting pressure distribution and wall streamlines at it = −14◦ . A corresponding picture of the surface flow is indicated, except that the clossflow separation at the outer trailing edge is not that distinct in the numerical results. The local pressure distribution at the outer spanwise position is plotted in Figure 8. Both graphs show a comparable distribution, but a noticeable deviation. Concerning the numerical distribution, the applied Spalart-Allmaras turbulence model is well known to underpredict turbulent trailing-edge separation and the simulation of the main wing surface as an inviscid wall lead to higher local incidences at the tailplane. The later effect was estimated by 0.2◦ . On the other hand discrepancies in the experiment reduced the local incidence angle. As a result of the driving gear clearance at the tailplane shaft the resulting tailplane pitching moment reduced the incidence angle by 0.6◦ . This explanation also has to be used for the deviations in the force coefficients, either the local normal force in Figure 9 or the rear-end lift in Figure 10. Note, that the crossflow separation in the numerical simulation does not lead to a significant loss of lift at the outer tailplane compared to the experiment. Also the lift derivatives with respect to the elevator incidence between experiment and simulation match. 0.4
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The spatial behaviour of the tailplane flow at two conditions of partly separated flow at the stabiliser can be attained by the Particle Image Velocimetry. Measurements for two stabiliser settings in the section at y/sh = 0.75 illustrate the distribution of the absolute value of the mean velocity |V | = |u| + |w| in Figures 11 and 12. Along prescribed lines, distributions of the mean velocity and the component of the Reynolds shear stress tensor u w are extracted. The positions are selected to cover conditions between attached and highly separated flow. These results are plotted in Figures 13 and 14 respectively, whereas each symbol represents one vector in the
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PIV result. Because of bias errors due to gradient effects vectors very close to the surface are omitted. Further details of the experimental setup are given in Grote et al. [3]. For attached flow, distributions of the mean velocity and the Reynolds stress are in good agreement. The boundary-layer edge is matched by the simulation as well as the peak in the Reynolds stress distributions and its distance above the surface. 0.3
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Fig. 14. Velocity distribution and Reynolds shear stress, y/sh = 0.75, α = −2◦ , it = −14◦ open symbols: PIV measurement solid line: numerical simulation
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Further downstream in Figures 13 (b) and 14 (b), the discrepancies between the experiment and simulation become more significant. The measured velocity profiles denote signs of a beginning separation by noticeably reduced near-wall velocities. The simulation indicates attached flow by the form of the velocity profile and by predicting a smaller boundary-layer thickness. The apparent discontinuities in the profiles of the velocity and the Reynolds stress at a height of ξ ≈ 0.03 in the numerical simulation result from the change in grid-cell type from prismatic close to the surface to tetrahedral above. This change generates an unsteady velocity distribution and jumps in the distribution of the Reynolds stress.
6 Conclusions The current work aims to qualify the aerodynamic design of a generic research model, optimised for maximum Reynolds number at the horizontal tailplane. Using our new wind tunnel configuration twice as large Reynolds numbers are obtained compared to a conventional full-span model. The characteristic flow phenomena up to the limits of tail stall for a common transport aircraft tailplane could be achieved. It was shown that the occurance of tail stall depends on the transition process at the tailplane. If transitional phenomena cause the appearance of a laminar separation bubble near the leading edge, a characteristic sudden stall is initiated by its burst. A more continuous process appears if early transition prevents the formation of a bubble. In this case the turbulent trailing-edge separation expands continuously up to the leading edge, forming a pair of vorticies. The numerical simulations show already a good correspondance with the experiments. Flow separation from the trailing edge can be reproduced. Local pressure distributions at the outer tailplane show good agreement with the experiments, if systematic effects in the error analysis are taken into account. For attached flow even the boundary-layer velocity profiles and the Reynolds shear stress profiles correspond well to the PIV data. Future work in the validation of numerical models for tailplane stall prediction should adress improved turbulence and transition models along with accounting for the complete viscous flowfields of the experiment.
Acknowledgement This work was funded by the Deutsche Forschungsgemeinschaft (DFG). All NavierStokes computations have been performed on the IBM pSeries 690 of the HLRN [4].
References [1] T. Gerhold. Overview of the Hybrid RANS Code TAU. In: N. Kroll et al. (Ed.). ”MEGAFLOW - Numerical Flow Simulation for Aircraft Design”. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Vol. 89, Springer, 2005, pp. 81–92. [2] A. Grote and R. Radespiel. Investigation of Tailplane Stall for a Generic Transport Aircraft Configuration. In: H.-J. Rath et al. (Ed.). ”New Results in Numerical and Experimental Fluid Mechanics”. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Vol. 92, Springer, 2006, pp. 50–58.
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[3] A. Grote and R. Radespiel. ”Studies on Tailplane Stall for a Generic Transport Aircraft Configuration”. AIAA-Paper 2006-655, 44th AIAA Aerospace Sciences Meeting and Exhibit, 9-12 January 2006, Reno, Nevada, USA, 2006. [4] Norddeutscher Verbund f¨ur Hoch- und H¨ochstleistungsrechnen, www.hlrn.de [5] A. Krynytzky. Conventional wall corrections for closed and open test sections. In: B. Ewald (Ed.). ”Wind tunnel wall corrections”. AGARD-AG-336, RTO/NATO, 1998.
URANS and DES Simulation of Flow Around a Rectangular Cylinder Claudio Mannini1 , Ante Soda2 , Ralph Voß2 , and G¨unter Schewe2 1
Department of Civil Engineering, University of Florence, Via S. Marta 3, 50139 Florence, Italy
[email protected] 2 Institut f¨ur Aeroelastik, DLR G¨ottingen, Bunsenstraße 10, 37073 G¨ottingen, Germany
Summary This paper deals with numerical simulation of the flow around a 1:5 rectangular cylinder. The Unsteady Reynolds-Averaged Navier-Stokes (URANS) and DetachedEddy Simulation (DES) computational techniques are employed. In the process the influence of various modelling parameters, such as turbulence modelling, and flow parameters, such as Reynolds number and incidence angle, is investigated. Simulations with both stationary and harmonically oscillating body are performed. Validation of computed results with experimental data shows that the URANS-based computational approach is capable of predicting the basic unsteady flow phenomena in the considered cases. These results are further confirmed by the DES method, which provides information about the instantaneous flow variables and offers a deeper insight into the flow physics.
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Introduction
Simulation of the flow around bluff bodies is still a challenge due to the massive separation, flow re-attachment, shear-layer transition and high unsteadiness of the wake. In a previous paper by the same authors [1], an overview of the computations concerning a square cylinder, a 1:5 rectangular cylinder and a realistic bridge section, has been presented. The current paper concentrates on the rectangular cylinder with a chord-to-thickness ratio B/H = 5.0 in the incompressible flow regime (M a = 0.1). This geometry has been used in the experimental investigation by Schewe [2] in the high-pressure wind tunnel at the DLR-G¨ottingen in Germany. This is an interesting study case because the flow, which separates from the surface due to the sharp geometry edges, is supposed to re-attach on the side surfaces of the cylinder. An unsteady separation bubble is formed at the surface, whose length is Reynolds-number dependent [3]. In addition, this profile is often considered to be a benchmark case for studies dealing with bridge aerodynamics and aeroelasticity. Simulation of the flow in the case of stationary and harmonically oscillating body are compared with experiments. Both 2-D Unsteady Reynolds-Averaged Navier-Stokes C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 36 –43 , 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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(URANS) and 3-D Detached-Eddy Simulation (DES) approaches are employed to compute the turbulent wake past the cylinder.
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Numerical Tool
The flow is simulated using the finite-volume unstructured solver DLR-Tau code [4]. It solves the compressible governing equations with the second-order accuracy in both space and time. The capability of the compressible TAU code to simulate low Mach number flows, has been demonstrated in [5]. In the current computations both the diffusive and the convective terms in the governing equations are modelled with the central differencing scheme employing the scalar dissipation. For the time-accurate simulations the dual-time-stepping technique is used, together with the explicit residual smoothing and a multi-grid technique based on the agglomeration of dual-grid volumes. All computations have been performed on the computer cluster at the DLR Institute of Aeroelasticity in G¨ottingen. In the URANS approach the turbulent Reynolds stresses, resulting from the averaging process, have to be modelled through the additional equations, the so-called turbulence modelling. The current investigation is performed employing two turbulence models. The first one is a well-known model of Spalart and Allmaras (S-A) [6], which solves one additional equation solved directly for the eddy viscosity. The second model, the Linearized Explicit Algebraic (LEA) model [7], is based on the solution of two additional equations, one for the turbulent kinetic energy and the other for the turbulent frequency. Two major drawbacks of URANS approach are the underlying assumption of turbulent kinetic energy production-to-destruction equilibrium and the assumption of isotropy of turbulent normal stresses (true only for pure shear dominated flows, for example flat plate). This often results in the over-prediction of eddy-viscosity production (too strong turbulent dissipation). Despite these limits the URANS approach is often appealing because relatively cheap computations on 2-D grids are possible. A more sophisticated method for the simulation of problems with massively separated flow is Detached-Eddy Simulation (DES), a recent technique introduced by Spalart in 1997 [8]. This technique combines the URANS approach, employed in the regions of attached flow, and the Large-Eddy Simulation (LES), in the regions of massively separated flow, where URANS is not sufficiently accurate. In the current DES approach the 1-equation S-A model is adopted (SA-DES), both as the RANS turbulence model in the boundary layer and as the sub-grid-scale (SGS) model in the separated regions, where the code acts in the LES-like mode. In the separated regions the S-A model acts similarly to the Smagorinsky’s SGS model in LES applications [9]. The only additional parameter with respect to the pure URANS simulation is the constant CDES , which defines the criterion to switch from URANS to LES, and consequently the percentage of the domain resolved with the two methods. For the current research the value CDES = 0.25 is chosen, instead of the recommended value 0.65 [10, 11], because of the unstructured domain discretization and the second-order accurate spatial discretization of the fluxes. A smaller value of
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DES constant should increase stability and enable the development of smaller eddies without damping them out [11, 12]. A wide range of DES computations using structured grids are reported in [11], whereas the first application of this technique with a second-order-accuracy structured code on hybrid grids can be found in [13]. In the current study the hybrid meshes are employed, characterized by the structured grid layers around the solid body (body-aligned quadrilateral cells), and the unstructured triangular elements in the remaining part of the domain. In order to ensure the grid-independent 2-D URANS flow solutions, the computational meshes with different levels of refinement are tested. The medium grid is presented in Fig. 1. For all computations the non-reflecting farfield boundary conditions are assumed at a distance 100 chord lengths away from the solid body, with the viscous wall boundary conditions imposed at the body contour. The mesh for the 3-D DES computation is obtained by extending the 2-D medium grid for one chord length into the spanwise direction. A ratio ∆y/∆x = 1.5, between the cell dimension in the span-wise and chord-wise directions, is chosen according to the DES grid convergence study in[12]. This gives 64 cells in the span-wise direction, and total of 1.3 million nodes and 2.0 million cells. Periodic boundary conditions are imposed at the lateral planes.
Fig. 1. Details of the 2-D RANS medium grid. Clearly visible are the wake refinement (left) and the hybrid grid structure (right). Grid contains 20,000 nodes and 32,000 cells.
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Results
3.1 Stationary Body The flow is simulated at different Reynolds numbers and for two angles of attack (α = 0◦ and 4◦ ), while keeping the rectangular cylinder stationary. The URANS approach in combination with the S-A turbulence model gives practically the steady solution for α = 0◦ , although the very small lift fluctuations allows the estimation of the Strouhal number. In the case of α = 4◦ the wake is unsteady. Comparing the URANS S-A results with the experiment [2], the Strouhal number and mean drag coefficient are overall correctly predicted, whereas the computed mean lift is close to experiments only at low Reynolds number (Fig. 2). In particular, the numerical simulation is not able to capture the significant increase in the lift coefficient due to Reynolds number effects. The 2-equation LEA turbulence model, on the other
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hand, predicts the unsteady flow also at zero-degree angle of attack and captures qualitatively the lift increase. It is worth noting that with the S-A model grid convergence is achieved with the medium grid (Fig. 1), whereas the LEA model gives a grid-independent solution only if the fine grid is used. It has been shown in [1] that pressure coefficient distributions computed with the URANS approach, especially with the LEA model, are in good agreement with experimental results for the rectangular cylinder at zero-angle of attack. In order to try to overcome some of the limits shown by the URANS approach, a 3-D DES computation is performed for the rectangular cylinder at zero-angle of attack and Reynolds number Re = U B/ν = 1.32 × 105 , where B is the chord length, and U the free-stream velocity. A non-dimensional time-step size ∆s = U ∆t/B = 3.4 × 10−4 is chosen together with 75 inner iterations. The expected Strouhal period of wake oscillation is resolved with more than 5000 timesteps, which satisfies the recommendations of Spalart [10] concerning the relationship between temporal and spatial discretization. In Fig. 2 the DES results are compared to the URANS and experimental results. Fig. 3 shows the DES computed lift, drag and moment coefficient time histories, as well as the power spectral density for the moment coefficient. 0.35
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It can be observed that, despite the DES computation being based on the S-A turbulence model, the resulting flow field is unsteady. In addition, the calculated time-signals of aerodynamic forces are not periodic, as in the case of URANS simulations, since finer flow features can be resolved with the DES approach. This is clearly shown by the instantaneous velocity flow field presented in Fig. 4 (left). As a result of that, there are several peaks appearing in the spectra of force coefficients. Nevertheless the dominant peak, corresponding to the Strouhal number of 0.11, agrees well with the experimental results [2]. The vorticity iso-surface, Fig. 4 (right), demonstrates that the DES simulation is able to capture the threedimensional development of the wake. Although these DES results are only preliminary, with longer time histories and additional analysis required for more detailed conclusions, the mean drag prediction in Fig. 2 seems to agree with experiments better than those obtained with both URANS computations.
Fig. 4. Results of the DES computation: snapshot of the velocity x-component flow field (left) and the iso-vorticity surface (right)
3.2 Oscillating Body The URANS simulations of the flow around the rectangular cylinder harmonically oscillating in pure heaving and pure pitching motion are performed by using the 1-equation S-A turbulence model. The body oscillates around a zero-degree angleof-attack mean position with an amplitude of oscillation of ±2◦ (to be interpreted ˙ , with h being the vertical for the heaving motion as an apparent angle of attack h/U displacement). In all computations the 2-D fine grid is used. Three reduced frequencies of oscillation K = ωB/U = 2.51, 0.84 and 0.42 are tested, where ω is the circular frequency of oscillation. The motion-dependent lift and moment acting on the cylinder can be expressed according to the well known Scanlan’s formulation [14]: ˙ h(t) B α(t) ˙ h(t) ] + K 2 H3∗ α(t) + K 2 H4∗ + KH2∗ B U U ˙ h(t) B α(t) ˙ h(t) ] + K 2 A∗3 α(t) + K 2 A∗4 + KA∗2 M (t, K) = qB 2 [KA∗1 B U U L(t, K) = qB[KH1∗
(1) (2)
URANS and DES Simulation of Flow Around a Rectangular Cylinder
41
where q is the free-stream kinetic pressure, while α and α˙ are angular displacement and angular velocity respectively. Parameters Hi∗ , A∗i are the flutter derivatives, which are supposed to be functions of profile geometry, reduced frequency of oscillation, mean angle of attack and, to a certain extent, Reynolds number [2]. 5
30
Exp. (Matsumoto, Ref. [15]) Exp. (Righi, Ref. [16]) URANS S−A
0
20
−5
10
H*
H*
2
0
1
−10 −15
−10
−20
−20
−25
−30
−30 0
5
10
15
20
−40 0
25
5
10
U
12
−20
8 4
0
H*
3
H*
16
−40
−60
−100 0
20
25
15
20
25
R
20
−80
15 U
R
4
0
Exp. (Matsumoto, Ref. [15]) Exp. (Righi, Ref. [16]) URANS S−A 5
10
15 UR
−4
20
25
−8 0
5
10 UR
Fig. 5. Comparison between measured and calculated flutter derivatives. UR = 2π/K is the reduced wind speed.
In Fig. 5 the numerical results for Re = 2.0 × 105 are compared with the experimental data by Matsumoto [15] and Righi [16]. Flutter derivatives were measured with a 1-DoF forced-vibration system and Reynolds number up to 2.0 × 105 in the first case, and with a 2-DoF free-vibration technique and Reynolds number up to 1.15 × 105 in the second case. The two experimental results agree only for the socalled “direct” flutter derivatives H1∗ , A∗2 and A∗3 . The functions H4∗ and A∗4 are difficult to be measured but their influence is small and they can be usually neglected. The numerical results are in good agreement with those reported in [15], for the high and the medium reduced frequency of oscillation. Conversely, larger discrepancies are evident for the lowest reduced frequency (K = 0.42 ⇒ UR = 2π/K = 15.0) and this could be due to the strong interference with vortex shedding, which was negligible at higher reduced frequencies, as clearly shown in Fig. 6. However, the lift force component in quadrature with the heaving motion (H1∗ ) is slightly overestimated also for K = 0.84 (⇒ UR = 7.5).
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0.5
0.5
K = 0.84
0.2
0.2
0.1
0.1
[−]
0.3
L
0 −0.1
0 −0.1
−0.2
−0.2
−0.3
−0.3
−0.4 −0.5
K = 0.42
0.4
0.3
C
C
L
[−]
0.4
−0.4
−2
−1
0
ωh/U [°]
1
2
−0.5
−2
−1
0
ωh/U [°]
1
2
Fig. 6. Numerically simulated lift coefficient vs. apparent angle of attack in the case of heaving harmonic motion for two different reduced frequencies of oscillation
4
Conclusions
The results of the computations performed with the 1:5 rectangular cylinder show that with the URANS approach the main global aerodynamic quantities, such as Strouhal number, mean drag and flutter derivatives, are predicted with acceptable accuracy. The great advantage of the URANS simulation is a relatively moderate computational cost but the price paid is the loss of instantaneous fluctuations of flow variables and the inability to fully resolve the zones of massive flow separation. The ongoing research shows the advantages of the more expensive DES computational technique, which is needed in order to expose the limits of the URANS approach and to better understand the unsteady physical processes governing the flow development around bluff bodies.
References [1] C. Mannini, A. Soda, R. Voß and G. Schewe. ”URANS and DES simulation of flow around bridge sections”. In Proc. 9th Italian National Conference on Wind Engineering IN-VENTO, Pescara, Italy, 2006. [2] G. Schewe. ”Influence of the Reynolds-number on flow-induced vibrations of generic bridge sections”. In Proc. Int. Conf. on Bridges, Dubrovnik, Croatia, 2006, pp. 351-358. [3] A. Okajima. ”Strouhal numbers of rectangular cylinders”. J. Fluid Mech. 123, 1982, pp. 379-398. [4] T. Gerhold, M. Galle, O. Friedrich and J. Evans. ”Calculation of complex threedimensional configurations employing the DLR-Tau-Code”. AIAA Paper 97-0167, 1997. [5] W. Haase, B. Aupoix, U. Bunge and D. Schwamborn. ”Flomania - a European Initiative on Flow Physics Modelling”. Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), Vol. 94, Springer, 2006. [6] P.R. Spalart and S.R. Allmaras. ”A one-equation turbulence model for aerodynamic flows”. AIAA Paper 92-0439, 1992.
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[7] T. Rung, H. L¨ubcke, L. Xue, F. Thiele and S. Fu. ”Assessment of Explicit Algebraic Stress Models in transonic flows”. In Proc. 4th International Symposium on Engineering Turbulence Modelling and Measurements, Ajaccio, France, Elsevier, 1999, pp. 659-668. [8] P.R. Spalart, W.-H. Jou, M.Kh. Strelets and S.R. Allmaras. ”Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach”. 1st AFOSR Int. Conf. on DNS/LES, Ruston, LA, 1997. In Advances in DNS/LES, C. Liu & Z. Liu Eds., Greyden Press, Columbus, OH. [9] M.L. Shur, P.R. Spalart, M.Kh. Strelets and A. Travin. ”Detached-eddy simulation of an airfoil at high angle of attack”. Engineering Turbulence Modelling and Experiments 4, Rodi and Laurence Eds., 1999, pp. 669-678. [10] P.R. Spalart. ”Young-person’s guide to Detached-Eddy Simulation grids”. NASA CR2001-211032, 2001. [11] M.Kh. Strelets. ”Detached Eddy Simulation of massively separated flows”. AIAA paper 2001-0879, 2001. [12] A. Soda. ”Numerical investigation of unsteady transonic shock/boundary-layer interaction for aeronautical applications”. Ph.D. Thesis, RWTH Aachen, Germany, 2006. [13] R.P. Hansen and J.R. Forsythe. ”Large and Detached Eddy Simulations of a circular cylinder using unstructured grids”. AIAA Paper 2003-0775, 2003. [14] E. Simiu and R.H. Scanlan. ”Wind effects on structures”. Third edition. John Wiley & Sons, Inc., New York, 1996. [15] M. Matsumoto. ”Aerodynamic damping of prisms”. J. Wind Eng. Ind. Aerodyn. 59, 1996, pp. 159-175. [16] M. Righi. ”Aeroelastic stability of long span suspended bridges: flutter mechanism on rectangular cylinders in smooth and turbulent flow”. Ph.D. Thesis, University of Florence, Italy, 2003.
Active Manipulation of a Rectangular Wing Vortex Wake with Oscillating Ailerons and Winglet-Integrated Rudders Ralf H¨ornschemeyer, Christoph Rixen, Sebastian Kauertz, G¨unther Neuwerth, and Rolf Henke Institute of Aerospace Engineering, RWTH Aachen University, W¨ullnerstraße 7, 52062 Aachen, Germany
[email protected] http://www.ilr.rwth-aachen.de
Summary This paper presents results of experimental investigations regarding the vortex wake of a rectangular wing with winglets in a water towing tank. The model comprises ailerons and additional rudders, which are integrated into the winglets. The ailerons and the rudders are able to oscillate around a static deflection to excite inherent short-wavelength instabilities in the vortex system. The Particle Image Velocimetry method is used to investigate the vortex wake up to about 40 spans behind the model. The results show that, depending on the preselected aileron and rudder deflections, an oscillation of correctly chosen frequency leads to a faster decay of the vortex wake in comparison to the statical case.
1
Introduction
Flight tests in the 1960s show that the main hazard in a vortex wake for an aircraft with a wingspan less than half the span of the vortex-generating plane is the induced rolling moment that takes effect upon flying into the vortex core [1]. This hazard is much higher the more concentrated and stronger the vortices of the wake are. However, if the vorticity is not concentrated in a small area of the core of the wing tip vortex, but extensively spread over the area of the wingspan, the threat posed to following aircraft is considerably reduced. This reduction can contribute to decrease the separation distances during landing and take off at airports. In order to achieve this goal, instabilities inherent in a system of vortices can be utilised. Such instabilities in a vortex system were first detected by Crow for a single pair of counter-rotating vortices [4]. Based on the work of Crow, Crouch and Fabre described the stability of two pairs of vortices [2, 5, 6]. Instabilities in a vortex system lead to a faster decay of the whole wake. An additional excitation of these inherent instabilities is able to further accelerate the decay. Crouch proposes the excitation of instabilities by the oscillation of control surfaces for a practical application [3]. This idea is pursued in this work which resumes investigations carried out by Kauertz. Kauertz examined the vortex wake of a rectangular C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 44–5 1, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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wing equipped with swept winglets. An inherently unstable vortex system was produced by the deflection of ailerons. The instabilities of this system were excited by the oscillation of winglet-integrated rudders [8]. However, excitation by means of winglet rudders alone leads to marginal success. Furthermore, lift fluctuations result from the oscillation, which would lead to problems, at the latest in a realistic application. Earlier investigations done by Haverkamp show good results using oscillating ailerons [7]. For the work presented here, the vortex wake of the rectangular wing used by Kauertz was investigated. In this study theoretical stability analyses deliver frequencies for an excitation of instabilities with the ailerons, which favour a fast decay of the vortex system. In this scenario the winglet-integrated rudders oscillate in opposite phase with a shift of 180◦ to the ailerons, serving to minimize lift fluctuations.
2 2.1
Experimental Setup
Towing Tank
The experiments have been conducted in the water towing tank of the Institute of Aerospace Engineering. The tank has a length of 9.0 m, a width of 1.5 m and a height of 1.0 m. The model has been suspended from a towing rig, which drives on rails mechanically decoupled from the test section. All experiments were carried out with a speed of 1.0 m/s. In the acceleration phase the rig covers a distance of approximately 1.5 m. The deceleration at the end of the tank is made within a distance of around 0.5 m. The measurement area in the centre of the towing tank allows the observation of the evolution of the vortex wake up to several spans behind the model without interferences caused by the starting or deceleration procedure. The sides and the bottom of the tank are paned and thus well suited for optical measurement methods. In order to minimize surface waves the water surface is covered with plastic mats. 2.2
Model
The used model consists of a straight rectangular wing with swept winglets. The rectangular section has a chord length of c = 0.085 m. The total span, including winglets, is b = 0.58 m and has a NACA 0014 profile. The winglets have a dihedral of ν = 45◦ and are provided with rudders. Furthermore, ailerons are attached to the rectangular part. The dimensions of the model are shown in Fig. 1. All control surfaces can be set to an arbitrary position in which a positive angle denotes a downward deflection. The winglet rudders and ailerons can be deflected periodically around the static position by an electric motor. The oscillation amplitude of the winglet rudders amounts to ±10◦ and those of the ailerons to ±5◦ . The frequency of oscillation is continuously adjustable up to 5 Hz. A sting, which is mounted to the centre of the wing, contains the linkage running for the motion and allows mounting the model to the towing rig. The sting has a length of 0.42 m and a NACA 0022 profile. Both model and sting are manufactured from solid aluminium.
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Fig. 1. Model with winglets and rudders
2.3
Particle Image Velocimetry
Particle Image Velocimetry (PIV) combines the advantages of a non-intrusive measuring procedure with a simultaneous planar resolution of the flow field. With this method, a light sheet is illuminated by a double pulsed laser. The applied 2C-PIV procedure is based on the principle of identifying the position of particles between two laser pulses, and determining the velocity components of the particles. In such a way the inference of the flow structure at this position is possible, if the particles have a good following behaviour. Therefore, it is necessary that the particles have nearly the same density as the flow medium. For the investigations in the water towing tank polyamide particles with a mean diameter of 55 µm and a density of ρ = 1.016 g/cm3 are used. The light sheet is produced by a Quantel Twins Ultra 120 double head pulsed Nd:YAG laser which has a maximum pulse energy of 120 mJ. The thickness of the light sheet is approximately 4 mm in the plane of measurement and becomes wider as it propagates vertically and is perpendicular to the towing direction. At one end of the tank a 1.4 MPixel CCD camera was positioned and recorded a field of 550 mm × 400 mm in the measurement area. The distance between camera and plane of measurement amounts to about 4 m. Camera and laser are operating with a frame rate of 4 Hz. So with a towing speed of 1 m/s and 100 recorded frames the vortex wake can be examined up to over 40 spans behind the model.
3
Theoretical Background
Within the conducted investigations the vortex wake is formed by a system of three vortex pairs directly behind the model (cp. Fig. 2). For the theoretical stability analysis of this system, the vortices are modeled as filaments. The velocity components induced by the vortex filaments are formulated according to the law of Biot-Savart. The equations take into account the interaction of the vortex filaments and the perturbation velocities, produced by the oscillation of the rudders. The formulation of the equations of motion mainly follows the derivations given by Crouch [2] and Fabre [5]. For the theoretical analysis of the evolution of the vortex system a time integration is used, which is described by Kauertz in [8].
Active Manipulation of a Rectangular Wing Vortex Wake
47
Fig. 2. Six-vortex-system
The assessment of the individual configurations is based on the determination of the rolling moment induced by the velocity field on a virtual rectangular following wing with 20% of the span of the generator wing (b/bF = 0.2). For the determination of the induced rolling moments the strip method is used. The induced rolling moment coefficient is computed as cRM,ind =
2 b2F
bF /2
−bF /2
cLα arctan
vz u∞
y dy .
(1)
Here bF corresponds to the span of the following wing and vz to the vertical component in the velocity field. The lift gradient is set to cLα = 2π.
4
Results
Using the velocity distributions measured with PIV, the maximum rolling moment induced to an imaginary following wing is calculated and plotted against the distance between the two wings. Figs. 3, 4 and 5 exemplify the evolution for an aileron deflection of δ = 20◦ and an angle of attack of α = 4◦ . The graphs represent progressions of the induced rolling moments on a following wing, which were each obtained from five measurements of the same configuration. The progressions of the rolling moments representing similar configurations are merged in each diagram and referenced to the CL -value of the corresponding experiment with static rudder deflection. The progression of the induced rolling moments at a stationary deflection of the rudders, i.e. without oscillation, is plotted as a black dashed line as a reference. The data plotted in red and marked with a circle shows the progression of the rolling moment with a stationary aileron deflection and an oscillating winglet rudder moving around the respective preset deflection. The green curve marked with triangles represents the progression of the rolling moment, with both winglet rudder and aileron oscillating. Winglet rudder and aileron oscillate phase-shifted around the respective preset deflection. The curves for oscillating winglet rudders and ailerons in Figs. 3, 4 and 5 start on a distinctively lower level compared to the reference configuration with stationary rudder deflection. Furthermore, over the range of the first five spans the gradient
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Fig. 3. Maximum induced rolling moment on a following wing for configuration χ = -10◦, δ = 20◦ , α = 4◦
Fig. 4. Maximum induced rolling moment on a following wing for configuration χ = 0◦ , δ = 20◦ , α = 4◦
Fig. 5. Maximum induced rolling moment on a following wing for configuration χ = 10◦ , δ = 20◦ , α = 4◦
Active Manipulation of a Rectangular Wing Vortex Wake
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of the graphs with oscillated ailerons and winglet rudders decreases faster than for the reference configuration. These advantages do not occur when only the winglet rudders are oscillating. In a distance of about 10 to 15 wingspans the gradient for the graphs with both rudders oscillating becomes smaller compared to the static reference. However, for the correctly chosen configuration of both rudders oscillating the induced rolling moment is less than the reference within the whole range of measurement. This results in an advantage up to a distance of x/b = 40.
Fig. 6. Vorticity distribution due to static winglet rudder deflection and static aileron deflection; χ = 0◦ , δ = 20◦ , α = 4◦
Fig . 7. Vorticity distribution due to oscillating winglet rudder and static aileron deflection; χ = 0◦ , δ = 20◦ , α = 4◦
Fig. 8. Vorticity distribution due to oscillating winglet rudder and oscillating aileron; χ = 0◦ , δ = 20◦ , α = 4◦
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Figs. 6, 7 and 8 represent the instantaneous distribution of the vorticity attained from the analysis of the left wing seen from behind for the configuration with a winglet rudder deflection of χ = 0◦ , an aileron deflection of δ = 20◦ , and an angle of attack of α = 4◦ . The figures on the left show the distributions in a distance of 0.1 wingspans while the figures on the right show the distributions in a distance of 4.8 spans. The vortices at the tip of the winglet (outer winglet vortex), at the bend (inner winglet vortex) and at the aileron are clearly recognizable 0.1 spans behind the model and are marked in Fig. 6 (cp. to Fig. 2). The high maximum value of the induced rolling moment for the static case in this plane (cp. Fig. 4) is caused by the distinctive vortex at the bend. The drop of the induced rolling moment is caused by the clockwise rotating vortex pair at winglet tip and the bend. At 4.8 wingspans behind the model these vortices are virtually side by side (see Fig. 6). A following wing, positioned in the core of the stronger vortex, experiences a counter rotating moment caused by the nearby second vortex. A short distance later, the two clockwise rotating vortices merge, marked in the graph of induced rolling moments by a rise in the curve (Fig. 4). An oscillation of the winglet rudder causes the location of maximum vorticity to shift back and forth between the vortices at the winglet tip and at the bend. Fig. 7 shows the situation where the winglet tip vortex represents the strongest concentration of vorticity. At 4.8 wingspans behind the model the winglet tip vortex and the vortex at the bend are still separated. In time they merge and form a stable vortex with highly concentrated vorticity. A case not analysed until now is the oscillation of winglet rudders and ailerons in counter-phase. The induced rolling moment shown in Figs. 3, 4 and 5 already declines within a few wingspans distance to a lower level compared to the reference case. The vorticity distribution (Fig. 8) represents an expansion of the wing tip vortex already 0.1 wingspans behind the model, meaning that a clear distinction between the vortices at the winglet tip and the bend can no longer be made. This effect on the vorticity distribution can be seen in all instantaneous measurements involving an oscillation of winglet rudders and ailerons. The additional oscillation of the ailerons hinders especially the vortex at the bend to a completely development like it can be seen without oscillation. This causes a decrease of the vorticity concentration in this area. Due to this at 4.8 wingspans behind the model the induced rolling moment of this configuration has reached a level that the other investigated configurations first reach 20 spans behind the wing. The vorticity in this plane is more extensively spread and considerably concentrated on a lower level than can be found with the reference configurations (cp. Fig. 6, 7 and 8 at x/b = 4.8). For the tested model with oscillating winglet rudders and ailerons a noteworthy reduction of the induced rolling moment can be obtained in the near field for the right configurations. However, only little advantage is apparent in a distance of 40 wingspans.
5
Conclusion
The results of the experiments demonstrate that by simultaneous oscillation of ailerons and of winglet-integrated rudders a significant reduction of the rolling moment
Active Manipulation of a Rectangular Wing Vortex Wake
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induced to a following wing can be achieved in the near field. At distances further than 40 wingspans behind the model evidence of the wake behaviour is not yet available. Future experiments planned to be conducted in a larger towing tank shall give evidence of whether there are influences of the size of the tank on the results. Moreover it is possible in a larger tank to investigate the vortex wake up to the far field (x/b ≈ 100).
Acknowledgements This work has been performed with funding by the Deutsche Forschungsgemeinschaft in the Collaborative Research Centre SFB 401 ”Flow Modulation and FluidStructure Interaction at Airplane Wings” of the RWTH Aachen University,Germany. The results are based on and compared to results gained within the project Innovative Hochauftriebskonfigurationen in the framework of the Luftfahrtforschungsprogramm 3 which was funded by the German government.
References [1] W.H. Andrews, G.H. Robinson, R.R. Larson: ”Aircraft Response to the Wing Trailing Vortices Generated by Large Jet Transports”. NASA Aircraft Safety and Operating Problems, Vol. I, NASA SP-270, May 1971, p. 115-26. [2] J.D. Crouch: ”Instability and transient growth for two trailing-vortex pairs”. J. Fluid. Mech., Vol. 350, 1997, pp. 311-330. [3] J.D. Crouch, G.D. Miller, P.R. Spalart: ”An Active-Control System for Breakup of Airplane Trailing Vortices”. AIAA Journal, Vol. 39, No. 12, 2001, pp. 2374-2381. [4] S. C. Crow: ”Stability Theory for a Pair of Trailing Vortices”. AIAA Journal, Vol. 8 No.12, 1970, pp. 2172-2179. [5] D. Fabre, L. Jacquin: ”Stability of a four-vortex aircraft wake”. Physics of Fluids, Vol. 12, No. 10, 2000, pp. 2438-2443. [6] D. Fabre, L. Jacquin, A. Loof: ”Optimal Perturbations in a four-vortex aircraft wake in counter-rotating configuration”. J. Fluid Mech., Vol. 451, 2002, pp. 319-328. [7] S. Haverkamp, G. Neuwerth, D. Jacob: ”Active and passive wake vortex mitigation using control surfaces”. Aerospace Science and Technology, Vol. 9, No. 1, 2005, pp. 5-18. [8] S. Kauertz, G. Neuwerth: ”Excitation of Instabilities in the Wake of an Airfoil with Winglets”. AIAA-2006-3470, Applied Aerodynamics Conference, June 5-8, 2006.
A New Actuator Disk Model for the TAU Code and Application to a Sailplaine with a Folding Engine Axel Raichle, Stefan Melber-Wilkending, and Jan Himisch German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology Lilienthalplatz 7, D-38108 Braunschweig, Germany
Summary In this paper a new actuator disk is presented which is consistent with the classical model of momentum theory of a zero thickness surface which imposes a discontinous pressure jump to the flow. The topology of the grid maintains fluxes conservative through the disk. The load of the disk is calculated by time averaging the sectional load of a rotating blade. The sectional load is calculated with the blade element theory for a propeller of a sailplaine with a folding engine and the actuator disk is applied to investigate the effect of the jet to the elevator.
1 Introduction Today propellers are still a very reliable and effective mean of propulsion for many aircrafts in the low and middle speed range. The jet of a propeller affects the flow around an aircraft and is of great importance to the stability and aerodynamic capability of the design. A numerical simulation of a rotating propeller requires an instationary calculation. This is often too time consuming for the calculation of the flow around a complete aircraft configuration. In addition the discretization of the complete blade geometry increases the grid size especially for viscous calculations. For many cases the flow around the blade is not of interest and the jet behind the propeller can be considered as stationary. This motivates to replace the rotating blades with a stationary and geometrical simpler model propeller.
2
The Actuator Disk Model
A propeller generates propulsion due to aerodynamic forces on the surface of the rotating blade. As a consequence reaction forces act on the fluid which increase the momentum and the energy of the flow. The model propeller must apply equivalent stationary forces to the fluid. To define these forces first the blade geometry has to be simplified. Because the flow around the blade is not of interest for the jet and the extension in circumferential direction is small compared to the extension in radial direction the blade is replaced with a line. Figure 1a shows a six-bladed propeller with one idealized blade. The surface force distribution of the blade is replaced with a line force distribution by C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 52–61, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
A New Actuator Disk Model for the TAU Code
dr
fs
fl
dϕ
ϕ
r ω (a)
(b)
l
Ft
fn
ρvn β
S
vt i vte
fr
αe
αi
vn ve vt ω r c
(c)
(d)
(e)
(f)
F
F
v
(g)
Fig. 1. Graphics of the aktuator disk model
v
(h)
d
ft
53
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A. Raichle, S. Melber-Wilkending, and J. Himisch
partial integration in circumferential direction. For propellers this distribution of sectional loads can be calculated with the blade element theory or it is determined else and read in as for rotors with deforming blades. The rotating lines define a disk. For the disk a stationary surface force distribution can be calculated which applies the same momentum to the flow as the rotating lines. This disk is denominated as actuator disk. From the point of view of the disk the rotating line force distribution can be interpreted as a synchronized switching of values of points in the same way as text is moving on LED displays. Figure 1b shows a line force distribution rotating with the angular velocity ω. The time of circulation for one revolution is T = 2π/ω. At the point (r, φ) the line force fr is active on a line element dr for the time dt. During this time the line element dr rotates with the angle dφ = ωdt and sweeps over the surface dS = rdrdφ. The line force fr applies the momentum dI = fr drdt to the surface dS. The application of momentum is periodic with T . For the time n dt the n blades apply the momentum n dI to the surface dS and for the time T − n dt nothing happens. An equivalent steady surface force fs which applies the same momentum during the time T is defined as fs dS = n
dI drdt drdφ rdrdφ n = fr n = fr n = fr n = fr dS. T T ωT 2πr 2πr
(1)
To calculate the integral force Fs of a surface cell the surface force fs is assumed to be constant and the integration reduces to the geometry integral n 1 dS. (2) Fs = fr 2π r S
The polygonial cells on the surface of the actuator disk are subdivided into triangles and the integral is calculated with a third order quadrature formula in a preprocessing step.
3 The Blade Element Theory The line force fr at a point of the actuator disk is the vector sum of the sectional lift l and the sectional drag d of the blade when it touches this point l=
ρ 2 v c Cl (α), 2
d=
ρ 2 v c Cd (α). 2
(3)
The sectional force depends on the geometry, the aerodynamic coefficients and the state of the flow. The geometry of the blade is mapped with a radial distribution of the blade twist β and the blade depth c. The values are read in from a table for nri sections at the radial positions ri and linearly interpolated in between. The aerodynamic coefficient functions Cl (α) and Cd (α) are approximated with polynoms Cl (α) =
dPl k=0
Clk αk ,
Cd (α) =
dPd
Cdk αk .
(4)
k=0
The radial distribution of the polynomial coefficients Cli and Cdi is also read in from a table for nrj sections at the radial positions rj and linearly interpolated in between. In
A New Actuator Disk Model for the TAU Code
55
general dPl = 1, dPd = 2 and nri > nrj . According to figure 1d the effective angle of attack of the blade is calculated αe = β − αi .
(5)
In the disk coordinate system the induced angle of attack and the effective velocity relative to the blade are
−vn αi = arctan (6) , ve = (ωr + vt )2 + vn2 . ωr + vt The normal component vn and the tangential component vt of the velocity v and the density ρ are evaluated from the state of the flow at this point. As reaction force to the fluid the negative sectional lift and drag are transformed into a normal and tangential force on the disk fn = (−l) cos(αi ) − (−d) sin(αi ), ft = (−l) sin(αi ) + (−d) cos(αi ).
(7)
4 The Grid Topology The actuator disk is implemented in the DLR hybrid RANS solver TAU. This code uses a finite volume scheme with a compact metric. The control volumes form a dual grid which is shown in figure 1f. If there is no load on the disk the points act like inner grid points and a dual grid cell should exist around each point. From the classical momentum theory the application of a surface force causes a jump of the state through the disk. To map a discontinuity the actuator disk needs two points at the same geometric position. To maintain the grid topology all points must be surface points because no edge connection can exist between two points at the same geometric position. These requirements are fulfilled with a construction as inner periodic boundary with the transformation identity. Figure 1e shows the topology of the grid. Due to the periodicity there exist pairs of points in the inner of the disk which are topological different but geometrical identical. The border points are fix points similar like axis points of a rotatory periodic boundary. Figure 1f shows a section through the dual grid of the disk. The periodic boundary cells complement one another to an inner dual grid cell with two points in the inner of the disk and with one point at the border line. The border cells are treated like ordinary grid cells and a surface force is set only set in the inner of the disk. Without restriction it is possible to construct disks with inner holes.
5 The Numerical Scheme A rotating blade causes upside low pressure and downside high pressure which sucts air upstream and blows air downstream into the direction of the average normal direction of the blade. The primary effect is an acceleration in axial direction which causes the jet and and the secondary effect is an acceleration in azimutal direction which causes the
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swirl. By neglecting the blade geometry and time averaging, the surface of the blade is replaced with the surface of the disk and the original orientation in azimutal direction is lost. Due to the zero thickness the rotating pressure field is idealized with a stationary pressure jump. Because pressure forces act only normal to a surface swirl cannot be generated with a pressure jump. Therefore the swirl is modeled with a jump of the tangential velocity through the disk. From the kinematics a jump makes only sense in azimutal direction which is closed in itself. Radial components of the tangential force should be omitted. The numerical scheme is splitted into a treatment of the normal component and the tangential component of the surface force. The state at the inflow is updated by calculating the fluxes and stresses over the surface of the control volume in figure 1 marked with a dashed line. The faces upside and downside of the disk are identical with opposite orientation. The difference of the fluxes are prescribed by the actuator disk. The mass flux is conserved, the difference of the momentum flux is the normal component Fn of the surface force Fs and the difference of the energy flux is the power Fn vni which is the inner product of the force Fn with the normal velocity vni at the inflow. The control volume can be replaced with the dual grid cell around this point if these contributions are split off and added as external contributions to the respective governing equations. The state at the outflow has to be calculated directly from the state at the inflow. If a flow is simulated and a force acts on the fluid in flow direction the fluid accelerates and the pressure lowers when approaching to the point the force is exerted. One cell behind this point there is a strong recompression followed by a continuous expansion. The state after the recompression moves towards the point the force is exerted with a finer grid and in the limit this is the state at the outflow. This state can be calculated analytically with a 1D gas dynamics orthogonal to the disk. First the new total energy is calculated with the raise of specific energy due to the application of the power Fn vni to the mass flux ρvni S through a cell on the surface of the disk. The energy equation relates to the mass specific energy of a particle. The ratio of the total temperature T0 to the temperature at the inflow Ti is
Fn κ − 1 vn2 i T0 + =1+ 2 . (8) Ti ci 2 ρi S The formula contains the velocity of sound at the inflow ci and the adiabatic exponent κ. With an isotropic change of state the critical state is calculated. The critical state ∗ is the state at which the velocity reaches the velocity of sound after an isotropic expansion from total conditions. Application of the continuity equation to the control volume of figure 1h maintains the stream density through the disk ρe vne = ρi vni . With this the ratio of the stream density ρe vne to the critical stream density ρ∗ c∗ at the new state is known from the state at the inflow. The same ratio must be reached with an isotropic expansion from the total conditions to the state at the outflow. This ratio can be expressed with the Laval number La = vne /c∗
1 2 κ−1 1 − κ−1 ρe vne vni κ+1 La = = La (9)
κ+1 ρ ∗ c∗ 1 − κ−1 2(κ−1) κ+1 2 T0 ci κ+1 Ti
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The equation must be solved iteratively. With the Laval number the temperature ratio between outflow and inflow and the velocity at the outflow are
Te 2 T0 κ−1 2 T0 Te La , 1− vne = ci = − . (10) Ti Ti κ+1 κ − 1 Ti Ti With the isentropic relations the pressure ratio and the temperature ratio between outflow and inflow are known 1 Te κ−1 ρe = , ρi Ti
k Te κ−1 pe = . pi Ti
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The calculation of the jump of the tangential velocity for a cell on the surface of the disk is shown in figure 1c. Because of the continuity equation the stream density ρvn is constant through the disk. With the application of the momentum equation and energy equation in tangential direction ρvn S(vte − vti ) = Ft , 1 ρvn S (vt2e − vt2i ) = Ft vt . 2
(12) (13)
With resolving (12) and division (13) by (12) the state at the outflow is vte = vti +
Ft , Sρvn
vt =
1 (vt + vti ). 2 e
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At the inflow the external force is the vector sum of the normal and tangential component Fs = Fn + Ft and due to the orthogonality the external power is the algebraic sum of the normal and tangential power Fs v = Fn vni + Ft vt . For vanishing flow through the disk the tangential velocity vte must be limited.
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Application
For a first validation of the actuator disk a constant pressure jump of p = 500P a has been applied to the disk of R = 1m with a farfield velocity of v = 50m/s. The pressure and the velocity distribution are compared with an analytic solution based on incompressible potential theory. Figure 2 shows the pressure distribution and figure 3 the velocity distribution along the x-axis orthogonal to the disk. The pressure distribution and the velocity distribution are in good agreement with the theoretical result. The twiggle of the axial velocity at the center of the disk comes from the compressible formulation of the algorithm. The pressure jump at the disk causes a jump of the density. To maintain the mass flux there must also be a jump of the axial velocity. The deviation behind the disk is caused by the loss of axial velocity due to the pressure jump. Also dissipation of the higher velocity behind the disk and the close distance of the farfield at x = 12m may have an effect. To demonstrate the applicability of the actuator disk for a complex configuration, the glider SB14 of the Akaflieg Braunschweig [1], Figure 4 was chosen. This glider has
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Fig. 4. SB14 with folding engine, pressure distribution on the surface, two planes of iso-velocity |V | = 26 m/s
Fig. 5. Pressure distribution on the elevator, top: without engine, bottom: with engine
been designed since 1989 as a successor of the SB8 with a wing span of 18 meters. The aim of this configuration was a minimization of the wetted surface and therefore the fuselage and the tail were built as small as possible. The maiden flight took part
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in 2002 and since this time the glider has been used amongst other activities for the extension and validation of the DLR TAU-code for the area of small Reynolds- und Mach-numbers including the prediction of the laminar/turbulent transition, [4]. To use the actuator disk a fictious electrical engine has been added, which has been designed for self-launch of the SB14, Figure 4. The propeller (design M. Hepperle, DLR Braunschweig) has a diameter of 2 meters. In this section the powered configuration is compared with the clean glider configuration in level flight. The deployment of all aerodynamic control surfaces was determined in a flight test to get a vanishing moment around each axis for the clean glider configuration. The unstructured hybrid mesh with about 16 million points was generated with the grid generator Centaur [6]. The flow simulation was done with the DLR TAU-code on 24 CPUs of an AMD-Opteron based Linux-cluster. To accommodate the small Reynolds- and Mach- numbers preconditioning, matrix-dissipation and transitionprediction was applied, [4]. In Figure 5 the pressure distribution on the elevator is compared for the powered and the clean glider configuration. For the powered configuration an unsymmetry in the pressure distribution can be seen. The reason is the swirl of the anticlockwise turning propeller which increases the local angle of attack on the left part and reduces the local angle of attack on the right part of the elevator.
7 Conclusion The new actuator disk model is an advancement in the numerical simulation of propellers. The input data set is directly related to the construction data set of the propeller. The numerical model has been derived in accordance with the physics of the flow and reproduces all characteristics of the analytical model of the actuator disk. The scheme has been proven to be stable for a complete configuration of a glider as a demonstrator for an industrial application. With minor modifications it is also possible to apply the actuator disk to helicopter rotors. For helicopter rotors the elasticity of the blades and the blade vortex interaction have to be taken into account to calculate aerodynamic forces. The distribution of sectional loads of the rotor disk is therefore calculated externally with a simplified aerodynamics. An interface has to be created which reads the output in form of a structured grid with polar coordinates and interpolates it to the unstructered grid of the actuator disk. Because the main flow acts orthogonal to the disk and in forward flight a helicopter rotor produces negative lift it can happen that the inflow direction changes locally. To get a stable algorithm the inflow and exhaust direction have to be determined and set locally.
References [1] Akademische Fliegergruppe Braunschweig e.V.: ”Akaflieg Braunschweig, 75 Jahre”. Akaflieg Braunschweig, p. 50-59, 1997. [2] Chaffin, Mark S.; Berry, John D.: ”Navier-Stokes Simulation of a Rotor Using a Distributed Pressure Disk Method”. American Helicopter Society 51st Annual Forum, Fort Worth, TX, 1995.
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[3] Le Chuiton, Frederic: ”Actuator disc modelling for helicopter rotors”. Aerospace Science and Technology 8 (2004) 285-297. [4] Melber-Wilkending, S.; Schrauf, G.; Rakowitz, M.: ”Aerodynamic Analysis of Flows with Low Mach- and Reynolds-Number under Consideration and Forecast of Transition on the Example of a Glider”. 14th AG STAB/DGLR Symposium Bremen, 16.-18. November, 2004. [5] Schwamborn, D., Gerhold, T., Heinrich, R.: ”The DLR TAU-Code: Recent Applications in Research and Industry”. In Proceedings of ”Eurpean Conference on Computational Fluid Dynamics”, ECCOMAS CFD 2006, Delft The Netherlands, 2006. [6] www.centaursoft.com
Design of a Retrofit Winglet for a Transport Aircraft with Assessment of Cruise and Ultimate Structural Loads Th. Streit1, J. Himisch2, R. Heinrich1, B. Nagel2, K.H. Horstmann1, and C. Liersch1 1
Institute of Aerodynamics and Flow Technology, DLR Braunschweig, Lilienthalplatz 7, 38108 Braunschweig, Germany
[email protected] 2 Institute of Composite Structures and Adaptive Systems Lilienthalplatz 7, 38106 Braunschweig, Germany
Summary In this work retrofit winglets are designed for a transonic aircraft. The used geometry is a generic twin engine aircraft. In a first step, winglets are designed using a lifting line method along with RANS solutions. L/D is optimized by taking into account a certain wing root bending moment reserve for the reference wing. The final analysis includes wing deformation studies by means of fluid-structure coupling. Therefore a finite element model has been developed with respect to standard loadings of certification authority. Using the fluid-structure coupling process RANS solutions with deformed wing shapes are obtained for cruise conditions in order to determine the influence of deformation on performance and for a 2.5g load case in order to evaluate ultimate structural loading. Comparing the results of the rigid wing with the deformed wing the wing root bending moment and the bending moment of the device is clearly reduced for the deformed wing. Thereby the advantage to which an implementation of the described method for future design processes would lead becomes apparent.
1
Introduction
Increasing fuel prices and environmental issues are key drivers for airliners and the aerospace industry to consider measures which reduce the fuel consumption and pollution of their aircraft. For existing civil aircraft the use of a retrofit winglet constitutes one of the measures most often used to improve fuel consumption. The design of a retrofit winglet continues to be a good example of a challenging multidisciplinary problem [1], [2], [3]. Under consideration of structural constraints, an appropriate design strategy combined with accurate CFD prediction tools are required to provide the best aerodynamic benefit for installing a winglet in an existing aircraft. In this work a conventional numerical design of a retrofit winglet for a transonic twin engine transport aircraft is performed. In a second step the impact of C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 62–70, 2007. © Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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deformations is determined using fluid-structure coupling. Especially winglets of large size or strongly loaded winglets lead to large wing deformations. These deformations on one hand alter the performance and structural loading at cruise and on the other hand show a strong influence on the structural loading for flight conditions under high aerodynamic loads (e.g. gusts, pull out of a dive, etc). Besides the structural load at cruise (fatigue), the ultimate structural loading at these flight conditions constitutes a further design criterium for a winglet. The design of a retrofitted winglet is performed with a lifting-line-method [4] in combination with the TAU-RANS flow-solver [5]. Retrofitted winglets are compared at cruise condition to a reference configuration with a Küchemann-tip. The used structure models are finite element models (FEM) created for the reference wing with the Küchemann-tip and the different winglets, using the DLR PARA_MAM Programme (PARAmetric simple and fast Mesh based Aircraft Modelling Tool) [6]. The automatic DLR-process chain [7] was used for coupling the structural and the flow computations. Coupled fluid-structure RANS solutions were obtained for the cruise and the n=2.5g load case. This allows a realistic and complete assessment of the designed winglets.
2 Geometry and Design Flow Conditions and Design Constraints A generic transonic wing body configuration is used. The wing has a doubletrapezoidal planform with leading edge sweep 27.43°, aspect ratio 9.35, projected taper ratio 0.21. Wing airfoil sections are designed for the 1g design cruise condition i.e.: M=0.78, h=36000 ft, 0.5< CL< 0.55. The reference configuration has a Küchemann-tip. Retrofitted winglets can change the wing geometry only outboard of 90% of wing semi span. A semispan increase of 5% is allowed. The increase of wing-root-bending moment (WRBM) is constrained to 4% based on the reference model. For the n=2.5g load case a lower altitude is chosen since at the cruise altitude the corresponding CL exceeds CLmax. Therefore the used flow condition for the n=2.5g case is M=0.754, h=22500 ft, CL=0.749. For this condition on one hand the structural load corresponds to n=2.5g and on the other hand the flow shows an incipient separation behind the shock. Furthermore the pull out of a dive will occur at a lower altitude compared to the cruise one.
3 Design of a Retrofit Winglet for the 1g Rigid Shape The idea of using winglets for a transport aircraft is to reduce the induced drag without increasing the structural stress as much as an increase of span for the same reduction of induced drag would do. Both values can be calculated by a three dimensional Lifting-Line-Method [3]. The big advantage is the short time needed for the calculation of a single geometry (in this case approximately 30 seconds). By writing an Excel-based input-file-generator it was possible to calculate several ten thousand different geometries. Therefore, the additional work for the implementation of an optimiser was not justified.
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The disadvantage is the non-consideration of friction-, wave- and interference drag. As already mentioned in Chapter 2, the increase of the wing-root-bending moment is constrained to 4% compared to the geometry with Küchemann-Tip. With respect to the effects of load reduction at the outer wing according to the deformation described in the following chapters this constraint shall not be deemed to be a hard limit. It is accounted as target for the whole assessment.
Fig. 1. Reduction of induced-drag for different Winglet-Sizes. Height related to half span.
The parameters were put into a matrix and each of the resulting geometries was simulated. Figure 1 show as example a part of the results of those calculations. Every point represents one geometry. Also a change in the twist distribution of the describing airfoils is defined as new geometry. To summarise the results it can be shown that large vertical winglets promise the best results. Lifting-Line Simulation Optimum TAUSimulation
Definition of additional constraints
TAUOptimization YES Improvement of planform necessary? NO Final TAU Optimization
Fig. 2. TAU-RANS result for one of the first obtained Winglets
Fig. 3. Procedure of interaction between TAU and Lifting-Line
The second program used was the RANS flow solver TAU [5]. Numerical results were obtained on an unstructured hybrid mesh with about 12 million mesh points
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having thirty-two prismatic layers for the resolution of the boundary layer. As turbulence model the one equation model “Spalart-Allmaras” was selected. A sufficient resolution of the near wall boundary layer mesh was checked in the solution, by assuring that y 1.3. Convergence to a stationary numerical solution was obtained within 4500 iterations. The first considered geometry was an optimum taken out of the Lifting-LineCalculation. Figure 2 shows the obtained pressure distribution result. It indicates an increase of wave-drag resulting from interference effects at the junction between the wing and winglet. Hence, there was nearly no drag reduction compared to the reference configuration. Consequently, it was decided to restart the Lifting-LineCalculations under the additional constraint of a larger blending. Simultaneously, the used airfoils where optimized to reduce the wave-drag at the junction due to shocks. Figure 3 shows the used procedure. In the process a TAU -Optimisation means the manual change of twist-distribution and airfoil selection to improve as example the interference effects at the junction. Whereas the change of planform is solely done with the Lifting-Line method. +
Fig. 4. Reduction of induced-drag for different Sharklike- geometries
Fig. 5. Pressure distribution of a Sharklike wing-tip-device
Beside the Winglet also a Shark-Fin [8] like geometry (see Figures 4-5) was optimised with the method presented above. In Figure 4 every point represents one geometry. Also a change in the twist distribution of the describing airfoils is defined as new geometry. Depicted are two set of points with different ratio between the increase of span and the increase of height of the device. By analysing the LiftingLine solutions it could be shown (as example see Figures 1 and 4) that in general Shark-Devices seem to have less capability to reduce the induced drag according to their lower height (see also [4]) compared to conventional winglets. Their advantage tends to be having fewer problems with interference effects. Comparing Figure 2 to Figure 5, the pressure minimum is pulled to the nose at the Shark-Device. This leads to less drag according to interference effects. As result of the presented procedure, device one to three, depicted in Figure 6, were taken for the investigations with the structure coupled method.
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Fig. 6. Wing tip devices used in the coupled fluid structure analysis
4 Fluid-Structure Coupling 4.1 Used Wing Tip Device Geometries Three winglet tip device configurations are used. They are shown in Figure 6. “Wing Tip Device 1” and “Wing Tip Device 2” are two of the promising designed retrofit winglets obtained within this work. In addition, a third winglet is considered: “Wing Tip Device 3”, which is a classical winglet of large size. Furthermore, coupled fluidstructure solutions are obtained for the reference configuration, which has a Küchemann-tip (basis). 4.2 Structure Model A Finite Element model was created for the reference wing using the DLR PARA_MAM programme (PARAmetric simple and fast Mesh based Aircraft Modelling tool) [6], which fits the parametric description of the inner structure into the aerodynamic flying shape defined by the aerodynamic surface mesh. The whole primary structure is modelled using eight node shell elements whereby the stringers are considered implicitly ('smeared') as stiffness equivalent layer of the wing's skin. The elastic properties of average aircraft aluminium are implemented. Nose and trailing section do not contribute to the wing's strength but have to maintain their shape in aeroelastic coupling. This is realised by very stiff ribs in these sections and a very flexible skin, which does not contain nodes between the ribs. An engine dummy serves for the introduction of engine mass and thrust as well as for feed back of engine deflections in coupled analysis. Further on, the weight of the landing gear and fuel mass of the two wing tanks are applied to the models. Inertial acceleration is applied. 4.3 Coupled Fluid-Structure Chain To take into account the interaction between fluid and structure within a simulation process accurately, a full automatic process chain [7] has been developed at DLR, allowing a coupling between CFD and computational structure mechanics (CSM). On CFD side the DLR code TAU is used. On structure side the commercial FEM code ANSYS [9] is used. Linear interpolation, conservative force interpolation as well as scattered data interpolation techniques based on radial basis functions constitute the interpolation components for the CFD-CSM data-transfer.
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Fig. 7. Wing Tip Device 1: deformed shapes VDPHLQFLGHQFH D JVKDSH
R
GHIVKDSH
VDPHOLIW &/ DWQ J JVKDSH
GHIVKDSH
Fig. 8. Skin friction lines and pressure distribution contours for upper wing with the Wing Tip Device 1 geometry : RANS solutions for deformed and rigid shapes for M=0.754, h=22.500 ft. On the left side comparison is made at the same incidence, on the right side comparison is performed at the same lift.
4.4 Results RANS solutions for deformed shapes were obtained for the 1g design cruise condition and for the 2.5g load case. Figure 7 shows a front view showing wing deformed shapes with the Wing Tip Device 1 geometry. It comprises the 1g cruise and the 2.5g load case computed with and without considering the inertial forces (i.e. the 1.5g centrifugal force which reduces the structural load). The influence of deformation is shown in Figure 8, which includes a comparison of pressure distributions and skin friction lines for the Wing Tip Device 1 case, obtained either using the deformed shape or the 1g rigid shape of the reference configuration. The impact of deformation is very large for the comparison at the same incidence, the results showing a different streamline pattern. The comparison at the same lift (here the n=2.5g lift value) does not show such large differences, however the results indicate a transfer of load from outboard sections to inboard sections. This results in a weaker shock at the outboard sections.
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deformed
rigid shape
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Fig. 9. Wing Tip Device 3 skin friction lines and pressure distribution contours: RANS solutions for rigid (left) and deformed (right) shapes for cruise and n=2.5g load case conditions
Figure 9 shows a comparison of pressure contours for deformed and rigid shapes for the Wing tip device 3 case. For the n=2.5g load case the rigid shape shows a shock along the complete winglet span, whereas in the deformed shape this shock is reduced to a small region located at the junction between winglet and wing. REF= deformed Basic at corresponding flight condition
' C D / C D_REF [%]
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Fig. 10. Drag reduction as function of wing root bending moment increase for rigid and deformed shapes
The relative decrease in drag as function of the relative increase in WRBM is shown in Figure 10. Results are given for cruise and for the n=2.5g load case and are related to the deformed basis configuration. The deformed shapes lead to a decrease in WRBM and an increase in drag in comparison to the rigid shapes. This means that for each device the deformed shapes show a displaced aerodynamic benefit optimum at the 4% WRBM constraint. From the three studied devices, the second one shows the best trade-off between structural load penalty and aerodynamic benefit, since for the deformed shape its WRBM increase comes close to the 4% WRBM structural
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reserve. In addition, at similar structural load the aerodynamic optimum of this device leads to a larger reduction of aerodynamic drag. The wing tip device configurations with the largest WRBM for the rigid shape show the largest decrease in WBRM for the deformed shapes. Especially for the n=2.5g load case the results for the deformed shape, lead to large differences in comparison to the rigid shape results, with a reduction of WRBM up to 5.55%. A further optimisation of the three devices at the cruise design point with deformed shape was not considered within this work. 0.025 REF=deformed basic at correponding flight condition
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Fig. 11. Bending moment of wing tip devices at cruise and for the n=2.5g case
The analysis of local load distributions results for cruise and the n=2.5g case shows that for the deformed shapes there is a load transfer from the outboard wing to the inboard wing. Especially for the wing outboard sections this leads to a roll moment reduction which is larger for the wing tip device configurations. These results are consistent with the reduction of WRBM and increase in drag obtained between the deformed shapes and rigid shapes. For the structural load analysis of the wing tip device configuration it is important to determine the bending moment of the device itself. Results for the tip device bending moment are given in Figure 11. The larger devices show larger bending moment values and their bending moment also show the largest decrease between rigid and deformed shape. For computing the device bending moment a reference point is selected which is placed at the leading edge of the intersection which separates the wing device to the unchanged wing.
5 Conclusions In this work retrofit winglets are designed for a generic twin engine transonic transport aircraft. The planform and aerodynamic load of the winglets is obtained by optimizing L/D for the cruise condition using a lifting line method along with RANS solutions. Airfoil design was used to reduce wave drag. RANS solutions were obtained with the DLR-TAU code. Typical constraints of a retrofit design are imposed, these are: a maximum wing root bending moment increase and a maximum
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wing span increase based on the reference wing, furthermore the wing can only be modified outboard of a specified station. The reference configuration has a Küchemann-tip. To have a realistic assessment of the designed winglets, fluidstructure coupling is performed in order to compute the performance and structural loading (fatigue) at cruise as well as ultimate structure loadings (i.e. gusts, pull out of a dive) for deformed shapes. The used structure models are finite element models (FEM) created for the reference configuration with the different wing tip devices using the DLR PARA_MAM programme. The coupled fluid-structure RANS solutions are obtained for the cruise condition and the n=2.5g load case. The n=2.5g load case simulates a pull out of a dive, therefore for the structural analysis aerodynamic, weight and inertial (centrifugal) forces are considered. Results show a large impact of the deformation on the structural load. For the deformed shapes the wing root bending moment and the device bending moment is reduced relative to the rigid shape results. Especially for the n=2.5g case this reduction is large (up to 5.55%). At cruise, this reduction of root bending moment goes along with a drag increase. Flow analysis shows that there is a load transfer from the outboard wing to the inboard wing. The shock obtained with the rigid shape in the winglet region for one winglet case for the n=2.5g load was greatly reduced for the deformed shape. This work shows that under consideration of structural constraints, an appropriate design strategy combined with accurate CFD prediction based on deformed shapes are required to provide the best aerodynamic benefit when installing a winglet into an existing aircraft.
References [1] A. Mann, The M-DAW Project, Modelling and Design of Advanced Wing Tip Devices, CEAS/KATNET Conference on Key Aerodynamic Technologies, Conference Proceedings, June 2005, Bremen. [2] J.-L. Hantrais-Gervois, M. Rapin, Aerodynamic and Structural Behaviour of a Wing th Equipped with a Winglet at Cruise, 44 AIAA Aerospace Science Meeting and Exhibit, January 2006, Reno, Nevada, AIAA 2006-1489 [3] B.B. Prananta, A. Namer, J.E.J. Maseland, J. van Muijden, S.P. Spekreijse, Winglets on Large Civil Aircraft: Impact on Wing Deformation, International Forum on Aerolasticity and Structural Dynamics IFASD 2005, 2005. [4] K.-H. Horstmann: Ein Mehrfach-Traglinienverfahren und seine Verwendung für Entwurf und Nachrechnung nichtplanarer Flügelanordnungen, DVFLR-FB 87-51, 1987. [5] N. Kroll, J.K. Fassbender [Hrsg.]: MEGAFLOW – Numerical Flow Simulation for Aircraft Design, Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), 89, Springer Verlag, Closing Preparation DLR Project MEGAFLOW, 2002 Braunschweig . [6] B. Nagel., M. Rose, H.P. Monner, R. Heinrich: An Alternative Procedure for FE-Wing Modelling, Deutscher Luft- und Raumfahrtkongress Nov. 2006, Braunschweig, Germany. [7] R. Heinrich, J. Wild, Th. Streit., B. Nagel: Steady Fluid-Structure Coupling for Transport aircraft , Deutscher Luft- und Raumfahrtkongress 2006, Braunschweig. [8] G. Heller, P. Kreuzer, S. Dirmeier, T. Streit: Aerodynamische Untersuchung verschiedener Flügelseitenkanten-Erweiterungen der Envoy 7, Jahrbuch DGLR, Bd. III, pp. 1629-1634, 2002. [9] http://www.ansys.com
Numerical Simulation of a Wing with a Gapless High-Lift System Using Circulation Control K.-C. Pfingsten and R. Radespiel Technische Universit¨ at Braunschweig, Institut f¨ ur Str¨ omungsmechanik, Bienroder Weg 3, 38106 Braunschweig, Germany
[email protected]
Summary Numerical 2D simulations with a RANS flow solver are conducted to find the aerodynamic sensitivities of a gapless high-lift system. The investigated high-lift configuration is an airfoil which utilises trailing edge blowing. A small fraction of the engine flow is used for circulation control. The air is blown from a slot directly upstream of the flap and thus the flow over the flap can bear large adverse pressure gradients without separation. It was found that the use of circulation control yields lift coefficients which are comparable or superior to those generated by conventional high-lift systems. The promising results of the 2D simulations motivate applications to a wing-body configuration. The results show that a gapless high-lift system equipped with circulation control has the ability to provide sufficient lift for take off, climb and landing.
1
Introduction
In recent years noise pollution from aircraft, especially around airports, has become a huge problem. Hence there is an increasing interest in reducing the noise emitted during take off and landing. The conventional high-lift systems, consisting of slats and slotted flaps, are a major contributor of airframe noise. Therefore a gapless high-lift system without slats has a potential of reducing the overall noise, emitted by an aircraft. Only with active flow control, like trailing edge blowing, a gapless high-lift device is capable of generating the high lift coefficients needed for take off and landing. For circulation control (CC) a small fraction of the cold engine flow (about 5%) is used for blowing. The bleed air is pipelined from the engine to a slot directly upstream of the flap and thus the flow over the flap can bear large adverse pressure gradients without separation. Thus a gapless high-lift device with CC is able to generate the required lift. The low drag coefficients during climbing, achievable with these powered high-lift system, could also allow the use of new low-noise trajectories, which would further reduce the noise exposure at ground level. The absence of slats might allow laminar flow in cruise flight, thereby reducing the drag in this C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 71–79 , 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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flight segment. Even with taking into account the additional system weight associated with the bleed air distribution for a gapless high-lift system, there is a chance of reducing the total weight of the aircraft and possibly the cost, because slats and fowler flaps are now redundant. First experiments to understand the principal of CC were conducted in the thirties of the last century [1], [2]. At Braunschweig Technical University systematic measurements and theoretical considerations for wings with blown flaps yielded lift increase versus neccesary momentum coefficients [3], [4]. In the seventies two demonstrator aircraft using CC to allow for short take off and landing abilities were built in the United States [9], [5] and went through extensive flight testing. Today a wide range of literature concerning CC is available [10]. This paper presents new results of 2D flow field simulations around profiles with trailing edge blowing at high-lift and analysis at cruise conditions in order to address aerodynamic design trades for CC applied to a modern transonic airfoil. Due to the promising results of the 2D numerical simulations further 3D simulations of a wing-body configuration are performed to analyse integration aspects for transport aircraft.
2
Basic Principle: The Coanda-Effect
Profiles with trailing edge blowing use the well known Coanda principle to generate high lift coefficients: A high velocity, tangentially blown air jet remains attached to a convex surface due to the balance between centrifugal forces and the sub-ambient pressure in the jet sheet. The Coanda-effect works best when the slot height is about 1% to 5% of the curved surface radius and the slot height is between 1 and 2 per mil of the chord length [5]. The driving parameter for the Coanda-effect is the dimensionless momentum coefficient Cµ of the jet, which is defined as follows: v m ˙ Cµ = 1 jet jet (1) 2 ρ v ∞ ∞S 2 It is important to notice that the increase of the lift coefficient is much higher than the used dimensionless momentum coefficient. The augmentation can be as large as eighty times the applied Cµ [10]. So the lift gain is due to flow separation control and supercirculation and does not arise because the momentum of the jet is directed downwards.
3 3.1
Two-Dimensional Investigation
Design Consideration
As a starting point for the 2D investigation of CC a modern generic transonic airfoil has been chosen. The 2D simulations are conducted for a cross section of the wing with a chord length of c = 6m which corresponds to the mean aerodynamic chord of the reference wing. Here the results for a dimensionless slot height of h/c = 0.001 are presented.
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To design an airfoil with CC first the x-wise position of the slot is defined, which will also determine the length of the flap. Upstream of this position the original upper surface is used. Downstream of the vertical slot the upper surface of the flap has to be shaped. To get a realistic flow at the slot exit a duct is designed which has a length of seven times the height of the slot. At the inflow position the duct is two times as high as the slot to ensure the jet is accelerated until it reaches the exit. In Figure 2 the shape of the duct can be seen. 0.1
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Fig . 1 . Different flap geometries for CC
Fig. 2 . Mesh for CC profile with large high-lift flap
There are two basic design constraints for the flap geometry. On the one hand the Coanda-radius should be as large as possible to exploit the Coandaeffect, on the other hand the profile should be as well optimized for cruise flight, which means the new geometry should be as close as possible to the original transonic airfoil. One possibility of meeting these design constraints is to hide the Coandaradius in the profile whilst in cruise flight. This flap is called large high-lift flap and can be seen together with the other investigated flap geometries in Figure 1. It has a length of cf lap /c = 0 3. As for all geometries the flap hinge is located at the lower surface. The only difference to the original transonic
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airfoil is that the upper surface downstream of the flap has been translated slightly downwards to create the slot. So in cruise there will be a step with the height of one per mil of the chord length, which will have only a marginal effect. It would be also possible to have an unaltered profile for cruise and open the slot only for the high-lift case, when the flap is deflected. This would result in two moving parts. One would be the flap and the other would be a small part of the upper surface upstream of the flap, which opens to create the slot. The other option is to design a dual-radius flap (Figure 1), which was first mentioned by Englar [5]. Here the upper surface of the flap is composed out of two circle segments. The first circle segment is inside the profile whilst in cruise flight. It is always chosen as large as possible, so the radius is about the local thickness of the profile at the slot position. Because the second segment is from a much larger circle, the changes of the upper surface in comparison to the basic profile are relatively small. But for dual-radius flaps of the original Englar approach the curvature of the second Coanda-radii is much larger than the curvature of the original upper surface. Therefore a profile with a dual-radius flap of this kind (1st generation) is shorter than the original profile. As the slope of the upper surface is much larger at the trailing edge than for the other flaps, this yields advantages for the high-lift case but has severe disadvantages in cruise flight. The flap length is cf lap /c = 0.05. A second generation of dual-radius flaps was also designed, which have a much better performance in cruise. Here the radius of the second segment is chosen in a way that the position of the trailing edge stays unchanged, so the new shape stays very close to the original profile and no scaling is necessary. In Figure 1 it can be seen that these flaps are thinner and slightly longer than the dual-radius flaps of the first generation. It has been necessary to increase the flap length to cf lap /c = 0.07 to achieve similar lift-coefficients as are generated by the dual-radius flaps of the first generation. 3.2
Numerical Flow Simulation
Numerical solutions of the Reynolds-averaged Navier-Stokes equations are computed using the DLR hybrid unstructured flow solver TAU [7], which is based on a finite volume scheme. The code processes meshes with different types of cells and combines the advantages of structured grids to resolve boundary layers with the flexible grid generation of unstructured grids. To accelerate the convergence to steady state, techniques like local time stepping, residual smoothing and multi-grid technique based on agglomeration of the dual-grid volumes are available. All computations were undertaken assuming the boundary layer to be fully turbulent and with the Spalart-Allmaras turbulence model, which has proven its capability of computing flow fields around profiles with CC, using simulations of the experimental results achieved by Novak [6] for an elliptical profile with CC, see also Swanson in [11]. The generation of the unstructured hybrid grids was done using Centaur. In the vicinity of the jet slot the grid is clustered to resolve the jet behaviour.
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Behind the trailing edge a wake plane with structured cells is added to get a better resolution at these sharp edges. In Figure 2 the mesh around a CC profile with large high-lift flap can be seen. For all geometries the total number of points is about 70000. A layer of 30 cells was used to form the structured grid close to the surface. 3.3
Results of 2D Simulations
The results for all three flaps in Figure 1 are compared in Figure 3. All results are for a slot height of h/c = 0.001. For take off and landing the large high-lift flap generates high lift coefficients up to Cl,max = 3.9 for a flap deflection angles of η = 40. For this flap the geometry in cruise does not significantly change compared to the basic profile, thus there is no drag penalty in cruise expected.
4
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Large high-lift flap: η=20° Large high-lift flap: η=30° Large high-lift flap: η=40° Dual-radius flap (1.gen.): η=20° Dual-radius flap (1.gen.): η=40° Dual-radius flap (1.gen.): η=60° Dual-radius flap (2.gen.): η=20° Dual-radius flap (2.gen.): η=40° Dual-radius flap (2.gen.): η=60° Reference profile
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Fig. 3 . Cl over α for different flap geometries (h/c = 0. 001) : Ma∞ = 0. 21, Re = 29 · 106 , Cµ = 0.04 Cµ = 0.03 for the large high-lift flap
It can be seen that the much shorter and lighter dual-radius flap of the first generation achieves lift coefficients of Cl = 3.3. The dual-radius flaps of this type change the geometry of the profile shape in cruise significantly, as discussed above, and this results in an 10% increase in cruise drag. This increase in cruise drag is caused by flow separation at the trailing edge. To further improve the cruise performance a second generation of dualradius flaps was developed. These flaps generate the same lift as the dualradius flaps of the first type, but have a much better performance in cruise. Due to the new upper surface of the flap the cruise drag in the design point of the profile (Ma∞ = 0.72, Cl = 0.4) is now increased by about 2 drag counts. A slightly larger flap is necessary to generate the same lift in low speed as achieved by the old dual-radius flap with its higher flow turning ability. In
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Fig. 4 . Dual-radius flap 2. generation (h/c = 0.001) : Ma∞ = 0.21, Re = 29 · 106 , cf lap /c = 0.07, η = 60◦ , α = 9◦ Cl = 3.3, Cd = 0.054, Cm = 0.48, Cµ = 0.04
Figure 4 the Mach number distribution of the flow field around an airfoil with a dual-radius flap is plotted at conditions close to maximum lift. Note that according to these calculations airfoil stall with CC is caused by reversed flow above the flap. Here the Coanda jet still follows the flap contour whereas flow reversal of the wake from the main wing takes place. The implementation of CC does not generate higher pitching moments than modern high-lift profiles with slat and fowler flaps. These results show the high potential of the CC concept in comparison to a conventional as well as other powered high-lift systems [8].
4 4.1
Three-Dimensional Investigation
Design Consideration
Due to the promising results of the 2D simulations a wing-body configuration with CC was designed and 3D flows were numerically investigated. A high wing aeroplane with a wing surface area of S = 244m2 , a wingspan of b = 48m and a passenger capacity of P AX = 260 is equipped with a slot for blowing, which is implemented on the partial wingspan from the central wingbox to the aileron. The slot has a height of 0.1% of the local chordlength and is positioned directly upstream of the flap. Like for the 2D simulations a convergent duct is used to get uniform flow in the slot exit. Due to its good overall high-lift performance the large hig-lift flap is used for the 3D configuration. As a reference the flow around a wing-body configuration with a large high-lift flap and no CC is also simulated.
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4.2
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Numerical Flow Simulation
The hybrid mesh for the wing-body configuration with CC consists of 25 · 106 nodes. Especially around the slot a high resolution of grid points is used. Due to the high demand in computing power the simulations were all computed on 32 cpus of the IBM p690 (Regatta) compute servers of the “Norddeutscher Verbund f¨ ur Hoch- und H¨ ochstleistungsrechnen” (HLRN). All computations were undertaken assuming the boundary layer to be fully turbulent using the Spalart-Allmaras turbulence model.
Fig. 5 . Cp -distribution and surface streamlines on a wing-body configuration : Ma∞ = 0.23, Re = 32 · 106 , η = 40◦ , α = 4◦ No blowing: CL = 1.55, CD = 0.204 With CC (h/c = 0.001): CL = 2.48, CD = 0.286, Cµ = 0.03
4.3
Results of 3D Simulations
In Figure 5 the Cp -distributions and surface streamlines on a wing-body configuration with and without CC are displayed. For the wing-body configuration without blowing the flow is fully separated along the upper surface of the flap, whereas the flow is attached when CC is applied. The increased circulation introduces additional suction along both Coanda surface and main wing. Independent of the flap deflection angles the dimensionless momentum coefficient is Cµ = 0.03, which corresponds to 5% of the total engine mass flow for the reference aircraft. In Figure 6 and 7 the results of the 3D simulations are summarised. The configuration achieves lift-coefficients of CL,max = 2.5 for a flap deflection angle of η = 40. This would be a setting for approach, hence the necessary drag CD = 0.29 is also generated. A deflection angle of η = 20 would supply sufficient lift, of around 2, for take off in combination with favourably small drag coefficients. In Figure 6 it can be noticed that blowing decreases the angle of attack for which maximum lift is produced. For higher angles of attack separation starts at the outer wing, which is not a problem of CC, but appears also for the reference aircraft. This is due to the wing taper and should be improved in subsequent wing design work.
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CL
Reference config.: η= 0° Reference config.: η=20° Reference config.: η=40° With CC: η= 0° With CC: η=20° With CC: η=40°
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Fig . 7 . CL /CD over CL for a wing-body configuration
Conclusion
Steady-state Reynolds-averaged Navier-Stokes calculations have been carried out for a supercritical 2D airfoil and a 3D wing-body configuration utilising CC. The profile demonstrates good performance at high-lift conditions. Lift coefficients of Cl = 3.9 are predicted. The data for the wing-body configuration with CC confirms the potentials of the concept to generate high lift in combination with high drag (η = 40◦ ) for approach and landing as well as keeping the drag low for good take off and climb performance (η = 20◦ ). For this flap geometry the airfoil shape in cruise is unaltered, as the Coanda surface is hidden inside the profile. While these simulations are very encouraging, additional calculations are needed to further define the optimum flap geometry and to determine the minimum total pressure and mass flow rates needed to achieve the necessary lift. Moreover, more experimental data to validate the numerical flow predictions are recommended.
Acknowledgements This work is part of the project “Innovative Hochauftriebskonfigurationen”, which is part of the research programme Lufo III of the BMWA. The authors thank the “Norddeutscher Verbund f¨ ur Hoch- und H¨ochstleistungsrechnen” (HLRN) for providing the necessary computational resources.
References [1] M.J. Bamber. “Wind tunnel tests on airfoil boundary layer control using a backward-opening slot”. NACA Report 385, 1932. [2] H. Hagedorn and P. Ruden. “Windkanaluntersuchungen an einem JunkersDoppelfl¨ ugel mit Ausblaseschlitz am Heck des Hauptfl¨ ugels”. Bericht A 64 der Lilienthal-Gesellschaft f¨ ur Luftfahrtforschung, 1938.
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[3] F. Thomas. “Untersuchungen u ¨ ber die Grenzschichtbeeinflussung durch Ausblasen zur Erh¨ ohung des Auftriebes”. Technical University Braunschweig, dissertation, 1961. [4] H. K¨ orner and R. L¨ ohr “Dreikomponentenmessungen am Modell eines leichten STOL-Flugzeuges mit Ausblasen in Fl¨ ugeltiefenrichtung”. Deutsche Forschungs- und Versuchsanstalt f¨ ur Luft- und Raumfahrt, DLR-FB 75-74, 1975. [5] R.J. Englar and R.A. Hemmerly. “Design of the circulation control wing STOL demonstrator aircraft”. AIAA Journal of Aircraft Vol.18, No. 1, 1981, pp. 5158. [6] C.J. Novak and K.C. Cornelius and R.K. Roads. “Experimental investigations of the circular wall jet on a circulation control airfoil”, AIAA Paper 87-0155, 1987. [7] T. Gerhold. “Overview of the hybrid RANS code TAU”. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Volume 89, 2005, (MEGAFLOW - Numerical Flow Simulation for Aircraft Design), pp. 81-92. [8] K.-C. Pfingsten and R. Radespiel and M. Kamruzzaman. “Use of upper surface blowing and circulation control for gapless high-lift configurations”. CEAS/KATnet Conference on Key Aerodynamic Technologies, 2005. [9] J.L. Loth. “Advantages of combining BLC suction with circulation control high-lift generation”. Progress in astronautics and aeronautics, Volume 214, 2006 , (Applications of circulation control technology) , pp . 3-21 . [10] R.D. Joslin and G. S. Jones. “Applications of circulation control technology”. Progress in astronautics and aeronautics, Volume 214, AIAA, 2006. [11] R. C. Swanson and C. L. Rumsey. “Numerical issues for circulation control calculations”. AIAA Paper 2006-3008, 3rd AIAA Flow Control Conference, San Francisco, June, 2006.
Shock Control Bumps on Flexible and Trimmed Transport Aircraft in Transonic Flow Benedikt K¨ onig, Martin P¨ atzold, Thorsten Lutz, and Ewald Kr¨ amer Universit¨ at Stuttgart, Institut f¨ ur Aerodynamik und Gasdynamik (IAG), Pfaffenwaldring 21, D-70550 Stuttgart, Germany {Koenig, paetzold, lutz, kraemer}@iag.uni-stuttgart.de
Summary Shock control bumps are a means to reduce wave drag that occurs at the upper limit of civil transport aircrafts flight envelope. An SCB was optimized for and applied to the rigid wing-body DLR F11 model. The effect of the SCB on the trim drag of the F11 configuration with an attached horizontal tail plane was investigated. Flexibility of the wing-body model was considered and the influence of aeroelasticity on the SCB performance was examined. RANS simulations with the DLR FLOWer code showed, that both the influence of trimming as well as of aeroelasticity is negligible for SCB design.
1
Introduction
Increasing the aerodynamic efficiency is a main design issue for civil transport aircraft, as this leads to a direct reduction of DOCs. Todays long-range transport aircraft cruise within the transonic speed regime in order to maximize the operational range as well as cruise speed. This speed regime is characterized by regions of supersonic flow that are terminated by shock waves. Practically, these shocks set the upper limit of the cruise speed by the associated drag rise. Therefore, shock control is one important means to further extend the cruising envelope to higher Mach numbers with improved aerodynamic efficiency. The Shock Control Bump (SCB) is a device to reduce the shock strength and thereby wave drag in transonic flow by inducing compression waves upstream of the shock. It was for example investigated numerically and experimentally within the Euroshock II project [7]. For the two dimensional case of an airfoil section it was shown by Sommerer that wave drag reductions of up to 75% are possible [6]. The effect of a three dimensional boundary layer of swept wings on the effectiveness of SCBs was investigated in a previous study [2]. It was concluded that the concept of SCBs is also applicable to three-dimensional flow over swept wings. Considering the application of SCBs on realistic transport aircraft configurations in free flight, the potential for drag reduction by SCBs is influenced by two more aspects that were not addressed so far. The changed pressure C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 80–87, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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dirstribution on the wing, altered by the SCB, first of all leads to an additional nose-down pitching moment. This needs to be compensated by a trimming moment, which in turn, influences drag. Secondly, the changed pressure distribution affects the loads on the wing structure. A non-rigid airframe will be aeroelastically deformed by these load changes. Because one-point optimized SCBs are sensitiv towards changes in flow conditions [6], these deformations, that affect shock position and strength on the wing, and are expected to directly influence the optimal SCB design. To investigate the sensitivity of the SCB on changing freestream conditions SCB optimizations for infinite wings were performed and analysed. In a next step, an SCB was optimized for a rigid DLR F11 wing-fuselage configuration. Its impact on aeroelastic deformation of a flexible model, on trim and on the related changes in aircraft drag were investigated.
2 2.1
Numerical Methods
Optimization Environment POEM
The Parallel Optimization Environment with Modular structure (POEM), developed by IAG [3], consists of three main modules. These are the optimizer module, the geometry module and the analysis module. The optimization tool applied in this work is the Downhill-Simplex optimizer contained in the commercial iSight software. A set of design variables is passed to the geometry module which generates the appropriate geometries for the design. The performance of this design is then evaluated using the block structured RANS solver FLOWer by DLR [1] which returns the corresponding objective function value to the optimizer. A finite volume RANS grid is generated automatically for every SCB design using Python scripts. For the following numerical investigations with FLOWer, the convective fluxes of the main equations were discretized in space applying a second-order central scheme according to Jameson. Closure of the RANS equations was achieved by application of the SST turbulence model. 2.2
Trimming and Aeroelastic Coupling
To account for the steady aeroelastic deformations of a transport aircraft in free flight, a staggered coupling procedure is applied. The CFD method used is again the DLR FLOWer code. For the aeroelastic computations the CFD solver and a reduced order condensed structural model are loosely coupled. Load and deformation transfer between two non-coinciding aerodynamic and structural meshes is based on the FEMSPLINE method by Mauermann [4]. The structural model is derived from a generic Nastran linear structural model with similar mass distribution and eigenmodes as a typical transport aircraft comparable to the DLR-F11
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configuration. It consists of 97 nodes for fuselage, wing and horizontal tail plane with a total of 582 degrees of freedom. As it is a free flight model, as opposed to a restrained wind tunnel model, none of the nodes are fixated, i.e. the model has a full six degrees of freedom for every node. It features not only the elastic eigenmodes but also the modes of the rigid body motion, and modal synthesis was used to split these modes [5]. For the elastic analysis of a free flying model, an equilibrium state of all external forces (a trimmed state) should be reached beforehand, to ensure that no resulting force acts on the center of gravity. Otherwise, the resulting model corresponds to a model with a virtual fixture at the center of gravity, which in general does not coincide with one of the structural nodes. The structural analysis is then done by solution of the system F = Kx with the constant stiffness matrix K, the vector of the loads acting on the structural nodes F , and the unknown deformation vector x. Based on the linear property of the model with the constant stiffness matrix K this computation may be reduced to the multiplication x = K−1 F with a pre-calculated inverse stiffness matrix K−1 . This operation is performed directly in the aeroelastic module and actual coupling to the Nastran application can be avoided. The trim procedure is nested as an inner loop into the aeroelastic coupling process. By evaluating the external aerodynamic loads as well as the internal mass distribution, lift and pitching moment changes ∆L and ∆My are calculated as the differences between the current state and a trimmed equilibrium condition. Using these increments, correction values ∆α and ∆η are derived for the angle of attack and the setting angle of the adjustable stabilizer, respectively, with the following approximate relations: cL,η cL,η + ∆L xc.p.,wing + cL,α xc.p.,htp ∆My 1 + cL,α , (1) ∆η = cL,η (xc.p.,wing − xc.p.,htp ) · q∞ Aref ∆α =
∆L/ (cL,η · ∆η) . cL,α · cL,η · q∞ Aref
(2)
∂cL L Herein, cL,α = ∂c ∂α , xc.p.,wing , cL,η = ∂η and xc.p.,htp denote the lift gradients and the x-coordinates of the center of pressure c.p. of the wing and the horizontal tail plane, respectively. The incompressible dynamic pressure is given by q∞ and the reference area is Aref . Underelaxation is applied to ∆η and ∆α to prevent oscillations. Rather than using a grid deformation algorithm to adopt the computational grid to the geometric modifications during the iterative coupling process, a new grid is generated automatically for every cycle. For this task, a set of scripts is available at IAG to automatically generate high quality structured grids for wing-body configurations with or without empennage, winglet and nacelles. The current work inlcudes a wing-body configuration with horizontal tail plane. To reduce grid generation time during the trim cycle, the horizontal tail plane is realized as a Chimera overlapping grid. Thus,
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only the relatively small grid for the htp is regenerated during each cycle and recombined with the larger background grid containing fuselage and wing.
3
Shock Control Bumps
In transonic flow, wave drag is generated along a shock front by the nonisentropic change of state of the fluid across the shock. The increase in entropy is roughly proportional to the third power of the shock strength, which varies along the shock front. This entropy gain comes along with total pressure losses and, thereby, wave drag. The shock strength depends on the pre-shock Mach number, which, usually, decreases with distance to the boundary-layer edge. Thus, the strongest part of the shock and, therefore, the largest total pressure losses appear slightly above the boundary-layer edge. Shock control bumps induce a local recompression upstream of the main shock in order to reduce the pre-shock Mach number, and, thereby, the wave drag. The recompression of the supersonic flow may be attained either by a succession of infinitely weak quasi-isentropic shocks, induced by concave curvature of the surface, or by inducing a λ-shock structure, where the main shock is split into two successive shock legs. With respect to the shape of shock control bumps, the first mechanism of recompression is obtained with a smooth and gradual concave ascending bump flank, whereas the second mechanism appears at a ramp-like upstream facing flank with a kink at the beginning of the shock control bump. In the following, the smooth contour is denoted SCB1, while the contour with kink is named SCB2. .
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Fig . 1 . Pressure distribution in the shock region for different shock control bump contours. Optimized with POEM, mesh refinement in the shock region.
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Polars of the aerodyFig . 2 . namic efficiency of different SCBcontour shapes. Computed without mesh refinement.
Taking the DA VA2 airfoil section as basis, one bump of each kind was optimized for a single off-design condition of the datum airfoil. The pressure distributions of the resulting bumps are shown in Figure 1. SCB1 exhibits a gradual recompression along its concave flank, which is followed by an
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expansion over the convex bump part up to the remaining shock. SCB2 starts with a kink, directly followed by a convex surface. As a result, the shock is split into a λ-shock. The front shock leg is a weak oblique shock which decelerates the supersonic flow gradually to a lower supersonic Mach number. At these flow conditions, a total drag reduction of 14.7% was achieved with SCB1 and 14.2% with SCB2. Concerning the application of shock control bumps to real finite wings, a main issue is the robustness of the bump with respect to varying shock position. Variations in the shock position show up along the wing span and result from changing flight conditions. Robustness to varying shock positions reduces the need for adaptation of the bump contour and position during flight. Figure 2 shows the aerodynamic efficiency verus the lift coefficient for the two bump contours. The diagram reveals that the SCB2 features better performance below the design lift coefficients, whereas the SCB1 shows a slightly larger drag reduction at the design point. The reason for the higher robustness of SCB2 is the fixing of the front shock leg at the kink of this contour. Thereby, the recompression due to the oblique shock is preserved in a wider range of lift coefficients and no excessive acceleration occurs over the bump crest.
4
Results for DLR F-11 Configuration
The application of SCBs on the outboard wing of the rigid F11 wing-bodyconfiguration was investigated, see Figure 3. Due to its improved robustness, a contour with kink at the beginning of the bump (SCB2) was selected. The position and height of the bump at the kink as well as at the tip of the wing were optimized with respect to total aerodynamic efficiency at constant angle of incidence. The grid for the F11 wing-body configuration consists of around 15 mio. cells. Onset flow conditions for the bump design point were chosen at a high cruise flight Mach number of Ma = 0.85 with a Reynolds number of Re = 40 · 106 . At these flight conditions maximum aerodynamic efficiency of the F11 is reached at a lift coefficient of cL = 0.44, which corresponds to an angle of incidence of α = 2◦ . This angle was increased to α = 2.5◦ for the bump design, which results in a lift coefficient of cL = 0 . 53. Based on the calculated aerodynamic center, a location for the center of gravity was set with a static stability margin of about 3%. In Table 1 aerodynamic coefficients of lift, drag and pressure drag and the aerodynamic efficiency of the following investigations are given. Application of the optimized SCB2 design on the wing-body configuration showed an increase in the total aerodynamic efficiency of 3%. The second and third columns of Table 1 give the results of the clean wing-fuselage model at the design point and the model with optimized SCB, both without stablizier. The gain in L/D is due to a reduction of total drag by approximately 2%, and an increase in lift of about 1%. Figure 4 shows the spanwise distributions of lift and drag coefficients on the outboard wing for the clean case and the case
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Table 1 . Simulation results for DLR F11 at Ma = 0.85 and Re = 40 · 106 Case cL cD cD,p L/D
clean w/o htp 0.52830 0.02282 0.01614 23.149
scb w/o htp 0.53435 0.02241 0.01569 23.849
clean w/ htp 0.49970 0.02184 0.01554 22.881
0.7
scb w/ htp 0.50492 0.02139 0.01509 23.606
scb elastic 0.53097 0.02209 0.01539 24.034
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with SCB2. Both, the increase in lift and the decrease in drag in spanwise direction can clearly be seen. It must be mentioned, that the increase in lift only results due to the constant AoA specified for this investigation. If constant lift is considered instead, a reduced AoA results due to the lift increasing effect of the SCB. The impact on the spanwise lift distribution, however, remains. DLR F11, Re=40·10 , Ma∞=0.85, α=2.5 , η=0.85 6
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Trimmed simulations were performed for the clean wing F11 model without SCB as well as for the model with SCB, both with rigid flight shape. The
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horizontal tail plane was added as compared to the previous results. Again, a constant angle of incidence was specified, thus, trimming was carried out with respect to the pitching moment My only. The equilibrium of lift and mass was not considered in this case in order to keep the constant AoA and maintain comparable flow conditions along the wing. In Table 1 the aerodynamic coefficients are given in rows four and five. A comparison between the untrimmed and the trimmed case without SCB shows, that due to the htp down force, trimming of the model at constant AoA leads to reduced total lift at the given flow conditions. However, a separate analysis of the wing proves, that the wing lift itself is hardly influenced by the htp. Due to the wing downwash, the htp down force yields a negative pressure drag in this case, which overcompensates the lift-induced drag of the htp. As a result, overall drag is reduced in the trimming process. However, the dominating fact is the lower overall lift and the resulting reduction in aerodynamic efficiency. The same is true for the trimmed configuration with SCB. Again, the predominant effects are the decrease in lift and the compensation of the induced drag of the horizontal tail plane by the wing downwash. No discernable influence of the htp on the SCB performance can be stated and the resulting benefit of the SCB on L/D remains 3% for the trimmed case. The wing loading, changed by the SCB, is expected to influence wing deformation, if structural flexibility is considered. Therefore, aeroelastic computations for the wing-body configuration of the F11 were conducted with the SCB2 as it was optimized for the rigid wing-body configuration. Trimming was not yet considered in the process, in order to reduce computational effort and to better seperate the various effects. Elastic deformations were computed based on the differences in the load distribution due to the SCB. A comparison of the pressure distributions to the rigid case is shown in Figure 5 for η = 0.85 on the outboard wing. For the underlying generic structural model, a change in twist of the wing tip of θ ≈ −0.06◦ resulted, due to the changed aerodynamic loads by the SCB. The increase in lift reduces by around 60% due to elastic deformations. A reduction in lift below the SCB design point leads to an increase in aerodynamic efficiency for the configurations with and without SCB. Figure 6 shows a schematic view of the lift polars for these configurations. While the efficiency of the elastic model is increased compared to the design point, it is still below the value for the rigid model with the same lift. This is a result of the design lift coefficient being chosen above the flight conditions for maximum aerodynamic efficiency of the F11. Therefore, the slight reduction in efficiency of the SCB as it is operating outside of its design point is overcompensated by the increase in L/D as the whole model operates closer to its own maximum efficiency condition. The relatively small reduction in SCB performance results from the more robust SCB2 design. It is concluded, that it is acceptable for future investigations to rely on a rigid wing shape for SCB design.
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A parameterization of shock control bump contours on finite wings of realistic transport aircraft configuration was developed and applied to the optimization of an SCB on the rigid F11 wing-body model. Total aerodynamic efficiency was increased by an SCB applied to the outboard wing of the DLR F11. A coupling module for simulation of trimmed free flight of an aircraft with flexible structure was implemented and applied. Computations of the trimmed aircraft with and without SCB were performed for the rigid F11 model and the influence of the SCB on trim drag was found to be negligible. Furtheron, aeroelastic computations of the model with SCB showed no significant effect of the aeroelastic wing twist due to the SCB on SCB performance itself. The main reason for that is the enhanced robustness of the SCB2 design over the original SCB1 design. Further investigations will concentrate on this property and explore the potential of robust SCB designs.
References [1] N. Kroll, C. C. Rossow, D. Schwamborn, K. Becker and G. Heller: MEGAFLOW – a Numerical Flow Simulation Tool for Transport Aircraft Design. In: Proceedings of the 23rd International Congress of Aeronautical Sciences, ICAS, 2002. Toronto. [2] M. Kutzbach, T. Lutz and S. Wagner: Investigations on Shock Control Bumps for Infinite Swept Wings. In: 2nd AIAA Flow Control Conference, Portland, Oregon, USA, June 28 - July 1, 2004. [3] T. Lutz, A. Sommerer and S. Wagner: Parallel Numerical Optimisation of Adaptive Transonic Airfoils. In: Fluid Mechanics and its Applications, Vol. 73, Kluwer Academic Publishers, 2003. Symposium Transsonicum IV, G¨ ottingen, Germany, September 2 - 6, 2002. [4] T. Mauermann: The FEMSPLINE-Method for Connecting Condensed Structural Models and Aerodynamic Models. Tech. Rep., Airbus Deutschland GmbH, Loads & Aeroelastics EGLG31, 2005. [5] M. Michael: Personal Communictation on Structural Models of Free Flying Aircraft. unpublished, 2006. Institut f¨ ur Flugmechanik und Regelungstechnik, Universit¨ at Stuttgart. [6] A. Sommerer: Numerische Optimierung adaptiver transsonischer Profile. Dissertation, Institut f¨ ur Aerodynamik und Gasdynamik, Universit¨ at Stuttgart, 2005. [7] E. Stanewsky, J. Delery, J. Fulker and P. de Matteis: EUROSHOCK II: Drag Reduction by Shock Boundary Layer Control. Notes on numerical fluid mechanics and multidisciplinary design, Vol. 80. Springer, 2002.
A Flat Plate Experiment for Investigations of Vortex Generator Jets at High Reynolds Number Jens Ortmanns, Christian J. K¨ ahler, and Rolf Radespiel Technische Universit¨ at Braunschweig, Institut f¨ ur Str¨ omungsmechanik, Bienroder Weg 3, 38106 Braunschweig, Germany
[email protected]
Summary A flat plate boundary layer experiment is designed to examine the physical mechanism and the performance of vortex generator jets at Reynolds numbers up to Reθ = 20000. The investigation of vortex generator jets at this flow state is of fundamental importance for the assessment of flow control at takeoff and landing conditions. First results show that the optimal skew angle is about β = 15◦ for the slot actuator. The formation of the longitudinal vortex structures can be described in detail by the derived vortex topology based on the measurement results.
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Introduction
It is well known that vortex generator jets can prevent flow separation due to increased mixing in turbulent boundary layers [1]. By generating longitudinal vortex structures high momentum fluid is shifted towards the near-wall region which causes the velocity profiles to assume a less critical shape. This flow control strategy can be applied to improve the lift performance of wings for instance [2, 3]. The application of pulsed vortex generator jets promises enhanced system performance as a result of nonlinear interaction between the induced vortex structures and the turbulent boundary layer [4]. In order to use these effects both, the flow state and the response behavior of the flow has to be understood [5]. An optimal frequency f and duty-cycle ∆ – the time of blowing relative to the period – can be estimated by analyzing the start-up and the decay of the dynamic process. The challenge of implementing effective dynamic vortex generator jets at real flight conditions in take-off and landing configuration of aircraft is associated with the change in boundary conditions due to the Reynolds number and large pressure gradients. This leads to the questions whether the observations made at low Reynolds numbers are still valid and what the natural frontiers of this flow control strategy are. Usually several actuator parameters determine the vortex generation. Changing the flow state to high Reynolds C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 88–95, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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number imposes new considerations: A fixed velocity ratio λ = wj /U leads to an increasing jet exit velocity wj (U is the free stream velocity) which is reasonably limited at the speed of sound. The actuator frequency has to rise if we assume that the optimal reduced frequency F + = f · U/l has a constant value (l is the considered length). Furthermore, the ratio of actuator geometry and boundary layer size changes dramatically compared to the boundary conditions at low Reynolds number. Based on these considerations it is likely that the results of low speed measurements are not directly transferable to flight conditions. In order to examine these questions a flat plate experiment with an adjustable pressure gradient was designed. The present paper describes the design and the performance of the experiment. Initial investigations on a common actuator design – a skewed slot – are presented, too.
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A Flat Plate Experiment for the MUB
The Modellunterschall Windkanal Braunschweig (MUB) is a closed circuit atmospheric wind tunnel with two test sections, see [6]. For the flat plate experiment the 1.3 × 1.3 m2 test segment was used which has a length of 5.7 m. The wind tunnel is powered by a DC voltage motor with 300 kW electrical power supply and a maximum free stream velocity of U = 55 m/s can be achieved. The turbulence level determined at U = 53 m/s is about 0.2 %. With a heat exchanger system (Q = 300 kW) the flow can be held thermally stable at 10 ◦ C over ambient temperature with an accuracy of ±0.5 ◦C. The flat plate is based on four smooth laminated wooden segments which can be adjusted relative to the tunnel floor. They are installed at a height of about 300 mm with an aluminum base frame. The first segment is equipped with an elliptical leading edge with an aspect ratio of 6 : 1 and the overall length is lp = 5700 mm, see figure 1. In the second part of the test section
Fig . 1 . Mechanical drawing of the flat plate experiment in the MUB. The distance between the plate and the displacement body and its angle of attack can be set individually.
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different inserts with individual experimental setups (pressure holes, actuators, etc.) can be applied. To generate a well defined pressure gradient an airfoil can be installed inside the test section as outlined in figure 1. The airfoil designed for the pressure variation is a Wortmann FX 63-137 airfoil with a chord length of lc = 1000 mm. Its suction side faces the direction of the plate surface. The distance between the leading edge of the flat plate and the lc /4-line is 4400 mm. It is connected with the wind tunnel by a movable frame which is guided by high precision shafts and can be fixed by a clamp connection. Thus, a continuous variation of height between the displacement body and the plate is possible. The airfoil model is constructed of fiber glass with wooden side ribs and has an embedded steal pivot on each side at the lc /4-line for the connection with the base frame. The angle of attack can be varied around this point of rotation. The model is equipped with ten pressure holes for comparison of the pressure distribution with the design data. The preliminary numerical investigations with the MSES-code have shown that it is possible to force the flat plate boundary layer to separate while the airfoil flow is still attached. The computations were done for the two element configuration “flat plate and displacement body” with solid wall boundary conditions according to the wind tunnel geometries, see [7]. Results for the flow state along the plate surface are shown in figure 2 for different configurations at a Reynolds number of Re = 16.5 · 106 based on lp and a Mach number of Ma = 0.147. The velocity distribution induced by the displacement body can be seen on the left hand side. By varying the parameter angle of attack and height the pressure gradient and therewith the load on the flat plate boundary layer can be changed. The negative values in the cf -graph in figure 2 indicate the boundary layer separation. Additionally, the rise of the kinematic shape factor H12 shows the onset of flow separation. The load of the boundary layer can be estimated by the distribution of Reθ (x/lp ). 3
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Fig . 3. Results of various measurements to quantify the flat plate boundary layer: Top left: Pressure measurement along the plate surface for installation at a zero pressure gradient; bottom left: Combination of µ-PIV and 3C2D-PIV results in the velocity profile at x = 4650 mm and right: Hot wire measurements to determine the velocity spectra at x = 4650 mm, z = 5 mm (z + = 280)
To determine the properties of the turbulent boundary layer along the flat plate various measurements were performed. By varying the base frame height the plate could be adjusted in the wind tunnel to ensure a pressure gradient which equals zero. The pressure distribution was determined by using a PSI 8400 pressure transducer system with 1-psi modules at 25 pressure holes. A result for a free stream velocity of U = 50 m/s is shown in figure 3 top left: Except for the first measuring point which is influenced by the flow around the leading edge the variations of the velocity distribution have a relative error of ∆e = ±0.4 % with respect to its mean value. For the determination of the velocity profiles at the test section Micro-PIV (µ-PIV) and Stereoscopic-PIV (3C2D-PIV) measurements were investigated. The µPIV setup was similar to the one applied in [8]. By identifying the velocity gradient in the near-wall region the wall-shear-stress was calculated to normalize the velocity profiles in viscous units. The results from the 3C2D-PIV measurements are the velocity distribution up to the edge of the boundary layer. Figure 3 bottom left shows the graphs for the boundary layer
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profiles at U = [25, 50] m/s which are in good agreement with the common literature. Based on these profiles the Reynolds number of the momentum thickness was calculated as Reθ = 12672 and 23953 for the two free stream velocities, respectively. Hot wire measurements were made to determine the frequency spectra of the boundary layer flow. Figure 3 right shows the result at z = 5 mm (z + = 280): After bandpass filtering the data the evolution of the frequency spectra agrees suitably well with comparable graphs, see e.g [9]. It can be seen that the frequency spectrum is completely resolved in the f −5/3 -range. Thus, it can be summarized that the boundary conditions of the flat plate experiment are very well suited for experiments at high Reynolds numbers.
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Vortex Generator Jets at High Reynolds Number
The systematic investigations on vortex generator jets starts with the determination of the skew angle effect at zero pressure gradient and leads to a detailed look into the vortex mechanism for the slot actuator, which has a size of h × b = 32 × 1 mm2 and a wall-normal blowing direction. The actuator position is x = 4650 mm behind the leading edge of the flat plate, which corresponds to Rex = [7.67, 15.34] · 106 for the two investigated free stream velocities U = [25, 50] m/s, respectively. Two velocity ratios λ = [2.5, 5] were chosen and controlled via the mass flux through a flow meter RA65 manufactured by Kirchner & Tochter. Oil flow visualization was done for the different flow parameters at the angles β = [0, ..., 90]◦ (∆β = 10◦ ) and β = 45◦ , additionally. The results indicate which configuration is most effective in shifting fluid from the outer flow towards the wall. As mentioned in [11] the footprint of the flow induced by the vortex generator jets is dominated by the flow around the jet orifice, the vortex separation line and the reattachment line, cf. figure 4. It can be
Fig . 4 . Oil flow visualization for the slot actuator at a skew angle of β = 10◦ at U = 50 m/s and λ = 2.5
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seen that the longitudinal vortex production is based on the flow around the actuator. The streamlines with high degree of curvature indicate a small recirculation region. With increasing distance from the orifice in main stream direction the degree of curvature decreases and in the far field the stream lines align closely with the separation line. A qualitative estimation of the vortex strength can be derived from the evolution of the stream line curvatures. The determination of the vortex structure was done by means of 3C2DPIV measurements in planes perpendicular to the main flow direction. Figure 5 shows the results for the slot actuator 50 mm behind the actuator at different skew angles based on 400 double images. The effect is clearly visible: With increasing skew angle β the dominant vortex structure rotates towards the wall. Therewith connected is the position of the blockage area and its size. In all cases the vortex structure shifts high momentum fluid to the near-wall region but taking the locally decelerated flow area into account the overall benefit is different. Considering this effect, the most effective skew angle for the slot actuator investigated here is about β = 10 − 20◦ . A description of
Fig . 5 . Induced vortex flow fields 50 mm behind the actuator for the slot actuator at U = 25 m/s and λ = 5. The velocity change in main stream direction ∆u is shown by subtracting the reference flow (+ = positiveˆand − = negative, ˆ only every second vector is displayed).
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the optimization of vortex generator jets at Reθ > 20000 with respect to the change of the wall-shear-stress distributions is given in [10]. There, an optimal skew angle of β ≈ 15◦ for the slot actuator is found which is in good agreement to the result observed in our work and this validates the present experimental setup. Furthermore the topology of the vortex system is shown in figure 6. The streamlines in the background are based on the 3C2D-PIV measurements with 2C2D-PIV measurements in an additional wall-parallel plane. From these streamlines in each plane the vortex flow was estimated. The early vortex structures are of different strength and as a result the weaker one moves around the dominant vortex in a spiral motion. So, two effects are overlaid: On the one hand the starting vortex structures induced by the shape of the orifice and on the other hand the overall spiral motion which is related to the skewness of the actuator relative to the incoming flow direction. The dominant vortex moves out of the wake of the actuator position according to its self-induced flow field. The fluid transport from the outer region of the flow is clearly visible in the streamlines in the third plane.
Fig . 6 . Vortex topology of a skewed slot actuator, β = 20◦
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A flat plate boundary layer experiment was designed and validated to study the performance of actuators at Reynolds number up to Reθ = 20000. First results from the investigations indicate that in case of the slot actuator the most effective skew angle is about β = 15◦ based on the estimation of the relative velocity fields. This is in agreement with the measurements discussed in [10]. With the application of 3C2D-PIV and µPIV measurements a detailed
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observation of the longitudinal vortex flow can be achieved. The transport mechanism is a result of two effects: The starting vortex structures associated to the jet orifice and an overall spiral motion as a result of the skewness transports high momentum fluid into the near-wall region. In the future the experimental setup will be used for detailed investigations on the interaction of the longitudinal vortex structures and the turbulent boundary layer at high Reynolds number and different streamwise pressure gradients.
Acknowledgments The authors gratefully acknowledge the DFG (Deutsche Forschungsgemeinschaft) for financial support.
References [1] J. P. Johnston, M. Nishi: “Vortex Generator Jets – Means for Flow Separation Control”. AIAA Journal, Vol. 28, No. 6, pp. 989-994, 1990. [2] J. C. Magill, H. R. McManus: “Exploring the Feasibility of Pulsed Jet Separation Control for Aircraft Configurations”. Journal of Aircraft, Vol. 38, No.1, January-February, 2001. [3] P. Scholz, J. Ortmanns, C. J. K¨ ahler, R. Radespiel: “Leading Edge Separation Control by Means of Pulsed Jet Actuators”. AIAA 2006-2850, 3rdAIAA Flow Control Conference, San Francisco, USA, 5-8 June 2006. [4] J. Ortmanns, M. Bitter, C. J. K¨ ahler: “Visualization and Analysis of Dynamic Vortex Structures for Flow Control Applications by Means of 3C2D-PIV”. 12th Int. Symp. on Flow Visualization, G¨ ottingen, Germany, 10-14 Sept. 2006. [5] A. Darabi and I. Wygnanski: “Active management of naturally separated flow over a solid surface. Part 1: The forced reattachment process. Part2: The separation process”. Journal of Fluid Mechanics, 510, pp. 105-144, 2004. [6] http://www.tu-braunschweig.de/ism/institut/wkanlagen/mub. [7] M. Drela: “A User’s Guide to MSES 2.95”. MIT Computational Aerospace Sciences Laboratory, 1996. [8] C. J. K¨ ahler, U. Scholz, J. Ortmanns: “Wall-shear-stress and near-wall turbulence measurements up to single pixel resolution by mean of long distance micro-PIV”. Exp. in Fluids, vol. 41, no. 2, pp. 327-341, 2006. [9] H. Schlichting, K. Gersten: “Grenzschichttheorie”. Springer-Verlag, 1997. [10] G. Godard, J. M. Foucaut, M. Stanislas: “Control of decelerating boundary layer. Part 2: Opimization of slotted jets vortex generators”. AST, 10, pp. 394-400, 2006. [11] X. Zhang, M. W. Collins: “Measurements of a longitudinal vortex generated by a rectangular jet in a turbulent boundary layer.”, Phys. Fluids, 9 (6), June 1997.
Measurement of Unsteady Surface Forces by Means of Piezoelectrical Copolymer Coatings J. Domhardt, J. Leuckert, I. Peltzer, and W. Nitsche TU-Berlin, Institut f¨ur Luft- und Raumfahrt, Marchstr. 12, 10587 Berlin, Germany
[email protected] http://aero.ilr.tu-berlin.de/
Summary The Pressure Sensitive Copolymer Coating (PSC) with piezoelectric properties is introduced as a new surface measurement technique. In contrast to conventional surface measurement techniques the PSC combines high spatial and temporal resolution, and can be applied to arbitrarily formed surfaces. The linear relationship between the mechanical pressure and the electrical voltage due to the piezoelectric effect is established by calibration measurements. Furthermore, first applications in transition experiments in incompressible as well as in compressible flows are presented. Finally, investigations of the flow around a wall mounted cylinder are presented.
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Conventional surface pressure measurement methods can be divided into two categories. Some are developed to reach a high temporal resolution, e.g. Kulites. The disadvantage of these methods is the limited spatial resolution. In contrast, the pressure sensitive paint technique (PSP [1]) is developed with respect to a high spatial resolution for arbitrarily formed surface applications. However, its application for dynamic flow phenomena is restricted. Therefore, the Pressure Sensitive Copolymer Coating surface measurement technique (PSC) is developed to combine the advantages of these conventional methods. The PSC is based on the piezoelectrical properties of a surface coating made of vinylidene fluoride and trifluoroethylene which can be sprayed on arbitrarily formed surfaces. This coating permits flow measurements with very high spatial as well as very high temporal resolution.
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Measuring Principle with PSC
The piezoelectric principle is based on the linear relationship between electrical polarisation and a mechanical force. There are two possibilities to use this effect in technical applications. On the one hand, a piezoelectrical material will be deformed mechanically if an electric field is applied (inverse piezoelectric effect). Therefore, C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 96–103, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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piezoelectrical materials are used as actuators, e.g. in active transition control investigations. There are also a multitude of piezoelectric sensor applications using the direct piezoelectric effect, where a mechanical deformation generates a local charge displacement in the piezoelectric material. In recent years, piezoelectric sensors based on foils made of vinylidene fluoride have been successfully applied to boundary layer investigations, especially for transition detection [2]. Such a piezoelectric sensor reacts to unsteady surface forces of the boundary layer with a local charge displacement. This charge displacement (Q), which is proportional to a pressure fluctuation (p ) and the fixed area of the sensor (Asensing ) is converted into a measurable voltage (V ) using a charge amplifier [3]. This linear relationship is shown in equation 1, where d33 is the piezoelectric constant of a normal charge displacement parallel to the pressure fluctuation, which depends on the piezoelectric material. V (Q) = d33 · p · Asensing
(1)
The piezoelectrical polyvinylidene fluoride is conventionally used as a foil and is therefore mainly limited to plane applications, whereas the PSC is developed for the application on 3D-surfaces as shown in Fig. 1 (a). Furthermore, the PSC is optimised with respect to the signal path compared to conventional piezo measurements. In a first step, a very thin multilayer circuit board was developed to separate the sensing element (first layer) and the electric taps (final layer). A benefit of using this kind of electrical connection is the shortened signal path. Additionally, a heating layer is included to increase the signal to noise ratio if necessary, as well as a ground layer to reduce the electrical noise. The described finished multilayer circuit board is bent and integrated into the model. As presented in Fig. 1 (b), the copolymer is coated onto the cambered upper layer and finally vacuum metallised. A pressure sensitive coating can be achieved by a discrete activation of the copolymer with a corona discharge. The technique therefore allows for an arbitrarily distributed matrix of pressure sensitive sensing elements.
Fig. 1. Principle of PSC: (a) Measurement principle, (b) Electronic details
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Calibration
Calibration investigations of PSC coatings were performed in order to validate the linear relationship between the pressure fluctuation (p ) and the measurement output voltage (V ) as presented in equation 2. Furthermore, these experiments were conducted in order to identify the calibration coefficient (C), which depends on the PSC coating itself and the electrical taps. Moreover, the calibration tests were aimed at the optimisation of the piezoactive area with respect to sensing shape and dimensions. V = C · p
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For this purpose, a calibration chamber was designed as shown in Fig. 2 (a). Pressure fluctuations generated by two actuators are fed into the chamber, which is equipped with both a pressure sensor serving as reference to the PSC coating and a thermocouple to monitor the temperature. The PSC is mounted onto the chamber, thereby locking it pressure tight.
Fig. 2. Calibration investigation of PSC: (a) Experimental setup, (b) Calibration results for different sensing areas
The PSC coatings were calibrated in a frequency range of 0. 2 kHz ≤ f ≤ 1. 0 kHz by generating a sinusoidal sweep. In Fig. 2 (b) the results of the calibration for various piezoactive areas are shown. The linear relationship between the pressure fluctuation and the measurement voltage is clearlyvisible. Moreover, an increase of the sensitivity due to an increase in the piezoactive area can be observed. At the moment , the described calibration technique is restricted to small PSC surfaces as shown in Fig . 2.(a) For the calibration of 3d shaped surfaces another calibration setup is currently developed.
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First wind tunnel experiments were performed in order to examine the reliability of PSC when applied to surf aceflo w measurements.Transition investigations with
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both flat and curved PSC coatings were carried out. Furthermore, PSC was applied to the flow around a cylinder stump for surface-wide measurements. 4.1
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In a first step, transition experiments with PSC coatings were performed in a low speed wind tunnel on a 2D laminar airfoil geometry, which has already been investigated in previous experiments [4]. The experimental setup, shown in Fig. 3 (a), consists of a NACA0008 nose in the range of 0.00 ≤ x/c ≤ 0.40 with a perturbation source at x/c = 0.3. The nose is followed by a plate insert equipped with various measurement techniques. The flat PSC probe was applied to this insert covering a range of 0.444 ≤ x/c ≤ 0.524. Additionally, the insert was equipped with a surface hot-wire array arranged at the same stream-wise positions to validate the results. The experiments were performed at free-stream velocities of 18m/s ≤ u∞ ≤ 24m/s (1.5 × 106 Rec ≤ 2.0 × 106 ), varying the angle of attack in the range of −3.0◦ ≤ α ≤ 0.0◦ .
Fig. 3. Incompressible transition investigations, (a) Experimental setup, (b) Frequency spectra of a selected surface hot-wire sensor at varying angles of attack (u∞ = 24m/s, x/c = 0.444)
The frequency spectra of a selected surface hot-wire sensor (x/c= 0.444) for a free-stream velocity of u∞ = 24m/s and different angles of attack are presented in Fig. 3 (b). The laminar boundary layer at this position is clearly recognisable by the low frequency level for α = −2.0◦ . In contrast, an increase in the magnitude as well as the fundamental Tollmien Schlichting (TS)-frequency range between 250Hz ≤ fT S ≤ 650Hz characterises the transitional boundary layer for α= −1.5◦. In addition to the fundamental TS-frequency the first and the second higher harmonic are also visible. Furthermore, the turbulent boundary layer state is clearly present for α = −1.0◦ , indicated by the increase in the magnitude level. In comparison, the frequency spectra of the PSC for the same flow conditions and stream-wise position are shown in Fig. 4 (a).
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Fig. 4. PSC results of incompressible transition investigations at u∞ = 24m/s: (a) Frequency spectra at varying angles of attack at x/c = 0.444, (b) Time traces at α = −1.5◦
The fundamental frequency range of 250 Hz ≤ fT S ≤ 650 Hz is again clearly detected for α = −1.5◦ and coincides with the reference sensor. Moreover, the increase of the frequency level by increasing the angle of attack to α = −1.0◦ is also shown. The deviations in the spectra of the surface hot-wire sensor and the PSC (Fig. 3 (b), Fig. 4 (a)) at a frequency range higher than 1kHz are due to the different measuring principles. While a wall mounted hot-wire detects the velocity fluctuations of the flow in the viscous layer by means of a thin heated wire, PSC directly detects pressure fluctuations in the boundary layer. However, both measuring principles are able to capture the fundamental Tollmien-Schlichting instabilities.
Fig. 5. Frequency spectra of PSC: (a) Varyious free-stream velocities at α = −1.5◦ and x/c = 0.493, (b) Unaffected flow in comparision to active perturbation at u∞ = 20m/s, α = −1.0◦ and x/c = 0.444
The time traces of PSC (u∞ = 24m/s, α = −1.5◦ ) presented in Fig. 4 (b) clearly indicate the TS wave packages. Furthermore, the convection velocity can be determined to uc = 13.2m/s calculated from the time shift of 7.5ms over a distance of x/c = 7.62%. In addition, the influence of an increase of the free-stream velocity
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on the TS-frequency range is shown in Fig. 5 (a), showing a shift of the fundamental range to higher frequencies. Fig. 5 (b) shows the influence of a monofrequent perturbation (a = 200mV , f = 370Hz) on the boundary layer at u∞ = 20m/s and α = −1.0◦ . The transitional boundary layer is forced to become turbulent due to this perturbation, indicated by the disappearance of the TS-frequency band and the increase in the magnitude level. 4.2
Transition Measurement in Compressible Flow
Within the framework of the EC-Project ”Fliret”, the influence of roughness on the laminar-turbulent transition was examined at the Pilot European Transonic Windtunnel (PETW). For this, a two-dimensional curved airfoil model was equipped with PSC including integrated heating and temperature monitoring. The PSC coating consists of 24 active elements of the dimensions 1.5 × 3.0mm (stream-wise × span-wise) to cover the range of 0.20 ≤ x/c ≤ 0.74, Fig. 6 (a). Due to the expected 2D character of the flow, six active elements were shifted to the stream-wise position of the first row to increase the spatial resolution in the range of 0.22 < x/c < 0.37. Furthermore, a third row was included to validate the 2D character of the flow. In addition to the PSC, the model was also equipped with Temperature Sensitive Paint (TSP) in the range of 0.10 ≤ x/c ≤ 0.85 by DLR. In order to measure the mean pressure distribution, the model is also equipped with pressure taps both on the suction as well as on the pressure side. The instrumented 2D PETW model is shown in Fig. 6 (a).
Fig. 6. Compressible transition investigations: (a) Experimental setup, (b) RMS values at varyious angles of attack and ambient flow conditions (M = 0.24, Re = 1 × 106 , Rz = 0.1µm)
The RMS values of PSC at ambient flow conditions (M = 0.24, Re = 1 × 106 ) and varying angles of attack for the reference roughness of Rz = 0.1µm are presented in Fig. 6 (b). The constant low RMS levels detected at α = 2.0◦ characterise the laminar boundary layer state. When increasing the angle of attack to α = 2.5◦ , an increase of the RMS values at x/c > 0.5 becomes visible indicating the transional
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boundary layer along with increased pressure fluctuations. A further increase of the angle of attack leads to an upstream shift in the rise of the RMS values characterising the upstream shift of transition due to the adverse pressure gradient. 4.3
Flow Around a Wall Mounted Cylinder
In addition to the transition experiments using PSC, the surface-wide flow measurement is another field of application. Therefore, it was applied to detect the unsteady flow phenomena at a wall mounted cylinder with a height to diameter ratio of H/D = 2 (H = 240mm, D = 120mm), which was pivot-mounted on a rotatable insert. The experiments were carried out in a low speed wind tunnel at the University of Rostock for a Reynolds number of ReD = 2 × 105 . The cylinder was equipped with a flat PSC coating covering a quarter of its top cover and a curved PSC device, which covered its surface with two sensing rows. Furthermore, the rotatable insert was equipped with PSC consisting of two active rows for the investigation of both incident flow and wake. Since all PSC probes were based on round sensitive elements and the cylinder was pivot-mounted, this arrangement allows a surface-wide measurement of the cylinder flow when assembling the separate measurements at varying angular positions. Fig.7 shows the standard deviation on and around the cylinder in comparison with the frequency spectra at selected positions measured with the PSC probe. Clearly visible is the increase in the standard deviation (Fig. 7 (a)) of position 3 in comparison with positions 1 and 2. Moreover, the frequency peak of f = 34Hz at position 3 in Fig. 7 (b) represents the characteristic frequency of the wake vortices. This result agrees well with numerical simulations and combined Laser-Doppler Anemometry (LDA) and Time Resolved Particle Image Velocimetry (TR-PIV) measurements [5].
Fig. 7. Investigations on a wall mounted cylinder at ReD = 2 · 105 : (a) Standard deviation (StdDev.), (b) Frequency spectra at selected positions of the rotary insert
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Pressure Sensitive Copolymer Coating was introduced as a new surface measurement technique with a high spatial and temporal resolution for the application on both flat and curved surfaces. In addition to the calibration of the PSC, a first technical application was shown to detect both unsteady surface forces as well as characteristic structures in transitional boundary layers in incompressible and compressible flows. Furthermore, the successful application of PSC to a surface-wide flow measurement at a wall mounted cylinder was presented.
Acknowledgements The research on Pressure Sensitive Copolymer coating technique is supported within the framework of the DFG priority programme ”Image based measuring techniques for the flow analysis” and the EC-Projects ”Flight Reynolds Number Testing (Fliret)” and ”Testing for Laminar Flow on New Aircraft (Telfona)”.
References [1] R.H. Engler, C. Klein, O. Trinks.: Pressure sensitive paint systems for pressure distribution measurements in wind tunnels and turbomachines. Measurement Science and Technology Vol. 11 pp. 1077-1085. 2000. [2] W. Nitsche, P. Mirow: Piezo-electric foils as a means of sensing unsteady surface forces. Experiments in fluids Vol. 7 pp. 111-118. Springer Verlag 1989. [3] S. Bauer, et al.: Piezo-, Pyro- and Ferroelectric Polymers. Piezoelectric Materials: Advances in Science, Technology and Applications Vol. 76, pp. 11-19, Kluwer Academic Publishers, Dordrecht 2000. [4] D. Sturzebecher, W. Nitsche: Active Cancellation of Tollmien-Schlichting Instabilities on a Wing Using Multi-Channel Sensor Actuator Systems. International Journal of Heat and Fluid Flow Vol. 24, Issue 4, pp. 572-583, Elsevier Science 2003 [5] O. Frederich, J. Scouten, E. Wassen, F. Thiele, F. H¨uttmann, M. Brede & A. Leder: Investigation of the Flow around a Wall-Mounted Finite Cylinder. To be published in: Journal of Turbulence.
Studies of Reynolds Number Effects on Wing Tip Vortex Positions by Means of Laser Light Sheet (LLS) and Background Oriented Schlieren (BOS) Technique Under Cryogenic Conditions D. Pallek and F. Klinge Deutsches Zentrum für Luft- und Raumfahrt Institut für Aerodynamik und Strömungstechnik, Bunsenstraße 10, 37073 Göttingen, Germany
[email protected]
Summary The Background Oriented Schlieren Method (BOS) and the Laser Light Sheet technique (LLS) were successfully adapted to the cryogenic European Transonic Windtunnel (ETW) and applied to wing tip vortex investigations. Applying optical measurement techniques to ETW is difficult, because of the extreme wind tunnel conditions (temperature, pressure, poor optical access). The near field of a half model of a large modern transport aircraft with landing configuration was investigated at different angles of attack and Reynolds numbers. The results show a small, but significant dependency of the wing tip vortex position on the Reynolds number for certain angles of attack. The differences are up to 3% of the half-span width.
1 Introduction Wake vortex turbulence induced by large transport aircraft is problematic for airport capacity because of its potentially hazardous impact on following aircraft. Safety rules defining minimum distances between aircraft landing and taking off are based on the strength and temporal duration of dissipation of the vortex turbulence. The wing tip vortex position is of major interest because interactions with vortices generated from, for example the elevator of the airplane are so designed to accelerate its decay. Experimental data are needed to enable quick and accurate vortex modelling. Up to now, these data come mostly from wind tunnels which cannot simulate the ‘real’ Reynolds and Mach numbers. It is usually assumed that this mismatch does not affect the results significantly. The objective of this work was to adapt the stereoscopic Background Oriented Schlieren Method and the Laser Light Sheet Technique to the conditions of a cryogenic wind tunnel, thus providing the experimental tools to be able to investigate the question of whether the Reynolds number has a significant impact on the position of a wing tip vortex. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 104–112, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007
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2 Principles 2.1 The Stereoscopic Background Oriented Schlieren Method (BOS) The density gradients, which are induced by the vortices in the wind tunnel flow can be visualised by the optical measurement technique BOS. The underlying principle of this technique is the deviation of light rays passing through density gradients with their direction of propagation perpendicular to the density gradient, and has been described in detail in [2]. The set-up for stereo BOS measurements requires only two cameras focused on two separate backgrounds, which consist of randomly distributed dot patterns applied by painting. For each camera at least two pictures have to be acquired: a reference image without the density effect and a measurement image, with the density effect to be measured. The principle set-up is shown in Figure 1. Background plane
Increasing density
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Without the density effect, one point of the background pattern is imaged on the image plane as indicated by the black line. With the density effect, the light rays from the background will be deflected by the angle İ and will be imaged onto another location in the image plane, as indicated by the dashed red line. Thus, due to the density gradients, the background pattern will be imaged on another position onto the measurement image. The local distance between the two images of the background pattern on the image plane is named ¨x. The displacement of this pattern can be detected by using standard cross correlation algorithms. The cross correlation algorithms developed for Particle Image Velocimetry (PIV) are well suited for this task. For the determination of the local position of axi-symmetrical density distributions (like vortices), a software package named VRIEDER was developed at DLR [1]. The experimental set-up requires two cameras with crossing camera axes forming an investigation volume where the imaged regions are overlapping. VRIEDER is especially suited for axi-symmetrical density objects without complicated structures. 2.2 The Laser Light Sheet Technique (LLS) LLS is a well known technique which often has been used for the visualisation of vortices in wind tunnels: The flow is seeded with particles, illuminated by a laser light sheet and imaged by a camera. By orienting the laser light sheet perpendicular to the
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flow, vortices can be seen as dark holes in the bright light sheet. This dark hole is due to the strong centrifugal forces in the vortex core, which carry the particles out of the centre.
3 Experimental Set-Up The experiments were carried out in the European Transonic Windtunnel (ETW) in Cologne, Germany, which is a pressurised wind tunnel operating under cryogenic conditions. This wind tunnel was utilised due to its capabilities of changing the Reynolds number without thereby changing the wing load over a wide range of Reynolds numbers up to ‘real flight’ Reynolds numbers. The ETW is a closed circuit wind tunnel of type “Göttingen” with a 2 m x 2.4 m test section (Figure 2). Due to the extreme environmental conditions (temperatures down to 110 K and pressures up to 450 kPa, mechanical vibrations) optical measurement techniques are difficult to implement in this wind tunnel. So the aim of the experiment described here was both to adapt the optical measurement techniques BOS and LLS for application to the ETW and to determine the effect of Reynolds number on the wing tip vortex position.
Fig. 2. The European Transonic Wind Tunnel ETW
Since the only components needed for stereo BOS experiments are cameras and backgrounds, their adaptation to the ETW could be achieved easily. For the cameras, heated housings providing room temperature had to be used. For the backgrounds a colour which resists the low temperatures without peeling off had to be found. For simplicity, extra metal plates were used as backgrounds; they could be painted outside the tunnel and were mounted for the test only. The conditions for LLS were more complicated: Since a seeding of the flow with DEHS (a commonly used and approved seeding liquid) was not possible due to a possible damage of the insulating material of the wind tunnel, other seeding particles had to be found. As water had already been used during another test in the ETW for a Doppler Global Velocity (DGV) experiment, these experiences could be used for the seeding in the ETW. The water droplets were produced by two different systems outside the wind tunnel and brought into the flow in the vicinity of the nitrogen inlet.
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Fig. 3. LLS set-up (top view of ETW test section and plenum)
Concerning the laser, the pressure shell is closed and a window does not exist. Additionally, the test section moves significantly with respect to the outer pressure vessel due to temperature differences of about 200 K. These facts do not allow a direct optical coupling between a laser outside the wind tunnel and a light sheet inside. So the laser had to be placed outside the pressure shell, the laser light being coupled into an optical fibre which was led via a junction through the pressure vessel wall and into the plenum (Figure 3). The light sheet was formed by a rotating octagon and a cylindrical lens. As the optical fibre does not allow high power densities due to heating problems, the usage of a high power pulse laser was not feasible. So a continuous wave ArgonIon-laser with 3 Watts output power was used for the illumination of the particles inside the flow. The light sheet was oriented almost perpendicular to the flow direction with a slight downwards component so as not to shine into the BOS cameras, which were mounted right across from the light sheet optics. The LLS cameras had to be mounted into the test section wall at a very oblique angle to the light sheet. To get sharp images, Scheimpflug-adapters had to be used,
Fig. 4. View through test section
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which laterally tilt the lens with respect to the image plane of the camera. Since the heated camera housings offer very limited space, a special downsized Scheimpflugadapter was constructed and used. Figure 4 shows the optical paths inside the test section for both BOS and LLS and the aircraft model: The green area indicates the projected laser light sheet, blue the viewing fields of the LLS cameras, and yellow the viewing fields of the BOS cameras. For the whole set-up the farthest downstream position possible in ETW was chosen. Three BOS cameras were mounted in heated housings attached to the test section walls. One camera was placed in the upper wall viewing down on a background plate mounted on the bottom wall about one half of a wing span behind the wing tip. At the same distance two BOS cameras were placed in the left wall of the test section viewing towards the backgrounds located on the opposite wall of the test section. For an investigation of a flow by means of BOS and VRIEDER at least two cameras viewing the same volume are needed. For the present experiment, two volumes were investigated, one defined by the top wall camera and the upper side wall camera, and the other by the top wall camera and the lower side wall camera. For both BOS and LLS measurements, CCD cameras with a resolution of 1376 by 1040 pixels and sensor cooling were applied. The cameras were connected to computers by fibre optics for storage.
4 Results The near flow field of a half model of a large modern transport aircraft with landing configuration was investigated at different angles of attack and Reynolds numbers. The model has a half-span width of 1.2 m and was mounted on the ETW half model
Fig. 5. Test envelope with three horizons of constant q/E
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cart provided with a balance and a turning table. The landing configuration of the wing was not changed during the tests. The flow velocity was kept constant over the total measurement at Mach = 0.2 (typical landing speed) and the temperature range varied between 285 K and 115 K. Thus, Reynolds numbers between 3 and 25·10 could be achieved. Figure 5 shows the test envelope of the measurements. The Reynolds numbers of the tests were adjusted in such a way as to keep the ratio of the dynamic pressure q to the Young’s modulus E of the model wing constant. Following this constraint, three different horizons of fixed q/E-values were tested (Fig. 5). So, for each horizon, constant wing deformations could be expected even at different Reynolds numbers. Nevertheless, at all measurement points the position of the wing was determined by the ETW deformation measurement system SPT.
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Fig. 6. Cross correlated BOS images. ǻx is colour-coded.
Fig. 7. Vortex position inside the observation volume at Į = 9° for different Re numbers
Figure 6 shows an evaluated BOS image of the lower side wall camera with vectors indicating the virtual movements of the background pattern due to density
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gradients within the observed flow volume (the span-wise displacement is colourcoded; the top wall with the mounted fuselage is located left, flow direction is from top to bottom of the plot). Even though the flow velocity and the associated density fluctuations are relatively small, the BOS set-up is sensitive enough to visualise the wing tip vortex clearly: At the left side of the blue field, the vectors are changing their orientation, which indicates the centre line of the vortex (dashed white line). By combining this result with the data obtained from the camera in the top wall, the spatial vortex position can be calculated by means of the VRIEDER algorithm. Figure 7 shows the vortex position inside the observation volume for different Reynolds numbers at Mach = 0.2 and an angle of attack of 9°. The positions of the vortex lines differ with the Reynolds numbers, and a trend is not observable.
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Figure 8 shows a de-warped LLS image with two dark holes marking the location of the vortices at an angle of attack of 2.6° and a Reynolds number of 4·106. The view is one half span downstream, the top wall is located at the top, and the projected position of the wing tip is marked with ŏ. Whereas the stronger lower vortex originates from the wing tip, the upper vortex seems to come from the aileron of the wing, whose position is marked by the red cross. By transforming the images into the wind tunnel coordinate system and detecting the vortex position, the location of these vortices could be determined with high accuracy. Figure 9 shows the position of the wing tip vortex for different angles of attack at three different Reynolds numbers. In accordance with the BOS results, there is obviously a small, but not negligible dependency of the vortex position on the Reynolds number for a given angle of attack (up to 3% of the half span width), whereas a systematic trend cannot be observed. The vortex trajectories are in good accordance with the BOS measurements (green line).
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Fig. 9. Vortex trajectories for three different Reynolds numbers (LLS: solid lines, BOS: dashed line), the numbers indicate the angle of attack
5 Conclusion The optical measurement techniques BOS and LLS were successfully applied in the ETW. With BOS, which can be easily implemented also under cryogenic conditions, it is possible to obtain the position of the wing tip vortex in space precisely by applying new processing algorithms, even for low Mach numbers where low density gradients occur. LLS is also feasible even in the cryogenic environment and under difficult seeding conditions. Based on the experience with the application of these measurement techniques to ETW, other optical measurement techniques such as Particle Image Velocimetry (PIV) seem to be applicable to ETW. Also using both techniques, the vortex positions were investigated at different Reynolds numbers as well as different wind tunnel conditions. The results show that vortex positions in the near field behind the wing have to be measured under ‘real flight’ Reynolds and Mach number conditions, because there is a certain effect of these similarity parameters on the vortex locations in space. To decide whether this behaviour of the vortex position is a pure Reynolds number effect or whether it may be related to modifications of the incoming flow, wing tip shape, etc. will require further experiments with simultaneous observation of the flow field near the wing.
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Acknowledgements The work was sponsored by the BMWA, embedded in the programme “LUFO III”. The authors would like to thank the team of the ETW for its dedicated cooperation, J. Klinner from DLR-AT for providing the LLS laser optics, and W. Kühn from Airbus Bremen for giving us the opportunity to participate in the tests of the aircraft model.
References [1] F. Klinge. “Verfahren zur Bestimmung des Verlaufs einer beliebig geformten Linie im Raum”. Deutsches Zentrum für Luft- und Raumfahrt e.V., Deutsches Patent- und Markenamt, DE 103 08 042 A1. [2] M. Raffel, H. Richard and G.E.A. Meier. “On the applicability of Background Oriented Optical Schlieren Tomography”. Experiments in Fluids, 2000, pp. 447-481.
Project ForMEx – A New CFD Approach for Transposition of Wind Tunnel Data Towards Flight Conditions Stefan Melber-Wilkending and Georg Wichmann German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology Lilienthalplatz 7, D-38108 Braunschweig, Germany
Summary In this paper a new approach of CFD supported wind tunnel testing is presented based on investigations of the DLR project ForMEx [1] - [3]. The numerical simulation and respectively the analysis of the wind tunnel experiment considering all geometrical and aerodynamic conditions show improvements of today’s wind tunnel testing techniques which is outlined in this paper for the wind tunnel DNW-NWB.
1 Introduction For the design of new aircraft configurations the wind tunnel experiment still represents an indispensable tool in order to predict the aerodynamic performance of single aircraft components as well as the overall configuration and respectively to validate numerical procedures. In this context extrapolation of the wind tunnel tests to free flight conditions within this process contains certain inaccuracies. The wind tunnel flow does not correspond to the free flight because of wall and model mounting effects. In order to minimize these influences to a large extent, data corrections of the wind tunnel tests are performed, which up to now are based on simple procedures and hand book methods. The wind tunnel measurements usually are performed with smaller models compared to the original, and the extrapolation to real conditions is done by each aircraft company using their own extrapolation procedures. Aerodynamic performance data resulting from the wind tunnel experiment therefore are still affected by certain systematic errors. During the last years advanced modern procedures for CFD flow simulation have been further developed. In particular by the use of unstructured codes for the flow simulation around complex configurations and geometries also complete wind tunnel flows can now be handled with the required accuracy and justified effort. Thus the critical examination of existing wind tunnel correction procedures and their improvement is made possible, leading to more reliable procedures for the prediction and extrapolation of the wind tunnel experiment to free flight. Within the DLR project ForMEx the numerical simulation and respectively the analysis of the wind tunnel experiment considering all geometrical and aerodynamic conditions is performed in order to improve the wind tunnel testing technique described above. In the process also model and model mounting C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 113–120, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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deformations are considered using flow/structure coupling methods. From the deviations detected by careful comparisons of the experimental data with the results of the numerical simulation of the experiment correction rules will be derived. On the one hand they will help to identify the limits of existing wind tunnel correction methods and possibly will lead to certain improvements, on the other hand they also will serve for validation and improvement of numerical methods. Based on the ForMEx project work the present paper describes the CFD potentials to support wind tunnel testing in the low speed wind tunnel DNW-NWB with a transport aircraft half model mounted in the test section.
2 Numerical Method The solution of the Reynolds-averaged Navier-Stokes equations (RANS) is carried out using the hybrid unstructured DLR TAU code [4]. For the closure of the Reynoldsaveraged equations the k-ω-SST turbulence model of Menter is used, which combines robustness with the applicability for partly detached flows. Due to the low Mach numbers and the resulting stiffness of the RANS equations, low Mach number preconditioning is used. Finally, the central JST-scheme in combination with 80% matrix dissipation assures numerical flow solutions with low numerical dissipation. To increase the convergence, an implicit time-integration (LU-SGS) is implemented in the TAU code.
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Strongly coupled with the experimental simulation of high lift configurations in the wind tunnel is the so called half model technique. It results from the demand of the same Reynolds number in free flight and wind tunnel experiment to get the flight physics as real as possible in the wind tunnel experiment. On this point the half model technique is introduced using the assumption of a symmetrical flow around the aircraft by cutting it on the symmetry axis along the fuselage and measure this configuration in the wind tunnel. Using this technique the model size can be doubled without changing the test section and getting a doubled Reynolds number holding all other parameters constant compared to the full span. A reduction in the quality of the measurements results from the increased wind tunnel interference resulting from the model volume and the mounting of the model on the tunnel floor or ceiling, in which the model is partially covered by the tunnel wall boundary layer. To reduce this influence and to reduce the disturbance of the symmetrical flow the fuselage is often mounted on a cylindrical extension of its symmetry cut called peniche or stand-off. But even using a peniche a completely symmetrical flow in the symmetry plane cannot be achieved, due to the horse-shoe vortex between peniche an tunnel wall. Because of this, the unsymmetrical flow cannot be eliminated by changing the peniche height and this behaviour always leads to a difference between a half model compared to the full span model measurement. Only the displacement of the peniche will vary in case of a peniche height change.
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3.1 Model and Wind Tunnel In this paper the ALVAST transport aircraft geometry in landing configuration is considered in the wind tunnel DNW-NWB. The ALVAST model is a generic configuration of a modern, twin-engine transport aircraft comparable with an AIRBUS A320 in scale 1:10. Beside the wing this landing configuration consists of a single slat and a single slotted flap which is split into an inner and outer part [5]. The low speed wind tunnel in Braunschweig (DNW-NWB) is an atmospheric wind tunnel of G¨ottinger design with a closed loop. The construction of the tunnel was finalized 1960 and the tunnel is integrated since 1996 in the German-Netherland wind tunnels (DNW). The test section has the size of 3.25 m x 2.8 m and reaches a flow speed of 90 m/s at an maximum drive-power of 1.4 MW. 3.2 Numerical Simulation of a Wind Tunnel The peniche plays an important role for the half model technique and therefore it also has been considered and simulated in the investigation described in this paper. On the peniche as well as on the fuselage the boundary layer of the model and the tunnel wall interacts. Therefore the tunnel wall on which the model was mounted has to be simulated viscid. The remaining walls can be treated inviscid to reduce the numerical effort. However with an increasing angle of attack and thus blockage a wind tunnel model has a remarkable influence on the boundary layer of the tunnel walls. Because of this all tunnel walls are treated viscous in this simulation. In principle the numerical simulation of a wind tunnel can include the complete tunnel with test section, diffuser, direction change, drive, settling chamber and nozzle. Indeed this would be an additional effort to simulate the intrinsic flow in the test section. Therefore it would be sufficient to simulate only the test section with an in- and outflow. But the shape of the boundary layer on the tunnel walls at the inflow is not known. By adding the nozzle and the settling-chamber to the simulated domain this problem can be solved because the flow straighteners in front of the settling chamber remove the boundary layer and for this reason the flow topology can be handled numerically at this station. The boundary conditions for the simulation of the in- and outflow serve at the same time for the control of the flow speed in the numerical wind tunnel. A detailed description can be found in [6] and shall not be repeated here. To change the angle of attack of the model in the wind tunnel during the experiment a turntable on the tunnel floor is used. To simulate this numerically in the current investigation the Chimera technique is used. Therefore the tunnel is meshed without the Table 1. Investigated Configurations configuration with peniche without peniche wind tunnel free flight A o o B o o C o o
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model, afterwards the volume is cut out in which the configuration is rotated at different angles of attack. In this volume an second grid is inserted including the model. To assure the communication between both grids on the boundaries an overlap was used [7]. The final grid consists of about 21 · 106 points. Three configurations have been used to identify the wind tunnel influence on the flow for a high lift configuration under consideration of the half model technique. In Table 1 these configurations are listed. By simulating with and without peniche and accordingly with wind tunnel and free flight a breakdown in the influence of a finite test section (wind tunnel influence) and the half model technique (peniche influence) can be done. The simulations were accomplished using the following freestream conditions: V = 60 m/s, Re = 1.435 · 106 with a reference length of l = 0.41 m. Further investigations have been carried out on the peniche gap, peniche height and influence of the wing-fuselage junction on maximum lift. This additional investigations are not shown in this paper, details can also be found in [6]. 3.3 Peniche and Wind Tunnel Effect To determine the wind tunnel influence and at the same time to distinguish it from the peniche influence in this section the so called ”difference pictures” are used. In this pictures the flow variables angle of attack, lengthwise and crossflow velocities of two configurations are shown in cuts perpendicular to the freestream and to the wing span direction. Thereby the values of the first solution are deducted from the solution of the second configuration. Thus these ”difference pictures” (Fig. 1 and Fig. 2) show the changes between two configurations which otherwise are difficult to detect. Comparing the local angle of attack for configurations with and without a peniche in the wind tunnel (conf. A&B) it can be found Fig. (1a) that the peniche leads to an increased local angle of attack on the inboard wing of about ∆α ≈ 1o whereas the outboard wing is not influenced. The influence of the tunnel walls in contrast (conf. B&C, Fig. 1b) results in an additional angle of attack on the complete wing span of about ∆α ≈ 0.5o . The superposition of both effects can be found accordingly between the configuration A&C, Fig. 1c. The reason for this peniche effect is by the additional blocking of the peniche in the flow field. This leads to an additional displacement of the flow leading to an increased flow speed and local angle of attack on the model. With increasing angle of attack this effect increases. Further on with increasing distance (e.g. along the wing span) this peniche effect decays. Around the fuselage an interplay between the peniche and wind tunnel effect can be found. Considering Fig. 2a the main influence of the angle of attack can be found in proximity of the fuselage especially in regions where the horseshoe vortex is located. The wind tunnel effect in contrast leads to an increased angle of attack in front of the model of about ∆α ≈ 1o and behind the configuration to an additional value of about ∆α ≈ 4o compared to the free flight, Fig. 2b. The reason is the downwash of the wing, which cannot spread out downwards because of the wind tunnel wall. Again these effects are superimposed on configuration A&C. Concerning the crossflow velocity the peniche influence decelerates the flow above and accelerates the flow below the fuselage, in both cases of about ∆V = ±1 m/s,
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Fig. 1. Difference pictures of (left to right): angle of attack, crossflow velocity in x- and z- direction. top (a): conf. A&B, mid (b): conf B&C, bottom (c): conf. A&C. Cut through the test section in flow normal direction (front of the wing). Colors: +2 (red), 0 (green), -2 (blue).
whereas the wind tunnel influence has no effect (see Fig. 1). Because of the spatial reduction of the peniche influence the crossflow velocity is mainly changed on the inboard wing. The lengthwise velocity is decelerated because of the peniche influence in front of the model and accelerated above the model because of the wind tunnel effect. These wind tunnel effects lead to a nearly constant acceleration of about ∆u = ±0.5 m/s on the complete wing span superimposed by the peniche influence on the inboard wing. In the same manner as before the effects are superimposed in configuration A&C. Concluding the peniche effect, the flow around the fuselage and the flow deflection are increased leading to an increased flow velocity and local angle of attack on the inboard wing. The strength of the peniche effect is therefore a function of the angle of
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Fig. 2. Difference pictures of angle of attack (top), crossflow velocity x- (middle) and z (bottom), left: (a) conf. A&B, right: (b) conf. B&C. Cut through the test section in flow direction at the peniche. Colors: +2 (red), 0 (green), -2 (blue).
attack and changes the lift rise, compare Fig. 3. Further on the configuration without peniche has a reduced lift coefficient of 2.6% also. The peniche effect can be found at all angles of attack because the displacement of the peniche always takes place. From the lift curves a change in the maximum angle of attack can be found with and without peniche. With peniche the maximum angle of attack is at α = 15o whereas without peniche at α = 15.5o. In this case the additional load on the inboard wing due to the peniche reduces the maximum possible angle of attack. If this behaviour is triggered by the peniche it is mainly decided were the flow separates, first on the inboard or outboard wing. In the first case the peniche increases the load on the inboard wing and intensifies the lift breakdown there. Using half model measurements this effect must be always keept in mind. Comparing the lift curves of configuration A&C (Fig. 3) it is clearly visible the one got from the wind tunnel simulation is shifted above the one of the free flight. The reason is the wind tunnel effect which leads to an increasing angle of attack and flow velocity. The peniche effect leads simultaniously to an increased gradient of the lift curve compared with the free flight. 3.4 Wind Tunnel Correction In Fig. 3 the corrected and uncorrected lift curves from the measurement in the wind tunnel DNW-NWB are shown with the corresponding numerical simulations in the tunnel and the free flight. Configuration A corresponds to the uncorrected measurement,
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configuration C the corrected one. The corresponding curves of the measurement and the simulation show a good agreement in the linear range. To get a more detailed assessment of the wind tunnel correction without regarding a measurement and with removing possible measurement errors the lift curves of the simulation in the wind tunnel are corrected with wind tunnel correction and compared to the lift curves of the free flight. The wind tunnel correction is well defined, if this corrected results correlate with free flight simulations. The results are shown in Fig. 3 for configuration A (in wind tunnel), uncorrected and corrected, and for comparison configuration C (in free flight). The corrected lift curve of the configuration matches the lift curve of configuration C with a slightly higher gradient and a little bit increased level. The corrected lift curve of configuration B has a slightly reduced gradient and a lower level compared with the free flight. Without peniche there is no additional displacement which can compensate the boundary layer of the tunnel wall. On the other hand with peniche the displacement increases with increasing angle of attack and leads to an only point-wise matching of the corrected wind tunnel measurement and the free flight values as a function of the peniche height. Overall the used wind tunnel correction shows a good agreement in the linear range of the lift curve with the numerical simulations. However the peniche effect is not corrected, especially in its spanwise variation. This leads to spanwise differences in the pressure distributions (not shown here, compare [5]). Furthermore caused by the variation of the displacement with the angle of attack the wind tunnel correction can only be applied for one angle of attack.
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4 Conclusion A new approach of CFD supported wind tunnel testing is presented in the present paper based on investigations of the DLR project ForMEx. ForMEx is aimed at more accurate wind tunnel correction methods and improved extrapolation to free flight conditions by using CFD techniques for the complete wind tunnel flow analysis. The results show that the numerical simulation is able to identify the limits of existing wind tunnel correction methods and thus can be used as basis to improve today’s wind tunnel testing techniques outlined here for the wind tunnel DNW-NWB. To achieve these results the test section including the nozzle together with the ALVAST high lift configuration as half model and the ALVAST full span model in free flight have been simulated in order to investigate the half model (peniche) effect as well as the wind tunnel influence. From this investigation the following statements can be outlined: The impact of wind tunnel on the flow around a half model can be divided in a peniche and a wind tunnel influence. The local angle of attack and the flow velocity is increased mainly in the inboard part of the wing due to the peniche influence. The wind tunnel wall effect also has an influence on the inboard wing, but with a smaller value. Therefore the wind tunnel influence can be mainly found at outboard wing parts whereas its effect is present over the complete cross section. When reducing the angle of attack of the model the corresponding effects also decrease.
References [1] Wichmann, G.: The objectives of DLR’s ForMEx Project. European Wind tunnel Association (EWA), Initial Joint Workshop, ONERA Toulouse, October 18-20, 2004. [2] Wichmann, G.: CFD Supported Wind Tunnel Testing - First Results of DLR Project ForMEx. EWA, European Windtunnel Association, 2nd Joint Workshop, 2006. [3] Melber-Wilkending, S.; Heidebrecht, A.; Wichmann, G.: A new approach in CFD supported wind tunnel testing. ICAS 2006-3.4.2, 25th International Congress of Aeronautical Sciences, 3-8 September 2006, Hamburg, Germany, 2006. [4] Kroll, N.; Rossow, C.-C.; Schwamborn, D.; Becker, K.; Heller, G.: MEGAFLOW - A Numerical Flow Simulation Tool for Transport Aircraft Design. 23rd ICAS Congress, Toronto, ICAS 2002, 1.5-10.5, 2002. [5] Kiock, R.: The ALVAST Model of DLR. DLR IB 129-96/22, 1996. [6] Melber-Wilkending, S.: Numerische Untersuchungen zur Aerodynamik von Hochauftriebssystemen an Transportflugzeugen unter Ber¨ucksichtigung der Windkanalsimulation. DLR FB, to be published. [7] Madrane, A.; Heinrich R., Gerhold T.: Implemetation of the Chimera method in the unstructured hybrid DLR finite volume Tau-Code. 6th Overset Composite Grid and Solution Technology Symposium, Ft. Walton Beach, Florida, USA, 8.-10. October, 2002.
Pressure and Heat Flux Measurements on the Surface of a Low-Aspect-Ratio Circular Cylinder Mounted on a Ground Plate T. R¨odiger, H. Knauss, U. Gaisbauer, and E. Kr¨amer Institute of Aerodynamics and Gas Dynamics, University of Stuttgart, Pfaffenwaldring 21, 70569 Stuttgart, Germany
[email protected] http://www.iag.uni-stuttgart.de
Summary The flow over a finite-height cylinder of aspect ratio H:D=2:1 has been studied by means of oil-film visualization, pressure and heat flux fluctuation measurements. The measurements were concentrated on the diameter-based Reynolds number of ReD = 2 × 105 . A novel heat flux gauge with a high spectral resolution was used to investigate the heat flux fluctuation on the surface of a slightly heated cylinder. Comparative investigations showed qualitative correlations between the pressure and heat flux fluctuations around the cylinder and on its free-end. Several flow features could be identified by means of time-resolved surface measurements.
1 Introduction The flow over a finite-height cylinder has been the subject of extensive studies by means of various experimental techniques for many years. A summary of previous work is given by Pattenden [1] in 2005. The reader is refered to this summary for references in a wide regime of Reynolds numbers and range of aspect-ratios. The mean flow topology at the surface of an low-aspect ratio cylinder can be divided in three areas: the horseshoe vortex system in the lower part, the middle part dominated by the separating shear layer and finally the flow over the free-end strongly influenced by the development of a tip vortex system. In the present work, oil-film visualization and static pressure measurements have been used to localize dominant flow features, but the main focus is placed on comparative heat flux and pressure fluctuation measurements around the cylinder and its freeend. Convective heat transfer studies of a finite cylinder are fairly sparse and mostly concerning the local but not time-resolved measurements of heat transfer characteristics. The correlation between heat flux and pressure fluctuations touches the subject of the analogy between heat and momentum transport. The oldest theory concerning this analogy was stated by Reynolds, who assumed that there is a complete analogy between these quantities. Hinze describes in [2] that if such an analogy existed, it would be possible to calculate the mean velocity distribution and to obtain a relation between the coefficients of heat transfer and wall friction. Evidence is also shown in [2] for the C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 121–128, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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correlation of wall pressure fluctuations with the mean wall shear stress, which closes the circle and would deliver a quantitative framework for the comparison of heat flux and pressure fluctuations under certain conditions. The goal of this paper is however not a quantitative comparison of heat and pressure fluctuation but to show a qualitative correlation of these quantities for the detection of flow phenomena on the surface of a finite cylinder.
2 Description of Experiments and Wind Tunnel 2.1 Wind Tunnel and Model The experiments were performed in a closed circuit wind tunnel with an open test section. The installed nozzle with an exit diameter of 1.4 m allowed maximum free-stream velocities of about 37 m/s. The turbulence level in the test section flow is approximately 0.8 %. The cylinder model had an aspect ratio of H:D=2:1 with a height H=240 mm and diameter D=120 mm. The x, y and z axes are in the streamwise, transverse and spanwise directions, respectively. The model was mounted on a ground plate, which had an elliptic leading edge 1220 mm upstream of the z-axis of the cylinder. The boundary layer (BL) on the ground plate developed naturally and the measured velocity profile indicates a BL thickness of δ/H = 0.092 at the model position. 2.2 Surface Pressure Measurements The model used for the pressure measurements has 79 pressure tappings along a generator of the cylinder with a distance of 3 mm. Another 39 pressure tappings are located along a center line on the top surface of the model with the same spatial resolution of 3 mm. The model could be rotated by a servo driven table with a minimal increment of 0.2◦ . For the static pressure measurements a Scanivalve-Module (DSA 3217) with 16 channels and a pressure range of 2500 Pa at a maximum scanning rate of 200 Hz was used. Elastic hoses of 1 m length connected the module to the pressure tappings. The model was rotated on the base through 360◦ at 5◦ intervals and refined at angles between 50 ÷ 90◦ and 270 ÷ 310◦ to an increment of 2◦ . Simultaneous pressure fluctuations were measured by a set of 12 single pressure transducers mounted inside the cylinder. The sensors used were piezoresistive gauges (Keller PD-4E) with a pressure range of 3000 Pa. The gauges were installed with a tube length of 60 mm between the gauge and the cylinder surface. This was the minimal possible distance in order to achieve a minimum in damping of the pressure fluctuations. The gauges were not evenly distributed along the generator of the cylinder but concentrated at the top and the bottom to obtain a finer spatial resolution at the free-end and in the region influenced by the ground plate. The gauges were located at z/H = 0.03; 0.05; 0.09; 0.17; 0.28; 0.4; 0.53; 0.66; 0.78; 0.91; 0.95; 0.98. The cylinder was rotated on the base through 360◦ at 1◦ intervals. On the free-end the pressure transducers were evenly distributed along the radius of the cylinder.
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2.3 Heat Flux Measurements The heat flux measurements were carried out by a new fast-response heat-flux gauge, the so-called Atomic Layer Thermo Pile (ALTP). The working principle of this sensor is based on the Transverse Seebeck effect. For a detailed description of the working principle, structure and calibration procedure of the sensor, the reader is referred to [3]. ALTP gauges with a film thickness between 500-700 nm and an active area of 2 × 0.4 mm2 show a spectral resolution up to 1 MHz. In the present experiments, a sensor with a sensitivity-stabilizing protective coating and an active area of 3 × 3 mm2 was used. The coating reduces the spectral resolution of the sensor to a certain degree (200-250kHz), which exceeds however the required spectral regime of the present investigations by more than one order of magnitude. The housing of the module was made of a ceramic insulator (MACOR) to achieve a mainly one dimensional heat flux through the substrate. The module has a flat surface with a diameter of 8 mm. Thus, a maximal roughness of 0.13 mm is formed in the surface of the cylinder. The ALTP were evenly distributed along the generator of the cylinder at z/H = 0.18; 0.38; 0.58; 0.78. On the free-end the ALTP gauges were located at r/R = 0.24; 0.34; 0.5; 0.58; 0.66; 0.8 along the radius of the cylinder, where R = D/2. In order to achieve a temperature gradient in the incompressible BL of the cylinder, the model was heated by an external heating system mounted below the cylinder model under the tunnel floor. A variable air flow rate between 4 ÷ 10 m3 /h is forced past a heating cartridge with a nominal maximum power of 1500 W. The air heats the wall of the cylinder to an excess temperature difference of approximately ∆T ≈ 20 ÷ 25 K at ReD = 2 × 105 . The temperature distribution measured by an infrared camera shows a wall temperature difference of ≈ 5 K between the free-end and the lower part of the cylinder. The cylinder and the heating system were insulated from the wooden base to minimize the heating of the wake and inflow BL. It was rotated on the base through 360◦ at 5◦ intervals after the equilibrium temperature of the cylinder was established. The effect of the convection on the topology compared to the unheated case was studied by comparative temperature fluctuation measurements with a wall-mounted hot film gauge used as temperature resistance thermometer (Dantec Flush Mounting Probe 55R45 operated in constant current mode). The local shift of the fluctuation peak is minimal and within the measuring accuracy of this technique. Thus, the influence of the convection on the BL on the surface is minimal. Possible effects on the wake of the cylinder, however can not be excluded.
3 Surface Flow Visualization Oil-flow visualization was carried out in order to study the basic topology of the flow around the circular cylinder wall at ReD = 2 × 105 . A mixture of neon green colored particles and oleic acid was used. The tunnel was run for several minutes until a steady flow pattern established. The pattern was then photographed with a standard high resolution digital camera with the model in situ. The images (Fig.1) give an insight in the present flow phenomena around the cylinder and at its free-end. In the front and side view the stagnation line and the line of separation respectively are clearly visible. The foot-print of the horse-shoe vortex on the ground can be clearly identified. The
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Fig. 1. Oil-film visualization images for ReD = 2 × 105 . Top row (from left to right): Front view, Side view (flow from left to right), Rear view. Bottom Row: Top views (flow from left to right): Floor of tunnel, Free-end of cylinder.
rear view displays the three dimensional signature of the wake region induced by the free-end flow. The top view shows the extent of the wake and several dominant vortical structures. The detailed view of the free-end illustrates the bases of two vortices on the surface as dominant foci and a crescent-shaped region in the rear half of the free end.
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4.1 Static Pressure Distributions To study and identify the flow topology of the cylinder flow in a wider Reynolds number range, static pressure measurements were carried out for the Reynolds numbers ReD = 2.3 × 105 , 2 × 105 , and 1 × 105 . Fig. 2 and 3 show the static normalized pressure distributions around the cylinder, where φ is the angular position of the pressure tapping row in reference to the stagnation line. Some typical features of the low aspect-ratio cylinder flow topology are visible in the distributions. At φ = 0◦ the pressure coefficient near the top is reduced due to the upwash over the tip. The influence of the horseshoe vortex in the lower region
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Fig. 2. Distribution of pressure coefficient cp around the cylinder at ReD = 1 × 105 (left), ReD = 2.3 × 105 (right)
z = 0 ÷ 0.1H of the cylinder is clearly visible by a kink in the pressure coefficient. In comparison to an infinite cylinder, the separation occurs further upstream on the surface. For example, at ReD = 2 × 105 an angle of 80◦ is expected for the infinite cylinder, the pressure measurement on the low-aspect ratio cylinder, however, yield an angle of about 65 ÷ 70◦ . This feature was also observed in [1] and is attributed to the higher pressure behind the cylinder created by the free-end flow entering into the recirculation region. At separation, the magnitude of the minimal pressure coefficient decreases with increasing Reynolds number from cp ≈ −0.8 at ReD = 1 × 105 to cp ≈ −2 at ReD = 2.3 × 105 . In addition, the separation line shifts further downstream and the region of minimal pressure moves towards the middle of the cylinder with increasing Reynolds number. For ReD < 2×105, the region of minimal pressure broadens towards the tip of the cylinder and the flow separation is delayed for several degrees. At the back of the cylinder, the pressure coefficient clearly rises towards the base. This is caused by the reverse flow of the wake impinging on the surface. For ReD = 2.3 × 105 , the BL at the surface is transitional and the turbulent BL sustains higher pressure gradients and the separation line moves further downstream. In the following the investigations will be concentrated on the critical Reynolds number ReD = 2 × 105 . Its flow topology will be studied in more detail by the comparison of pressure and heat flux fluctuations. 4.2 Pressure and Heat Flux Fluctuation Distributions Fluctuation distributions are of special interest for the detection of BL transition on a surface. Close to the critical Reynolds number, the BL layer over the surface of the cylinder is laminar and transition is occurring in the shear layer immediately after the separation. Thus, strong fluctuation should appear in this region. Fig.3 displays the distribution of pressure fluctuations around the cylinder. The maximum fluctuations occur at about 95◦ . Towards the free-end the maximum is shifted downstream which correlates with the delayed separation observed in the static pressure distributions. In the back of the cylinder the magnitude of fluctuations increases
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Fig. 3. Distribution of pressure coefficient cp (left) and pressure fluctuation cp (right) around the cylinder at ReD = 2 × 105
Fig. 4. Distribution of normalized heat flux fluctuations q /qSP (left) and power spectra at plane 5 z/H = 0.78 (right) around the cylinder at ReD = 2 × 10
towards the base. The distribution in Fig. 4 shows the maximum heat flux fluctuations at the same angle of about 95◦ . The maximum is however more pronounced in relation to the pressure fluctuations. This might be caused by the damping of the flexible pressure hoses. Time-resolved wall pressure measurements by surface-mounted micro electromechanical pressure sensors confirm the existence of this more pronounced pressure peak on a cylinder of the same geometry [4]. Fig. 4 displays power spectra around the cylinder at a horizontal plane of z/H = 0.91. Frequencies up to 1 kHz are plotted; no significant heat flux fluctuations could be detected beyond this frequency. The spectral distribution shows a peak at about 95◦ caused by an increase of the intensity level in a frequency range up to 800 Hz. Beyond that point the range of dominant frequencies decreases again but stays above the level of the laminar BL before transition. This spectral distribution is qualitatively very similar to a typical pressure fluctuation spectrum. In general, a qualitative correlation of the pressure and heat flux fluctuation around the cylinder is visible and the detection of transition by means of heat flux and pressure fluctuation is successfully demonstrated.
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5 Free-End Flow The flow over the free-end is dominated by separation from the sharp leading edge. The oil-film visualization shows the complex 3-D recirculation region on the top of the cylinder. Two foci dominate the front part and mark the origin of two counterrotating vortices. The rear part of the free-end shows a crescent-shaped footprint. The topology of this zone has been discussed controversially in [1] and [5]. The oil-film visualization seems to confirm the flow pattern proposed by Roh and Park [5], who suggest the existence of a separation saddle point between two attachment nodal points in this region. The static pressure distribution in Fig.5 (left) displays an extensive low pressure region in the front part of the free-end. The pressure minimum is located in the center of the two foci transporting fluid away from the surface. Towards the rear part the surface pressure rises significantly and an area of higher pressure is formed in its second half. The static pressure distribution does not capture the character of this highly unsteady region very well. The fluctuation measurements in Fig. 5 (right) illustrate much better the crescent-shaped footprint observed in the oil-film images. This figure compares the normalized pressure fluctuations (upper half) with the heat flux fluctuations measured by ALTP gauges (lower half). Both types of fluctuations a’ are normalized to an interval [0, 1] for better comparison, i.e. the fluctuations were transformed to a ratio of differences according to the relation (a − amin )/(amax − amin ). The images show maximum fluctuations along a radius of about r/R = 0.6 in the rear part of the freeend. The region of high pressure fluctuations is smaller than the one covered by the ALTP measurements. The measured heat flux fluctuations extend further upstream and reach from the center towards the edges of the free-end. The pressure fluctuations, however, are more concentrated towards the center line. The features are probably caused by the higher temporal resolution of the ALTP in comparison to the pressure transducers. The observation might indicate that the dominant frequencies at the edge of this unsteady region are shifted towards higher values. In general, the results point out the importance of time-resolved surface measurements for the detection of unsteady flow phenomena.
Fig. 5. Distribution of pressure coefficient cp (left) and comparison of normalized pressure and heat flux fluctuations (right) on the free-end surface at ReD = 2 × 105
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6 Conclusions The flow topology of an finite-length circular cylinder with the aspect ratio 2 was used to study the qualitative correlation between heat flux and pressure fluctuation. Static pressure measurements in a Reynolds number range between ReD = 1 ÷ 2.3 × 105 allowed an identification of the major flow phenomena in combination with oil-film visualization at ReD = 2 × 105 . These measurements helped to identify dominant features in the fluctuation distributions of heat flux and pressure. A correlation of the maximum fluctuations was found at the transition of the separated shear layer around the cylinder. The spectrum of the heat flux fluctuation showed an increase of the intensity level in a frequency range up to 800 Hz at transition location. The pressure and heat flux fluctuation peaks in the rear part of the free-end were in good agreement and showed the importance of time-resolved measurements for the study of unsteady flow phenomena.
Acknowledgements This research was supported by the German Research Foundation (DFG) within the priority program 1147 and the project KN 490/1.
References [1] Pattenden, R.J.,; Turnock, S.R.; Zhang, X.: ”Measurements of the flow over a low-aspectratio cylinder mounted on a ground plane”. Exp. Fluids. 39, 2005, pp. 10-21. [2] Hinze, J.O.: ”Turbulence”. 2nd Ed. Mac Graw Hill, 1976. [3] Roediger, T.; Jenkins, S.; Knauss, H.; v. Wolfersdorf, J.; Gaisbauer, U.; Kraemer, E.: ”TimeResolved Heat Transfer Measurements on the Tip Wall of a Ribbed Channel Using a Novel Heat Flux Sensor - Part I: Sensor and Benchmarks”. J. Turbomach. 1128, 2006. [4] Wolter, A.; Leder, A.; Berns, A.; Obermeier, E.: ”Einsatz mikroelektromechanischer Drucksensoren (MEMS) in Kombination mit laseroptischen Messverfahren in der experimentellen Str¨omungsmechanik”. Lasermeth. Str¨omungsmesstechn. 2006, pp. 30.1-30.8. [5] Roh, S.C.; Park, S.O.: ”Vortical Flow over the free end surface of a finite circular cylinder mounted on a flat plate”. Exp. Fluids. 34, 2003, pp. 63-67.
An Investigation into Internal and External Force Balance Configurations for Short Duration Wind Tunnels Matthew Robinson, Jan Martinez Schramm, and Klaus Hannemann Spacecraft Section, German Aerospace Center, Bunsenstraße 10, Göttingen, 37073 Germany
[email protected]
Summary An investigation of both an internal and an external stress wave force balance that measures forces and moments on a re-entry type configuration was undertaken. Simulations performed using a finite element code suggest superior recovery of the applied forces and moments with an external force balance configuration. This is hypothesized to be due to the increased balance stiffness and decoupling between force components that can be achieved with an external type configuration. The influence of an external force balance as compared to an internal force balance on the model loads was determined to be negligible.
1 Introduction Force measurement in short duration wind tunnels is reliant on non-conventional measurement techniques. These techniques are required in order to account for the dynamic behavior of the model and its support structure since static equilibrium between the model and support structure is rarely established in the short test times available. One method that enables force measurement in short duration flollws is called the stress wave force measurement technique. This method relies on the measurement and analysis of stress waves. Such a technique requires careful modeling and force balance design in order to obtain suitable accuracies in the recovered loads. For such stress wave force balances, several balance configurations are possible for a particular test model. For example, depending on the model configuration either an internal or an external force balance configuration may be more geometrically suitable however one may provide a higher degree of accuracy of the recovered forces and moments than the other. In this paper, several force balance designs are investigated by simulating the dynamic response of a test model and force balance using a finite element analysis (FEA) solver. The test model chosen was the Pre-X configuration (see Fig. 1) positioned at 45o angle of attack. A number of stress wave force balance configurations were investigated. These are shown schematically in Fig. 2 using a generic rectangular model. From these configurations, balance designs 02 (denoted balance B) and 08 (denoted balance A) are discussed in this paper. Balance A is an internal type force balance with four short, stiff stress bars (that measure the normal force and pitching moment) and one C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 129–136, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Fig. 1. Schematic of Pre-X HEG model (dimensions in millimeters)
Fig. 2. Possible stress wave force balance configurations
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large axial stress bar to measure the axial force (co-ordinate system defined in Fig. 2). This balance has already been manufactured and tested (see [1]). Balance B is an external type balance with two normal stress bars (that measure the normal and pitching moment) and one axial stress bar (to measure axial force). Although reasonable results were obtained for some of the other configurations illustrated in Fig. 2, superior results were obtained for balance B (of the external configurations) and a stiffer version of balance A denoted balance C (of the internal configurations). Balance C (not shown in Fig. 2) was developed towards the end of these investigations, however since balance A was already manufactured, the results from this balance are presented and compared to those of balance B. Nevertheless, balance B was more accurate than balance C. 1.1
Force Recovery Technique
The stress wave force measurement technique involves measuring stress waves which propagate and reflect through the model and support structure. Upon flow arrival, stress waves propagate through the model at the speed of sound of the material and subsequently enter a stress bar which is instrumented with a strain gauge to record the time history of strain. If the model and support structure produce linear strains due to linear forces, then the dynamic behavior of the system can be modeled as timeinvariant, casual, and linear with an output y (t) (the resulting strain signal), being related to the applied aerodynamic load, u (t), via an impulse response function, g(t), as described by the convolution integral, t y(t) = g(t − τ )u(τ )dτ. (1) 0
The aerodynamic force in an experiment can be determined by the deconvolution of the strain signal with the impulse response function. A program named DECON [1], written in the C programming language solves Eq. 1 for n components. The program also includes several other tools such as file manipulation, data averaging and statistics. Also included is a Newtonian solver for force/moment prediction which is further described in Sec. 2.2. Since the deconvolution scheme is iterative in nature, the deconvolution can be started from an initial solution. It has been found that using a starting solution based on either a Newtonian or a computational fluid dynamic (CFD) prediction of the forces and moments, scaled with the time history of Pitot pressure, offers the best accuracy. Further, conditioning the impulse and output signals to account for differences in force magnitudes between the different components also increase the accuracy of the recovered signals.
2 Finite Element Modeling 2.1 Finite Element Simulations The finite element package ANSYS V10.0 was used to model the dynamic behavior of the model and force balance prototypes. A direct linear solution was
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performed using the PCG conjugate solver. Where possible, brick elements were utilized. Material damping was not included. Example FEA meshes for balances A and B are shown in Fig’s. 3 (a) and (b). To form the impulse response function for each balance, unit step point calibration loads were applied to the Pre-X model at positions along the model. The moment calibration loads can be seen in Fig. 3 (b) while the single axial calibration load can be seen in Fig. 3 (a). The strains due to these calibration loads were monitored at each stress bar and then combined together using superposition. Details regarding this process can be found in [2].
(a) Balance A mesh & axial calibration load. (b) Balance B mesh & moment calibration loads.
Fig. 3. FEA mesh examples and calibration loads
Surface interface elements were applied to the external model surface. These surface element numbers and their node locations were outputted and imported into the NFORCE solver (see Sec. 2.2). The calculated pressure on each element was then re-imported into ANSYS. The distributed pressure loading was then simulated and the corresponding strain signals from each of the stress bars were combined and deconvolved with the impulse response function in order to gauge the accuracy of each balance prototype. 2.2 Force Prediction The prediction of forces and moments on a wind tunnel model is an important and sometimes critical exercise. For the initial balance design investigations, a simple Newtonian prediction was made using the NFORCE suite in DECON, however detailed CFD simulations were also performed as presented in section 3.3 to investigate the influence of the support structure on the model forces. Further details of the Newtonian code are provided in [1]. Simulations were performed with the model o inclined at 45 angle of attack. Free stream parameters were P=540Pa, T=890K, ȡ=0.00213kg/m3 with a total enthalpy of Ho=21MJ/kg.
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3 Configurations 3.1 Calibration Responses The strains (due to predominately longitudinal stress waves) induced by the pure axial, pure normal and pure moment calibration loads at the monitored locations on each stress bar were outputted. These were combined to minimize cross-coupling between the dif ferent components in the program DECON. Fig. 4 (a) shows the responses for balance A while Fig. 4 (b) presents those for balance B. It is seen from the comparison of Fig’s. 4 (a) and (b) that the responses in the normal and pitching moment components are less coupled and also have a higher frequency response for balance B. The uses of other directional strains (such as shear) were also investigated with balance A, but these endeavors yielded similar results.
(a) Balance A calibration.
(b) Balance B calibration.
Fig. 4. Calibration responses
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3.2 Distributed Loading The Newtonian pressure distribution (see Sec. 2.2) was applied to the FEA models and the corresponding time histories of strain were outputted from each stress bar for both balances. Deconvolution of the combined outputted strain signals with the impulse response function formed from the calibration step responses was performed. The recovered loads are compared with the applied loading in Fig. 5. The accuracy of the recovered forces, both in terms of magnitude and response time, is seen to be greater for external balance B compared to internal balance A. For balance B, the resultant force was recovered to within 0.9% of the applied force (note that the difference in load magnitudes between Fig’s. 5 (a) and (b) is due to the model for balance B being inclined at 5ƕ relative to the axial bar).
(a) Balance A.
(b) Balance B.
Fig. 5. Recovered distributed loading
3.3 Influence of Balance on Model Loads Unless a completely isolated model is used (i.e. no connections exist between the model and the wind tunnel), the flow around the model and hence the model forces and moments will be affected. In this paper, to quantify the interference effect of the model support structure (i.e. the force balance), CFD simulations were performed for balances A and B (note that to simplify the modeling, the geometry of balance A was used for balance B but without the two normal stress bars). Two grids (using the same mesh generation spacing inputs) were created using the commercial grid generator Centaur. The final grids representing balances A and B had 2.9 and 4.3 million points respectively. As only an approximation of the influence of the stings on the model loads was of interest, simulations were performed with an ideal gas assumption. In total, six adaptations which added roughly 40% more points per adaptation and a final adaptation of 5%, were performed. Each mesh exploited the symmetry plane and had a structured sub-layer of 28 cells on the model and stings to accurately resolve the isothermal laminar boundary layer. The final adapted grids of balances A and B and contours of Mach number can be seen in Fig. 6.
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(a) Balance A grid.
(b) Balance A Mach number contours.
(a) Balance B grid.
(b) Balance B Mach number contours.
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Fig. 6. CFD simulations for balances A and B Table 1. Comparison of recovered forces and pitching moment between balances A and B Quantity
Balance A Fx Fz My [N] [N] [Nm] Newtonian -381.4 604.4 -138.7 CFD -380.4 619.0 -145.9 Error [%] 0.3 -2.4 -4.9
Balance B Balance A / B Comparison Fx Fz My Fx Fz My [N] [N] [Nm] Error [%] Error [%] Error [%] -381.5 605.7 -138.6 0.0 0.2 -0.1 -377.9 618.4 -145.9 -0.7 -0.1 0.0 1.0 -2.1 -5.0 -
The recovered forces and pitching moment are presented in Table 1 (note that M'y is defined about (0,0) and F'z is aligned to the free-stream velocity vector as shown in Fig. 6). A maximum difference of 0.3% is obtained for the resultant force, while a difference of 0.1% is obtained for the change in center of pressure (based on chord length).
4 Conclusions The designs of two stress wave force balances are presented. FEA simulations revealed that superior accuracy of the recovered force and moment loads was obtained
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with an external balance configuration. The increased balance accuracy is hypothesized to be due to the greater decoupling and stiffness possible with an external type configuration. The influence of an external configuration on the model forces and moments as compared to an internal balance was found to be negligible when the model is considered to be a blunt-body.
References [1] Robinson, M.J., Hannemann, K.: Short duration force measurements in impulse facilities. In: 25th AIAA Aerodynamic Measurement Technology and Ground Testing Conference, San Francisco, California (2006) [2] Robinson, M.J., Mee, D.J., Tsai, C.Y., Bakos, R.J.: Measurement of three components of force on a large scramjet in a shock tunnel. Journal of Spacecraft and Rockets 41(2004) 416–425
Multiple Discipline Take-Off Weight Minimization for a Supersonic Transport Aircraft U. Herrmann DLR, Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7, D-38108 Braunschweig, Germany
[email protected]
Summary Starting from a mixed fidelity multiple discipline analysis process for high-speed
transport aircraft, a multiple discipline optimisation scenario is set-up aiming to obtain minimum weight configurations. A new, more realistic objective function driving the optimisation is implemented. A realistic flight mission for a supersonic transport aircraft (from take-off to landing) is modelled using that process. For supersonic cruise conditions –were most flight time is spent-the disciplines aerodynamics and structures are modelled at high fidelity while other contributing disciplines and flight phases are modelled less accurate. For a generic long-range mission the impact of the supersonic cruise-speed on the minimal take-off weight (enabling to cruise this long range mission) is addressed.
1 Introduction The “after Concorde” age is facing constant requests for increased mobility but at the same time bears fundamental debates whether a high-speed transport successor is needed and whether it could be operated in an environmentally acceptable way. The environmental concerns are related to emissions and noise, but concentrate on noise aspects, both at supersonic (boom) as well as low-speeds. Beyond complying with environmental constraints there is the need for supersonic a/c to represent an economic business case. While this holds for any successful aircraft (a/c), the need for significant performance “jumps” is even more essential for supersonic a/c because of the high costs related to supersonic travel. The design process of an a/c involves many disciplines with often contradicting requirements and constraints. To obtain a new a/c design, a balance of the conflicting requirements as well as compliance with constraints is needed. Considering the main disciplines in a closely coupled way, improves the efficiency of the design process, as solutions deviating from requirements and constraint noncompliance are easily detected. The high demand to boost the a/c efficiency requires the design to approach physical limits. This can only be done reliably with high fidelity models as they usually incorporate all physical characteristics of the involved disciplines. The Airbus – EREA project CISAP [1] developed a core multi discipline analysis (MDA) suite [2] that was used by the partners to build their specific MDO processes. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 137–144, 2007. © Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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The application was aiming at range maximisation of new supersonica/c. Although any range increase is a global measure for the a/c’s economics, butnot necessarily this increased range is of practical economic interest. Therefore the optimisation task is changed here to minimize the a/c’s MTOW – being a more proportional measure of a/c related costs.
2 The DLR MDO Process The common MDA suite (developed in CISAP) represents a generalized long-range mission, see Fig. 1 and is applicable for a/c cruising at varying supersonic Mach numbers.
Fig. 1. Long-range mission with sub-, trans- and supersonic legs; Mach number over range
Fig. 2. Datum aircraft with four engines for supersonic cruise at M=1.6
Such a missions consists of several flight phases: take-off and climb out, transonic cruise (A), further climb and acceleration to supersonic cruise and landing at the scheduled destination (B). Furthermore reserves are modelled for 30min hold (C) and 250nm travel to an alternate destination (D). As most of the fuel is burned during the supersonic cruise part of the mission, high fidelity tools are applied to compute the a/c cruise characteristics regarding the contributing main disciplines. The considered supersonic a/c features a fuselage of 89m length, four engines, a double trapezium planform and transports 250 passengers over a distance of 5000nm. A datum CISAP configuration [3] cruising at M=1.6 is given in Fig. 2. The layout of the MDO process realized at DLR is depicted in Fig. 3. The MDA (middle box) is combined with two independent optimisers (grey boxes) to form the DLR MDO process. Both optimisers drive different sets of design variables. These two sets of design variables are given in Fig. 4. The figures top half shows geometric variables defining a double trapezium wing. This planform is parameterized by the indicated nine variables: leading edge sweeps ĭ, local spans S, ǻx and ǻz shifting the wing relative to the fuselage.
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Fig. 3. DLRs mixed fidelity MDO process driven by two independent optimizers (outer grey boxes)
At each of the three local chords C variables for the cross-sectional thickness and twist angles exist. The shape of the wing cross-sections (scalable) as well as the fuselage shape is fixed. The parameterised wing is completely described by 15 variables. Furthermore the a/c’s MTOW is a variable.
Fig. 4. Design variables of the wing geometry (left) and internal structural elements (spars and ribs in red; lumped stringers in green)
The Synaps Pointer Pro (SPP) environment [4], left grey box in Fig. 2, drives the geometric top-level design variables and calls the MDA. The bottom part of Fig. 3 depicts the elements of the wings internal structure. Depending on the variation of the planform, the parametric structural layout is changed accordingly. Beams and spars (red) and stringers (green) are indicated in Fig. 3. These elements are grouped to reduce the number of variables (115).
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The second optimisation, right grey box in Fig. 2, is the NASTRAN [5] internal structural optimizer. This optimiser drives the local element thicknesses to their minimal values. It is called from within the MDA during the iterative estimation of the wing weight. SPP relies on the SubPlex optimiser [6] to drive the top-level variables. For each of the changed variable sets the MDA is called to estimate the range of this a/c variant. The sequence of MDA modules is processed in the following manner: First the geometry and mesh generation (MegaCads [7]) is performed. (a) A 2.5g pull-up manoeuvre at the end of cruise represents the load case. To compute the wings structure withstanding this aerodynamic load, the lift for this manoeuvre is first guessed by the weight and balances module because the local element thicknesses and thus the wing weight is unknown. (b) For this manoeuvre lift the flow is computed by the DLR FLOWer [7] code. (c) The NASTRAN optimiser now minimizes the element thicknesses of the wing structure withstanding the aerodynamic manoeuvre load. The weight of the primary wing structure is then computed based on the new element thicknesses. Based on this updated wing weight the steps (b) and (c) are repeated until conver gence is obtained (ǻCL between two subsequent steps < 0.0002). The converged wing weight defines the correct cruise lift and the supersonic cruise performance is estimated. The engine is sized based on a fixed runway length. The flight mechanics module assesses the aircraft take-off and landing performance as well as the static stability and trimmability of the configuration. All constraints are checked. In case constraints are broken the cruise drag is penalized. Finally the mission range for this a/c variant is estimated. Details regarding the basic MDA [2], the DLR MDO process [9] and the used constraints [10] have previously been reported, but typical numbers of the numerical effort are repeated. A single flow solution accounts for 3-5 minutes on a PC, a single NASTRAN computation to about 1 minute. Due to the lift / weight balance loop, see Fig. 3, the MDA estimation of one variable change results in 20 to 40 minutes of PC compute time.
3 Objective Function Adaptation The original DLR MDO used in CISAP optimised a/c with fixed MTOW regarding their maximal range. Range improvements were mainly enabled by reductions in wing weight. To keep the MTOW constant for a reduced wing weight, more fuel was loaded in case enough tank volume is present. Furthermore better aerodynamic characteristics help to increase the range, as drag and fuel burn are correlating. Lower drag furthermore leads to smaller engines of lower weight. Maximising the cruise range of the a/c is one measure for the design improvement. Assessing the operating efficiency of that a/c is not too easy. A direct measure would be the direct operating costs. Unfortunately no model for this cost measure is currently available. Nevertheless a close correlation between costs and aircraft weight does exist. Minimising MTOW thus means to minimise the overall costs. Furthermore the requested range can be fixed to a realistic value.
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Based on this approach a new objective function is developed. This new function (OF) is constructed to obtain an a/c configuration of minimal MTOW for a given target mission: OF = MTOW*(1 + PF *((R –RT)/RT)2). The OF depends on the maximal take-off weight and the amplified (PF) range (R) deviation from the requested target mission (RT). The alluring advantage of this formulation is that it does not require changes of the MDO process. The associated disadvantages are: Ɣ Ɣ
MTOW is design variable and at the same time objective which may cause trouble to the overall optimiser. Because the original MDO process deals with a/c of fixed MTOW, no MTOW dependent scaling (landing gear,..) is so far implemented.
4 Multiple Discipline Optimisation Results Previous experience in range maximization of supersonic a/c reveals both stable and good performance of the SubPlex optimizer [6] especially as the design space has been experienced as noisy. Because of this positive experience the review of the overall functionality of the MDO process with the new objective formulation is performed again relying on the SubPlex optimiser.
Fig. 5. Range and MTOW change for a high and a low penalisation of the range deviation during a MTOW minimization for 5000 nm target range (11 variables)
To be able to check the functionality of the MDO process with the new objective function, first an appropriate value of the amplification factor for the range deviation needs to be found. This amplification factor drives the balance between the MTOW minimisation and the agreement to the target range. The convergence of two optimisation runs featuring different PF factors are compared in Fig. 5. For these optimisations a sub-set (11) of the global variables has been used. The target range of 5000nm was selected and is sufficient representative of a long-range mission. Fig. 5 shows the mission range in blue and the MTOW in red over the number (I) of objective function evaluations. Results obtained using a high penalty factor PF (PF=200) are depicted in the left hand graph of Fig. 5.
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The optimiser tries to stick to the target. Deviations are strongly amplified so that the sensitivity of the MTOW variable reduces and it’s value stays unchanged. Reducing the PF-factor (PF=10) changes the behaviour. MTOW is now reduced significantly while an increased deviation from the required target range is developing. The value of the range deviation factor is finally chosen to keep the range deviation within about 50nm. The analysis of the new objective function is continued to check it’s capability to find a “minimum” within a (somewhat) known design space. The design space of interest here is made up by 11 global (geometric) variables plus the maximum takeoff weight.
Fig. 6. Range maximization for three diffe- Fig. 7. Wing planform change at M=1.6 due rent MTOW pre-requisite for the minimal to MTOW minimisation for a target range of MTOW optimization (target range: 5000nm) 5000nm
The design space is briefly explored by three “classical” range maximisations starting from different MTOW’s, see Fig. 6. For the three MTOW values of 300, 320 and 340 to the range of the a/c is optimised using the geometric variables. Range values of the start configurations are given in green, values for the optimised configurations are printed in red. The obtained range increment seems to be independent of the take-off weight, but the maximal values increase with MTOW up to 5500nm. Please note that the considered cruise speed is M=1.6. From this design space exploration it is concluded, that an a/c is likely to exist, that has an MTOW below 320to but is able to cruise 5000nm. The next optimisation using of the new objective function tries to find a minimal weight a/c for that target range. About 900 objective function evaluations are needed to drive the start configuration (MTOW=320 to; range=4800nm) to the optimum (MTOW=312.2to ; range=4986nm). The result is fully in line with the expectations and is indicated as a blue diamond shaped symbol in Fig. 6. The associated a/c planforms are given in Fig. 7. The start configuration is depicted in the figures upper half. The final configuration features a wing of reduced size. The structural load path is straightened reducing the wing box weight. Furthermore the aerodynamic performance is increased. Both effects help to obtain the presented result.
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Fig. 8. MTOW minimization for M=1.5: aerodynamic performance and achieved range over the number of OFE
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Fig. 9. Minimal MTOW and associated range over supersonic cruise Mach number
A similar optimisation is conducted for the lower cruise Mach number of 1.5 but using the full set (15) of geometric variables. The optimisation is allowed to converge more (1500 objective evaluations (OFE)) and the convergence behaviour is depicted in Fig. 8. MTOW, mission range and L/D are given over OFE. The take-off weight is increased first to reach the target range. Later during the optimisation the wing weight is reduced and the aerodynamic efficiency is increased. That enables to reduce the MTOW to about 316to while keeping the range very close to the target value. This exercise has been repeated for the cruise Mach numbers of 1.6 and 1.7 and thus explores the expanded design space (cruise Mach number added). The results are summarised in Fig. 9. In addition the obtained ranges are given and they are found to be very close to the posed target of 5000nm. It needs to be noted, that the presented optima are all found very close towards the end of the optimisation runs, maybe indicating limited convergence due to the OFE number constraint. Finally the cruise Mach number is added as a design variable and a longer optimisation (OFEmax=2000) is started to prove the existence of a weight minimal a/c for a cruise Mach number around 1.6. The computed result of the final optimisation is added to Fig. 9 and has converged to a cruise Mach number of 1.61 at a slightly lower MTOW of about 309to. The functionality of the new objective function is shown. The capability to arrive at minima within known design spaces is indicated. Although the presented quantitative results are impacted by the used modelling (i.e. mesh density,..) in the process, the presented trends are reliable. Thus a/c can be optimised for specific missions but not necessarily the absolute range and weight values are correct. Thus the primary application for the presented MDO process is linked to technology evaluation.
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5 Conclusion Starting from an existing MDO process able to achieve range maximised a/c, the used objective function is changed to improve the realism of the overall MDO process at DLR. A new objective function being close to a cost related approach is formulated. It represents a weighted balance between weight minimisation and forced target range compliance. The used MDO process is described and the new objective function is implemented without requiring changes to the described process. The objective functions implementation is checked and tuned. Finally the functionality of the MDO process driven by the new weight minimisation objective function is tested. For this two different design spaces are explored in advance and optima close to the indicated minima in the design space are found indication full applicability of the chosen approach. The main application of this version of the DLR MDO process will be devoted to technology assessment regarding overall a/c performance improvement.
References [1] Herrmann, U., CISAP: “Cruise Speed impact on Supersonic Aircraft Planform -a Project Overview”, AIAA 2004-4539, 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, 2004. [2] Laban, M., “Multi-Disciplinary Analysis and Optimization of Supersonic Transport Aircraft Wing Planforms”, 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, AIAA 2004-4542, 2004. [3] Torenbeek, E., Jesse, E., Laban, M., “Conceptual design of a Mach 1.6 European Commercial Transport”, AIAA 2004-4541, 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, 2004. [4] Synaps Ingenieur-Gesellschaft mbH, SynapsPointer Pro V2.50, Synaps IngenieurGesellschaft mbH, Bremen, Germany, 2002. [5] MSC.NASTRAN 2001 Quick Reference Guide, MSC.Software Corporation, USA, 2002, www.mscsoftware.com. [6] Rowan, T. H., “Functional Stability Analysis Of Numerical Algorithms”, Ph.D. Thesis, Department of Computer Sciences, University of Texas at Austin, USA, May 1990. [7] Brodersen, O., M. Hepperle, A. Ronzheimer, C.-C. Rossow and B. Schöning, “The Parametric Grid Generation System MegaCads“ In: Soni, B.K., J.F. Thompson, J. Häuser and P. Eisemann (Ed.): “5th International Conference on Numerical Grid Generation in Computational Field Simulation“. National Science Foundation (NSF), 1996, pp. 35353562. http://www.megacads.dlr.de [8] Kroll, N., C.-C. Rossow, K. Becker and F. Thiele, “MEGAFLOW – A Numerical Flow Simulation System“, Aerospace Science Technology, Vol. 4, pp. 223-237, 2000. [9] Herrmann, U., “MDO Design and Aerodynamic Off-Design Analysis of a Mach=1.6 Aircraft”, Paper 17 of the KATnet/CEAS Conference, 20-22 June 2005, Bremen, Germany. [10] Herrmann, U., Frhr. von Geyr, H., Werner-Westphal, Ch.: “Mixed Fidelity Multi Discipline Optimization of a Supersonic Transport Aircraft”, AIAA 2005-534, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, Jan. 2005.
Three-Dimensional Discontinuous Galerkin Codes to Simulate Viscous Flow by Spatial Discretization of High Order and Curved Elements on Unstructured Grids Christian Lübon and Siegfried Wagner Universität Stuttgart, Institut für Aerodynamik und Gasdynamik, Pfaffenwaldring 21, 70569 Stuttgart, Germany
[email protected]
Summary The need of high order boundary discretization in case of a high order code, e.g. a Discontinuous Galerkin (DG) Code, has already been demonstrated in the literature. Bassi and Rebay extended the DG method to solve the NavierStokes equations for laminar and 3D turbulent flow. In the present paper an extension will be provided to include both three-dimensional flows and curved elements to properly represent three-dimensional bodies with curvature. Some results, like flows around a sphere and around an Onera M6 wing, are calculated and compared with experiments.
1
Introduction
The Discontinuous Galerkin (DG) method combines ideas from the finite element and finite volume methods, the physics of wave propagation expressed by Riemann problems and the accuracy obtained by high-order polynomial approximations within elements. It was originally developed for hyperbolic conservation laws in 2D flow [1,9,14] and sometimes 3D flow [13] including the Euler equations. However, in real life applications the flow is in most cases turbulent and 3D. The original development of DG methods was devoted to the Euler equations that contain only derivatives of first order. The brake through for solving the Navier-Stokes equations with derivatives of second order was presented by Bassi and Rebay [2]. Still another big step was to be done, namely to extend the algorithms for the handling of turbulent flows. The Reynolds averaged Navier-Stokes (RANS) equations had to be solved. Thus, the algorithms had to be extended to include turbulence models [3,4,5]. Since there is not only one turbulence model, but several, a big effort was necessary to gain experience with these models and to find out which one is best. This was done for two-dimensional flow by Landmann et al. [10] who also demonstrate the big improvement in accuracy and stability of numerical procedures by applying curved elements in this case. This was already shown for the Euler equations [1,9,13,14]. The present paper C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 145–153, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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is based on the experience by Landmann et al. [10] and extends the algorithms to three-dimensional laminar flows which is by far not straight forward.
2 2.1
Discontinuous Galerkin Schemes
Basic Equations
The Navier-Stokes equations can be written in the following compact differential form ∂U + ∇ · Fi (U ) − ∇ · Fv (U, ∇U ) = 0 (1) ∂t Here U is the vector of conservative variables, Fi and Fv are the convective and diffusive flux functions, respectively. The next step is to handle high order derivatives. According to Bassi and Rebay [2] we first reformulate this equation as a first-order system introducing the gradient of the solution ∇U as a new additional independent unknown Θ. Replacing ∇U with Θ we get ∇U − Θ = 0
(2)
∂U + ∇ · Fi (U ) − ∇ · Fv (U, Θ) = 0 ∂t
(3)
We now apply the DG approach resulting in the equations for an element E vk ΘdE − vk · Uh · n dσ + ∇vk · Uh dE = 0 (4) E E ∂E ∂Uh vk · Fi · n dσ − ∇vk · Fi dE vk dE + ∂t ∂E E E =Euler − vk · Fv · n dσ + ∇vk · Fv dE = 0 (5) E ∂E =N avier−Stokes
n
where Uh = U (x, t) = k=1 Uk (t)vk (x) is the approximation for the numerical solution Uk , and the n shape or basis functions vk are a base for the polynomial space P k . We choose polynomial functions vk according to Karniadakis [8], which form a hierarchical and orthogonal basis with diagonal mass matrix. The orthogonality reduces computational effort since all mass matrices are diagonal and their inversion is trivial. Our scheme can be advanced explicitly as well as implicitly in time. The explicit time integration, used in this paper, is performed with one-step RungeKutta type schemes of first to fourth order accuracy. 2.2
High-Order Boundaries
In engineering problems, especially in aerodynamics, walls are most often curved. Discretization by triangles or tetrahedrons might be sufficiently accurate for
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most problems, but there are some striking examples where this strategy fails. When computing the inviscid flow field around a circular cylinder or a sphere at a moderate Mach number using unstructured grids and classical triangles or tetrahedrons at the boundary, we get a highly unsteady solution and in some cases separation at the corners and edges between boundary elements. In the framework of the Euler equations, without viscosity, we would rather expect to obtain the steady potential flow field around the cylinder in the limit of vanishing Mach number. For the two-dimensional case Bassi and Rebay [1] have shown for the first time that they indeed are able to obtain the correct solution if they use curved triangles with a polynomial interpolation on the cylinder to take into account the correct physical shape of the wall. Now we present a new procedure for three-dimensional curved edges. Therefore, for a high order code and applied aerodynamic problems a curved boundary discretization is mandatory. The boundary discretization has to be continuously differentiable, that means that there are no discontinuities in the surface normals.
3 3.1
Numerical Procedure for Isoparametric Tetrahedron Elements Normal Vectors
Before constructing a mapping formula, we must give a definition of the curved boundary wall. The usually only available information are the boundary vertices itself, which stretch the boundary elements. With this information we can calculate a normal vector for each element. Afterwards at each boundary vertex a weighted average normal vector is calculated, figure 1. As weighting functions we use two alternatives, firstly the angle between the sides of the adjacent element and secondly the area of the adjacent elements, but for non-distorted elements this makes little difference. 3.2
Polynomial Interpolation
The spatial discretization will be done by tetrahedrons with triangular faces. We can formulate a polynomial interpolation procedure that satisfies nine conditions: p(ξ, η) = a1 + a2 · ξ + a3 · η + a4 · ξ 2 + a5 · ξ · η +a6 · η 2 + a7 · ξ 3 + a8 · ξ 2 · η + a9 · η 3
(6)
p(vi ) = 0,
i ∈ 1, 2, 3
(7)
∇p(vi ) = n(vi ),
i ∈ 1, 2, 3
(8)
Here p is the interpolation polynomial in the local ξ-η face coordinate system, vi (ξ, η) are the vertices and n(vi ) are the partial derivatives at each point, computed with the normal vector information, described in the previous section. After having solved this linear set of equations we get the wrapping surfaces of all walls in our flow field, figure 1.
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Fig. 1. 3D polynomial interpolation at a sphere
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Spatial Transformation
According to the finite element framework we have to change our integration domain from physical space to the numerical ξ-η-ζ system. The mapping from the numerical space to the physical x-y-z coordinate system is approximated by polynomials of the following form: u(ξ, η, ζ) =
−i−j N N −i N i=0 j=0
αijk · ξ i · η j · ζ k
(9)
k=0
In the tetrahedral case we use a cubic transformation polynomial with 20 transformation points P. We must now compute the transformation matrix like in the finite element case [11]: ⎡ ⎤ u(P1 (ξ, η, ζ)) ⎢ u(P2 (ξ, η, ζ)) ⎥ ⎥ A−1 = ⎢ (10) ⎣ ⎦ ... u(P20 (ξ, η, ζ)) After having inverted this matrix we solve the following equation α = A · ue
(11)
where ue is the point vector in physical space. With the coeffcients α we get the relation between the numerical and the physical system. ⎡ ⎤ ⎤ ⎡ −i−j αxijk N N −i N x(ξ, η, ζ)) ⎣ αyijk ⎦ · ξ i · η j · ζ k ⎣ y(ξ, η, ζ)) ⎦ = (12) i=0 j=0 k=0 αzijk z(ξ, η, ζ)) With this transformation unfortunately we lose the nice property of the Jacobian of the transformation to be constant within each tetrahedron. Because of our DG
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discretization (4,5) we have to evaluate some integrals over tetrahedral elements in the physical space. With our transformation Jacobian we are able to do this via a Gaussian cubature formula: (13) f (x, y, z)dxdydz = f (ξηζ) · |J(ξ, η, ζ)| dξdηdζ Because of this Jacobian the Gaussian cubature scheme becomes more expensive compared to the non curved case with a linear mapping scheme. The increase of the computational cost is neglectable because the whole domain contains only a few curved wall cells.
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Three Dimensional Tetrahedron Boundary Mapping Scheme
If, as presupposed, the tetrahedrons wall side is on the outline, we cannot guarantee that the interior faces of the curved tetrahedron are flat like in the much simpler two-dimensional case, figure 2. Therefore, we must put a second layer of curved cells around our primary curved wall cells to avoid holes and overlappings in the discretization domain, figure 5. The process is the same as for the curved wall cells, but this time we get a curved inner surface. This surface is constructed by mapping the known transformation points from the wall cells to the curved inner cells. Afterwards again the cubic transformation follows, as described in (9).
Fig. 2. Curved Cells, 2D case with straight edges and the 3D case with a curved boundary triangle and curved inner faces
In high Reynolds number simulations, we need additional layers to use cells with high aspect ratios [10]. For thin boundary layers also hexahedral elements might be better suited.
6 6.1
Results
Potential Flow Field Around a Sphere
The first example is the inviscid flow past a sphere. When computing this flow field at a moderate Mach number using unstructured grids and classical triangles
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cp 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2
Fig. 3. Pressure distribution of order 2-4, classical triangle boundary discretization (top) and curved boundary discretization (bottom)
at the boundary, we get a highly unsteady solution and in some cases separation at the boundary edges. Within the framework of the Euler equations, without viscosity, we would rather expect to obtain the steady potential flow field around the sphere in the limit of vanishing Mach number. The following figures shows a comparison between flat triangle boundaries and the curved boundary discretization technique, figure 3. For the flat triangles we see separated flows and unphysical effects, caused by the bad spatial resolution. To avoid compressibility effects, we use a low Mach number of M a∞ =0.1 and are able to reach the minimum of the pressure coeffcient cp,min =-1.25 in the curved case, predicted by symmetric potential theory. 6.2
Laminar Flow Past a Sphere
The next example shows the steady laminar flow past an isolated sphere, which may be seen as a simplified case of a general family of body flows. For very low Reynolds numbers an exact solution exists, the Stokes solution with fore and rear symmetry. At Reynolds numbers, between 20 and approximately 210, a separated, steady and axisymmetric solution exists, for Reynolds > 210 instabilities appear and cause unsteady flow like in the two-dimensional counterpart, the Von Karman vortex street. For our numerical investigation, we choose the experimentally often measured case with Re=150 with a small separation area and steady flow. To get a mesh independent solution we use different grid levels, the finest grid with 240 000 cells and the coarsest grid with 14 cells in circumferential direction on the sphere, one cell in the boundary layer and 17 000 cells in the
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Table 1. Comparison of the separation area for the laminar flow behind a sphere at Re=150 cd xs xc present results: coarsest grid - O 5 0.68 1.21 0.80 finest grid - O 5 0.69 1.20 0.81 Johnson/Patel [7] 1.21 0.83 1.19 Tomboulides [12] Donley [6] 0.69
yc
θs
0.38 120 0.37 118 0.33 121
Fig. 4. Velocity distribution and streamtraces for the laminar flow around a sphere
whole computational domain. As illustrated in table 1 and figure 4 the flow separates from the surface of the sphere and rejoins at the point xs on the symmetry axis to form a separation bubble centered at (xc ,yc ). The outstanding result is that the new procedure of the present paper predicts these parameters including the drag coefficient with high accuracy even when applying a very coarse grid. 6.3
Onera M6
The last example, an airplane wing, shows the advantages and some disadvantages of a curved tetrahedral boundary discretization. Like the much simpler case, the sphere, our curvation algorithm constructs a really good surface interpolation in smooth regions, like the leading edge. However, the surface of the wing differs in two points from the sphere, it contains a pointed trailing edge and a circular wingtip. Since the trailing edge should remain pointed we use here classical non curved triangles, but the tip edge should show the real boundary contour. But this is not possible. If we look at the constructed normal vectors and the very distorted tetrahedral surface cells we recognize that we get degenerated cells in this region, figure 6. An improvement of this degeneration problem could
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Fig. 5. Curved wallcells and curved inner cells
Fig. 6. Nearly degenerated Tetrahedron
Fig. 7. Curved and backward straight trailing edge
be the usage of prismatic elements, which are better suited for the discretization of the boundary layer, anyway. This leads to a global hybrid discretization scheme for example with tetrahedral and prismatic elements which is state of the art for unstructured finite volume solvers. Figure 7 shows the partly resolved trailing edge of the wingtip.
7
Conclusion
This new boundary method is currently being used in our in-house CFD solver SUNWinT (Euler and Navier-Stokes). We have shown that a classical polygonal
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boundary description ruins the accuracy of the high order method in the case of applied aerodynamics specifically in three-dimensional flow. The importance of curved boundaries in the case of turbulent flows will be the objective of our further investigations, we have shown this for the two dimensional case [10]. In the future we are going to extend our code to include curved prismatic elements to solve the shown problems. Furthermore, we need to increase our convergence rate by including multiple speedup schemes, like a matrix free implicit algorithm and an hp multigrid scheme.
References [1] Bassi F., Rebay S., High-order accurate discontinuous finite element solution of the 2D Euler equations, J. of Comput. Physics, vol. 138, pp. 251-285, 1997. [2] Bassi F., Rebay S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. of Comput. Physics, vol. 131, pp. 267-279, 1997. [3] Bassi F., Rebay S., A high-order discontinuous Galerkin method for compressible turbulent flows, Discontinous Galerkin methods, pp.77-88, Springer, 1999. [4] Bassi F., Crivellini A., Rebay S., Savini M., Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations, Computers and Fluids, vol. 34, pp. 507-540, 2005. [5] Bassi F., Crivellini A., Di Pietro D.A., Rebay S., A high-order discontinuous Galerkin solver for 3D aerodynamic turbulent flows, European conference on Computational Fluid Dynamics, ECCOMAS CFD, 2006. [6] Donley E., The Drag Force on a Sphere, UMAP Journal, vol. 12, no. 1, pp.47-80, 1991. [7] Johnson T. A., Patel V. C., Flow past a sphere up to a Reynolds number of 300, J. of Fluid Mechanics, vol. 378, pp. 19-70, 1999. [8] Karniadakis G. E., Sherwin S., Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, 2004. [9] Krivodonova L., Berger M., High-order accurate implementation of solid wall boundary conditions in curved geometries, J. of Comput. Physics, vol. 211, pp. 492-512, 2006. [10] Landmann B., Kessler M., Wagner S., Krämer E., A parallel Discontinuous Galerkin code for the Navier-Stokes equations, AIAA 2006-111, 2006. [11] H. R. Schwarz, Methode der finiten Elemente, Teuber, vol. 47, pp. 66-82, 1980. [12] Tomboulides A. G., Direct and large-eddy simulation of wake flows: flow past a sphere, PhD Thesis Princeton University, 1993. [13] Vegt J. van der, Ven H. van der, Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows, II. Efficient flux quadrature., Comput. Meth. Appl. Mech. Engrg., 191, pp. 4747-4780, 2002. [14] Vegt J. van der, Ven H. van der, Slip flow boundary conditions in discontinuous Galerkin discretizations of the Euler equations of gas dynamics, Fifth World Congress on Computational Mechanics, 2002
The Space-Time Expansion DG Method Frieder L¨orcher , Gregor Gassner , and Claus-Dieter Munz Institut f¨ur Aerodynamik und Gasdynamik, Universit¨at Stuttgart, Pfaffenwaldring 21, D-70569 Stuttgart, Germany {loercher,gassner,munz}@iag.uni-stuttgart.de
Summary In this paper the recently developed space-time expansion discontinuous Galerkin (STE-DG) approach for the two dimensional unsteady compressible Navier-Stokes equations is presented. The basis of the scheme is a weak formulation of the NavierStokes equations, where special care of the second order terms is taken. The spatial polynomial of the DG approach is expanded in time using the so called CauchyKovalevskaya (CK) procedure. With a polynomial of order N in space the CK procedure generates an approximation of order N in time as well yielding a scheme of order N+1 in space and time. The locality and the space-time nature of the presented method give the interesting feature that the time steps may be different in each grid cell. Hence, we drop the common global time steps and propose for a time-accurate solution that any grid cell runs with its own time step determined by the local stability restriction. In spite of the local time steps the scheme is conservative, fully explicit, and as in the DG approach the polynomial order could be chosen arbitrarily, the scheme is theoretically of arbitrary order of accuracy in space and time for transient calculations.
1 Introduction The class of discontinuous Galerkin (DG) finite element schemes is an interesting candidate to construct high order accurate schemes for solving the fluid flow equations in complex three-dimensional domains. Usually, high order time discretization of DG schemes for unsteady computations is done separately from space discretization by the method of lines approach using a Runge Kutta scheme (RK-DG). The fully explicit ADER-DG schemes, as presented for the Euler equations in [3], provide the time approximation in one single step with the same order of accuracy in time as the space discretization. The scheme presented in this paper is based on a Taylor expansion in space and time at the barycenter of a grid cell which results in a space-time polynomial in the grid cell from one time level to the other. Within the FV framework this idea has already been proposed by Harten et al. in [6] for the one-dimensional Euler equations. Once the space-time polynomial is computed using the Cauchy-Kovalevskaya procedure, the other steps for the construction of the scheme are straightforward. All integrals are approximated by Gauss quadrature formulae. The fluxes between the C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 154–161, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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grid cells are approximated by any appropriate numerical flux evaluated at the spacetime Gaussian points of the element interfaces. We call the presented scheme the space-time expansion discontinuous Galerkin scheme with the abbreviation STEDG. The STE-DG scheme is fully explicit and has to satisfy a quite similar time step restriction as the RK-DG or the ADER-DG schemes. Especially for problems with a large disparity in element sizes and wave speeds this may result in an excessive number of time steps being taken in regions of low activity, as common schemes using a global time step have to use the most restrictive time step over the whole computational domain. In particular this strongly shrinks the efficiency of adaptation by choosing the degree of the polynomials locally or by local mesh refinement. The choice of a local time step would cure this inefficiency. The space-time approximation of the STE-DG method gives the possibility to realize such a fully time-consistent and conservative local time-stepping in a straightforward way and for arbitrary order of accuracy. We allow the time steps to be different in each element, and each element may run with its individual time step determined, e.g., by the local stability condition. As the local time-stepping algorithm introduces a small computational overhead only, this strategy strongly enhances the efficiency for problems in which small time steps are needed in some regions only.
2 The Weak Formulation of the 2D Navier-Stokes Equations In two space dimensions the Navier-Stokes equations may be formulated in the conservation form as (1) Ut + ∇ · F (U ) = ∇ · F v (U, ∇U ) with the viscous flux Flv (U, ∇U ) = Dl1 (U )Ux1 + Dl2 (U )Ux2 .
(2)
To derive the weak formulation we first multiply the Navier-Stokes equations by a test function Φ = Φ(x) and integrate over an arbitrary space-time cell Ωin := Qi × [tn , tn+1 ]: (Ut + ∇ · (F (U ) − F v (U, ∇U ))) · Φ dxdt = 0. (3) Ωin
Following the derivation in [5], we get the weak formulation of the NavierStokes equations as Ut · Φ dxdt + (Fn − Fnv ) · Φ dsdt − (F − F v ) : (∇Φ) dxdt Ωin
+ ∂Ωin
∂Ωin
h(U, n, ∇Φ) dsdt = 0,
Ωin
(4)
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where the subscript n means differentiation in normal direction. The additional flux h is given by h(U, n, ∇Φ) := U · Fnvv (U, ∇Φ) − [U · Fnvv (U, ∇Φ)]IN T
(5)
with the transposed or adjoint fluxes Flvv (A, ∇B) = DT1l (A)Bx1 + DT2l (A)Bx2 , Fnvv (A, ∇B) := F1vv (A, ∇B)n1 + F2vv (A, ∇B)n2 .
(6)
3 The Space-Time Expansion Approach for the 2D Navier-Stokes Equations We now introduce the approximate solution Uh = Uh (x, t) which is at each fixed time level a piecewise polynomial in space. In the grid cell Qi it is represented by the polynomial Ui given as a linear combination of basis functions φi,l = φi,l (x), which span the space of polynomials of degree N with support Qi . Ui (x, t) =
N
ˆi,l (t)φi,l (x). U
(7)
l=0
ˆi,l (t), l = 1, .., N are the time dependent degrees of freedom (DOF) in the Here, U grid cell Qi . The temporal evolution of the degrees of freedom is represented by discrete values at the different time levels. E.g., at the time level tn we have Ui (x, tn ) =
N
ˆi,l (tn )φi,l (x) =: U
N
l=0
n ˆi,l U φi,l (x).
(8)
l=0
We insert Uh into the weak formulation (4) and restrict the test function to polynomials of maximal degree Ni choosing them equal to the basis functions φi,l = φi,l (x). As the approximate solution may be discontinuous across cell interfaces, we have to use numerical fluxes Gn , Gvn and gnvv . We choose for Gn the HLLC flux (see for example [9]) and for Gvn and gnvv viscous fluxes as derived in [5]. Using orthonormal sets of basis functions, the temporal evolution of the DOF of an element Qi for one time step is given by
ˆ n+1 U i
−
ˆn U i
n+1 t i
=−
n+1 t i
( F − F v ) : ∇Φdxdt + tn Qi i
( Gn − Gvn ) · Φdxdt tn ∂Qi i
n+1 t i
gnvv dxdt.
+ tn ∂Qi i
(9)
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All integrals are computed numerically using Gaussian quadrature formulae. As the ˆ n could give the arguments for the flux functions only at tn , we still have known U to define how to get these arguments for the whole time interval [tn , tn+1 ]. This is done in the STE approach introduced in [8] by a Taylor expansion in space and time. A natural choice is to expand the space time Taylor series in the center xi of each element Qi , using data from time level tn U (x, t) = U (xi , tn ) +
Ni ∂ 1 ((t − tn ) + (x − xi ) · ∇)j U (xi , tn ). j! ∂t j=1
(10)
With this space time Taylor series it is possible to approximate U and ∇U at all space time points (x, t) ∈ Qi × [tn ; tn+1 ], which is needed for the evaluation of the numerical fluxes. The space time derivatives of U at (xi , tn ) needed for the construction of (10) have to be computed using the so-called Cauchy-Kovalevskaya procedure.
4
Local Time Steps
The STE-DG scheme allows a time-accurate fully conservative local time-stepping algorithm with marginal computational overhead, see [8]. It allows to evolve in time with different time steps in each grid cell. 4.1 The Local Time-Stepping Algorithm We give up the assumption that all grid cells run with the same time step and therefore we do not have any longer a common time level. Let us denote the actual local ˆ n represent the solution time level in grid cell Qi by tni . The degrees of freedom U i at tni in this grid cell. Furthermore, each cell may evolve in time with its local time step ∆tni which has to satisfy the local stability restriction. With ∆tni , the next local time level in Qi is given as = tni + ∆tni . (11) tn+1 i ˆ i from level tn to tn+1 , the right hand side of equation In order to evolve the U i i (9) has to be evaluated for each Qi . To guarantee that this is done in a proper way, the succession of evolving elements has to be controlled. ˆ ∗i DOF without physical meaning obtained by evaluation of We denote by U equation (9), whereby the surface flux integrals are not or only partially taken into account. As an example we sketched a sequence of four time steps with three adjacent grid cells in figure 1 and start from a common time level t0 = 0. After the determination of the local time steps, which are assumed to be different in our example, the space-time Taylor expansions are calculated in each element. This results in an approximate solution in the space-time cells Qi × [t0i , t1i ] - in the example for i = 1, 2, 3.
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Fig. 1. Sequence of steps 1-4 of a computation with 3 different elements and local timestepping
The space-time polynomials are stored. After this step the degrees of freedom 0 ˆ U i at the time level t0i are not needed any longer and may be overwritten in the ˆ ∗ . Contributions of the flux integrals onto this variable will be computer code by U i added subsequently. When all contributions are gathered, the DOF are completely ˆ ∗ become U ˆ 1. updated and U i i ˆ ∗ we compute in order to evaluate the right hand side of equation (9) The first U i is given by ti+ 1 = tni and ti− 1 = tni for the surface fluxes, that is, these fluxes 2 2 are not yet taken into account. The flux and source volume integrals rely only on the local space-time polynomials and can easily be calculated for each element Qi . In our example we have the situation depicted in the upper left corner of figure 1. For each element, a space-time polynomial exists and the volume integral contribuˆ ∗ for each element Qi . Next, the surface flux tions are already contained in the U i contributions involving neighboring grid cells have to be considered. The local timestepping algorithm relies on the following evolve condition. The update of the DOF can only be completed, if tn+1 ≤ min tn+1 , ∀j : Qj ∩ Qi = ∅ i j
(12)
is satisfied. In our example, the only grid cell satisfying this condition is Q2 . So Q2 is evolved to t12 . To do so, the flux contributions at the right and left cell interface ˆ ∗ . The flux integrals are calculated have to be computed and added to the local U 2 using Gauss quadrature formulae from t = t02 to t = t12 at the right interface ∂Q2+ 12 and the left interface ∂Q2− 12 . The arguments for the numerical flux functions at
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time Gaussian points are obtained from the left and right space-time polynomials. In order to get a conservative and efficient scheme, the flux contributions computed ˆ ∗ and simultaneously to the corresponding for the evolution of Q2 are added to U 2 ∗ ∗ ˆ and U ˆ . If both flux contributions are added to U ˆ ∗ , the update is neighbors U 1 3 2 completed and the DOF of Q2 at the new time level t12 are known. We can now proceed as in the first time step. A new space-time polynomial is constructed on ˆ 1, Q2 × [t12 , t22 ] and the volume integral contribution is added to the local DOF U 2 ˆ ∗ . We are then in the situation sketched in the upper right corner now named by U 2 of figure 1. Now Q1 satisfies the evolve condition and can be advanced to t11 . As before, the volume integral contribution was already computed and added. But in this case, ˆ ∗ during the also a part of the flux contributions has already been added to the U 1 previous evolution of Q2 . So only the missing flux contributions, which are sketched ˆ ∗1 in order to get the in the lower left corner of figure 1, have to be added to the U ˆ 1 . Namely, on the interface ∂Q 1 , the flux integral has to be computed with U 1
1+ 2
a quadrature formula from t12 to t11 . As before, the flux integral computed on this ˆ ∗ . The time interval, for ˆ ∗ , but also to the U shared interface is not only added to U 1 2 which the flux contribution at the interface shared by an element Qi and an adjacent ˆ ∗ to U ˆ n+1 , is generally element Qj has to be computed when evolving U i
i
] = [max(tni , tnj ), tn+1 ] [tij , tn+1 i i
(13)
In this manner, the algorithm continues by searching for elements satisfying the evolve condition (12). So all elements are evolved in a suitable order by evaluating the different terms of the right hand side of equation (9) for each element in an effective order. At each time, the interface fluxes are defined uniquely for both adjacent elements, making the scheme exactly conservative. The presented local time stepping algorithm minimizes the total number of time steps for a computation with fixed end time. In numerical experiments presented in [5], we found that the STE-DG scheme is about 1-2 magnitudes faster than a Runge-Kutta global time stepping scheme, depending on the problem.
5
Numerical Results
5.1 Flat Plate Boundary Layer Results of a simulation of the flat plate boundary layer at M a∞ = 0.1 and Re∞ = 5000 are presented. The computation has been done with polynomial degree 4. In figure 2, the computed skin friction coefficient cf is compared to the incompressible blasius solution. One can see a very good agreement of the results.
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Fig. 2. Computational mesh (left) and computed skin friction coefficient for the simula tion of a flat plate boundary layer
5.2 Airfoil Gust Response P roblem In this section, we present results of a numerical simulation of acoustic response of a Joukovsky airfoil to a sinusodial gust. This AIAA benchmark problem was proposed at the 4th Computational Aeroacoustics (CAA) workshop on benchmark problems (category 3, problem 1, case 2). This example was computed using hp-adaptivity in the vicinity of the airfoil, that is, the local element size and polynomial degree are adapted to the flow solution automatically. On the left of figure 3, the computational grid with local polynomial degrees is shown. For the transport of acoustic waves in the farfield, large elements can be used, whereas in the vicinity of the airfoil much smaller cells have to be used in order to resolve the flow phenomena. The local time stepping algorithm guarantees very efficient computation of this farfield region due to locally large time steps. Indeed, the computational effort spent in the farfield is less than one percent of the entire CPU time.
Fig. 3. Computational mesh (left) and acoustical pressure contours
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On the right of figure 3, a snapshots of the contours of the acoustical pressure is shown. The results are in good agreement with results of Dumbser [2]. The entire computation took 8 hours on a single AMD Opteron 270 processor.
6 Conclusions In this paper we proposed an explicit discontinuous Galerkin scheme (called the STE-DG scheme), which is of arbitrary order of accuracy in space and time for the solution of the compressible Navier-Stokes equations. The locality of the STE-DG discretization and the space-time approximate solution allows to use different time steps in different elements which is bounded by the local stability restriction only. The proposed local time-stepping STE-DG scheme seems to have a large potential in combination with hp-adaption, as the time step restriction may strongly vary as a local function of element size, wave speed, and polynomial degree. As numerical example, a flat plate boundary layer was computed at M a = 0.1.
References [1] F. Bassi and S. Rebay: ”A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations”. Journal of Computational Physics 131, 1997, pp. 267-279. [2] M. Dumbser: ”Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains”. Shaker Verlag, Aachen,2005 [3] M. Dumbser, C.-D. Munz: ”Arbitrary High Order discontinuous Galerkin schemes”. Numerical Methods for Hyperbolic and Kinetic Problems, IRMA Series in Mathematics and Theoretical Physics, EMS Publishing House, 2005, pp. 295-333. [4] G. Gassner, F. L¨orcher, C.-D. Munz:”A Contribution to the Construction of Diffusion Fluxes for Finite Volume and Discontinuous Galerkin Schemes”, J. Comput. Phys., 2006, doi:10.1016/j.jcp.2006.11.004 [5] G. Gassner, F. L¨orcher, C.-D. Munz:”A discontinuous Galerkin scheme based on a spacetime expansion. II. Viscous flow equations in multi dimensions”, submitted to J. Sci. Comp., 2006 [6] A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy: ”Uniformly High Order Accurate Essentially Non–oscillatory Schemes III”. J. Comput. Phys. 71, 1987, pp. 231-303. [7] O. Inoue, N. Hatakeyama: ”Sound Generation by a Two-Dimensional Circular Cylinder in a Uniform Flow”. Journal of Fluid Mechanics 471, 2002, pp. 285-314. [8] F. L¨orcher, G. Gassner, C.-D. Munz: ”A discontinuous Galerkin scheme based on a space-time expansion. I. Inviscid compressible flow in one space dimension”, J. Sci. Comp., 2007, doi:10.1007/s10915-007-9128-x [9] E.F. Toro: ”Riemann Solvers and Numerical Methods for FluidDynamics”. Springer, 1997.
The Parallel Mesh Deformation of the DLR TAU-Code Thomas Gerhold1 and Jens Neumann2 1
2
DLR, Institute for Aerodynamics and Flow Technology DLR, Institute for Aeroelasticity, Bunsenstr. 10, 37073 Göttingen, Germany
Summary Mesh deformation for unstructured grids is usually solved employing algorithms based on a spring analogy or on a linear elasticity analogy. These methods require solving discrete equations for the given grid using iterative methods. The computational effort for these methods is not negligible compared to the effort required by the flow solver. In order to optimise the efficiency of the computational chain for coupled simulations of fluid structure interaction an alternative algebraic method was developed for the DLR TAU-Code. The TAU-Code is parallelized by domain decomposition using the message passing concept based on MPI such that each process works on one partition of the grid only. In order to employ the grid deformation on the grid partitions distributed over the processes, a parallel method is required. This paper describes the parallel grid deformation method and shows its efficiency and robustness for the DLR F6 geometry. Further applications are shown in order to demonstrate the grid deformation for dynamic aeroelastic simulations.
1 Introduction In many fluid mechanic simulations deformations of surfaces need to be handled. The source of surface geometry displacements can be for example the structural response of a body to loads. One way of handling surface displacements is via mesh deformation. In this approach all interior grid point locations are computed as a function of the surface displacements, such that the grid remains valid. A valid deformed grid contains grid cells which are moved and deformed without changing the grid connectivity and without the occurrence of cell collapsing or cell overlap. For unstructured hybrid grids, deformation methods are usually based on tension spring, torsion spring or linear elasticity analogy. An overview was recently given in [4], according to which only the linear elasticity method is stable enough for large scale deformations of 3D hybrid Navier-Stokes grids. The grid deformation method developed for the TAU-Code [3] is based on an alternative method, which is robust and very efficient with respect to CPU-time requirements in sequential mode. Nevertheless it has been parallelized to allow for simulations on parallel computers with distributed memory without a limitation of the application size due to the memory limits on the single computational nodes. This paper presents the parallel performance of the algorithms employed in the grid deformation method. Furthermore, results of dynamic aeroelastic simulations are presented for the demonstration of the applicability of the method for relevant use cases. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 162–169, 2007. © Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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2 Mesh Deformation Method Input for the mesh deformation are an unstructured hybrid grid and displacement vectors for the moving surface grid points. For the remaining surface grid points parameter input is required to define their boundary treatment. Surface points can be either fixed, move freely (e.g. on far-field boundaries), or they are projected onto a surface (e.g. on a symmetry plane) after computing the point movement. Output of the mesh deformation is a new grid with updated coordinates for all grid points, while the grid connectivity remains fixed. An algebraic method is employed as a first step. First, all grid points are sorted with increasing distance to the deformation surface. Then a single loop over this list of points is performed. The new displacement for each point is a scaled average from the displacements of the neighbour points which have a smaller distance and which are thus already updated. This strategy leads to advancing the front of displacements as iso-surfaces of equal distances, which was found to be of advantage in order to transport the information orthogonal to the surfaces into the interior of the grid. The average of the displacement (D) of a point (i) is weighted by the length of the connecting edges (L) to the contributing neighbours (j):
Di
¦W j
j
Dj
¦W
j
with
Wj
1 Lj .
j
The scale is a function of the ratio between the local displacement and the cell sizes around the grid point. It is used to reduce the displacements by a fraction of the local cell sizes in order to decrease the mesh deformation with increasing distance from the deformed surfaces. Therefore for each grid point a minimum height of the surrounding elements is computed and stored as length scale. If the local point displacement (D) is smaller than a fraction (e.g. K = 0.1) of this length (L) the local displacement is reduced by D = D * D / (K * L). In parallel mode this step is computed on each grid partition. The partitions have an overlap of one cell layer. The displacements are computed in the interior of the domains only. The values on the overlap points are updated by communication with the neighbouring partition having the matching points in the interior. In order to allow for the information transport over the partition boundaries the algorithm is repeated until the displacements in the overlap regions do not change any more in between two steps. As a result the single loop needed in sequential mode has to be repeated typically between 20 to 50 times, depending on the specific case. The impact on the parallel efficiency is shown further below in figure 3. For the efficient implementation of the above described method an efficient method for computing distances of grid points to the nearest surface point is needed. The method employed here stores (for each grid point) the coordinate of the nearest surface point, which is initially set to a coordinate far away (several multiples of the maximum grid size). For each surface point the coordinate is set initially to its own coordinate, which is already the correct value. In a loop over all interior points each point checks if the surface coordinate of one of its neighbours is nearer than its own surface coordinate and overwrites its value if this is the case. This loop is repeated until no point is updated any more. To minimize the number of sweeps the points are sorted such that the surface point neighbours come first, then the next neighbours and so on, which requires most often less than 15 or 20 sweeps in the sequential mode. In
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parallel mode the surface coordinates are communicated after each sweep for all points in the overlap regions, which increases the number of required sweeps. The impact on the runtime is shown further below in figure 3. The way the grid point displacements are averaged and scaled is somewhat arbitrary. Modifications of this procedure can change the robustness of the method and may even improve it. However, our experience is that modifications allow for relative small improvements only. When the displacements become too large the algebraic deformation reaches its limit, i.e. the first cells of the grid start to collapse. It has been observed that this limit can be extended considerably when accepting the collapse of a few more cells only. Thus, in a second stage of the deformation, an algorithm is employed to repair collapsed cells. To do so, each region in the grid containing collapsed cells is marked such that it is bounded by valid cells only. With the shapes of this boundary in the deformed and the non-deformed grid a transformation can be computed applying a volume spline interpolation [1], which allows rebuilding the collapsed cells as images of the original ones. As long as these regions remain small the additional computational costs for the local volume spline is low. Because the computational costs for a volume spline increase quadratically with the number of points regions with collapsed cells are split into parts having less than 1000 points. The spline technique is applied to all parts separately. This allows handling larger regions of collapsed cells with acceptable computational costs. Regions in which the local repair is not successful are enlarged by adding an additional layer of surrounding cells, before repeating the spline procedure in a loop until all cells are valid. The method is not only used to repair collapsed elements, it is also applied to skew elements which degenerate below a certain limit (due to the deformation). The grid repair stage is applied in parallel mode inside the grid partitions. After each sweep of the loop repeating the spline procedure the new coordinates on the partition boundaries are updated by communication. This can change the partition boundaries during the loop due to collapsed cells in the overlap region. As a result the number of iterations for the repair can change depending on the form or the number of grid partitions.
3 Demonstration of Robustness and Efficiency The application of the mesh deformation method of the TAU-Code on the DLR F6 geometry is shown in figure 2. The generic wing deformation is prescribed by vertical displacements of the wing surface computed as a function of the span wise position. The hybrid Navier-Stokes grid for the plain geometry is available from the 2 AIAA CFD Drag Prediction Workshop. It is composed of 2.5 million points, 3.4 million tetrahedra and 3.6 million prismatic cells. Figure 1 shows the elements which collapses after employing the algebraic method without the repair stage for different wing tip displacements. Figure 2 shows all grid deformations performed with and without applying the spline procedure. The worst inner angle in the tetrahedral part of the grid is plotted versus the size of the deformation. It can be seen that applying the local grid repair allows to increase the wing tip deflection by a factor of ten while keeping the grid valid. nd
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Wing tip deformation Collapsed tetras Collapsed prisms
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Wing tip deformation Collapsed tetras Collapsed prisms
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Wing tip deformation Collapsed tetras Collapsed prisms
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Fig. 1. Hybrid mesh for DLR F6 after deformation with the algebraic method. Grid units are 60 for the outer chord length and 580 for the span of the wing.
For the demonstration of the parallel deformation method the F6 grid with pylon nacelle is used. It is composed of 3.75 million points, 4.88 million tetrahedrons and 5.7 million prismatic cells. Because of the increased grid size it is more suited for relevant measures of the parallel performance with a larger number of processes. Figure 3 shows the timings for the main algorithmic parts of the method for different numbers of processes. The wing tip deformation was fixed to 240 grid units (4 times the outer chord length) for all computations, which is considered to be a large wing tip displacement of a realistic size. In addition to the timings of the grid deformation, the CPU time for ten 4W-multigrid cycles of the solver with one-equation turbulence model is plotted. For steady state simulations for static aeroelasticity in which the deformation is applied after 500 iterations this line represents two percent of the CPU time required by the solver. The plot shows that the runtime of the deformation is negligible for all steady state computations. For time accurate simulations ten multigrid cycles are considered to represent a quarter of the computational time needed for one physical time step. This scale shows that the deformation needs less than about 15 percent of the overall CPU-time up to 8 processes. This fraction increases to about 25 percent when increasing the number of processes to 32. The use
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of more processes for this grid size is not useful, because with less than 100000 points per grid partition the speed up of the flow solver employed in multigrid mode becomes too small for using the computer resources effectively.
Fig. 2. Deformation of the hybrid mesh for DLR F6 without (black) and with spline repair (red) for elements with inner angles less than 2 degree. The worst angle inside a tetrahedra in the mesh is compared against wing tip deflection.
Fig. 3. Timings for the main algorithmic parts of the deformation on a 2 GHz Opteron dual core cluster
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Using the deformation on more than eight processes, the grid repair stage becomes the CPU-time dominating algorithm. This is due do the fact that the load for repairing collapsed elements in not balanced over the grid partitions. The partition needing most CPU-time for repair of locally collapsed elements dominates the runtime, which is reduced slightly only by further increasing the number of processes. The costs of the repair stage increase with the number of collapsed cells, which increases with the size of the wing tip displacements. Because the chosen test deformation of half the span of the wing is already quite large for realistic cases, smaller fractions of the overall CPU-time can be expected for the grid deformation. Applying the half size of the wing tip deflection reduces the deformation runtime for 32 processes by approximately 20 percent.
4 Applications For the following aeroelastic simulations, the grid deformation method described above has been used to solve the dynamic mesh problem. The displacements of the surfaces are computed by the response of the structural models on the aerodynamic loads. For the data exchange between the aerodynamic surfaces and the structural model different interpolation methods based on radial basis functions are available. The interpolation algorithm is based on the equivalence of the virtual work and thus the conservation of the energies on both sides of this coupled problem is ensured [2].
(a)
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Fig. 4a and 5b. Wing tip displacement during a three dimensional flutter simulation in the time domain - start from the non-deformed state (AMP-Wing)
In the first example an unsteady fluid structure interaction of the AMP-Wing has been simulated for flow conditions in the transonic flutter regime which results in a three dimensional limit cycle oscillation. Large amplitudes cause wing tip deflections of 12 percent of the wing span, which is shown in figure 4 and 5. The aerodynamic mesh is a hybrid RANS-mesh with 25 semi-structured layers of prisms around the viscous walls and tetrahedrons in the farfield. The CFD grid had one million points, 27600 of them on the surface, while the NASTRAN FE model employs 75 nodes resulting in 225 translational degrees of freedom.
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Fig. 6a and 7b. Deformation of a chimera mesh wing–flap configuration (a) and maximum wing and flap deformations and integrated forces over the time (b)
The second application is for a generic wing/flap geometry in which the structural deformation of the geometry is the response of the aerodynamic loads varying due to the flap deflection, which is shown for one snap shot in time in Figure 6. A generic finite element model is applied for this test case used to check the functionality of the mesh deformation for grids with chimera overlap. The simulation was started for the non-deformed shape with zero angle of attack and zero flap angle. Then the flap angle was turned down to 10 degrees during one second. With increasing lift the aerodynamic loads cause an elastic deformation. After this time the flap angle was kept constant, such that the system stabilizes itself to an aeroelastic equilibrium state. In figure 6 the non-deformed and the deformed aerodynamic meshes are shown. For the deformed mesh the pressure distribution over the wing and flap surfaces are also depicted. The gradients over the simulation time of the deformations for the trailing edge of the wing tip and the leading edge of the flap and the integral forces are shown in figure 7. Admittedly only the elastic deformation is shown but not the rigid body deformation at the trailing edge of the flap due to the flap rotation. The vibration in the time history is the first modal frequency of the dynamic finite elements model.
Fig. 8a and 9b. Non-symmetric deformation of a generic X-31 wind tunnel model during a flight manoeuvre (a) and maximum wing deformations over the time (b)
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The last application is a simulation of a flight manoeuvre of a generic aeroelastic model of an X31 aircraft. The finite elements model has 819 discrete translational degrees of freedom. During the manoeuvre unsteady non-symmetric loads on the aircraft lead to non-symmetric deformation of the structure. This is shown in Figure 8 for the generic finite elements model. The aerodynamic mesh is composed of 19.4 million tetrahedrons, 3.44 million nodes and 261792 nodes on the surface. Figure 9 shows the time history for the wing tip displacements of the left and the right wing during the manoeuvre. It can be seen that due to the different aerodynamic manoeuvre loads the displacements on the left and right hand side differ from each other.
5 Conclusion A parallel algebraic mesh deformation method has been presented and demonstrated for coupled aeroelastic simulations in time or frequency domain. The method requires negligible costs for static aeroelastic simulations. For the dynamic mesh problems the costs can become relevant in parallel mode, due to a non load balanced repair algorithm for collapsed cells.
References [1] Beckert, H. Wendland, Multivariate Interpolation for Fluid-Structure-Interaction Problems using Radial Basis Functions. Aerospace Science and Technology, Art. No. 5125, 2001. [2] J. Neumann, G. EInarsson, A. Schütte, Multidisziplinäre Simulation eines rollended, generischen Deltaflügels – Kopplung von Aerodynamik, Flugmechanik und Strukturdynamik, In: Proceedings, Deutscher Luft- und Raumfahrtkongress 2005, Friedrichshafen, 26.-29. September 2005. [3] D. Schwamborn, T. Gerhold, R. Heinrich, The DLR TAU-Code: recent applications in research and industry, Proceedings of ECCOMAS CFD 2006, Egmond aan Zee, Netherlands, 5.-8. September 2006. [4] Z. Yang, D.J. Mavriplis, Unstructured dynamic meshes with higher order time integration schemes for the unsteady Navier-Stokes equations, AIAA 2005-1222, 41th AIAA Aerospace Sciences Meeting and Exhibit, January 10-13, 2005/Reno, NV.
Simulation of Oscillating Airfoils and Moving Flaps Employing the DLR-TAU Unsteady Grid Adaptation A.D. Gardner, K. Richter, and H. Rosemann DLR AS-HK, Bunsenstr. 10, 37073 G¨ottingen, Germany
[email protected]
Summary The use of unsteady adaptation with the DLR-TAU Navier-Stokes solver is presented as a method of improving the modelling of flows where the aerodynamic performance of a body is determined by the action of moving localised regions of high-gradient flow. Examples are presented of transonic limit cycle oscillation and dynamic stall. First results indicate that good grid convergence can be achieved without necessarily requiring that the flow around the specific airfoil is well understood in advance.
1 Introduction The modelling of oscillating systems is an inherently computationally-intense problem, due to the non-parallelisable job of computing successively dependant time steps. Oscillating airfoils have been extensively investigated using linearised codes [1], which lend themselves to fast computation so long as the flow remains attached. The use of simplified Navier-Stokes methods [2] can cover even these cases, but require extensive testing to ensure their applicability to the regime under consideration. Reynolds Averaged Navier Stokes (RANS) solvers are significantly more robust, but are computationally intensive, even when they are considered in 2D. With the increase in computing power available, it is likely that these codes will become the standard tool for the computation of oscillatory motion in the near future. The use of grid adaptation on an unstructured mesh is a useful method of ensuring grid convergence of Navier-Stokes codes in cases where the aerodynamic performance of a body is strongly influenced by moving localised regions of high-gradient flow, without requiring that the user manually select regions of the flow for refinement.
2 CFD Tools Computations presented in this article were performed with the DLR-TAU code [3]. A finite-volume solver was used on a hybrid unstructured 2-D grid consisting of a number of structured layers close to the surfaces and a field discretised with triangular cells (Figure 1), generated using the CENTAURTM [4] unstructured grid generator. A farfield radius of 50 chord lengths was used with a farfield vortex correction. All computations were fully turbulent, using the Spalart-Allmaras turbulence model with the Edwards modification (SAE) [5]. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 170–177, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Fig. 1. Density contour lines for VC-Opt after 1205 adaptation steps with moving flap
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Fig. 2. Block diagram for the unsteady adaption with deformation
2.1 Grid Adaptation The DLR-TAU adaptation module [6,7] is used to resolve fine structures by subdivision of the tetrahedral/triangular cells in the field based on a weighting function, or to adapt the structured prismid/square layers near the wall to a target y+ by movement of the wall-tangent cell boundaries. The use of grid adaptation in an unsteady computation requires some preparation because of the tendency of particular combinations of settings to manifest themselves in incorrect adaptation after a large number of adaptation steps. As long as some broad limitations are prescribed, a computation with grid adaptation has the advantage that a grid-converged solution can be reached without necessarily requiring that the flow around the specific airfoil is well understood in advance. The general method for an unsteady computation with adaptation is as Figure 2, with a stationary computation followed by a number of unsteady steps. The insertion of all movable grid points into the wake was hindered in the computations presented here by setting a “minimum edge length”, below which the adaptation was disabled, and by the use of a constraining “cut-out box” to ensure that refinement occurred only in the region of the airfoil. Adaptation used the default rules for the weighting function, equally weighting normalised gradients of flow speed, density, total enthalpy and total pressure, and the normalised square root of edge length. 2.2 Flap Movement by Deformation The TAU deformation tool is used to adjust a volume grid based on a surface deformation. The movement of surface points is smoothed into the main grid by an iterative wavefront method. This tool can be used to simulate the deployment of a gapless flap on an airfoil. The source code of the deformation tool was altered such that a linear surface deformation of type: ∆z = ∆zmax (x − xmin ) could be specified for xmin ≤ x ≤ xmax . In this way a gapless flap deployment angle (δ) was simulated, which was accurate for small angles. This method was simple, but caused a discontinuity in the surface gradient of the airfoil at the flap hinge point. The grid was deformed after every time step, after the adaptation had taken place. This meant that the grid was not deformed from the start grid after each time step,
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as would be desirable from the standpoint of grid quality, but underwent a chain of deformation and adaptation steps. The grid showed no appreciable decrease in quality from the continuous action of adaptation and deformation, and Figure 1 shows the grid after 1205 unsteady adaptation and deformation cycles.
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Test Case: Dynamic Stall
On the retreating side of a helicopter main rotor, the blade passes through incidences of up to α=20◦ at freestream Mach numbers of up to Ma=0.4. Under these conditions the flow separates dynamically and “Dynamic Stall” appears. The flow separates over the leading edge during the up-stroke and a strong dynamic stall vortex rolls up. During the shedding of this vortex, the lift of the airfoil increases considerably compared to the stationary stalled airfoil, along with the drag and pitching moment. Shortly below maximum incidence, lift breaks down dramatically as the dynamic stall vortex passes the trailing edge. During the down-stroke, periodic vortex shedding appears until the flow re-attaches at a moderate angle of attack. The computations do not show the levels of non-periodicity seen in experiments, which require averaging of data over many cycles. The difference between the numerical result for a single cycle and the mean numerical result is small, when compared with the differences between the different solver settings presented in this article. The joint ONERA-DLR project “Dynamic Stall” is investigating simulation and control of dynamic stall processes [8,9] both experimentally and numerically. An adequate numerical simulation of the dynamic stall phenomenon makes high demands on the grid characteristics. The grid needs to be very fine in a very large region where the development and shedding of vortices takes place. The DLR-TAU unsteady grid adaptation was used instead of grid pre-refinement for the first time. An oscillating OA209 airfoil was simulated with the DLR-TAU code at Ma=0.31, Re=1e6 and α = 9.83◦, ∆α = 9.08◦ with a reduced frequency (based on the chord length) of ω=0.1. At least three periods needed to be computed to reach a stable hysteresis. The simulation of the dynamic stall process needed to be performed with a temporal resolution of 1000 time-steps/period and 100 inner iterations per time step, which was fine when compared with the oscillating transonic airfoil presented below. Three dynamic stall simulations were performed using different grids. A standard grid without additional refinement had 26000 grid points and a pre-refined grid with much better resolution in the wake region consisted of 50000 grid points. The third grid was based on the standard grid but with an additional 24000 points introduced by the adaptation leading to 50000 grid points in total. The grid adaptation was performed after every second time step leading to 500 adaptations/period. Figures 7a to 7c picture these grids at the minimum incidence (α=0.75◦), showing that the adaptation puts most of the additional points into the wake. While the spatial resolution of standard and refined grids was constant during the oscillation, the resolution of the adapted grid changed considerably. Figures 7d to 7f show the adapted grid at different incidences, indicating very different flow situations. The grid resolution had considerable influence on the predicted airfoil performance. Figures 3 and 4 picture the hysteresis curves of lift and pitching moments respectively. While flow was attached, during both up-stroke and down-stroke, hardly any differ-
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ences could be observed between the grids. However when the flow separates major discrepancies occur. The resolution of the standard grid was generally seen to be too low for an adequate dynamic stall simulation since the peaks in cl and cm were underpredicted when compared to the refined and adapted grids. For the lift prediction, refined and adapted grids reached a converged solution in the up-stroke. In the downstroke the processes of vortex shedding and reattachment were predicted differently by all three grids. Here, the movement of the solutions between the standard and refined grids indicate that the converged solution will be expected to be near the prediction of the adapted grid. The cm -prediction was similarly affected by the grid resolution. However, in contrast to the lift behaviour, the prediction of the cm -peak showed a change in the peak characteristic in the adapted grid. While standard and refined grids showed a simple single peak, the adapted grid predicted a double peak. The precise aerodynamic mechanisms leading to this double peak are not yet completely clear and will be investigated in the future. Comparison with experimental results is currently under way to resolve these differences.
4 Test Case: Transonic Airfoil The usage of unsteady grid adaptation in dynamic stall simulations showed promising results with respect to grid convergence in vortex-dominated flows. For the further investigation of the unsteady adaptation, a test case with shock-dominated transonic flow was chosen. The supercritical profile VC-Opt was used, as it had previously been thoroughly investigated for steady flows [10,11]. This section describes work in the DLR High Performance Flexible Aircraft (HighPerFlex) project, and the DLR-ONERA project to investigate Control Surface Dynamics (COSDYNA) in transonic flow. The addition of a moving flap to the VC-Opt airfoil pictured in Figure 8, means that up to three strong shocks affect the flow at any given time. Such a transonic airfoil with forced
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Fig. 5. Time-stepping convergence for the transonic airfoil VC-Opt
Fig. 6. Grid convergence for the transonic airfoil VCOpt
body oscillations and flap oscillations where strong shocks are present requires both the shocks and the wake region to be well resolved. Numerical simulations were performed for a typical limit cycle oscillation, modelled as unsteady flow over the rigid airfoil connected to the rigid flap, using prescribed harmonic pitch motion for both surfaces. The gapless flap movement in cruise flight was modelled by the deformation of the surface in the unsteady computation. The moving areas of high-gradient flow mean that savings in computational time, along with an improvement in accuracy, can potentially be achieved by using the TAU unsteady grid adaptation. The grid used is shown in Figures 8c to 8f. A starting grid of 50000 points (Figure 8c) was statically refined to 150000 points (Figure 8d), before the unsteady computation was started. Surface deformation, adaptation to y+ =1 and field adaptation were undertaken at every time step. Time-stepping convergence for the transonic airfoil VC-Opt for Ma=0.80, Re= 2e6, α=0.0◦ , ∆α=0.2◦ , δ=0◦ , ∆δ=1.0◦ , ω=0.4 is shown in Figure 5 for the drag coefficient. Results showed that using temporal resolutions of 25 and 50 time steps per period were insufficient since discrepancies can be seen compared to the results obtained with 100 and 200 time steps/period (which give similar results). The results also showed that the computations become stable after three cycles. Based on these findings, 200 timesteps/period were used, with 100 inner iterations/time step for all grid convergence test cases. Grid convergence (Figure 6) for Ma=0.80, Re=2e6, α=0◦ , ∆α=2.0◦ , δ=0◦ , ∆δ= 2.0◦ was less clear. Reduced frequencies (based on the chord length) were ωα=0.4 and ωδ =0.8. The amplitude was increased somewhat in this computation, after comparison with the computations for dynamic stall presented above, so as to ensure that a maximum number of instationary features were available to be resolved by all grids. The coarse grid had 183000 points, pre-refined around the airfoil (Figure 8a) and the fine grid had 294000 points pre-refined in the same way (Figure 8b). For the coarse and fine pre-refined computations, y+ adaptation was undertaken, but no field adaptation. These were compared with a case where both y+ and field adaptation were used. The lift and pitching moment
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Fig .7. Grids for the airfoil OA209, Ma=0.31, Re=1e6 and α = 9.83◦ , ∆α = 9.08◦ with ω=0.1 (a) Standard coarse starting grid (b) Pre-refined grid (c) α=0.75◦ after two cycles, (d) α=15.16◦ , up-stroke, (e) α=18.90◦ , (f) α=15.16◦ , down-stroke
coefficients indicated that the grid adaptation achieved the same result as a pre-refined grid with double the number of cells. In contrast, the coarser pre-refined grid was too coarse, with different results in cl and cmy than the other two grids. The drag curves showed inconclusive convergence results. Although the drag results obtained using unsteady grid adaptation appear to be converged during the up-stroke of α, differences were observed during the down-stroke of α, which indicate inconsistent behaviour, possibly unrelated to the convergence level. Comparison with experimental results and further numerical investigations are currently under way to resolve these differences.
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Fig. 8. Grids for the transonic airfoil VC-Opt, for Ma=0.80, Re=2e6, α=0◦ , ∆α=2.0◦ , δ=0◦ , ∆δ=2.0◦ , ωα =0.4, ωδ =0.8. (a) Coarse pre-refined grid (b) Fine pre-refined grid (c) Start grid for the adaptations (d) Unsteady grid after steady cycles (e) and (f) First and second examples of unsteady adapted grids after >1000 time step.
5 Conclusion The use of unsteady grid adaptation is promising for the investigation of oscillating airfoils. Initial investigations indicate that good grid convergence can be achieved without necessarily requiring that the flow around the specific airfoil is well understood in advance. Work is continuing, both in transonic limit cycle oscillation and dynamic stall to
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compare the simulations with experimental test data which is still being evaluated for both cases, and further numerical work will investigate higher order turbulence models and unsteady 3-D simulations in the near future.
References [1] Voss, R., Geissler, W.: Investigation of the Unsteady Airloads with Oscillating Control in Sub- and Transonic Flows, Proceedings DGLR International Symposium on Aeroelasticity 1981, Nuremberg. DGLR-Bericht 82-01, pp. 65-80 (1982) [2] Saitoh, K., Voss, R., Kheirandish, H.: Numerical study of nonlinearity of unsteady aerodynamics for NLR7301 profile , IFASD 2005, Munich, Germany, 28 June - 01 July 2005 [3] Gerhold, T., Friedrich, O., Evans, J., Galle, M.: Calculation of Complex Three-Dimensional Configurations Employing the DLR-TAU-Code AIAA 97-0167, 1997. [4] www.centaursoft.com [5] Edwards, J.R., Chandra, S.: Comparison of Eddy Viscosity-Transport Turbulence Models for Three-Dimensional, Shock-Separated Flowfields, AIAA Journal, V34, N4, 1996. [6] Alrutz, T.: Chapter 7: Hybrid Grid Adaptation in TAU, in MEGAFLOW: Numerical Flow Simulation for Aircraft Design Results of the second phase of the German CFD initiative MEGAFLOW, pp 109-116, 2002. [7] Alrutz, T, Orlt M.: Parallel dynamic grid refinement for industrial applications, ECCOMAS 2006 , The Netherlands, Sept. 5-8, 2006. [8] Geissler, W., Dietz, G., Mai, H., Bosbach, J., Richard, H.: Dynamic Stall and its Passive Control Investigations on the OA209 Airfoil Section, 31st European Rotorcraft Forum, Florence, Italy, 2005 [9] Geissler, W., Haselmeyer, H.: Investigation of dynamic stall onset, Aerospace Science and Technolgy, No. 10, pp. 590-600, Elsevier, 2006 [10] Richter, K.: Druckverteilungsmessung am 2D-Profil VC-Opt mit split Flaps im Transonischen Windkanal G¨ottingen, Internal Report IB-224-2004-C-01, DLR, 2003. [11] Richter, K., Rosemann, H.: Experimental investigation of trailing-edge devices at transonic speeds, Aeronautical Journal of the Royal Aeronautical Society, pp185-193, 2002.
RANS Simulations with One and Six Degrees of Freedom Rigid Body Motions Thorsten Schwarz Institut f¨ur Aerodynamik und Str¨omungstechnik, DLR, Lilienthalplatz 7, 38108 Braunschweig, Germany
[email protected] http://www.dlr.de/as
Summary A one rotational degree of freedom and a six degree of freedom flight mechanics module are coupled to DLR’s RANS solver FLOWer. The algorithms used to solve the equations of motion are described. The 1-DOF module is validated for a freely rolling delta wing. A very good agreement with experimental data is observed. The capability of the 6-DOF module is demonstrated by a simulation of a store separating from a wing. The computation uses the overset grid technique to handle the movement of the bodies in relative motion.
1 Introduction The numerical simulation of the response of an air vehicle to unsteady airloads requires to couple a Computational Fluid Dynamics (CFD) method to a flight mechanics module. The CFD-method solves the Reynolds-averaged Navier-Sokes (RANS) equations for a given position and orientation of the aircraft. The computed aerodynamic forces and other applied forces are now used by the flight mechanics module to solve the equations of motion. In the general case this requires to solve the Euler-Newton equations for six degrees of freedom movement (6-DOF). In case of a constrained motion the number of degrees of freedom may be reduced. The change in position and orientation of the vehicle is now used again by the CFD-solver to compute the airloads for the next time step. By applying this procedure in turn the complete trajectory of an air vehicle can be simulated. Many examples in aerospace require such modeling, for example the simulation of store separation from an aircraft, aircrafts with slung loads, the response of an aircraft to a gust and simulations of flight control systems. In this paper the implementation of a flight mechanics module into the flow solver FLOWer is described. The module allows for simulations of bodies with one and six degrees of freedom, respectively. Examples of applications are the simulation of a freely rolling delta wing and the simulation of a store separation event. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 178–185, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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2 Flow Solver DLR’s flow solver FLOWer solves the Reynolds-averaged Navier-Stokes equations with a second order accurate finite volume discretization on structured, multi-block grids [7]. In the present work, the central scheme with scalar dissipation of Jameson is used to compute the fluxes through the cell interfaces [5]. Turbulence is modeled by the two-equation Wilcox k-ω model [12]. Unlike the main flow equations, Roe’s scheme is employed to compute the turbulent convective fluxes. The discretized equations are integrated in time using an explicit five-stage Runge-Kutta method. The solution process makes use of acceleration techniques like local time stepping, multigrid and implicit residual smoothing. Turbulence transport equations are integrated implicitly with a DDADI method [4]. For unsteady simulations, the implicit dual time stepping method [6, 8] is applied. FLOWer is parallelized based on MPI and is optimized for vector computers. The simulation of several bodies in motion can be simplified by using the overset grid technique [2, 11]. It allows to generate individual grids for each moving body and to embed them into a background grid. This allows to arbitrarily move the grids during a flow simulation without affecting the grid quality. Communication among the grids is established by interpolation techniques. If some grid points of an overset grid are inside a solid body, these grid points are flagged and excluded from the flow simulation.
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For the unsteady simulation of an aircraft’s response to applied loads the flight mechanics module is called by the CFD-solver each time the flow is to be computed for a new time step t + ∆t. The flight mechanics code calculates the new position xt+∆t of the vehicle based on its old position xt and on the airloads F t at time t. The algorithm to integrate the equations of motion has to be chosen with respect to the degree of freedom of the vehicle’s motion. In FLOWer a one degree of freedom (1-DOF) rotation around a given axis or a full six degree of freedom (6-DOF) motion can be simulated. 3.1 1-DOF Module The equation of motion for a one degree of freedom roll around an axis is given by ϕ¨A · IA = MA
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The approach chosen in FLOWer to solve Euler’s equations of motion follows the method presented by Murman et al. [9]. In order to transform the angular velocity ω b into a change of orientation the orientation is expressed by quaternions. Quaternions specify an axis a and a rotation ϕ around this axis ⎤ ⎡ ⎤ ⎡ cos ϕ2 q0 ϕ ⎢ q1 ⎥ ⎢ ax sin ⎥ 2 ⎥ ⎥ ⎢ (9) q=⎢ ⎣ q2 ⎦ = ⎣ ay sin ϕ ⎦ 2 ϕ q3 az sin 2 The elements of a quaternion remain the same when the frame of reference is changed and it is |q| = 1. A small change in orientation always results in a small change of the quaternion. This is the reason for preferring them instead of using Euler-angels, which may be subject to discontinuities and singularities. The change in orientation due to rotation is computed by ⎤ ⎡ −q1 −q2 −q3 1 ⎢ q0 −q3 q2 ⎥ ⎥ ωb q˙ = ⎢ (10) 2 ⎣ q3 q0 −q1 ⎦ −q2 q1 q0 In order to compute the unknown ω b in Eqn. (8) and q in Eqn. (10) a fourth order Runge-Kutta scheme is applied. After solving Eqn. (10) the quaternion must be normalized |q| = 1. The transformation matrix T needed to transform vectors from the body fixed frame to the inertial frame of reference is given by ⎤ ⎡1 2 2 q1 q2 − q3 q0 q1 q3 + q2 q0 2 − q2 − q3 (11) T = 2 ⎣ q1 q3 + q2 q0 12 − q1 q1 − q3 q3 q2 q3 − q1 q0 ⎦ q1 q3 − q2 q0 q2 q3 + q1 q0 12 − q1 q1 − q2 q2
4
Results
4.1 Freely Rolling Delta Wing The 1-DOF module is validated by the simulation of the free rolling motion of a cropped delta wing. The model has a sharp leading edge of 65◦ sweep angle, see Fig. 1. Experimental data for this model were obtained during the DLR program AeroSUM [10] at the 1 m × 1 m transonic wind tunnel DNW-TWG in G¨ottingen. During the experiments, the model was put into the wind tunnel with varying incidence and roll angles. After releasing the model, the subsequent free roll motion was measured. The motion was damped by mechanical friction which was in the range of 2 Nm to 4 Nm [1]. For the numerical flow simulations the same grid as in Ref. [1] was used. The grid represents the delta wing and the wind tunnel support strut, see Fig. 2. The grid is of C-O topology with 2.9 · 106 cells in total. The unsteady flow computations
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Fig. 1. AeroSUM wind tunnel model
Fig. 2. Computational grid for AeroSUM model , every other grid line is shown
were run based on the results of steady flow computations. In order to converge the dual-time stepping method, 50 inner iterations were used. Viscosity effects were treated by the k-ω turbulence model with vortical flow correction [3]. The test cases considered here correspond to an inflow Mach number of 0.85 and a Reynoldsnumber of Re = 4.9 · 106 based on the length of the fuselage. For the numerical flow simulations the mechanical friction of the strut was set to 3 Nm. The first simulated test case corresponds to an incidence angle of α = 9 ◦ and an initial roll angle of ϕ0 = 40◦ . In this position, the vortices close to the leading edge are attached to the wing, see Fig. 3, left. Two flow simulations were run with time step sizes of 1 ms and 2 ms, respectively. The computed roll angle with respect to time is shown in Fig. 3, center. The results are in very good agreement with the experiment, especially for the smaller time step. The difference in the final roll angle is only 1.6◦ . The variation of the roll moment with time is plotted in Fig. 3, right. At its final position the model still experiences a moment of 0.98 Nm pushing it to zero roll angle. But this is too small to overcome the friction of the bearing.
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The second test case with an incidence angle of α = 17◦ and an initial roll angle of ϕ0 = 60◦ is demanding for the flow solver because of a pronounced vortex dynamics. The use of a k-ω turbulence model including vortical flow correction has been essential to predict these effects correctly. In the model’s initial position the vortex on the left wing detaches, see Fig. 4, left. The computed roll angle with respect to time is in good agreement with experimental data, see Fig. 4, center. The last oszilation beginning at t = 0.3 s is not captured by the computations because the moment of 2.8 Nm does not overcome the friction of 3 Nm, see Fig. 4, right. 4.2 Store Separation The capabilities of the 6-DOF module are demonstrated by the simulation of a store separating from a wing. The generic configuration is set up by a M6-wing under which the store is placed at z/s = 0.8 in spanwise direction and y/s = 0.18 under the wing, see Fig. 5. The store is rotationally symmetric and has the shape of a NACA0012 profile. The length, mass and principal moments of the store are defined to match the properties of an attack munition. The computational grid is set up by an overset grid system where the individual grids for the wing and store are embedded into a Cartesian background grid, see Fig. 6. The Navier-Stokes grid system is relatively coarse with 929344 grid cells in total.
Fig. 5. Generic wing-store configuration
Fig . 6 . Chimera grid for wing-store configuration
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For the test case considered here, the inflow Mach number is 0.8 and the angle of attack of the wing and store is 0◦ . The flow computation was performed to represent a total time of 1.2 s with a time step size of 5 ms. 50 inner iterations were applied to converge the dual-time stepping method. Turbulence was modeled with the k-ω turbulence model. The computed flight path of the store is shown in Fig. 7. The components of the trajectory in downstream and downward direction as well as the pitch angle are plotted in Fig. 8. After the release of the store it begins to fall downwards due to gravity. At the same time the store starts to rotate nose downwards about its mass center. Due to the rotation the drag of the store increases resulting in an increasing acceleration in downstream direction. 20
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Conclusion
The capabilities of DLR’s flow solver FLOWer have been extended for the prediction of trajectories. Two flight mechanics modules have been implemented in order to compute rigid body motions with one rotational degree of freedom and six degrees of freedom, respectively. The 1-DOF module allows to include mechanical friction. In the 6-DOF module a fourth order Runge-Kutta method is used to solve the non-linear Euler-equations of motion. Singularities during the prediction of the orientation of the aircraft are avoided by using quaternions. The 1-DOF flight mechanics module is validated for a freely rolling delta wing. The computational results are in good agreement with wind tunnel experiments. The capabilities of the 6-DOF module have been demonstrated for a generic test case involving a store separating from a wing. In the current implementation the coupling of the flight mechanics modules with the CFD solver is first order accurate in time. Therefore, small time steps are required to predict trajectories accurately. Future activities will be devoted to extend the coupling to second order accuracy. This will allow for larger time steps which
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will reduce CPU-time. It is also planned to extend the 1-DOF module for simulations where the rotation axis is subject to angular acceleration. Finally, the validation of the 6-DOF module will continue.
Acknowledgements The autor acknowledges the work of Mr. Wei Ren. Mr. Ren implemented the first version of the 6-DOF module into FLOWer within the course of his master’s thesis.
References [1] Arthur, M.T., Allen, M., Ceresola, N., Kompenhans, J., Fritz, W., Boelens, O.J., Prananta, B.B.: Exploration of the Free Rolling Motion of a Delta Wing Configuration in Vortical Flow. Proceedings of the NATO Symposium on Flow Induced Unsteady Loads and the Impact on Military Applications, (RTO-MP-AVT-123), Paper 14, Budapest, Hungary, April 25-28, 2005 [2] Benek, J.A., Steger, J.L., Dougherty, F.C. : A Flexible Grid Embedding Technique with Application to the Euler Equations. AIAA-Paper 83-1944, 1983 [3] Dol, H.S., Kok, J.C., Oskam, B.: Turbulence Modelling for Leading-Edge Vortex Flows. AIAA-Paper 2002-0843, 2002 [4] Fassbender, J.K. : Robust and Efficient Computation of Turbulent Flows around Civil Transport Aircraft at Flight Reynolds Numbers. Aerospace Science and Technology, Vol. 9, 2005, pp. 672-680 [5] Jameson, A., Schmidt, W., Turkel, E.: Numerical Solutions of the Euler Equations by Finite Volume Methods using Runge-Kutta Time-Stepping Schemes. AIAA-Paper 811259, 1981 [6] Jameson, A.: Time Dependent Calculations Using Multigrid, with Applications to Unsteady Flows Past Airfoils and Wings. AIAA-Paper 91-1596, 1991 [7] Kroll, N., Rossow, C.-C., Becker, K., Thiele, P.: The MEGAFLOW Project. Aerospace, Science and Technology, Vol. 4, 2000, pp. 223–237 [8] Melson, N.D., Sanetrik, M.D., Atkins, H.L: Time-Accurate Navier-Stokes Calculations with Multigrid Acceleration. Proceedings of the 6th Copper Mountain Conference on Multigrid Methods, NASA Conference Publication 3224, 1993, pp. 423-439 [9] Murman, S.M. , Aftosmis, M.J., Berger, M.J.: Simulation of 6-DOF Motion with a Cartesian Method. AIAA-Paper 2003-1246, 2003 [10] Sch¨utte, A., Einarsson, G., Madrane, A., Sch¨oning, B., M¨onnich, W., Kr¨uger, W.R.: Numerical Simulation of Manoeuvring Aircraft by CFD Aerodynamic and FlightMechanic Coupling. RTO-AVT Symposium 2002, Paris, France, April 22-24, 2002 [11] Schwarz, T.: The Overlapping Grid Technique for the Time-accurate Simulation of Rotorcraft Flows. 31st European Rotorcraft Forum, Florence, Italy, September 13-15, 2005 [12] Wilcox, D.C.: Reassessment of the Scale-Determining Equation for Advanced Turbulence Models. AIAA Journal, Vol. 26, No. 11, November 1988
Improvement of the Automatic Grid Adaptation for Vortex Dominated Flows Using Advanced Vortex Indicators with the DLR-Tau Code M. Widhalm1 , Andreas Sch¨utte1 , Thomas Alrutz2 , and Matthias Orlt2 1
DLR, Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7, D-38108 Braunschweig, Germany
[email protected],
[email protected] 2 DL R, Institute of Aerodynamics and Flow Technology, Bunsenstrasse 10, D-37073 G¨ottingen, Germany
[email protected],
[email protected]
Summary Vortex dominated flows appear in many flow simulations such as wake turbulence of an aircraft or a delta wing at a high angle of attack. For detailed investigations of vortex breakdown, vortex interactions or tracing vortex cores, an automated grid adaptation with suitable vortex indicators is essential. Physical indicators, e.g. the vorticity magnitude or the total pressure loss, are in most cases not sufficient for correctly identifying a vortex core. This paper presents advanced vortex core indicators which properly identify a vortical structure independent of the flow case. These vortex indicators are tested in typical flow applications to determine the right cut-off value which is important for an automated adaptation procedure. A grid refinement for a delta wing testcase in combination with the newly introduced vortex indicators will demonstrate the improvements compared to the standard pressure loss indicator.
1 Introduction Over the past few years physical vortex indicators, e.g. the magnitude of vorticity or total pressure loss were used with the DLR Tau-Code [4] grid adaptation for vortical structure refinements. Recently, vortex identifications based on the kinematics implied by the velocity gradient tensor ∇V have been proposed in the literature and implemented in the Tau-Code. In addition, the normalized helicity as a vortex indicator has also been introduced. These indicators are local or point-methods where a function can be evaluated grid point by grid point. According to a criterion based on the point values it can classify each point being inside or outside a vortex. For resolving vortex dominated flows, an automated grid adaptation used with flow independent vortex indicators will be an efficient approach. Adapting only the relevant vortical structures is the main issue for many applications and will be presented here for a delta wing testcase. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 186–193, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Chong et al [3] used the critical point theory to describe the topological features of flow patterns by forming a local Taylor series expansion of the flow field. The solution trajectories can be related and classified with three matrix invariants. In a fluid flow the velocity gradient tensor ∇Vi,j , computed from the nondimensionalised velocity vector, can be decomposed into a symmetric (S) and antisymmetric (Ω) part: ⎞ ⎛ ∇Vi,j
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Second Invariant Q. Hunt et al. [5] defined a vortex as the region with a positive second invariant, Q > 0. The second invariant is derived from the characteristic Eq. 4 and is defined as written in Eq. 5. Kinematic Vorticity Number Nk . Truesdell [12] defined the kinematic vorticity number to measure ”the quality of rotation”. He defined Nk as: Nk =
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Melander and Hussein’s [8] investigations of vortex core dynamics identified the core as a region with Nk > 1. Nk is non-dimensionalized by the magnitude of strain rate and identifies vortices with large and small vorticity as long as the quality of rotation is the same for both. λ2 Criterion. Jeong and Hussain [6] introduced a definition of a vortex in terms of the eigenvalues of the symmetric tensor S 2 + Ω 2 . The vortex core is defined as a region where two negative eigenvalues of S 2 + Ω 2 appear. Note, that S 2 + Ω 2 is symmetric and the eigenvalues are real values. The vortex core is identified with the requirement: λ1 ≤ λ2 ≤ λ3
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3 Normalized Helicity Hn - Stream Vorticity In comparison to the indicators above Levy, Degani and Seginer [7] introduced the normalized helicity. Two major shortcomings of the above indicators are the inability of indicating the swirl direction of the vortex and they are unable to differentiate between primary and secondary vortices. To locate and identify coherent structures the normalized helicity is used: v · rotv = cos α. (8) Hn = |v | · |rotv | High values of helicity reflect regions with high velocity and vorticity whenever both vectors get close to parallel. The cosine between the two vectors provides a sign and shows the direction of swirl in relation to the streamwise velocity field. Hn differentiates between primary and secondary vortices because it is directly related to the velocity vector. Hn was succesfully used for a delta wing at a high angle of attack by Alrutz and R¨utten [1].
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Cut-off values control the representation of the vortex core by iso-surfaces. Each grid cell inside the iso-surface will be refined by the adaptation procedure. Due to their invariant behavior these thresholds almost remain constant for a wide range of applications. But, as mentioned by Miliou [10], some indicators, e.g λ2 criterion, are very sensitive. Fig. 1, 2 and 3 show the λ2 criterion for a low aspect ratio wing with RAE 2822 airfoil sections. Fig. 1 indicates the λ2 iso-surfaces very close to the theoretical value of zero. Far behind the wing many grid cells are visible with a very small vortex activity and they are not important for the main vortex structure at the wing’s side edges. Refining all these grid cells would add points not required for vortex identification. Fig. 2 shows the λ2 iso-surface at a cut-off value of 0.001. Most of the grid cells far behind the wing are not considered anymore. Some grid cells are still marked where the vortex core is weak. Fig. 3 shows a stable vortex core on each side of the wing.
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Fig. 1. Iso-surface with λ2 val- Fig. 2. Iso-surface with λ2 val- Fig. 3. Iso-surface with λ2 values ≤ 0 ues ≤ -0.001 ues ≤ -1.0
5 The Automated Flow and Grid Adaptation Chain 5.1 Gathering Experience with Different Vortex Indicators The EC 145 is a midsize two engine helicopter. Experimental measurements have shown a separation at the back door below the tail rotor boom under certain flow conditions. Subsequently, a flow simulation was carried out to investigate the separation behind the helicopters fuselage in more detail. The computation was performed with a Mach number of 0.2081 and a Reynolds number of 4.33 million. The angle of attack is zero degrees. The main and rear rotor were modeled by actuator discs. In Fig. 4-9 the iso-surfaces of the indicators at different cut-off values are shown. The vorticity magnitude |ω| and the total pressure Ptot values are found through inspection of visualized iso-surfaces. The invariant indicators and Hn had to be changed slightly from their theoretical value and in these cases the iso-surfaces indicate the region of the grid which will be refined. The figures are shaded with the first component of the vorticity vector to detect the rotation direction, except the normalized helicity Hn which provides the sign of swirl directly.
Fig. 4. Iso-surface of |ω| = 100
Fig. 5. Iso-surface of Ptot = 89.000
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Fig. 6. Iso-surface of Q = 0.1
Fig. 7. Iso-surface of λ2 = −1.0
Fig. 8. Iso-surface of Nk = 1.1
Fig. 9. Iso-surface of Hn = ±0.9
All the indicators were able to detect the separation at the tail rotor boom. The normalized helicity Hn was able to split the swirl action from the rotor discs. Q, Nk and the |ω| detected vortices coming from the rotor discs. All approaches except Hn erroneously detect vortex structures on the fuselage. Finally, only Hn identifies the important sections at the rear fuselage for an efficient adaptation of the separation. 5.2 Use of Vortex Indicators in the Tau Adaptation Tau adaptation uses an edge based local refinement strategy. The main steps are the indication of edges to be divided considering a current solution and the subdivision of all elements which are needed to get a valid grid with respect to the indicated edge subdivisions [2]. Usually the edge indication uses differences of solution values, e. g. Ptot or |ω|, or differences of gradients as sensors in order to minimize these differences which are supposed to be large in regions with large local errors.
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Under some circumstances the differences of physical variables produced by the flow solver in the region of vortices are not large enough to resolve these flow phenomena. A possible explanation could be, that the initial grid resolution is too coarse for the flow solver to simulate the vortices sufficiently. So a flattened solution prevents the edge indication step from refining some vortex regions. Some simulations with pre-refined grids [11] seem to back this observation. On the one side it may be hard to produce pre-refined initial grids for an unknown flow and on the other side it will be very expensive in terms of computational costs to make pre-refinements in all regions which are possible to contain unresolved vortices. Additionally, this method contradicts the idea of an automated grid adaptation. So the use of vortex indicators could be an alternative approach. The vortex indication of Tau adaptation uses one of the sensors described above for the edge indication. All edges with a point which is found to be in a vortex region are considered. The needed target point number is found by scaling these edges with a power of its length and marking all edges with an indicator larger than a limit. 5.3 Adaptation Results for a 65◦ Delta Wing with Rounded Leading Edges The flow field around a delta wing is well suited as a typical aerodynamic application for vortex dominated flows. The initial grid of the delta wing has a size of 1.8 million grid points. The flow conditions are a Mach number of 0.4, a Reynolds number of 3 million and an angle of attack of 13 degrees. The indicators are computed with the velocity vector V which is nondimensionalised by (M ∗ γ ∗ pr /ρr ) where M is the Mach number, pr and ρr are the reference pressure and density. In this case the flow topology is different to a sharp leading edge case where two primary vortices are formed right from the apex at the leading edge. In a rounded leading edge case two primary vortices on each side of the wing rotate in the same direction, an inner weaker and a stronger outer vortex. The formation of the inner vortex first occurs close to the apex. The stronger outer vortex is formed further downstream. Without going into more detail of the flow physics, this case is well suited to estimate the ability to accurately resolve different kinds of vortex structures within the flow field.
Fig. 10. Adapted grid with P tot (left) and Hn (right) at 80 per cent cord length
The impact of the grid adaptation is seen after several adaptation steps in Fig. 10 left by using the total pressure Ptot and in comparison in Fig. 10 right by using the normalized helicity Hn . The final grid using Ptot contains about 10 million and for Hn about 5 million grid points. The resulting grids are compared in a cut plane at 80 per
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Fig. 11. Pressure distribution on the delta wing surface measured by PSP (left) and flow computation after several adaptation steps with the Ptot (middle) and Hn indicator (right)
cent of cord length. It is first of all seen that the main flow features (i.e. the vortices) are correctly predicted by the solver and detected by both adaptation indicators, as seen in Fig. 10 for the left and right cut plane. It is evident that more flow structures are detected by the Hn indicator and these flow structures are more concentrated in the vortex region than in the case of using Ptot . One can see that in the case of Hn ( right Fig. 10) in addition to the inner and outer vortex as well the horse-shoe vortex generated by the strut is well detected in contrast to the Ptot case while introducing only the half number of grid points during the adaptation procedure. In the case of Ptot adaptation the addition of new grid points is concentrated on the strong outer vortex where the total pressure gradients are high. In case of Hn adaptation the addition of new grid points is more balanced and weak vortex structures will also be detected and refined. In Fig. 11 the pressure distribution of both CFD solutions in comparison to experimental data obtained by PSP (Pressure Sensitive Paint) measurements is shown. The center and right figure show the solutions of Ptot and the normalized helicity Hn adaptation. As discussed before, both numerical solutions are predicting the main flow features correctly compared to the experiment. In both solutions the inner vortex is predicted too weakly. However in the case of Hn adaptation the inner vortex is better refined as discussed before. In both cases the outer vortex is predicted to be too strong and appearing too far upstream. This over-prediction of the outer vertex is mainly dependant on the two equation turbulence models used. Each of the available models have many different modifications or corrections implemented to make allowance for different physical phenomena. An improvement might be a higher order turbulence model solving this vortex dominated flow.
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6 Conclusion The advanced indicators with different cut-off values were presented to accurately identify vortex cores. From the proposed indicators the normalized helicity Hn is a very powerful and reliable indicator for complex flows. The Galilean invariant indicators Q, Nk and λ2 are able to identify vortex cores but they tend to identify strong shear flows as well and finding the appropriate cut-off value becomes more sensitive. Thus it appears that the applicable cut-off values need further investigations for each indicator. A very effective and efficient way is demonstrated for vortex core refinement on the delta wing. The advantage of the specific vortex core refinement is essential and was the proposed aim for implementing the mentioned indicators. Another feature with Galilean invariant and Hn in comparison to physical indicators is the refinement of the vortical structure only. Nevertheless, it seems practicable that both physical and advanced indicators are brought together in the grid adaption as sensors for refinement.
References [1] Alrutz T., R¨utten M.: Investigation of Vortex-Breakdown over a Pitching Delta Wing applying the DLR TAU-Code with Full Automatic Grid Adaptation. 35th AIAA Fluid Dynamics, 6-9 June, Toronto, 2005. [2] Alrutz T., Orlt M.: Parallel dynamic grid refinement for industrial applications. In Proceedings of ECCOMAS 2006, Egmond aan Zee, The Netherlands, September 5-8 [3] Chong M.S., Perry A.E., Cantwell B.J.: A general classification of three-dimensional flow fields. Phys. Fluids, A 2, 765, 1990. [4] Gerhold T., Friedrichs O., Evans J., Galle M.: Calculation of Complex three-dimensional configurations employing the DLR TAU. AIAA-97-0167, 1997 [5] Hunt J.C.R, Wray A.A., Moin P.: Eddies, stream and convergent zones in turbulent flows. Center for Turbulent Research Report CTR-S88, p. 318, 1988. [6] Jeong J., Hussain F.: On the identification of a vortex. J. Fluid Mech., pp. 69-94, 1994. [7] Levy Y., Degani D., Seginer A.: Graphical Visualization of Vortical Flows by Means of Helicity. AIAA Journal, Vol. 28, No. 8, 1990. [8] Melander M.V., Hussain F.: Polarized vorticity dynamics on a vortex column. Phys. Fluids, A 5, 1992, 1993. [9] Metcalfe R., Hussain F., Menon S.,Hayakawa M.: Coherent structures in a turbulent mixing layer. Springer, A 5, 1985. [10] Miliou A., Mortazavi I., Sherwin S.: Cut-off analysis of coherent vortical structure identification in a three-dimesnional external flow. Comptes Rendus Mecanique, pp. 211-217, 2005. [11] Sch¨utte, A.; Einarsson, G.; Sch¨oning, B.; Raichle, A.; M¨onnich, W., Neumann, J.; Arnold, J.; Alrutz, T.: Prediction of the Unsteady Behavior of Maneuvering Aircraft by CFD Aerodynamic, Flight-Mechanic and Aeroelastic Coupling. RTO AVT-Symposium Budapest, April 2005. [12] Truesdell C.: The kinematics of vorticity. Indiana University 1953.
Adjoint Algorithms for the Optimization of 3d Turbulent Configurations Richard P. Dwight and Jo¨el Brezillon German Aerospace Center (DLR), Lilienthalplatz, 7, D-38108 Braunschweig, Germany
[email protected],
[email protected]
Summary The solution of the discrete adjoint equations for an unstructured finite volume compressible Navier-Stokes solver is discussed. In previous work fixedpoint iterations taken from the non-linear method - suitably adjointed - were applied to the adjoint problem. Here it is seen that there are often situations in which these iterations can not be expected to converge. To address this the Recursive Projection Method is developed as a stabilizer, and then used to perform an eigenmode analysis of attached and separated flow on a single geometry, allowing identification of flow regions that were unstable under the basic iteration. Finally an adjoint based optimization with 96 design variables is performed on a wing-body configuration. The initial flow has large regions of separation, which are significantly diminished in the optimized configuration.
1
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The adjoint equations are enjoying increasing importance in the field of computational aerodynamics, as emphasis shifts from the modelling of physical phenomena to their control and optimization. Adjoint algorithms are necessary to evaluate cost-function sensitivities with respect to a large number of design variables with an effort only weakly dependent on their number. But solving the adjoint equations is a problem at least as hard as solving the original flow equations, and often significantly harder. One possibility is to apply the same iterative techniques used for the non-linear problem, suitably adjointed, to the adjoint problem. This technique guarantees adjoint convergence if the non-linear problem converges asymptotically [1] - but this is often not the case. For this reason the Recursive Projection Method (RPM) is studied as a stabilizer, Section 2. The approximate eigenmodes that are calculated within RPM are useful for identifying the cause of divergence of the original iteration, and in our particular cases the cause is found to be flow separation, see Section 3. The algorithms described are finally applied to optimization of a 3d turbulent wing-body configuration. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 194–201, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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We forego a detailed description of the compressible Navier-Stokes equations and their unstructured finite volume discretization in the DLR TAUCode, referring the interested reader to [2, 3], and note only that the method uses the Jameson-Schmitt-Turkel (JST) flux with scalar artificial dissipation and the one-equation Spalart-Allmaras turbulence model with Edwards modification.
2
The Recursive Projection Method
Despite the guarantees regarding convergence provided by the theory of adjointed fixed-point iterations (FPIs) [1] there are regularly situations in which it is possible to obtain a reasonably converged solution of the non-linear equations, but not of the corresponding adjoint equations. This can occur for three reasons: either a) the non-linear solution is not sufficiently converged, or b) there is a discrepancy between the linear and non-linear problem due to some approximation of the Jacobian, or c) the FPI applied to the non-linear problem does not converge asymptotically itself. In an effort to understand and mitigate these phenomena, we consider the Recursive Projection Method (RPM), originally developed by Schroff and Keller in 1993 as a means of stabilization of unstable procedures [4]. Since then it has been applied in aerodynamics for convergence acceleration [5], and stabilization of linear frequency domain solvers [6]. Let the (linear) adjoint system be written Ax = b. The idea of RPM is to regard the transient solution of the linear problem as a sum of eigenvectors of the relaxation operator Φ = (I − M −1 A) where M is some iteration operator, e.g. LU-SGS with multigrid. The application of Φ to an approximate solution then corresponds to a product of each eigenvector with its corresponding eigenvalue. Divergence of the iteration implies that there is at least one eigenvalue of Φ with modulus greater than unity. Assuming that the number of such eigenvalues is small, and that the space spanned by their eigenvectors is known, call it P, then it must be possible to solve the projection of the problem onto this low dimensional subspace using some expensive but stable method, while solving the projection onto the complimentary subspace Q using the original FPI iteration, which is known to be stable there. Newton-Raphson is typically used on P. The space of dominant eigenvectors is determined as the calculation progresses, by applying the principle that the difference between successive applications of the FPI on Q form a power iteration on the dominant eigenvalues of Φ restricted to Q. In more detail: consider a relaxation operator written N (x) = (I − M −1 A)x + M −1 b.
(1)
Let V be an orthonormal basis of P, then orthogonal projection operators onto P and Q may be written respectively P = V V T and Q = I − V V T .
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Further define xP = P x, xQ = Qx. Then the RPM iteration may be written = QN (xn ), xn+1 Q
(2)
xn+1 = xnP + (I − P ΦP )−1 [P N (xn ) − xnP ] , P
(3)
xn+1 = xn+1 + xn+1 P Q .
(4)
where The derivative term in the Newton iteration (3) may be written V (I − H)−1 V T , where H is a square matrix of size dimension of P, whose inversion is cheap if P is of low dimension. To determine the basis itself consider the sequence & ' n−k+1 K = ∆xnQ , ∆xn−1 , (5) , . . . , ∆x Q Q − xnQ are readily available from the iteration on Q. Using where ∆xnQ = xn+1 Q the linearity of Φ it is easy to show that ∆xnQ = QΦQ∆xn−1 , so K is a Krylov space for QΦQ. Asymptotically this subspace will tend to contain the dominant k eigenvectors of QΦQ, as they are the components of the solution most amplified by repeated application of the operator. Via QR factorization an orthogonal basis for K is obtained, and these vectors are added to V when the power iteration becomes sufficiently converged. For more information see [1].
3
Influence of Separation on Linear Convergence
To demonstrate the effect unsteady flow phenomena can have on the convergence of the adjoint problem, and to evaluate RPM, we consider two cases: the RAE 2822 aerofoil Case 9 and Case 10 [7]. The only differences between these two cases are the Mach and Reynolds numbers, but Case 9 is fully
RAE Case 9
101
RAE Case 10
101
Non-linear Adjoint Adjoint + RPM
Residual
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6000
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Fig. 1. Non-linear and linear convergence histories for the two cases
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-1
-0.5
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-1
-0.5
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Fig. 2. Dominant eigenvalues of the linear iterations on the two cases
attached and Case 10 has a large region of shock induced separation. Convergence histories of the flow solutions are shown in Figure 1 where LU-SGS with multigrid with exactly the same settings is used in both cases. The attached case eventually reaches a region of asymptotic convergence, but Case 10 enters a limit cycle after about 1000 iterations and converges no further. However, the lift and drag are well converged, and an engineer might reasonably be satisfied with these values. In fact the parameters of the FPI were specially chosen such that this situation would occur, in a attempt to model a circumstance which is common for complex geometries. The exact cause of the limit cycle is generally difficult to determine with any certainly. The lack of non-linear asymptotic convergence means that one sufficient condition for convergence of the linear problem has not been met. Figure 1 shows the linear convergence (without RPM), and as expected, the adjoint of Case 9 converges and that of Case 10 diverges. The engineer who then wishes to optimize Case 10 (perhaps to remove the separation) with an adjoint-based method can not, even though she can reliably obtain force coefficients. This is clearly an undesirable situation. Applying RPM then speeds up the convergence of Case 9 and brings Case 10 to convergence, see Figure 1, consistent with the framework developed in the previous section. The discontinuities in the convergence are the iterations at which the basis is extended. As a side effect of RPM the dominant eigenvalues and eigenvectors of Φ (the LU-SGS smoothed multigrid iteration applied to each case) are approximated, Figure 2. As must be true - given the behaviour of the linear convergence - all eigenvalues of Case 9 lie within the unit circle, and therefore all modes converge. Four eigenvalues of Case 10 lie outside, and the eigenmodes related to these eigenvalues are amplified at each iteration of the scheme. Effectively the diverging components of the problem have been isolated.
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RAE Case 9
RAE Case 10
Fig. 3. The dominant eigenvectors of each of the two cases; dark regions indicate large values. The corresponding eigenvalues are highlighted in Figure 2.
Plotting the eigenvectors themselves then allows identification of those regions of the flow responsible for slow convergence and divergence respectively. Each eigenvector has a similar structure to a solution vector, with five complex components at each grid point; in Figure 3 the L2 norm at each point is taken, representing the overall size of the vector in all components at that point. The two eigenvectors shown correspond to the eigenvalues marked a and b in the previous figure, and it should be noted that they therefore show different things: for Case 9 parts of the field that are most slowly damped, for Case 10 parts of the field that are diverging. From these eigenmodes much information may be gleaned that is ordinarily unavailable. For example it becomes clear that in Case 9 the convergence rate of the iteration is limited by the convergence of the flow in the regions of the shock and upper surface negative pressure gradient. Devising an FPI that improves the convergence somewhere else, e.g. near the stagnation point, would result in no overall improvement. Similarly in Case 10 the cause of the non-linear limit cycle instability has been positively identified as the separation, in particular the largest recirculating region immediately behind the shock. Any treatment of the instability, in either the non-linear or linear problem, must necessarily involve this region. In both modes dark spots appear in the field under the aerofoil where no special physical features are present, indicating mesh or discretization problems. Of course the eigenmodes depend on the FPI, and so the analysis relates to physical phenomena only over the discretization. On the other hand this is a feature that makes the analysis useful in the study of FPIs. Further work will apply eigenmode analysis to attempt to systematically categorize the behaviour of some common FPIs with respect to certain flow features, the goal being to quantify the local influence of, for example different directional
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multigrid coarsening algorithms. The wider use of Krylov methods also introduces the need for an FPI that is a good Krylov preconditioner, though not necessary a good multigrid smoother. Where previously Fourier analysis was an essential tool for studying multigrid smoothers, we expect eigenmode analysis to be useful for studying Krylov preconditioners.
Fig. 4. Stabilization of DLR-F6 adjoint computation with RPM, showning the ρ-residual restricted to P and Q, and the dimension of V . Parameterization of the wing with a free-form deformation box with 84 paired nodes. Twist is parameterized separately with 12 variables.
4
Gradient-Based Optimization
The adjoint method is now applied to the drag minimization of the DLR-F6 wing-body configuration at Mach 0.75, a Reynolds number of 3 × 106 , and CL = 0.8, at which conditions the case has a large region of separated flow in the junction between the upper surface of the wing and the fuselage, as well as along most of the length of the wing. For the adjoint problem on this geometry, the standard method of adjointed LU-SGS with multigrid alone was unconditionally unstable, and applying RPM was necessary to obtain a converged solution, see Figure 4. The corresponding eigenvalues in the same figure show that in total 8 unstable modes were found. The optimization algorithm used is conjugate-gradients (CG), as in [8], where the angle-of-attack is varied to constrain the lift. The surface of the computational grid is shown in Figure 4, and is coarse, but sufficient to resolve the separation mentioned. The 84 paired nodes of a free-form deformation bounding box are also shown, whose vertical positions, as well as 12 additional wing twist variables were used to parameterize the wing. The pairing of nodes constrained the wing thickness. With such a large number of design variables
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Fig. 5. Convergence of the F6 drag-minimization optimization. Comparison of the region of corner separation before and after optimization.
only gradient-based optimization is viable, and only the adjoint method can deliver the gradient efficiently. Note that since the bounding box passes inside the fuselage, the wing-body junction also varies, and this is accounted for by the geometry and grid generation process. The metric sensitivities needed in the gradient calculation are evaluated by finite-differences on the mesh deformation. The convergence of the optimization is shown in Figure 5, the horizontal axis shows the number of calls to the flow solver (both linear and non-linear), thereby approximately representing computational effort. Symbols indicate gradient evaluations. After 32 solver calls CG was unable to reduce the drag further, giving a final reduction of about 10 drag counts. In contrast a similar optimization with 42 parameters produced a reduction of only 8 counts on this mesh (in a similar CPU time) [8], emphasizing the need for a comprehensive parameterization. The optimization reduced the region of corner separation considerably, Figure 5, while not completely eliminating it, which is unlikely to be possible within the design space considered, as it does not allow deformation of the body.
5
Conclusions
The influence of separation on the convergence of the non-linear and the adjoint problem has been examined with the use of eigenmode analysis and the recursive projection method. This method was shown to stabilize the linear calculation in situations where the use of the exact adjointed FPI for the nonlinear problem was unstable. The resulting adjoint code was applied to the optimization of a wing-body configuration, whereby the region of separation was considerably reduced.
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References [1] Dwight, R., Brezillon, J., Vollmer, D.: Efficient algorithms for solution of the adjoint compressible Navier-Stokes equations with applications. In: Proceedings of the ONERA-DLR Aerospace Symposium (ODAS), Toulouse. (2006) [2] Gerhold, T., Galle, M., Friedrich, O., Evans, J.: Calculation of complex 3D configurations employing the DLR TAU-Code. In: American Institute of Aeronautics and Astronautics, Paper AIAA-97-0167. (1997) [3] Dwight, R.: Efficiency Improvements of RANS-Based Analysis and Optimization using Implicit and Adjoint Methods on Unstructured Grids. PhD thesis, School of Mathematics, University of Manchester (2006) [4] Schroff, G., Keller, H.: Stabilization of unstable procedures: The Recursive Projection Method. SIAM Journal of Numerical Analysis 30 (1993) 1099–1120 [5] G¨ ortz, S., M¨ oller, J.: Evaluation of the Recursive Projection Method for efficient unsteady turbulent CFD simulation. ICAS 2004 (2004) [6] Campobasso, M., Giles, M.: Stabilization of a linear flow solver for turbomachinery aeroelasticity by means of the recursive projection method. AIAA Journal 42 (2004) 1765–1774 [7] Cook, P., McDonald, M., Firmin, M.: 6. Number 138 in AGARD–AR. In: Aerofoil RAE 2822 — Pressure Distributions and Boundary Layer and Wake Measurements. AGARD — Advisory Group for Aerospace Research & Development, Neuilly-sur-Seine, France (1979) [8] Brezillon, J., Brodersen, O., Dwight, R., Ronzheimer, A., Wild, J.: Development and application of a flexible and efficient environment for aerodynamic shape optimisation. In: Proceedings of the ONERA-DLR Aerospace Symposium (ODAS), Toulouse. (2006)
Enhanced Runge-Kutta/Implicit Methods for Solving the Navier-Stokes Equations C.-C. Rossow DLR, Institut für Aerodynamik und Strömungstechnik Lilienthalplatz 7, D-38108 Braunschweig, Germany
[email protected]
Summary Recently, Runge-Kutta/Implicit methods were proposed for the solution of the Euler and Navier-Stokes equations, allowing a reduction of computation times by about half an order of magnitude compared to methods presently in use. In this contribution the efficiency of such methods is further enhanced, first by reducing the number of Runge-Kutta stages, and second by introducing direct line-solves in the direction of maximum stiffness. Compared to a well tuned, standard reference code, for high-Reynolds number flows computation times are reduced by more than an order of magnitude.
1
Introduction
Even with the tremendous progress in computer performance, the challenge to establish fast and reliably converging algorithms for solving the governing equations of fluid flow still persists. When solving the Navier-Stokes equations at high Reynolds numbers, the resolution of thin boundary layers requires highly stretched meshes with very high cell aspect ratios. The large disparity in the spectral radii of the corresponding coordinate directions results in high stiffness of the discrete equations, thus severely deteriorating convergence of the numerical method. The problem of slow convergence is further aggravated when dealing with methods using dual time-stepping or adjoint-based optimization, where mere convergence of total forces is inadequate, and a sufficiently low level of the residual in the discrete equations becomes mandatory. Recently, Runge-Kutta/Implicit methods were proposed for efficient solution of the Euler and Navier-Stokes equations [8, 9]. These methods showed superior performance with respect to standard solution techniques: the number of cycles required for a converged solution was reduced by about an order of magnitude, and corresponding CPU times by a factor of 5. The present work aims at further enhancing the efficiency of these methods. On the one hand, the number of stages of the basic explicit Runge-Kutta time integration scheme will be reduced. On the other hand, the solution of the implicit system used for the smoothing of residuals will be augmented by directly solving along lines in the direction of maximum discrete stiffness. The enhancements of the method will be assessed by computing turbulent, compressible and incompressible flow around airfoils at various Reynolds-numbers. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 202–209, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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2
203
Governing Equations
We consider the two-dimensional compressible Navier-Stokes equations. For a control volume fixed in time and space, the system of partial differential equations in integral form in a Cartesian reference frame is given by:
³³
Vol
&
& wW dV wt
&
³ F n dS
0 ,
(1)
S
where W >U , Uu, Uv, UE @T represents the vector of conservative variables, F is the & flux-density tensor, and Vol, S, and n denote volume, surface, and outward facing normal of the control volume. The flux density tensor F may be split into an inviscid, convective part Fc and a viscous part Fv : F
Fc Fv ,
(2)
where Fc and Fv are given by
Fc
& ª Uq º &» « & « Uuq& pi&x » « Uvq pi y » « & » «¬ UHq »¼
ª « « « « «§¨ uW vW k xy «© xx ¬
, Fv
&
0
&
W xx i x W xy i y & & W xy i x W yy i y wT · & § ¸i x ¨¨ uW xy vW yy wx ¹ ©
º » » » » & · wT ¸¸i y » k wy ¹ »¼
,
(3)
& & & q is the velocity vector with Cartesian components u, v, and i x , i y denote the unit vectors in direction of the Cartesian coordinates x and y. U , p, H , T represent den-
sity, pressure, total specific enthalpy, and temperature, k is the coefficient of thermal heat conductivity, and W xx ,W yy ,W xy are the viscous stress tensor components. In order to close the system given by eq. (1), the equation of state (4) p / U R T is used with R as specific gas constant.
3
Basic Solution Scheme
The basic solution scheme uses a cell centered, finite volume space discretization on structured meshes [7]. A semi-discrete form of equation (1) may be written as: & wWi , j wt
1 Voli , j
K
¦F
A,k
Sk
0,
(5)
k 1
where the indices i,j denote a computational cell in the flow field, Vol is the volume of cell i,j, FA is the flux density vector corresponding to the direction normal to a cell face S, and K represents the maximum number of cell faces of the control volume with k as running index. Using Flux Difference Splitting (FDS) [6], the convective part of the flux density vector Fc,A normal to a cell interface reads: Fc,A
& 1 L 1 Fc ,A FcR,A AA ' W , 2 2
(6)
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where FcL;A and FcR,A are the left and right states of the inviscid flux density vector, and AA is the corresponding flux Jacobian containing the absolute eigenvalues. &
The expression 'W denotes differences in conservative variables on the left side L & & & & and right side R of a cell interface, giving 'W W R W L , and AA 'W is expressed in terms of an interface normal Mach number M 0 [7]. Discretization of the viscous part of the flux density tensor Fv,A is performed by central difference operators [5, 7]. Time integration of Eq. (5) is achieved by a Runge-Kutta/Implicit method, which combines an explicit Runge-Kutta scheme with the solution of an implicit system of equations [8, 9]. Using a purely explicit Runge-Kutta scheme, conservative & variables W are updated by: & Wi,(m) j
& & (m) Wi,(0) Ri,(mj 1) , j D
(7)
where superscripts (m) denote the stage count with m running from 1 to the maximum number of stages, and D ( m ) is the stage coefficient of the (m)-stage. The & vector Ri(,mj1) represents the conservative residuals evaluated with the variables of the previous (m-1)-stage: & Ri(,mj 1)
& m1
K
Gt i , j Voli , j
¦ F W S A,k
,
k
(8)
k 1
where Gti , j is the local time step of cell i,j. Following Refs. [8, 9], at each stage the explicit Runge-Kutta time stepping of Eq. (7) is augmented by residual smoothing, where the implicitly smoothed residual ~& R is obtained by evaluating a fully implicit operator: § Gt ¨ I i, j ¨ Vol i, j ©
K
¦A
A,k
k 1
· ~& S k ¸ Ri, j ¸ ¹
& Gt i, j Ri, m1 j Voli, j
K
¦A
A,k
~& S k R NB (k ) ,
(9)
k 1
where NB(k) indicates the neighbor of cell i,j corresponding to face k, I is the identity matrix, and the two matrices AA and AA represent the positive and negative parts of the flux Jacobian AA corresponding to the direction normal to a cell face S: wFA & , wW
AA
AA
0.5 AA AA
,
AA
0.5 AA AA
.
(10)
~&
The residuals R obtained from the solution of Eq. (9) are then used in the RungeKutta framework given by Eq. (7) for updating conservative variables: & Wi (, mj )
~& & Wi (, 0j ) D (m) Ri, j .
(11)
In the actual implementation, eq. (9) is transformed to primitive variables, which allows substantial memory reductions when storing the matrices AA and AA [8, 9]. The implicit system given by eq. (9) is solved only approximately using 1-3 lexicographic Symmetric Gauss-Seidel (SGS) sweeps [9]. With the approximate
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solution of eq. (9) as a means for residual smoothing in the Runge-Kutta scheme of eq. (11), the CFL-restriction of the basic explicit method is essentially removed, and CFL-numbers in the order of 1000 may be used [9]. Time integration is further enhanced by employing multigrid following the ideas of Jameson [4]. The influence of turbulence is modeled according to Baldwin and Lomax [1].
4
Enhancements to the Basic Runge-Kutta/Implicit Method
In Ref. [9] it was demonstrated that, in comparison to a standard reference code, the use of a 3-stage Runge-Kutta/Implicit scheme with only one Symmetric GaussSeidel (SGS) sweep reduced computation times by about a factor of 5 for solving sub- and transonic flows. In this work, first the efficiency of the Runge-Kutta/Implicit method will enhanced by reducing the number of stages to two. Since the coefficient of the second and last stage is always fixed to D (2) 1.0 for consistency, only the coefficient of the first stage D (1) remains as a free parameter. Numerical experimentation showed that only a very narrow range of values of 0.2 D (1) 0.3 is effective. Second, a different strategy for solving the implicit system will be employed. Instead of using the Symmetric Gauss-Seidel scheme, a direct line-solve will be employed along the direction of maximum stiffness when solving eq. (9). For structured meshes with C-topology, the direction of maximum stiffness is defined by the gridlines being perpendicular to the solid walls of the airfoil and normal to the wake center line. Since in the C-meshes used in this study these gridlines are directly identified by the j-direction in the computational domain, implementation of a line-solve into the framework of the present code is straightforward.
5
Computational Results
A 4-level W-cycle is employed in the multigrid algorithm, where corresponding coarse meshes are created by successively omitting every second gridline. On the finest mesh, the second order Flux Difference Splitting space discretization of Refs. [8, 9] is employed, which on coarse meshes is reduced to first order. Converged solutions are identical to results of Refs. [8, 9] and will not be reported here. All computations are performed on the same SGI Octane workstation as in previous investigations [8, 9]. 5.1 Compressible, Turbulent Flow at Moderate Reynolds Numbers First, the two proposed enhancements of the Runge-Kutta/Implicit method are applied to the flow around the RAE 2822 airfoil at free-stream conditions of the transonic Case 9 M f 0.73; D 2.31q; Re 6,500,000 from the investigations of Cook, McDonald and Firmin [2]. The same C-mesh with 320x64 cells as in Refs. [8, 9] is used, and computations are stopped when the L2 -norm of the residual of the continuity equation is decreased by 13 orders of magnitude. Figure 1 shows the effect of reducing the number of stages in the basic RungeKutta scheme from three to two. In this figure, ‘Res(r)’ denotes the L2 -norm of
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the residual of the continuity equation and ‘Cl’ is the airfoil lift coefficient. The 2stage scheme (rk2-i3) yields almost identical convergence rates as the 3-stage scheme (rk3-i3) used in Ref. [9]. In both cases, 3 SGS sweeps are employed for approximately solving the implicit system of eq. (9), and a CFL number of 1000 was used. With the 2-stage scheme, CPU time is reduced by almost 30% compared to the 3-stage scheme, see Table 1. A reduction of the SGS sweeps from three to one further reduces computational effort despite an increase in the number of total iterations, as was already demonstrated in Ref. [9]. Note that this holds in a similar manner for both, the 3-stage (rk3-i1) and the 2-stage (rk2-i1) scheme, see Table 1 by comparing CPU times of schemes ‘rk3-i3’, ‘rk3-i1’, ‘rk2-i3’, and ‘rk2-i1’. Table 1 . Compar ison of time-integr ation tr ategies for tr ansonic Case 9 (Mesh: 320x64)
Strategy
rk3-i3 rk3-i1
---(3 SGS) (1 SGS)
rk3-jline (1 SGS)
CPU [sec]
No. of iter.
746
137
rk3-i3
---(3 SGS)
6x106
854
180
rk3-jline (1 SGS)
6x106
661
6
100x10
2016
rk3-jline (1 SGS)
100x106
1318
---(3 SGS)
6x106
512
602 592
134
---rk2-i3 (3 SGS)
512
132
rk2-i1
439
175
(1 SGS)
rk2-jline (1 SGS) ---rk5/3-i (reference)
Table 2. Reynolds-number var iation with differ ent time - integr ation str ategies for subsonic flow (Mesh: 368x88)
445
130
Strategy
rk3-i3
rk2-i3
1559
(3 SGS)
rk2-jline (1 SGS) rk2-i3
3008
Reynolds-No.
rk2-jline (1 SGS)
6
477
100x106
1416
100x106
932
6x106
5341
6
17025
6x10
(3 SGS)
.
---rk5/3-i (reference)
rk5/3-i (reference)
CPU [sec]
100x10
Next, the effect of using a direct line-solve in the direction of maximum stiffness, is investigated. The line-solve in j-direction is applied for all gridlines with index i const , and the computational domain is covered by sweeping once in ascending and once in descending order through all i-indices. Convergence histories are displayed in Figure 2 for a 3-stage scheme, and computation times can be obtained from Table 1. With the line-solve strategy (rk3-jline), the number of iterations is similar to that of the scheme with 3 SGS sweeps (rk3-i3), and computation times are almost identical to the case with one SGS sweep (rk3-i1). Using the 2-stage scheme in combination with the line-solve shows the same trends as with the 3stage scheme, see Table 1 and compare schemes ‘rk2-i3’, ‘rk2-i1’, and ‘rk2-jline’. Thus, for moderate Reynolds numbers, the most effective strategies are the 2-stage scheme with either 1 SGS sweep or with a line-solve in j-direction. Both strategies yield the same computational efficiency, and CPU time is reduced by factors of 78 compared to a reference code (rk5/3-i), see Table 1. This reference code is a well
Enhanced Runge-Kutta/Implicit Methods for Solving the Navier-Stokes Equations mesh 320x64 Re=6,500,000 CFL=1000
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Case 9
Mf = 0.73 D = 2.80 q
1.00
Cl
-2
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mesh 320x64 Re=6,500,000 CFL=1000
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-10
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100
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Fig. 1. 3 and 2-stage RK/Implicit scheme Fig. 2. Different 3-stage RK/Implicit strategies
tuned representative of methods presently in use and employs a 5-stage RungeKutta scheme with scalar implicit residual smoothing and an evaluation of the numerical dissipation at every odd stage to reduce computational effort [5, 7]. 5.2 Compressible, Turbulent Flow at High Reynolds Numbers
Second, efficiency is investigated at high Reynolds numbers. Starting from the subsonic conditions of Case 1 M f 0.676; D 1.93q; Re 5,700,000 in the investigations of the RAE2822 airfoil by Cook, McDonald, and Firmin [2], at fixed Mach number and angle of attack, Reynolds number is varied by more than an order of magnitude from Re 5.7 x10 6 to Re 100 x10 6 . The C-meshes from Ref. [3] with 368x88 cells are used, where the meshes were adapted to the corresponding Reynolds numbers, leading to cell aspect ratios varying from about 3,000 to over 50,000. Following Ref. [8, 9], a pure second order reconstruction of variables without limiting is employed on the finest mesh. Analogously to the previous investigations, computations are stopped when the L2 -norm of the residual of the continuity equation is decreased by 13 orders of magnitude. In contrast to the previous case, here a reduction of the number of SGS sweeps to one was not possible and three sweeps are used, indicating that a lower limit in the approximate solution of eq. (9) is reached on these meshes. Varying the Reynolds number by one order of magnitude, the number of iterations increases for the basic method by about a factor of 2.3 with Reynolds number, see results denoted ‘rk3i3’ in Figure 3. These results are identical to those presented in Ref. [9]. With the line-solve strategy, the number of multigrid cycles is reduced from about 240 to 190 for the highest Reynolds number, and the number of multigrid cycles increases only by a factor of 1.9 with varying Reynolds number, see results marked by ‘rk3jline’. For the highest Reynolds number, the line-solve strategy reduces CPU time by about 30% compared to the basic Runge-Kutta/Implicit method, see Table 2. Using a 2-stage scheme, the trends are similar to the 3-stage scheme, see Figure 4 and Table 2 with results denoted by ‘rk2-i3’ and ‘rk2-jline’. Computational effort is reduced by almost 30%, and with respect to method ‘rk3-i3’, the 2-stage scheme with line-solve strategy (rk2-jline) reduces computational effort for the highest Reynolds number by about a factor of 2. Compared to the standard reference code
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C.-C. Rossow mesh 368x88 RAE 2822 Re=6 -100x10 6 Mf = 0.73 CFL=1000 D = 2.80 q
100
Res(r)
1.00
Cl
10-2
10
mesh 368x88 RAE 2822 Re=6 -100x10 6 Mf = 0.73 CFL=1000 D = 2.80 q
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(rk5/3-i), the 2-stage scheme with line-solve strategy reduces computation times by more than an order of magnitude. For the case with the highest Reynolds number of Re 100 x10 6 , where the effect of discrete stiffness due to high aspect ratio cells is most severe, CPU time is reduced by almost a factor of 20. 5.3 Incompressible, Turbulent Flow at H igh Reynolds Numbers
Last, incompressible flow around the RAE2 2822 airfoil at various Reynolds numbers is computed. Except for the free-stream Mach number, the on-flow conditions are identical to those used in the subsonic compressible cases, and the same computational meshes are used. From Figures 5 and 6, where the results for 3-stage and 2-stage schemes with 3 SGS sweeps and with line-solve are presented, the same trends as for the compressible computations are observed: the line-solve strategy alleviates the effect of directional stiffness at the highest Reynolds number more efficiently than the three SGS sweeps, leading to a substantial reduction in computational effort. The number of iterations required for the 2-stage scheme is similar to that of the basic 3-stage scheme. For the incompressible case the L2 norm of the residual of the continuity equation is only decreased by eight orders of magnitude instead of thirteen orders to avoid round-off errors [9].
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Two enhancements to Runge-Kutta/Implicit methods where proposed. First, the number of Runge-Kutta stages was reduced from three to two. Computational results confirmed that convergence properties did not deteriorate. CPU time savings of about 30% were achieved with respect to the original Runge-Kutta/Implicit method with three stages. Compared to a well tuned reference code which serves as a representative for today’s state-of-the-art codes, the use of two stages in the Runge-Kutta/Implicit time integration reduced computation times by factors of 78. Second, a direct line-solve was used in the direction of maximum stiffness when solving the fully implicit system for residual smoothing. At moderate Reynolds numbers, the efficiency of the line-solve was comparable to a scheme where the implicit system was solved by only one Symmetric Gauss-Seidel sweep. At very high Reynolds numbers, the line-solve showed superior efficiency and robustness, and compared to the well tuned standard reference code, computation times were reduced by more than an order of magnitude.
References 1. Baldwin, B. and Lomax, H., "Thin Layer Approximation and Algebraic Turbulence Model for Separated Turbulent Flows", AIAA-Paper 78-257, 1987. 2. Cook, P. H., McDonald, M. A., and Firmin, M. C. P., "Aerofoil RAE2822 Pressure Distributions and Boundary Layer and Wake Measurements", AGARD-AR-138, 1979. 3. Faßbender, J., "Improved Robustness of Numerical Simulation of Turbulent Flows around Civil Transport Aircraft at Flight Reynolds Numbers", Ph.D. Thesis, Technical University of Braunschweig, http://opus.tu-bs.de/opus/volltexte/2004/579, 2004. 4. Jameson, A., "Multigrid Algorithms for Compressible Flow Calculations", in Second European Conference on Multigrid Methods, Cologne, 1985, ed. W. Hackbusch and U. Trottenberg, [Lecture Notes in Mathematics, Vol. 1228], Springer Verlag, Berlin, 1986. 5. Radespiel, R., Rossow, C.-C., and Swanson, R.C., "An Efficient Cell-Vertex Multigrid Scheme for the Three-Dimensional Navier-Stokes Equations", AIAA J., Vol. 28, No.8, 1998, pp.423-459. 6. Roe, P.L., "Approximate Riemann Solvers, Parameter Vectors and Difference Schemes", J. Comput. Phys., Vol. 43, 1981, pp.357-372. 7. Rossow, C.-C., "A Flux Splitting Scheme for Compressible and Incompressible Flows", J. Comput. Phys., Vol. 164, 2000, pp.104-122. 8. Rossow, C.-C., "Convergence Acceleration for Solving the Compressible Naver-Stokes Equations", AIAA J., Vol. 44, 2006, pp.345-352. 9. Rossow, C.-C., "Efficient computation of Compressible and Incompressible Flows", J. Comput. Phys., Vol. 200, 2007, pp.879-899.
Navier-Stokes High-Lift Airfoil Computations with Automatic Transition Prediction Using the DLR TAU Code A. Krumbein1 and N. Krimmelbein2 1
Deutsches Zentrum für Luft- und Raumfahrt e.V., AS-NV, Bunsenstraße 10, D-37073 Göttingen, Germany
[email protected] 2 Technical University of Braunschweig, ISM, Bienroder Weg 3, D- 38106 Braunschweig, Germany
[email protected]
Summary A Reynolds-averaged Navier-Stokes solver, a laminar boundary-layer code and different transition prediction methods for the prediction of Tollmien-Schlichting and cross flow instabilities were coupled for the automatic prediction of laminar-turbulent transition on general 3-dimensional aircraft configurations during the ongoing flow computation. The procedure is applied to a two-dimensional three-element high-lift airfoil configuration which is characterized by the existence of laminar separation bubbles using different operation modes of the procedure.
1 Introduction The modelling of laminar-turbulent transition in Reynolds-averaged Navier-Stokes (RANS) solvers is a crucial issue when high quality simulation results for aircraft shall be produced. Especially the simulation of flows around high-lift systems of aircraft may result in significant errors when the transition points are of insufficient accuracy or are not taken into account at all. High-lift systems very often involve multi-component wings (e.g. slat, main wing, and flaps) and may have very high levels of total circulation. Because all components of the high-lift system are in close interaction with one another the total circulation and the complete flow field is affected by the transition line on any of the components. Although the overall lift value may be predicted with satisfactory accuracy slight deviations between the real and the computed pressures can lead to large errors in the computed overall drag value. It could be shown that the overall pressure drag of a high-lift configuration, which dominates the drag value of the configuration as a whole as well as the drag of every single element, is composed of a balance of very large positive and negative contributions, such as the suction forces at the noses or the resistance forces in the coves and the trailing edge regions. The contribution of one single element may be one order of magnitude larger than the resulting overall drag of the complete configuration. Thus, a relative error of 5% of the computed drag on the slat upper side may result in a change of 50% for the overall drag value [1]. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 210–218, 2007. © Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Another aspect of taking into account transition is that in many cases the high potential of higher order turbulence models can be made use of only when the areas of laminar-turbulent transition are known and deployed in the computational procedures with sufficiently high accuracy. Thus, in modern computational fluid dynamics (CFD) tools a robust transition modelling must be established together with reliable and effective turbulence models. Only if the transition locations are taken into account with sufficient accuracy all physical characteristics of the flow field can be reproduced in such a way that the demanding quality requirements are satisfied. For the design process of wings, there exists the demand for a RANS-based CFD tool that is able to handle flows automatically and autonomously with laminarturbulent transition. Existing transition prediction methods vary from empirical transition criteria via the local, linear stability equations based on small disturbance theory or non-local, linear and non-local, non-linear stability methods using the parabolized stability equations over large eddy simulations to direct numerical simulations of the Navier-Stokes equations. Empirical transition criteria and the eNmethod [2],[3] based on local, linear stability theory and the parallel flow assumption represent state-of-the-art methods for the prediction of transition onset in many industrial applications. Although they do not account for a number of fundamental aspects in the transition process eN-methods are used in aircraft industry most frequently for design purposes covering transition due to Tollmien-Schlichting (TS) and cross flow (CF) instabilities. Because there are no other practical methods presently available for industrial applications eN-methods together with the two-Nfactor method and empirical criteria for transition mechanisms which are not covered by the eN approach (e.g. bypass and attachment line transition) are going to be used further on for the design of aircraft wings and wing systems even for a future laminar wing of transport type aircraft. Recently the unstructured/hybrid RANS solver TAU [4] of the Deutsches Zentrum für Luft- und Raumfahrt, German Aerospace Center (DLR) has been provided with a general transition prediction functionality which can be applied to general 3dimensional aircraft configurations. The developments and first technical validation steps have been carried out at the Institute of Fluid Mechanics (ISM) of the Technical University of Braunschweig (TU-BS), [5],[6]. The TAU code is used together with the laminar boundary-layer method in [7] and the local linear stability code in [8]. These two codes and an infrastructure part of the TAU code are components of a so called ‘transition prediction module’ that is coupled to the RANS solver and that interacts with the RANS solver during the computation in a very similar way as it is documented in [9]. For a long time it was necessary to use transition database methods in order to apply the eN -method for transition prediction in a fully automatic way so that the transition location iteration could be executed without intervention (automatic) by the user of the RANS code and without a priori knowledge of the transition characteristics of the specific flow problem (autonomous). Now the fully automated local, linear stability solver in [8] is available using a frequency estimator for the detection of the relevant regions of amplified disturbances for TS instabilities and a wave length estimator for CF instabilities. In this paper the coupling structure between the TAU code and the transition prediction module is outlined and the transition prediction strategy is described together with the different operation modes of the transition prediction module which can be selected by the user. The main objective is to demonstrate the different
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characteristics of the different operation modes and their impact on the computational results which are obtained for a two-dimensional three-element high-lift airfoil configuration which is characterized by the existence of laminar separation bubbles for the flow case presented. The computational results are compared to experimental findings.
2 Transition Prediction Coupling On the one hand, the transition prediction module consists of an infrastructure part inside the flow solver which performs preprocessing operations necessary for each step of the transition prediction procedure, for example the extraction of the surface pressure distribution from a wing section. On the other hand, the module contains a number of additional components which basically execute the transition prediction. These additional components are a laminar boundary-layer (BL) code for swept, tapered wings [7], two eN -database methods, one for TS and the other for CF instabilities [10],[11] and a local, linear stability code [8]. With respect to the calculation of the laminar BL parameters the coupled system can be run in two different modes: Either the TAU code communicates the surface pressure distribution of the configuration to the laminar BL method, the laminar BL method then computes all of the BL parameters that are needed for a selected transition prediction method and the transition prediction method determines new transition locations that are given back to the RANS solver (BL mode 1). Or the TAU code computes the necessary BL parameters internally and communicates them directly to the transition prediction method (BL mode 2). Also with respect to the transition prediction (PD) method, the system can be run in two different modes: Either the two eN -database methods (PD mode 1) or the local, linear stability code (PD mode 2) can be used for the determination of transition points due to TS or CF waves. This coupled structure results in an iteration procedure for the transition locations within the iterations of the RANS equations. The structure of the approaches using the two different BL modes is outlined graphically in Fig. 1.
Fig. 1. Coupling structure: BL mode 1 (left) and BL mode 2 (right)
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During the computation, the RANS solver is stopped after a certain number of iteration cycles usually when the lift has sufficiently converged, that is when pressure oscillations have been damped to a sufficiently low degree. Then the transition module is called, geometrical data are processed and all laminar viscous data – basically the velocity profiles in streamwise and crossflow direction and their 1st and 2nd derivatives – are calculated either by the BL code or by the TAU code itself. Then, either the two eN -database methods or the stability code analyze the laminar boundary layer and try to determine a transition point. For BL mode 1 this is possible only when the transition point is located upstream of the separation point predicted by the BL code because the BL code terminates when a separation is detected. If a transition point due to TS or CF instabilities was found it is communicated back to the RANS solver. If no transition point due to TS or CF instabilities upstream of the laminar separation point could be found the laminar separation point is used as approximation of the real transition point. This is an attempt to predict transition due strictly to the presence of separation bubbles. This approach often yields a good approximation of the real transition point when transition does not occur before the laminar boundary layer separates, particularly for low Reynolds number flows. For BL mode 2 the laminar BL data are calculated beyond the point of laminar separation which is detected by the TAU code inside the RANS computational grid. Thus, transition inside laminar separation bubbles can be detected without relying on an approximation. Practically, the determination of transition inside laminar separation bubbles is only realizable using PD mode 2 because the eN-database methods lack parameters for the base flow profiles in laminar separation bubbles. These steps are done for the upper and lower sides of all specified wing sections. When all new transition locations have been communicated back to the RANS solver, each transition location is slightly underrelaxed to damp oscillations in the convergence history of the transition locations. Then, all underrelaxed transition points – they represent a transition line on the upper or lower surface of a wing element in form of a polygonial line – are mapped onto the surface grid of the configuration applying a transition setting algorithm subdividing the surface of the geometry into laminar and turbulent regions, and the computation is continued. In so doing, the determination of the transition locations becomes an iteration process itself. With each transition location iteration step the underrelaxation factor is reduced until a converged state of all transition points has been obtained.
3 Computational Results The system was applied to the A310 take-off configuration [12] consisting of slat, main airfoil and flap defined by M = 0.221, Re = 6.11 x 106 and Į = 21.40°. According to [13], as value for the limiting N-factor of the TS-database method NT = 9 was used. In the experiments [12] the following transition locations were determined on the upper sides of the slat, (xTupp/c)slat = 0.15, and the flap, (xTupp/c)flap = 0.345. On the main airfoil upper side the transition location was not measured, but the location of the upper side kink – the point where the slat trailing edge is located when the configuration is undeflected – is useful as a point of
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orientation,(xkinkupp/c)main = 0.19. On the lower sides, the transition points were not measured. In the computations a standard one-equation turbulence model was applied. The computations were started with free stream initialization and were carried out for two different computational grids [14] exhibiting different grid densities. In the fine grid, the grid resolution was highly increased compared to the coarse grid in streamwise as well as in wall normal direction in the structured parts which resolve the boundary layers. While the coarse grid (grid 1) consists of about 22,000 primary grid points the fine grid (grid 2) has about 122,000 points, Fig. 2. Because in the experiments a laminar separation bubble on the slat upper side caused transition, the computations using grid 2 were intended to resolve the laminar separation bubble and to investigate the impact on the transition locations caused by the different modes of the transition prediction module.
Fig. 2. Computational grids: grid 1 (above), grid 2 (below)
The three different combinations of modes which are currently available in the TAU code for 2-dimensional cases were applied: a) BL mode 1 & PD mode 1, b) BL mode 1 & PD mode 2 and c) BL mode 2 & PD mode 2. For grid 1, the transition prediction procedure which was started with initial transition points located almost at the upper side trailing edges of the particular elements (on the lower sides, fully laminar flow was assumed up to the trailing edges) was run with a pre-prediction phase of 1000 iteration cycles where the laminar separation points which occur in the RANS grid are used as transition points in order to stabilize the computation. The pre-prediction interval was 20 iteration cycles. Then, the transition prediction iteration was started using a prediction interval of 500 iteration cycles. While for the mode combinations a) and b) the results are identical, for mode combination c) it was found that the transition location iteration could not converge because the computational grid was not fine enough. This outcome is due to the following: Up to the pressure minimum, the boundary layer is strongly accelerated and stable. Downstream of the pressure minimum the boundary layer is retarded and becomes unstable. The different stability property is caused by a change in the velocity profiles which need sufficiently many grid points in streamwise direction for their proper development from one station to another. If there are too few grid points, the simulated change in the velocity profiles is not represented correctly and does not reflect the physics. The computed profiles are still stable, so that no transition point can be found. This situation is improved if sufficiently many grid points are used.
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Fig. 3. cp- (left) and cf-distributions (right) for grid 1
In Fig. 3, the cp- and cf-distributions for grid 1 are shown. As expected, the results from the computations with predicted transition (PD) yield more negative pressure levels on the upper sides of all elements than the fully turbulent (FT) results. This effect is pronounced in the suction peak areas. The comparison of the cf-distributions clearly shows the transition from laminar to turbulent flow on all elements. On the main wing element transition occurs directly upstream of the kink, on slat and flap upper side the predicted transition points are located upstream of the experimentally determined locations. Fig. 4 shows the results for grid 2. For grid 2 all three mode combinations yield converged results and again the results of the mode combinations a) and b) are identical. For the computations with transition, a separation bubble on the slat upper side is reproduced. For a) and b), where the transition point inside the separation bubble was approximated using the laminar separation point from the BL code, the resulting bubble is of too small extent and strength due to the fact that through the approximation the turbulence production starts too far upstream, so that the separation bubble can not fully develop. For mode combination c), where the stability analysis is carried out inside the laminar separation bubble, the extent and strength of the bubble show a good qualitative agreement with the experimental pressures. Whereas for a) and b) the predicted transition location on the slat upper side is not very different from that in grid 1, for mode combination c) the measured transition point now is reproduced with excellent accuracy. The transition point on the main wing element is determined downstream of the kink using a) and b) and more upstream of the kink than it was the case for grid 1 when combination c) is used. In the transition region on the main element, the interaction between transition and the influence of the kink leadto a relatively strong change between the cf-distributions from mode combinations a) and b) on the one hand and from c) on the other hand. On the flap upper side the transition point resulting from a) and b) shows almost the same deviation from the measured value as for grid 1. For mode combination c) however a visible downstream shift of the transition point occurs decreasing the gap between the former computational results and the experiment without showing a separation. In
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Fig. 4. cp- (above) and cf-distributions (below) for grid 2
Figs. 5 and 6, the velocity profiles in the region of the separation bubble on the slat and in the area of the transition point on the flap are shown. The corresponding transition locations are marked. A close inspection of the grid and the velocity profiles in the transition region on the flap upper side reveals that the boundary layer is shortly before separation. Possibly, a separation existed on the flap during the experiment which can not be resolved by the cell density in this area of grid 2. Very probably, a finer streamwise grid resolution in this area will lead to a better result and yield an improved prediction of the transition location on the flap.
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Fig. 6. Velocity profiles in the flap transition region, scales independent
4 Conclusion The TAU code coupled to a newly developed transition prediction module was applied to a two-dimensional three-element high-lift airfoil configuration which is characterized by the existence of laminar separation bubbles. The prediction of the transition location was carried out in a fully automatic manner during the ongoing RANS computation so that no intervention of the user is needed. It could be shown that when the computational grid is fine enough transition locations inside laminar separation bubbles can be predicted with high accuracy while the separation bubble itself can be reproduced well with respect to its extent and strength.
Acknowledgements This work has been carried out within the project “RANS Simulation of Reynolds Number Effects on Airfoil Stall”, a collaboration between the Institute for Aerospace Research of the National Research Council Canada, the ISM of the TU-BS and the DLR Institute of Aerodynamics and Flow Technology. The project was managed by TU-BS and funded by the Helmholtz-Gemeinschaft Deutscher Forschungszentren. The development work within the TAU code was carried out by TU-BS within the German Research initiative MEGADESIGN.
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References [1] Rudnik, R., Ronzheimer, A., and Schenk, M., „Berechnung von zwei-und dreidimensionalen Hochauftriebskonfigurationen durch Lösung der Navier-Stokes Gleichungen“, Jahrbuch 1996 - Bd. II der DGLR – JT 96-104, Deutscher Luft- und Raumfahrtkongreß, DGLR-Jahrestagung Sep. 1996, pp. 717-726. [2] Smith, A.M.O., Gamberoni, N., “Transition, Pressure Gradient and Stability Theory“, Douglas Aircraft Company, Long Beach, Calif. Rep. ES 26388, 1956. [3] van Ingen, J.L., “A suggested Semi-Empirical Method for the Calculation of the Boundary Layer Transition Region“, University of Delft, Dept. of Aerospace Engineering, Delft, The Netherlands, Rep. VTH-74, 1956. [4] Kroll, N., Rossow, C.-C., Schwamborn, D., Becker, K., and Heller, G., “MEGAFLOW A Numerical Flow Simulation Tool For Transport Aircraft Design”, ICAS Congress 2002 [CD-Rom], ICAS, Toronto, Canada, 2002, pp. 1.105.1-1.105.20. [5] Nebel, C., Radespiel, R., and Wolf, T., “Transition Prediction for 3D Flows Using a Reynolds-Averaged Navier-Stokes Code and N-Factor Methods“, AIAA-2003-3593. [6] Krimmelbein, N., Radespiel, R., Nebel, C., “Numerical Aspects of Transition Prediction for Three-Dimensional Configurations“, AIAA-2005-4764. [7] “COCO – A Program to compute Velocity and Temperature Profiles for Local and Nonlocal Stability Analysis of Compressible, Conical Boundary Layers with Suction”, ZARM Technik Report, November 1998. [8] Schrauf, G., “LILO 2.1 User’s Guide and Tutorial”, Bremen, Germany, GSSC Technical Report 6, originally issued Sep. 2004, modified for Version 2.1 July 2006. [9] Krumbein, A., “Automatic Transition Prediction and Application to 3D Wing Configurations“, in print at Journal of Aircraft, DOI: 10.2514/2254; also AIAA Paper 2006-914, June 2006. [10] Stock, H. W., Degenhardt, E., “A simplified eN method for transition prediction in twodimensional, incompressible boundary layers“, Zeitung für Flugwissenschaft und Weltraumforschung, Vol. 13, 1989, pp. 16-30. [11] Casalis, G., Arnal, D., “ELFIN II Subtask 2.3: Database method – Development and validation of the simplified method for pure crossflow instability at low speed”, ELFIN II - European Laminar Flow Investigation, Technical Report n° 145, ONERA-CERT, Département d’Études et de Recherches en Aérothermodynamique (DERAT), R.T. DERAT nº 119/5618.16, December 1996. [12] Manie, F., Piccin, O., Ray, J.P., “Test Report of the 2D Model M1 in the ONERA F1 Wind Tunnel”, GARTEUR AD(AG-08), TP-041, 1989. [13] Arthur, M. T., Dol, H., Krumbein, A., Houdeville, R., Ponsin, J., “Application of Transition Criteria in Navier-Stokes Computations”, GARTEUR AD(AG-35), TP-137, 2003. [14] Wild, J., private communications, DLR, Institute of Aerodynamics and Flow Technology, Braunschweig, Germany, Nov. 2003 & Nov. 2005.
Manipulation of Attachment Line Transition by Geometry Modification at the Slat of a Multi-element Airfoil Jochen Wild1 and Holger Dettmar2 1
DLR Braunschweig, Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7, D-38108 Braunschweig, Germany
[email protected] 2 CeBeNetwork GmbH, Flughafenallee 26, D-28199 Bremen, Germany
Summary This study shows the application of the criterion of Pfenninger for the prediction of the influence of geometric variations at the slat of a multi-element airfoil on the occurrence of attachment line transition (ALT). Additionally the use of criteria for the prediction of relaminarization based on the results of 2D RANS computations is examined. Flow calculations are performed for a representative wing section of a three-dimensional swept and tapered high-lift wing for which the occurrence of ALT is assumed at higher Reynolds numbers. Systematic geometry variations are performed to investigate the influence on the predicted onset of ALT and the influence on aerodynamic characteristics in order to identify possibilities for the design of a slat with delayed ALT onset. From this the most sensitive parameters for ALT transition and aerodynamic performance have been identified.
1 Introduction The aerodynamic performance of a high-lift wing in terms of lift coefficient is coupled to the development of the boundary layer on the slat as it influences the effective curvature through viscous displacement. Hereby the occurrence of ALT on the slat of a high-lift wing can be responsible for a limit of the aerodynamic performance with increasing Reynolds number. Without ALT the boundary layer starting from the attachment line downstream to the suction peak is laminar due to the acceleration of the flow and transition occurs slightly behind the suction peak at the earliest. In this case, with increasing Reynolds number, the boundary layer thickness decreases resulting in a higher effective curvature and therefore more suction and an increasing lift coefficient. In contrast the occurrence of ALT results in an overall turbulent boundary layer with increased thickness and therefore reduced effective curvature. This leads to reduced suction at the leading edge of the slat and a reduced high-lift performance in terms of lift coefficient is observed. The existence of the phenomenon of ALT was first discovered by Gray [7]. Pfenninger conducted further investigations with the emphasis on avoiding ALT [10][11][12].These experiments led to a criterion, described below, that was C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 219–226, 2007. © Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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assumed to characterize the occurrence of ALT. Experimental investigations by Poll [13][14] and Arnal and Juillen [1][2] validated this criterion for the assumption of infinite swept wings. Newer experiments of Séraudie et al. [15] also verified the criterion for swept finite high-lift wings at high angles of attack. In addition to the wind tunnel experiments mentioned above flight tests also show the relevance of the criterion [6][10]. Another phenomenon when dealing with accelerated flow is the occurrence of relaminarization. When the flow is highly accelerated it is possible that existing turbulence is damped so much, that the flow gets laminar again. Major experimental research on relaminarization on multi-element wings has been conducted by van Dam et al. [5], Bourassa et al [3] and Arnal and Juillen [1]. In contrast to the ALT criterion up to now there is no agreed universal criterion for relaminarization. But all formulations for such a criterion are based on the work of Launder [9]. In this work we will investigate different formulations suitable for the application to relaminarization prediction based on 2D flow calculations. Starting point of this investigation are measurements of the ProHMS high-lift wing-body half-model of a transport aircraft in the cryogenic wind tunnel Cologne DNW-KKK for a Reynolds number range of Ref=1.4u106…6.2u106 [17]. With increasing Reynolds number the maximum lift coefficient of the high-lift configuration decreased after having reached a maximum at a medium Reynolds number of Ref=3u106. This led to the assumption that ALT occurs above Ref=3u106. Since transition measurements were not performed, the attempt is made to predict the occurrence of ALT by numerical methods. Instead of computing the whole three dimensional flow field, here the attempt is made to use only two-dimensional flow calculations of selected wing sections. There are at least two reasons for this approach: a) the two-dimensional flow calculations are cheaper in terms of computational resources and easier to set up; b) due to the fact that high-lift design is still mainly performed based on twodimensional computations a validated prediction method based on this data can be easily implemented into the design process, giving hints to avoid ALT.
2 Attachment-Line Transition and Relaminarization Prediction The prediction criterion for the onset of ALT of Pfenninger [10] used within this work distinguishes between the flow normal to the leading edge and the crossflow as shown in Figure 1. It correlates the crossflow velocity outside the boundary layer wf with the acceleration of the flow out of the stagnation line expressed by the gradient of the edge velocity wUews. Pfenninger formulates an attachment line boundary layer Reynolds number wf ReTa .l . 0.405 (1) wU e Q ws with the cinematic viscosity Q, the local arc length s in the coordinate system normal to the leading edge and the velocity component at the edge of the boundary layer normal to the leading edge Ue.
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Fig. 1. Decomposition of the local flow velocity into components normal to the leading edge and crossflow
A number of experiments on swept wings and cylinders (e.g. of Pfenninger [12] and Poll [14]) verified a lower limit of ReT,a.l.,krit.=100 below which no ALT occurs. Experiments of Arnal [1] with a swept wing at incidence showed, that due to the limited extent of a wing an upper critical value of ReT,a.l.,krit.=133 exists. As can be seen from equation (1) the only term that takes into account threedimensional flow is the crossflow velocity wf outside the viscous flow, which for an infinite swept and untapered wing is constant wf Qf sin M (2) with the onflow velocity Qf and the sweep angle M. All other terms of equation (1) correspond to the flow components normal to the leading edge. By assuming that the effects of tapering and the limited span of the wing only play a minor role for the most part of a high aspect ratio wing, this criterion for ALT can be based on the evaluation of 2D calculations of the flow normal to the leading edge, as shown by Wild and Schmidt [18]. In the occurrence of ALT the turbulent flow coming from the attachment line can get laminar again due to the high acceleration around the leading edge of the slat. This has been shown by measurements in wind tunnel (Bourassa et al. [3]) or flight experiment (van Dam et al. [5]). A criterion for the existence of relaminarization was first given by Launder [9] for two-dimensional flow. Similar to the ALT criterion the relaminarization parameter K is based on the local velocity gradient at the outer edge of the boundary layer wUews. Q wU e K u2 D (3) U e2 ws For an incompressible flow using Bernoulli’s equation and the boundary layer assumptions the relaminarization parameter can also be formulated nondimensionalized (Bourassa et al. [3]) 3
KC p 2 D
1 1 ª 1 º 2 wc p « » 2 Re «¬ 1 c p »¼ ws
(4)
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where s* is the arc length along the contour non-dimensionalized with the reference wing chord. This formulation is especially helpful when using results from numerical flow simulations, because no dimensionalization is necessary. When looking on a swept wing the 2D formulation of K is no longer valid due to the cross flow components. The extension to 3D flow made by Viswanath et al. K u3 D
Q wU e qe2 ws
cos 2 \
(5)
can also be formulated using non-dimensionalized values 3
KC p 3 D
1 1 ª 1 º 2 wc p cos\ « » 2 Re «¬ 1 c p »¼ ws
(6)
where for the infinite swept wing the crossflow angle \ is calculated by cos\
Ue U e2 wf2
(7)
In the literature the critical value of K where relaminarization occurs is given with K=1u10-6…5u10-6, although Bourassa et al. [12] are questioning the use of this value for the prediction of relaminarization due to their results from measurements.
3 Flow Calculation For the calculation of the flow the structured DLR RANS-solver FLOWer [8] is used. It solves the unsteady compressible Reynolds-averaged Navier-Stokes equations in applying an explicit 5-stage Runge-Kutta time-stepping method. Turbulence modeling is done using the Spalart-Allmaras model with Edwardsmodification. The turbulence equations are solved using a fully implicit scheme that allows for high CFL numbers. In order to minimize grid dependencies of the flow solution, separate grids are generated using the DLR grid generator MegaCads [4] for each wing section at each Reynolds number. In particular the boundary layer resolution is adjusted in order to have approximately the same resolution in terms of the number of cells in the boundary layer and to obtain a value of the dimensionless wall distance y on the order of 1. For this investigation representative wing section of the ProHMS high-lift model is used. It is located approximately at the mid span of the wing part, far enough from any model tracks, so that 2D flow assumptions are most likely to apply. In order to perform 2D calculations comparable with the 3D flow the assumption of an infinite swept wing is used. For this purpose the normalization of the wing section into a coordinate system normal to the leading edge is applied (see Wild and Schmidt [18]).
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The flow was calculated for a free stream Mach number of Mf=0.13 and a Reynolds number of Ref=6u106 at two angles of attack D =15° and D=17.5°. The results discussed here belong to the lower angle of attack, but the tendencies for the higher angle of attack are identical.
4 Geometry Variation Parameters The geometry generation procedure used divides the clean airfoil by splitting curves into the main wing and the high-lift elements. For the slat the splitting curve forms the rear part of the slat as well as the so called fixed leading edge of the main wing. With this procedure it is guaranteed that the high-lift system can be retracted forming exactly the clean wing airfoil.
Fig. 2. Parameters used for the geometric changes of the slat
Figure 2 shows the parameters used for the geometric variations of the slat. The variation of the splitting curve (left) is done by a movement of the upper and lower end points xo and xu along the contour of the clean airfoil as well as with a free movement of the nose point xn and yn of the fixed leading edge (FLE). This procedure allows a flexible description of the splitting curve with only four parameters. The coordinate system for the deflection of the slat is shown on the right. The origin of the axis is defined by the intersection of the clean airfoil reference chord line and the vertical tangent to the fixed leading edge. The reference point at the slat for the definition of the deflection is defined to be at the upper trailing edge. The rotation of the slat is measured towards the retracted position.
5 Results When evaluating the geometry variations not only the influence on the Pfenninger criterion is of interest. It is also essential to have a look at the variation of the lift coefficient, since the prevention of ALT is useless if the aerodynamic performance is reduced too much. In Figure 3 on the right hand side the variation of ReT,a.l. is shown for the three deflection parameters, while on the left the influence on the lift coefficient is shown. In the figure for ReT,a.l. the horizontal lines depict the critical region between ReT,a.l.,krit.=100 and ReT,a.l.,krit.=133. Concerning the position variables it is
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visible that the reference slat position is optimal in terms of lift coefficient, while the lift coefficient could be increased by reducing the deflection angle. For the attachment line transition it is observed, that the value of ReT,a.l. can be reduced by opening the slat gap by either moving the slat upward or forward, but both is accompanied by losses in lift coefficient. Also the increase of lift by less rotation of the slat is not favorable for attachment line transition. The effect that is responsible for the reduction of ReT,a.l. can be found in a movement of the stagnation point at the slat more towards the leading edge. By this it is closer to the suction peak, resulting in a higher acceleration of the flow out of the stagnation point, which is reducing the ReT,a.l. accordingly to equation (1).
Fig. 3. Influence of deflection parameters on the lift coefficient and the attachment line Reynolds number for D = 15°
Fig. 4. Influence of shape parameters on the lift coefficient and the attachment line Reynolds number for D = 15°
For the shape parameters, shown in Figure 4, it is observed that approximately no reduction of the ReT,a.l. can be achieved. But for the extension of the upper slat trailing edge an increase of lift is observed, combined with almost no change of ReT,a.l.. The most promising shape variation for eliminating attachment line transition would therefore be a combination an increased slat gap together with the upper slat trailing edge extension. The first would reduce the risk of ALT while the second would recover losses in lift coefficient without affecting ALT. For one selected combination of 'xs=3.5%c and 'xo=5%c a reduction of the ALT criterion down to ReT,a.l..=106 was achieved with a nearly constant lift coefficient. Finally Figure 5 shows the influence of the most promising geometric parameters on the K-parameter for relaminarization. It can be seen that the trends correspond
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Fig. 5. Influence of the most promising parameters on the K -criterion for a =15°
to the ones observed for ALT, where a forward deflection has a favorable effect and the slat trailing edge extension shows no influence.
6
Conclusion
This study dealt with the possibilities to influence the attachment line transition and relaminarization by geometric changes of the slat of a multi-element airfoil. 2D calculations have been performed to evaluate the Pfenninger criterion for ALT-prediction and a 3D formulation the Launder K-criterion for relaminarization. It has been shown that especially the extension of the slat together with an adopted setting is able to reduce the risk of ALT. It has been observed that the prediction criterion for relaminarization gives favorable tendencies for the same geometric changes as for the attachment line transition.
References [1] Arnal, D., Juillen, J.: Leading-edge contamination and relaminarization on a swept wing at incidence. In Cebeci, T., ed.: Numerical and Physical Aspects of Aerodynamic Flows. Springer-Verlag Berlin Heidelberg (1990) 391-402. [2] Arnal, D., Juillen, J., Reneaux, J., Gasparian, G.: Effect of wall scution on leading edge contamination. Aerospace Science and Technology 1 (1997) 505–517. [3] Bourassa, C., T. O. Flint and R. C. Nelson: A High-Lift Building Block Flow: Turbulent Boundary Layer Relaminarization, Final Report, University of Notre Dame (2000) [4] Brodersen, O., Hepperle, M., Ronzheimer, A., Rossow, C.C., Schoning, B.: The parametric grid generation system MegaCads. In Soni, B., Thompson, J., Hauser, J., Eiseman, P., eds.: 5th International Conference on Numerical Grid Generation in Computational Field Simulation, National Science Foundation (NSF) (1996) 353–362. [5] van Dam, C.P., Vijgen, P.M.H.W., Yip, L.P. and Potter, R.C.. Leading Edge Transition and Relaminarization phenomena on a subsonic high-lift system. AIAA Paper No. 933140 (1993). [6] van Dam, C., Los, S., Miley, S., Roback, V., Yip, L., Bertelrud, A., Vijgen, P.: In–flight boundary–layer state measurements on a high–lift system: Slat. Journal of Aircraft 34 (1997) 748-756.
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[7] Gray, W.: The effect of wing sweep on laminar flow. TM 255, RAE (1953). [8] Kroll, N., Rossow, C.C., Becker, K., Thiele, F.: MEGAFLOW – A numerical flow simulation system. ICAS Proceedings 1998–2.7.4 (1998) [9] Launder, B.E.: Laminarization of the turbulent boundary layer by acceleration. Technical Report 77, MIT Gas Turbine Lab (1964). [10] Pfenninger, W.: Flow phenomena at the leading edge of swept wings. AGARDograph 97, AGARD (1965) Part IV. [11] Pfenninger, W., Bacon jr., J.W.: Amplified laminar boundary layer oscillations and transition at the front attachment line of a 45° swept flatnosed wing with and without boundary layer suction. In Wells, C., ed.: Viscous Drag Reduction, Plenum Press New York (1969). [12] Pfenninger, W.: Laminar flow control – laminarization. Report AGARDR–654, AGARD (1977). [13] Poll, D.: Three–dimensional boundary layer transition via the mechanism of "attachment line contamination" and "cross flow instability". In Eppler, R., Fasel, H., eds.: Laminar–Turbulent Transition. SpringerVerlag Berlin Heidelberg New York (1979) 253–262. [14] Poll, D.: Some observations of the transition process on the windward face of a long yawed cylinder. Journal of Fluid Mechanics 150 (1985) 329–356. [15] Sdraudie, A., Perraud, J., Moens, F.: Transition measurement and analysis on a swept wing in high lift configuration. Aerospace Science and Technology 7 (2003) 569–576. [16] Viswanath, P.R., Mukund, R., Narashimha, R., Crouch, J.D.: Relaminarization on Swept Leading Edges Under High-Lift Conditions, AIAA Paper 2004-99 (2004) [17] Wild, J., Puffert–Meissner, W., Sitzmann, M., Lekemark, L.: Messung des FNG Hochauftriebs–Modells bei hohen Reynoldszahlen im Kryogenischen Windkanal Koln. DLR–IB 124–2003/37, DLR (2003). [18] Wild, J. and Schmidt, O.T.: Prediction of attachment line transition for a high-lift wing based on two-dimensional flow calculations with RANSsolver. Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), 92, Springer Verlag (2004) 200–207.
Using CryoTSP as a Tool for Transition Detection and Instability Examination at High Reynolds Numbers U. Fey, Y. Egami, and C. Klein Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR), Bunsenstrasse 10, 37073 Göttingen, Germany
Summary The method of temperature-sensitive paint (TSP) adapted to cryogenic wind tunnels, i.e. cryoTSP is used to detect the natural laminar-turbulent transition of boundary layers in high speed flows. Besides pure transition detection, cryoTSP images provide information on the transition zone, the characteristics of transition, and on the type of instability. A method is proposed to determine the transition zone out of cryoTSP result images for 2D-wings, where the transition process is caused by Tollmien-Schlichting instabilities.
1
Introduction
In modern cryogenic wind tunnels like the European Transonic Windtunnel (ETW), measurements are usually performed with unfixed transition since it is possible to achieve full-scale flight Reynolds numbers with the subscale models. Hence, the detection of laminar-turbulent boundary layer transition is a basic interest in cryogenic testing. The well-known techniques for transition detection in ‘warm’ wind tunnels like infrared thermography (IRT), or the use of thermographic liquid crystals (TLC) are not applicable in cryogenic testing. The decrease in radiated heat energy and the shift to longer wavelengths make it difficult to use common IR systems for temperatures T < 200 K, and today’s available TLC stop working for T < 250 K. On the other hand, there exist temperature-sensitive paints (TSP) with high sensitivity in the temperature range covered by cryogenic wind tunnels (300 > T > 100 K), so-called cryoTSP. 1.1
Temperature-Sensitive Paint and Hardware
The cryoTSP paint used here was developed by JAXA (Japan Aerospace Exploration Agency) and covers the special needs for cryogenic testing in large, industry-scale facilities [1]. For image acquisition, high sensitivity CCD cameras with large dynamic range (at least 12bit) have been operated. Illumination of the TSP paint has been realized by different light sources which were adapted to the given situation of the particular (cryogenic) wind tunnel. Figure 1 shows the DLR hardware equipment for cryoTSP measurement which has been successfully adapted to the European Transonic Windtunnel (ETW), the large, low-speed cryogenic wind tunnel of the German-Dutch Wind tunnels in C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 227–234, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007
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Cologne (DNW-KKK) and in the Ludwieg-type cryogenic wind tunnel in Göttingen (DNW-KRG). The hardware is described in more detail in [2],[3],[4]. During design and development of our cryoTSP system we often used the pilot facility of the ETW wind tunnel (PETW), representing a scaled version of ETW (1:9). PETW enables the same total temperatures Tt, total pressures pt, and Mach numbers M as used in the ETW, but dimension of the test section is about 27 cm x 22 cm (compared to 240 cm x 200 cm for the “real” ETW).
Fig. 1. Mobile cryoTSP hardware of DLR as adapted to different cryogenic wind tunnels. Cameras and lamps for ETW tests must be mounted in heated boxes since they have to be operated in cryogenic atmosphere.
1.2
Working Principle of CryoTSP
Our cryogenic paint consists of particular temperature-sensitive molecules (luminophores) incorporated into a transparent polyurethane binder. When illuminating these luminophores by light within a certain wavelength range, this light will be absorbed and instantly emitted by the molecules, but at shifted wavelengths. The intensity of the emitted light is temperature-dependent. In the case of our cryoTSP, more light is emitted by the luminophores in cold areas, and less light is emitted in the warmer areas. Isolation of excitation light and emission light is realized by the use of suited optical filters for excitation light source and camera. To detect the transition location on a wind tunnel model surface, a large enough temperature difference between laminar and turbulent parts of the boundary layer must exist. In the following, we describe how such a temperature difference can be created.
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Method of Temperature Steps in Cryogenic Testing
In cryogenic wind tunnels, a decrease in flow temperature - dTt /dt (so called negative step) can be established by injecting more liquid nitrogen (LN2) than is needed in case of continuous injection (used to maintain constant flow temperature). On the other hand, stopping LN2 injection completely will cause an increase in flow temperature (positive step), generated by the heat input of the fan. Thus, the model is heated by the flow during a positive temperature step +dTt /dt. Underneath a turbulent boundary layer the model surface (which is covered by cryoTSP) is heated faster caused by a larger convective heat transfer coefficient N compared to the laminar boundary layer. During a temperature change dTt /dt, the resulting temperature difference Twall,tur – Twall,lam on the model surface is recorded by a CCD camera as an intensity image of the cryoTSP layer. If not otherwise stated, we used positive temperature steps during transition measurement because this method has been proven to have the least influence on the instability leading to transition [2],[3]. Especially, this means that the laminar region appears darker compared to the turbulent region when using positive steps. It has to be mentioned, that the N-factor of the cryogenic wind tunnel may be decreased by increased LN2 injection (caused by a higher turbulence level and/or increased noise), while completely stopping the LN2 injection may improve the flow quality, at least in small cryogenic wind tunnels like PETW. 1.4
Evaluation of CryoTSP Images
The procedure of cryoTSP image acquisition is the following: first, several images are taken under stable conditions (i.e. constant flow temperature). They will serve as reference images later, where no temperature difference seen on the model surface is caused by the different type of boundary layer (nevertheless, intensity variations caused by viewing angle or irregular incidence light may be seen). Next, a temperature step dTt /dt is initiated in the flow and additional images are taken while the flow temperature is changing. By dividing these step-change images by a reference image, the temperature difference between the laminar and turbulent regions is highlighted. Furthermore, inhomogeneous light distributions are levelled out. Depending on measurement conditions, sometimes it is beneficial to average reference and step-change images for increased signal-to-noise ratio. The resulting intensity values of the ratio image (step-change image divided by reference image) are around unity. Hence, they have to be multiplied by a certain factor and the resulting grey-values (of the new b/w image) may be shifted. The multiplier, as well as the shift value, has to be adapted to the certain experimental situation (i.e. total temperature, total pressure, temperature gradient, exposure time) to get best results. However, if cryoTSP result images are compared among each other in the following figures, they are all evaluated in the same way. By calibrating the cryoTSP it would be possible to calculate the absolute temperature distribution on the surface. But this recommends some temperature sensors to be in contact with the cryoTSP layer, making the setup for TSP measurement more complex. Nevertheless, a lot of analysis can be carried out without knowing the absolute temperatures on the surface.
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2
Analysis of CryoTSP Result Images
For analysis of the transition zone as detected by cryoTSP, grey-level distributions of the processed images have to be investigated. This can simply be done by estimating the transition location by eye, or by detailed analysis of line cuts. Since we use high spatial resolution CCD cameras, it is possible to evaluate the transition point with high accuracy. It is furthermore possible to map the 2D images onto a 3D numerical grid of the model by the use of markers. This technique provides the opportunity to determine transition points as 3D co-ordinates, but spatial resolution is restricted to the given grid spacing (usually smaller than pixel spacing). This paper will show only the “simple” analysis of 2D results with the benefit of maintaining the full pixel resolution of the images, but with the drawback of missing 3D surface mapping. Hence, the chord-wise direction in the following figures showing line cuts is named x*/c since it does not accurately represent the exact x/c location. In a more sophisticated 2D evaluation, it is nevertheless possible to apply nonlinear corrections to allow for image distortion (caused by short focal length optics and by curvature of the wing’s surface) by calibrating the camera views with point-grids. 2.1
Detection of Transition Zone on 2D Wings
Figure 2 shows transition patterns by cryoTSP on a 2D wing model which was investigated in the DNW-KKK wind tunnel. The model was a bulky airfoil made out of carbon-fiber reinforced plastics. Span-wise length was 2.40 m and chordwise length was 0.3 m. The carbon-fibre shows good thermal properties (moderate heat capacity and low thermal conductivity). Therefore it was possible to use positive steps with rather small temperature gradients (dTt /dt |.sec fRr transition detection. The paint was sprayed on a limited area in the middle of the model (with respect to span-wise direction), going around the upper and lower surface of the wing. Hence, leading and trailing edge regions can be seen completely in the cryoTSP images (i.e. x/c = 0 up to x/c = 1 is covered by TSP). A laminar separation bubble is indicated by the darkest regions in (a) to (c) in Fig.2. This is caused by the span-wise vortex preventing the surface from being cooled effectively by the flow. A similar effect occurs when the flow separates from the surface completely for large angles of attack D (D is shown in Fig. 2). Separation seems to start at D = 10º (which can be seen by the decreasing temperature in the line cut rather than in the image) and is clearly pronounced for D = 15º (dark area for x*/c > 45% in (f) ). Especially line cut (d) in figure 2 demonstrates that the temperature across the transition is similar to the skin-friction curve for partly laminar and partly turbulent boundary layers. For wp/wx = 0, Prandtl number Pr = 1, and Twall = const., the wall-friction coefficient c’f and heat transfer coefficient N are connected by Reynolds analogy: N = c’f / 2 · ue · O/Q (ue: velocity at the edge of the boundary layer, O: thermal conductivity, Q: kinematic viscosity). Another test was performed in the PETW wind tunnel, where we used a supercritical 2D airfoil section for transition measurement. The model was made out of
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Fig. 2. TSP result images showing transition on a bulky 2D wing in DNW-KKK. (a): large, negative angle of attack (D = -5º) exhibiting strong laminar separation bubble and turbulent re-attachment. (b),(c): size of the separation bubble becomes smaller with increasing D. (d): direct transition on wing. (e): transition location has moved upstream. (f): transition appears close to the nose (x/ctr < 5%), separation occurs for x/c > 45%. Surface-temperature Twall is given in arbitrary units (au). Mach number M = 0.25, Reynolds number Re = 2.5 Mio. Flow temperature Tt = 220 K. Tunnel pressure pt = 100 kPa.
stainless steel and TSP was applied only within a limited ”pocket” (120 Pm in depth) which was milled into the upper surface beforehand. After filling up this pocket with the cryoTSP, the paint was polished and the junction between the blank, metallic areas and the TSP area was made smooth. The purpose of this test was an investigation of the influence of different leading edge roughness on transition. In Figure 3, the change of transition patterns with increasing angle of attack D is shown for M = 0.24 and Re = 3 Mio, for constant leading edge roughness. The location of the cryoTSP pocket with respect to the model is sketched in (c), and the extent of the TSP with respect to x/c is shown in (d). The line cuts clearly exhibit that there is no well defined transition point, as may be expected by purely visual expertise of the images, but a more or less extended transition zone which reflects the physical mechanism behind transition. Compared to the bulky carbon-fiber model in figure 2, where the turbulent region is represented by a more or less extended plateau (preceded by a local maximum), the raise in temperature on the line cuts shown in figure 3 seems to continue all the way up to the trailing edge of the model. This can be caused by heat conduction from the blank metallic areas of the model: near the thin trailing edge, the warmup (by the positive temperature step) on the lower side is transferred faster to the upper side by metallic heat conduction, compared to regions of more thickness. This effect can be observed in all line cuts for x*/c > 0.7 shown in figure 3.
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Fig. 3. Transition patterns for different angles of attackDon a supercritical airfoil tested in PETW. A limited “pocket” on the wing’s upper surface is covered by cryo TSP, ranging from x*/c = 0.1 to 0.87 as explained in (c) and (d). Turbulent wedges seen in the images are caused by local roughness on the TSP layer. The dot-symbols on the curves represent the transition location as predicted by the Xfoil code (see text). Chord-length c = 100 mm; M = 0.24, Re = 3 Mio. Flow temperature Tt = 190 K, tunnel pressure pt = 317 kPa.
However, this increase in temperature caused by heat conduction from the lower side (and from the blank part of the trailing edge on the upper side for 0.87 < x/c < 1) can be distinguished from the temperature increase caused by transition to turbulence. At least, simple inspection of the b/w images by eye gives a good idea of the location of boundary layer transition. But how can the “exact” transition location be measured? For the line cuts shown in figure 3, despite the different slopes of temperature increase and different extent of transition region, one can detect a nearly linear part in the temperature curve, highlighted by the vertical dashed lines in the figure. We propose to define this linear part in the line cuts as the “TSP transition zone”, because it most suitably coincides with the darkopale change seen in the images. The location of the transition zone can be calculated by applying a linear fit to the described linear range of the temperature curve (dT/d(x*/c) = const.) and defining start and end of transition where the real curve T(x*/c) deviates by a certain ' x/c). It has to be shown in future cryoTSP image analysis whether the statement “there is a linear part in the (relative) TSP temperature curves across transition” holds
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true in general and whether the TSP transition zone as defined above really agrees with the fluid mechanics transition zone for Tollmien-Schlichting instability. We additionally calculated the transition location for the 2D airfoil under study with the well-known Xfoil code of Mark Drela [5]. The predicted transition locations x/ctr using an N-factor of 10 are shown in figure 3 as dot-symbols. Drela’s code predicts the onset of transition (i.e. the linear stage of disturbance growth), since it is based on linear stability analysis. This is in good agreement with the transition curves shown in the figure.
3
Appearance of 3D Instability in TSP Images
We conducted measurements on a sectionalised 2D and 3D model in the DNWKRG wind tunnel: two identical airfoil sections were separated by end-plates and fastened together. The single airfoil segments were polished with different sandpaper, leading to a different roughness of the TSP surface. The aim of this test was to check the influence of paint roughness on Tollmien-Schlichting instability (TSI) and crossflow instability (CFI) in cryoTSP testing. Some result images are shown in Figure 4. Despite the influence zone adjacent to the separating end-plates, a large enough area can be seen where natural transition takes place. On the 36ºswept 3D-model in (a) and (b), CFI-dominated transition is to be expected.
Fig. 4. 3D- and 2D-airfoil-sections tested in DNW-KRG. Different roughness was applied to the cryoTSP on the section’s surfaces (lower wing sections are more rough than upper wing sections). Mach number M = 0.71. (a) Re = 2.1 Mio, Tt = 280 K. (b) Re = 2.8 Mio, Tt = 280 K. (c) Re = 3.0 Mio, Tt = 220 K.
Compared to the 2D-model ((c) in fig.4), where the instability is represented by a smooth increase in grey-level, the transition patterns on the swept 3D model are characterized by a saw-tooth shape (see (a) and (b), especially the zoomed area in (b)). So it seems possible to distinguish between 2D Tollmien-Schlichting instability and (stationary) crossflow instability on a 3D wing. This was also proven in an ETW test, where TSI dominated transition areas and CFI dominated transition
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areas could be seen on a full model’s swept wing in parallel, but on different spanwise locations [3]. What is seen cannot be the primary crossflow vortices, since their span-wise wave length is of the order of the boundary layer thickness, thus being too small to be seen in these TSP images. On the other hand, the patterns can be clearly distinguished from roughness-induced bypass transition. Therefore, the saw-tooth transitions as seen in figure 4 are proposed to be some kind of ”downstream footsteps” of the initial 3D instability, generated by merging of primary vortices. But more studies are needed to verify this assumption in detail.
Aknowledgements The authors would like to thank the team members of the three wind tunnel for their constant support during testing and preparation. Special thanks go to ETW for their interest in the improvement of the cryoTSP method and partial funding of the cryoTSP research. Part of this work was carried out within the TELFONA and FLIRET Projects, funded by the 6th European Framework Program.
References [1] Y. Iijima et al. “Optimization of Temperature-Sensitive Paint Formulation for Large-Scale Cryogenic Wind Tunnels”, In 20thInternational Congress on Instrumentation in Aerospace Simulation Facilities (ICIASF). Göttingen, Germany, August 25.–29., 2003, pp 70 – 77. [2] U. Fey et al. “Transition Detection by Temperature Sensitive Paint at Cryogenic Temperatures in the European Transonic Windtunnel (ETW)”, 20th International Congress on Instrumentation in Aerospace Simulation Facilities (ICIASF). Göttingen, Germany, August 25.–29., 2003, pp. 77 – 88. [3] U. Fey, Y. Egami, and R.H. Engler: “High Reynolds Number Transition Detection by Means of Temperature Sensitive Paint”, In 44th AIAA Aerospace Sciences Meeting and Exhibit. Reno, NV, USA 2006, AIAA 2006-514. [4] Y. Egami, U. Fey, Ch. Klein, M. Sitzmann and J. Wild. “Transition Detection on High-Lift Devices in DNW-KKK by means of Temperature-Sensitive Paint“. In 12th International Symposium on Flow Visualization. Göttingen, Germany, September 10–14, 2006. [5] M. Drela and M.B. Giles: ”Viscous-Inviscid Analysis of Transonic and Low Reynolds Number Airfoils”, AIAAA Journal 25 (10), October 1987, pp. 1347-1355.
Direct Numerical Simulation of a Short Laminar Separation Bubble and Early Stages of the Bursting Process Olaf Marxen and Dan Henningson Department of Mechanics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden
[email protected]
Summary Direct numerical simulation of a pressure-induced short laminar separation bubble developing on a flat plate has been carried out. Transition in this bubble was triggered by small disturbance input with a fixed frequency and fixed spanwise wave number. The resulting short bubble was shown to be converged in time to a statistically steady state, while possessing essential features of short laminar separation bubbles as reported in the literature. In the present case disturbance input is required to maintain a short bubble. Switching off this disturbance input yields a growing separation bubble. This phenomenon is denoted as bubble bursting, since indication is found that the bubble develops towards a long-bubble state.
1 Introduction Transition to turbulence in a separated boundary layer often leads to reattachment of the turbulent flow within a short distance, resulting in the formation of a laminar separation bubble (LSB). Such a scenario is typical for a pressure-induced short LSB, e.g. found on a (glider) wing in free flight, or for an experiment where the region of pressure rise is preceded by a favourable pressure gradient that damps out unsteady perturbations [1]. The laminar part of the bubble usually features stagnant flow which expresses itself in a pressure plateau. Yet only a limited influence of the bubble on the pressure distribution is visible. Typically, the size of the laminar and turbulent portions of the bubble is quite similar. In the region of an adverse pressure gradient, disturbance waves are subject to strong amplification. Their saturation marks the location of transition to turbulence. In environments with a low level of disturbances fluctuating in time, the transition process for short LSBs is usually governed by strong convective amplification of these disturbances, i.e. a Kelvin-Helmholtz-type instability. All these features have been confirmed in numerous investigations of laminar-turbulent transition in short laminar separation bubbles in the past: both experimental studies [1, 2, 3] as well as direct numerical simulations (DNS) [4, 5, 6, 7, 8] have been conducted.
Present address: Center for Turbulence Research, Stanford University, CA 94305, USA.
C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 235–243, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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However, the short laminar separation bubble is not the only type of LSB that exists. Owen and Klanfer (1955) [9] were the first to distinguish between short and long separation bubbles. This initial classification was based on the bubble length in comparison to the chord of an airfoil. Tani (1964) [10] introduced a classification that depends on the (upstream) influence of the separation bubble, for instance on the overall pressure distribution. This influence is supposed to be local for short and global for long bubbles. Gaster (1966) [11] carried out an experimental investigation to settle characteristics of both bubble types, and to establish parameters that shall allow prediction under which (external) circumstances a short or a long separation bubble appears. A switch-over between the two types is denoted as bursting. On an airfoil, bursting can lead to stall of the airfoil. Prediction of bursting, corresponding to a bursting criterion, remains an unsolved problem, since none of the proposed criteria has found general acceptance so far [12]. The usual definition of bubble bursting refers to the phenomenon of sudden change in final (mean) state with only minor changes in governing parameters and not to the dynamical process of the actual bursting of a short to a long bubble. In the present article, this latter view point shall be taken instead: with the change of one parameter, increasing deviation in time from the short-bubble state shall be demonstrated and denoted as bubble-bursting. It will eventually lead to a long bubble. Time-accurate three-dimensional direct numerical simulations of a long LSB or the bubble-bursting process have not yet been reported. Present simulations concentrate on the latter aspect and cover early stages of the bubble-bursting process of a short LSB. The aim of such a simulation is a better understanding of the occurrence of bursting by investigating the role of the transition process for this phenomenon. This is expected to lead to a better prediction of bubble bursting in the future.
2
Configuration
We restrict ourselves to pressure-induced laminar separation on a flat plate, as done in many experiments, see e.g. [1, 3, 11], and some numerical simulations, e.g. [7]. A streamwise pressure gradient is designed in a way that we have a favourable pressure gradient followed by an adverse one. Such a flow is believed to be a more realistic model of the flow over an airfoil than a pure deceleration while at the same time it is easier to realise in a laboratory experiment. Before discussing DNS results obtained from the three-dimensional incompressible time-dependent Navier-Stokes equations, we consider a two-dimensional slip flow (potential flow) for the same basic configuration. The slip-flow result does not only serve to generate an initial condition for DNS, it is also a good way to visualise the configuration. Above all, it is useful for comparison with DNS results, e.g. with respect to wall-pressure distribution and streamlines in the vicinity of the wall. 2.1 Slip Flow A slip flow can be obtained from Laplace’s equation in two dimensions x, y. At y=0 we place a (slip) wall, i.e. impose a no-through-flow condition. For simplicity,
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here we assume periodicity in streamwise direction x. This turns out to give a good approximation to constant streamwise velocity uslip (and vanishing wallnormal velocity vslip ) at inflow xif l =−320 and outflow xof l =986.6 in the present case. At a distance ymax =160 from the plate/wall, we prescribe a distribution of the streamwise velocity based on the following formula: u(ymax ) = 1 + A1 · exp(−B1 x2 ) + A2 x2 · exp(−B2 x2 )
(1)
y [-]
with the choice A1 =10.0, A2 =−41.0, B1 =200.0, and B2 =12.0. The coordinate system is chosen such that the maximum of this distribution is located at x=0. Details of the computation of the slip flow can be found in [13]. From the resulting slip-flow field, the streamfunction can be obtained by wall-normal integration of the streamwise velocity, giving a good overall visualisation of the configuration (Fig. 1).
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Fig . 2. Coefficients for surface pressure cp (left) and contours of stream function Ψ (right). Slip-flow results using eq. (1) (dashed line), slip-flow results for a cylinder above a wall (dotted line) and time-averaged DNS results with disturbance input (solid).
The resulting slip-flow field uslip , vslip resembles the one created by a cylinder above a wall – the latter possibly modeled by a dipole of appropriate strength at (x, y)=(0, 173.5) and its counterpart mirrored at the wall. In Fig. 2, right, it can be seen that streamlines corresponding to both solutions are identical close to the
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wall. However, the present slip flow possesses the advantage that it results in almost constant streamwise (and vanishing wall-normal) velocity at the inflow and outflow boundary for the chosen streamwise domain size. This would not be the case for a flow field created by dipoles with a strength which gives a comparable pressure distribution on the plate, as it can be seen quite well from Fig. 2, left. In that figure the pressure for the cylinder solution is seen to not remain constant downstream of x > 400 unlike the streamwise-periodic slip-flow solution using eq. (1). 2.2
Boundary and Initial Conditions for DNS
At the inflow boundary xif l =−320 a Blasius similarity solution is prescribed with a Reynolds number based on the displacement thickness Reδ∗ =515.02. All quanti∗ ties are non-dimensionalised using a reference length Lref =1.16 · δinf low and the free-stream velocity U∞ at the inflow. The (global) Reynolds number amounts to Re=600, corresponding roughly to the local Reynolds number based on the displacement thickness at x=0. Upstream of the outflow boundary a damping zone smoothly returns the flow to a steady laminar state. The outflow boundary is located either at xof l =441.905 (case lowres) or at xof l =333.061 (case highres) with the damping region starting at xst,oBZ =402.7216 (293.8776, respectively). The useful region of the integration domain extends up to x ≈ 380 (270). The wall-normal height ymax was chosen to be 160. At the upper boundary, vorticity components are set to zero and u is set according to eq. (1), i.e. it is the same as in case of the slip flow. In the homogeneous spanwise direction z the flow is assumed periodic with a spanwise extent given by λz =2π/γ0 , where γ0 = 0.12 corresponds to a width of λz =52.360. Simulations are performed with deterministic disturbance input. A pair of oblique time-harmonic perturbations is triggered via blowing and suction at the wall through a disturbance strip. The streamwise location of the strip is x ∈ [1.5238, 15.4558] while the amplitude of the disturbance input amounts to Av =0.5 · 10−4 for the wallnormal velocity of each oblique wave. To break spanwise symmetry, the forcing amplitude of one of the two oblique waves is increased by Av =10−8 . The fundamental frequency is β0 =1/T0=0.3 and is among the most amplified frequencies according to linear stability theory for the present flow (see Fig. 5 in [13]). The fundamental spanwise wavenumber for the disturbances is γ0 =0.12. The initial flow field uIC , vIC is composed of the slip-flow solution uslip , vslip (see above) and a Blasius boundary-layer solution uBL , vBL (wIC ≡ 0). A way to generate an initial solution would be to match the slip-flow solution uslip , vslip with a Blasius boundary layer solution uBL , vBL . Here we apply a simplified approach by making use of the fact that uBL outside the boundary layer is equal to 1, while inside the boundary layer uslip (y) ≈ const.=uslip |y=0 . Thus, for the initial condition uIC we use: uIC = uBL · uslip (2) Note that at the inflow uslip ≡ 1 and thus no simplification was made there.
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For the wall-normal velocity we require continuity equation to be fulfilled, and ∂v . thus we have ∂y = 0, since ∂u ∂x =0 at the wall. While this condition is fulfilled by the boundary-layer solution vBL , it is not fulfilled by the slip-flow solution vslip . We try to match the wall-normal derivatives instead of the flow variable itself and integrate in the wall-normal direction. If we introduce a simplification by assuming low domains only (small ymax ) and moderate streamwise changes in the streamwise ∂uslip |y=0 1) we finally arrive at: slip-flow velocity at the wall ( ∂x vIC = vBL + vslip Again, at the inflow
∂uslip ∂x |y=0
(3)
≡ 0 and thus no simplification was made there.
3 Numerical Method and Resolution for DNS The three-dimensional time-dependent incompressible Navier-Stokes equations are solved numerically applying a vorticity-velocity formulation. Computations are carried out using the code n3d, which is a solver for highly accurate simulations of transitional and turbulent flows, developed at the Institut f¨ur Aerodynamik und Gasdynamik (IAG), Universit¨at Stuttgart. It is based on a discretisation suggested and carefully analysed by Kloker (1998) [14], and thereafter slightly refined, extended, and optimised by many others. Further details of the code n3d, i.e. the numerical method and the implementation, can be found in [15]. The method uses finite differences of fourth/sixth-order accuracy on a Cartesian grid for downstream (NMAX) and wall-normal (MMAX) discretisation. Grid stretching in wall-normal direction allows to cluster grid points near the wall. In spanwise direction, a spectral ansatz is applied (KMAX + 1 modes) for complex modes (IMAX=2). An explicit fourth-order Runge-Kutta scheme is used for advancing a vorticity-transport equation in time with LPER time steps per fundamental period T0 . The solution of a Poisson equation for the wall-normal velocity is obtained by a direct method based on a Fourier expansion. Table 1. Overview on two different resolutions used for DNS Case MMAX NMAX KMAX + 1 IMAX LPER lowres 769 3501 72 2 900 highres 1537 6001 136 2 1800
Two different resolutions were used in the present study (Tab. 1). Fig. 3 compares results obtained for these two resolutions. Good agreement is seen for timeaveraged quantities (Fig. 3, left) as well as for quantities Fourier-analysed in time (Fig. 3, right). Deviation in mean quantities is only seen slightly downstream of reattachment (i.e. from x ≈ 100 onwards) and in the Fourier analysed quantities downstream of disturbances saturation (i.e. from x ≈ 75 onwards). While the latter deviation can likely be ascribed to the random character of the flow once it transi tions, differences in the mean flow are due to slightly insufficient averaging intervals.
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In any case, the region containing the separation bubble and the important amplification region of mode (1, ±1) are not affected. Differences for mode (2, 0) around separation are due to a small remaining non-periodicity that aliases the computation of Fourier modes. Nevertheless, we can claim to have shown that the simulation of the LSB is resolution independent.
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Fig . 3. Coefficients for surface pressure cp (left) and amplification curves for the max(h,k) imum (in y) streamwise velocity fluctuation |ˆ u max | from a Fourier analysis in time (wavenumber coefficient h) and spanwise (k) direction (right). DNS results with high resolution (solid) and low resolution for two different time intervals T1 (dotted) and T2 (dashed). In the left figure: T1 : t/T0 ∈ [90 − 100] while T2 : t/T0 ∈ [65 − 75]. In the right figure T1 : t/T0 ∈ [96 − 100] while T2 : t/T0 ∈ [71 − 75].
4 A Short LSB and Early Stages of Its Bursting Time-averaged streamlines from DNS with disturbance input are shown in Fig. 2, ¯ visualises the short right. Very close to the wall, the mean dividing streamline ψ=0 LSB. Displacement of streamlines away from the wall caused by the boundary layer can be seen quite well when comparing DNS to slip-flow solutions. The short LSB possesses typical features of a short bubble. We observe a pressure plateau and only local deviation from a slip-flow cp (Fig. 2, left). Amplification of the triggered perturbation mode (1,1) agrees perfectly with linear stability theory (LST) for a convective-type disturbance (Fig. 3, right), indicating a Kelvin-Helmholtz instability. Laminar separation takes place at x=34 and turbulent reattachment happens in the mean at x=89 (Fig. 3, left), while transition takes place roughly at x ≈ 65 − 70. Thus the laminar and the turbulent portion of the LSB are quite similar in size. Convergence in time to a statistically steady state can be estimated from Fig. 3. In that figure, results are given for the lower resolution analysed over two different time intervals T1 , T2 . Good agreement of results from both time intervals up to slightly downstream of reattachment for the mean quantities and up to transition for the disturbance components is seen. Fig. 4, left, qualitatively underlines that the bubble size remains the same over time.
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Fig . 4 . Contours of spanwise-averaged spanwise vorticity ωz =0 at the wall as a function of time with disturbance input (left) and for a simulation where disturbance input was switched-off at t/T0 =46 (right). The dotted black line in the right figure corresponds to the solution with disturbance input for better comparison.
Switching off disturbance input at t/T0 =46 initiates a continuously growing separation bubble (Fig. 4, right). This process of deviation from the intially statistically steady state shall be denoted as the bubble-bursting process. The bubble is expected to develop towards a long-bubble state similar to those observed by Gaster (1966) [11]: indication is that the reattachment line moves downstream much quicker than the separation line moves upstream, while transition roughly remains at the same position, resulting in a shift in relation between laminar and turbulent portion.
5 Conclusions The present article deals with direct numerical simulation of a pressure-induced laminar separation bubble on a flat plate. Such a separation bubble is caused by prescribing a favourable-to-adverse pressure gradient distribution – shown to be similar to the one created by a cylinder above a wall – via the streamwise velocity at a certain constant distance away from the wall. Transition in the short LSB is triggered by a small-amplitude time-harmonic oblique pair of waves. The resulting LSB exhibits typical features of a short LSB. In particular, only a local deviation in cp from slip-flow is seen as well as a roughly equal portion of laminar and turbulent flow inside the LSB. Furthermore, disturbance amplification was shown to be in favourable agreement with linear stability theory. The resulting short LSB is demonstrated to be converged with respect to grid resolution in the region containing the LSB, while differences are still seen downstream of it, which is, however, not relevant for the present study. In addition, the short LSB was converged to a periodic and thus statistically steady state. It was demonstrated that disturbance input, though quite small, is essential to maintain a short bubble in the present case – unlike e.g. in simulations of Spalart & Strelets (2000) [4]. In particular, switching off this input resulted in a continuously
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growing laminar separation bubble, a process that is here denoted as bubble bursting. The fact that the transition position remained constant while the turbulent part grows faster than the laminar part is taken as indication that the bubble develops towards a long-bubble state. This clearly underlines the high importance and central role of the transition process for the occurrence of bubble bursting. Only an early stage of the bursting was simulated here. The study shall be continued in the future by following the bursting process further in time towards a much larger LSB in order to confirm the hypothesis that a long LSB develops (as indicated by results [13]). It is expected that physical mechanisms leading to a delayed reattachment in the long-bubble state can be clarified in this way.
Acknowledgements OM gratefully acknowledges financial support of this research by the Deutsche Forschungsgemeinschaft (German research foundation) under grant MA 3916/11 and computational time by the H¨ochstleistungsrechenzentrum (HLRS) Stuttgart, project long lsb. Furthermore, he thanks Ulrich Rist and Markus Kloker, IAG, Uni Stuttgart for providing the DNS code n3d.
References [1] Watmuff, J.H.: Evolution of a wave packet into vortex loops in a laminar separation bubble. J. Fluid Mech. 397 (1999) 119–169 [2] H¨aggmark, C.: Investigations of disturbances developing in a laminar separation bubble flow. PhD thesis, TRITA-MEK 2000:3, Royal Institute of Technology, Stockholm (2000) [3] Lang, M., Rist, U., Wagner, S.: Investigations on controlled transition development in a laminar separation bubble by means of LDA and PIV. Experiments in Fluids 36 (2004) 43–52 [4] Spalart, P.R., Strelets, M.K.: Mechanisms of transition and heat transfer in a separation bubble. J. Fluid Mech. 403 (2000) 329–349 [5] Alam, M., Sandham, N.D.: Direct numerical simulation of ‘short’ laminar separation bubbles with turbulent reattachment. J. Fluid Mech. 410 (2000) 1–28 [6] H¨aggmark, C.P., Hildings, C., Henningson, D.S.: A numerical and experimental study of a transitional separation bubble. Aerosp. Sci. Technol. 5 (2001) 317–328 [7] Marxen, O., Lang, M., Rist, U., Wagner, S.: A Combined Experimental/Numerical Study of Unsteady Phenomena in a Laminar Separation Bubble. Flow, Turbulence and Combustion 71 (2003) 133–146 [8] Wissink, J., Rodi, W.: DNS of a laminar separation bubble affected by free-stream disturbances. In Friedrich, R., Geurts, B., M´etais, O., eds.: Direct and Large-Eddy Simulation V. Volume 9 of ERCOFTAC Series., Proc. 5th internat. ERCOFTAC Workshop, Munich, Germany, Aug. 27-29, 2003, Kluwer Academic Publishers, Dordrecht, Boston, London (2004) 213–220 [9] Owen, P.R., Klanfer, L.: On the Laminar Boundary Layer Separation from the Leading edge of a Thin Airfoil. Technical Report No. Aero 2508. In: A.R.C. Technical Report C.P. No. 220, 1955, Royal Aircraft Establishment, UK (1953)
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[10] Tani, I.: Low-speed flows involving bubble separations. Prog. Aerosp. Sci. 5 (1964) 70–103 [11] Gaster, M.: The structure and behaviour of laminar separation bubbles. Number CP–4 in AGARD (1966) 813–854 [12] Diwan, S.S., Chetan, S.J., Ramesh, O.N.: On the bursting criterion for laminar separation bubbles. In Govindarajan, R., ed.: Laminar-Turbulent Transition, 6th IUTAM Symposium, Bangalore, India, 2004, Springer, Berlin, New York (2006) 401–407 [13] Marxen, O., Henningson, D.: Numerical Simulation of the Bursting of a Laminar Separation Bubble. In Nagel, W.E., J¨ager, W., Resch, M., eds.: High Performance Computing in Science and Engineering 06, Transactions of the HLRS 2006, Springer (2006) 253–267 [14] Kloker, M.: A Robust High-Resolution Split-Type Compact FD Scheme for Spatial Direct Numerical Simulation of Boundary-Layer Transition. Appl. Sci. Res. 59 (1998) 353–377 [15] Meyer, D., Rist, U., Kloker, M.: Investigation of the flow randomization process in a transitional boundary layer. In Krause, E., J¨ager, W., eds.: High Performance Computing in Science and Engineering 03, Transactions of the HLRS 2003, Springer (2003) 239–253
Experimental Investigations of Controlled Transition in a Laminar Separation Bubble at an Axisymmetric Diffuser L. Hoefener and W. Nitsche Technical University Berlin, Institute for Aeronautics and Astronautics (ILR) Marchstr. 12-14, 10587 Berlin, Germany Tel. : 49-30-314-22954, Fax: 49-30-314-22955
[email protected]
Summary Results of an experimental study of controlled laminar-turbulent transition in a smooth axisymmetric diffuser at 7800 < ReD1 < 10600 based on the inlet diam D /2 eter D1 and the bulk velocity um = D21 0 1 u(r)dr are presented. The inlet flow is an incompletely developed laminar pipe flow with a clear boundary-layer shape (boundary layer thickness δ99 /D1 ≤ 0.3). The smooth diffuser contour causes the formation of a closed pressure-induced laminar separation bubble, which is notedly receptive for small-scale disturbances. Upstream of the diverging geometry, controlled periodic perturbations are introduced into the boundary layer. The instability of the local velocity profiles with an inflection point causes a massive growth of instability waves within the shear layer leading to a turbulent breakdown of the laminar flow. For two different perturbation modes, the resulting mean flow field as well as the velocity fluctuations are measured by means of a Laser Doppler Velocimetry (LDV) and Particle Image Velocimetry (PIV) system in detail.
1
Introduction
For many technical Aerospace applications laminar-turbulent transition and separation play an important role, e.g. for the laminar wing design, optimization of highlift devices or in turbomachinery. Despite the enormous progress that has been made in the past decades the physical mechanisms contributing to formation, instability and breakdown of the separated shear layer are not yet fully understood. A laminar boundary can sustain only a small adverse pressure gradient without separation. Therefore, a pressure induced laminar separation bubble often develops at sufficiently divergent surfaces. Already within the attached laminar boundarylayer, small-scale instability waves, so-called Tollmien-Schlichting (TS)-waves, develop and grow downstream according to the linear stability theory[6]. At the separation point these instabilities propagate to the detached shear layer. The resulting local velocity profiles with an inflection point within the laminar separation layer[3] C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 244–251, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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have a strong destabilizing effect on the laminar flow. Thus, even small-scale disturbances cause a massive growth of instability waves[8]. Along the shear layer the instabilities are heavily forced and sub- and higher-harmonic modes are generated. Finally, a non-linear interaction of different modes leads to the laminar-turbulent transition[1]. The arising large-scale structures of the primary TS-wave cause an increased lateral mixing and momentum transfer within the shear layer. The onset of the transition is associated with a massive increase in the thickness of the boundary layer. Eventually, the strong momentum transfer towards the wall generates the turbulent reattachment of the flow, which recovers slowly towards a turbulent equilibrium boundary layer.
laminar boundary layer
u∞
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Fig. 1. Schematic depiction of separation at the axisymmetric model geometry
The investigated inlet flow shows an incompletely developed laminar pipe flow having a clear laminar boundary layer shape (δ99 /D1 ≤ 0.3). The smooth diffuser contour with a maximum opening angle of approximately 9◦ and an expansion ratio D2 /D1 of 1.6 causes a widespread pressure-induced laminar separation (s. Fig. 1). The closed separation bubble starts near the beginning of the expansion (x/D1 ≈ 0) and stretches across the test-section. The natural instabilities of the unbiased baseflow grow within the shear layer and trigger the transition several inlet diameters downstream. In order to study the influence of different perturbation modes, controlled disturbances were introduced at x/D1 = −3 by means of a membrane actuator. These perturbations raise the instability waves within the separated shear layer and lead to a controlled transition and a final turbulent reattachment.
2 2.1
Experimental Setup
Test Section and Measurement Techniques
All investigations were performed in a closed-circuit cylindrical pipe facility at the ILR. The inlet pipe has a diameter of D1 = 0.05 m and within the test section the geometry is expanded to an outlet diameter of D2 = 0.08 m. In order to allow optical access, the test section wall as well as the pipes are made of Plexiglas. Since the working fluid is a specific oil with a refraction index that matches that of the Plexiglas at a temperature of T =23◦ C, interference-free optical measurements can be performed. The present experiments were carried out using Laser Doppler Velocimetry (LDV) and Particle Image Velocimetry (PIV). The two-component LDV,
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Fig. 2. a) Solenoid driven membrane actuator, b) azimuthal modes
mounted on a three-way traverse unit, allowed detailed point-measurements. Complementary, the PIV-system enables velocity field measurements in the light sheet plane. Additionally, near-wall velocity fluctuations were measured using surface hot-wire anemometry. 2.2
Actuator
A solenoid-driven membrane actuator was designed to generate the controlled perturbations (s. Fig. 2). This actuator is build with a flush mounted rubber membrane and four separate air chambers in circumferential direction. The highly elastic membrane is driven by four powerful solenoids via a connecting rod. All four parts can be triggered individually using an oscillating input signal. The application of periodical signals, even with an azimuthal phase shift, allows the realization of several perturbation modes. In particular the axisymmetric mode (in-phase m = 0) and the first helical mode (anti-phase mode m = 1) have been examined. The membrane deflection was measured using a laser vibrometer in order to approximate calculate the mean perturbation parameter cµ =
3 3.1
Amembrane Ainflow
2
· v membrane . u2 m
Results
Baseflow
The baseflow, a laminar pipe flow, shows a normalized boundary layer thickness δ99 /D1 = 0.3. Due to the diverging wall surface, a pressure-induced laminar separation bubble deploys close behind the leading edge of the diffuser (x/D1 ≈ 1.4). With a low Reynolds number (Fig. 3 a, normalized axial velocity u/um , ReD1 ≈ 7800, Reδ1 ≈ 560) the rather stable flow core does primarily not sway and an extremely wide-spread bubble develops[2,5]. With a gaining, over-critical Reynolds number, the natural disturbances cause an increased amplification of instabilities of the laminar flow (s. Fig. 3 b, ReD1 = 10600). The growth of these dominant 2d instability waves results in a massive rise of the velocity fluctuations and results in enhanced RMS-values within the detached shear layer (Fig. 3 c). Subsequently, a massive lateral momentum transfer deploys, which causes an exchange of slow near-wall fluid into the core flow and a concurrent entrainment of fast fluid into the
Experimental Investigations of Controlled Transition
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3.2
The present investigation was mainly focused on the topological differences of the flow field caused by the introduction of different azimuthal modes at x/D1 = −3 by means of the membrane actuator. A sine wave test (sweep) was performed, where the frequency response of the actuation signal at increasing frequency is recorded using a traversable hot-wire probe at x/D1 = −0.2. The result in Fig. 4 a yields to an instability range of the most amplified TS-waves around f ≈ 18 Hz. Therefore, the chosen perturbation frequency results in a Strouhal number StD1 = fu·Dm 1 0.18 ≤ StD1 ≤ 0.35. Moreover, the Fig. 4 b shows the disturbance velocity profile urms /uδ 99 and phase φ for the perturbed flow (StD1 = 0.18, cµ ≈ 2.96 · 10−6 ). E
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Fig. 5. Time traces of streamwise staggered surface sensors
Therein, the typical Eigenform of the disturbed laminar boundary layer deploys, featuring two RMS-peaks, one close to the wall and another further away, divided by an phase-shift of ∆ φ = 180◦ [7]. Furthermore, the downstream amplification of convective TS-wave packets along the separated shear layer is illustrated in Fig. 5. The amplitude of the TS-wave clearly rises from x/D1 ≈ 5 up to x/D1 ≈ 10 by more than one order of magnitude. In addition, a time shift of the associated packet arises, originating from the convection velocity. 3.3 Comparison of I n-phase (m = 0) and Counter-P hase (m = 1) Perturbation This section presents results obtained for a Reynolds number ReD1 ≈ 7800. The controlled perturbations were placed at the fundamental frequency of the TSinstabilities (StD1 = 0.35) at a magnitude of cµ ≈ 2.9 · 10−4 .
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Fig. 7. Counter-phase perturbation (m = 1), ReD1 = 7800, velocity profiles of normalized axial velocity u/um (a) and normalized RMS-values urms /um (b), (LDV-measurement)
In comparison to the base flow (Fig. 3 a), the extension of the separation bubble in Fig. 6 a is clearly reduced for the axisymmetric (m = 0) disturbance case (reattachment at xr /D1 ≈ 5.5). The disturbances cause a faster growth of the TSinstabilities already upstream of the diffuser. Along the separation bubble, these instabilities are amplified within local high-shear zones. Sequentially, three characteristic peaks of the RMS-values urms /um develop (Fig. 6 b), one close to the wall and two at the edge of the detached shear layer[4]. Further downstream, the intensity of the fluctuations grows quickly and the oscillations spread out wedge-like. At the same time, the RMS-level saturates non-linearly for x/D1 > 5 leading to the final stage of transition. Complementary to this, Fig. 6 c clarifies the onset and downstream development of large-scale instability waves by means of the normalized lateral velocity v/um . Synchronous to the controlled perturbations, alternating regions of inbound (v/um < 0) and outbound (v/um > 0) orientated fluid emerge and grow in streamwise direction. The disturbances give rise to a strong lateral momentum transfer and the resulting mixing is the driving force for the turbulent reattachment. The dissipation of the fluctuations heads slowly for a fully turbulent pipe flow. Moreover, the controlled disturbances with the counter-phase mode m = 1 (Fig. 7 a) lead once more to a reduced reattachment length up to xr /D1 ≈ 4.5. Already upstream of the separation, a local maximum of the radial RMS-profile
Fig. 8. Streamwise maximum amplification distribution
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develops in the outer part of the boundary layer close to the wall. The amplified TSwaves convect downstream up to the point of separation, where they float into the detached shear layer. Within the high-shear zones inside of the bubble, the destabilizing properties of the local velocity profiles with an inflection point give rise to a strong amplification of the instability waves and the radial RMS-profiles show three local maxima again (Fig. 7 b). The non-linear saturated primary TS-waves cause a strong unsteady vortex shedding and interact with the solid wall in the reattachment zone. Following the recirculation zone, the local RMS-peaks of the radial profiles merge, forming a single, prominent wedge. In order to quantify the spatial growth of the disturbances, the normalized amplification curve was determined. As shown in Fig. 8, the instability wave grows exponentially in downstream direction and saturates inside the reattachment area. This strong growth sets off close to the separation within the region of adverse pressure gradients at x/D1 > 1.8 (axisymmetric mode m = 0) and x/D1 > 2.3 (anti-phase perturbation m = 1), respectively. Between 3 ≤ x/D1 ≤ 5, both curves feature the same gradient but for m = 0 finally higher amplitudes are reached and the decay starts earlier. The reason for this behavior can be found in the different transition paths of both cases. For the symmetric perturbation, the transition scenario is dominated by the initially two-dimensional development. In contrast, the anti-phase disturbances of mode m = 1 cause an accelerated interaction of two- and three-dimensional modes and a faster formation of small-scale secondary structures. Thus, the momentum transfer is intensified leading to a rapid turbulent reattachment of the flow.
4
Conclusion
Results of an experimental study on controlled transition in a laminar separation bubble on an axisymmetric smooth diffuser have been presented. The chosen geometry generated on the base flow with an incompletely developed laminar pipe flow a large laminar bubble. For an investigation on the impact of different azimuthal disturbance modes, controlled perturbations were introduced into the flow already upstream of the diffuser by means of a segmented membrane actuator. A boundary layer measurement was performed in order to evaluate the excitation of the TS instability waves. The extension and stability of the recirculation zone were found to depend strongly on the inlet Reynolds number ReD1 . For clearly over-critical ReD1 ≈ 10600, the natural instabilities were amplified and resulted in characteristic peaks with increased RMS-levels urms . An introduction of controlled disturbances proved to amplify the TS-instabilities, with their amplitude growing for more than one order of magnitude within the separated shear layer. The disturbances spread out wedge-like in downstream direction and the resulting lateral mixing is the driving force of the turbulent reattachment. A comparison of axisymmetric (m = 0) and anti-phase (m = 1) disturbances showed distinct differences of the emerging amplification curves. While symmetric perturbations have mainly boosted the fundamental 2d instability waves, the counter-phase induction has caused a premature interaction of 3d modes. The resulting breakdown of the large-scale vortices into small-scale structures led to a
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forced transition and a reduction of the recirculation zone for the azimuthal mode m = 1 compared to the in-phase perturbation.
Acknowledgement The research described herein was financially supported by the German National Science Foundation (Deutsche Forschungsgemeinschaft, DFG) within the scope of the Priority Program (Schwerpunktprogramm, SPP “Transition”), contract number Ni 282/14.
References [1] M. Alam and N.D. Sandham. Direct numerical simulation of ’short’ laminar separation bubbles with turbulent reattachment. Journal of Fluid Mechanics, 410:1–28, 2000. [2] F. Bao and U.C. Dallmann. Some physical aspects of separation bubble on a rounded backward-facing step. Aerospace Science and Technology, 8:83–91, 2004. [3] A.V. Dovgal, V.V. Kozlov, and A. Michalke. Laminar boundary layer Separation: instability and associated phenomena. Prog. Aerospace Sci., 30:61–94, 1994. [4] C.P. Häggmark, A.A. Bakchinov, and P.H. Alfredsson. Experiments on a two-dimensional laminar separation bubble. Phil. Trans. R. Soc. Lond. A, 358:3193– 3205, 2000. [5] L. Hoefener, W. Nitsche, A. Carnarius, and F. Thiele. Experimental and numerical investigations of flow separation and transition to turbulence in an axisymmetric diffuser. In H.J. Rath, C. Holze, H.-J. Heinemann, R. Henke, and H. Hönlinger, editors, New Results in Numerical and Experimental Fluid Mechanics V, Contributions to the 14th STAB Symposium Bremen, 2004, volume 92, pages 217–224. Springer-Verlag, 2006. [6] Y.S. Kachanov. Physical mechanisms of laminar-boundary-layer transition. Annual Review of Fluid Mechanics, 26:411–482, 1994. [7] M. Lang, O. Marxen, U. Rist, and S. Wagner. A combined numerical and experimental investigation of transition in a laminar separation bubble. In S. Wagner, M. Kloker, and U. Rist, editors, Notes on Numerical Fluid Mechanics, volume 86 of Recent Results in Laminar-Turbulent Transition - Selected numerical and experimental contributions from the DFG priority programme "‘Transition"’, pages 149–164. Springer-Verlag, 2003. [8] U. Maucher. Numerische Untersuchungen zur Transition in der laminaren Ablöseblase einer Tragflügelgrenzschicht. PhD thesis, IAG der Universität Stuttgart, 2002.
Development of a Sensor-Actuator-System for Active Control of Boundary Layer Instabilities in Compressible Flows Marcus Engert, Andreas P¨ atzold, and W. Nitsche Technische Universit¨ at Berlin, Institut f¨ ur Luft- und Raumfahrt, Marchstr. 12-14, 10587, Germany
[email protected] http://aero.ilr.tu-berlin.de
Summary This paper describes the development of an active control system for delaying laminar-turbulent transition in compressible flows. Convective boundary layer instabilities are detected during their linear amplification stage by a reference sensor and reduced downstream through destructive interference of artificial counterwaves. By means of highly sensitive surface sensors the transitional boundary layer for Mach numbers up to Ma=0.5 was investigated in detail. An actuator was developed for the generation of counterwaves with appropriate amplitude, wavelength, frequency range and phase. After integration of reference sensors, actuator and error sensors into a slim wing model, artificially generated boundary layer perturbations were attenuated as a first test.
1
Introduction
The laminar-turbulent transition of a boundary layer of an unswept wing, caused by Tollmien-Schlichting waves (TS-waves), can be delayed by using Active Wave Control (AWC), [1, 5]. Incoming TS-instabilities are identified during their linear growth by a reference sensor and cancelled out downstream via wave superposition with the help of an actuator. The actuator is driven by a realtime controller which aims to minimize the error signal at the end of the control path, see Fig. 1. Various types of actuation were suggested in the past, for example a slot-type suction/blowing system [1], traveling bumps at the wall [2], membrane actuators [5] and piezo bending actuators [3]. For effective transition delay it is necessary to ensure a repeatable 2-dimen sional boundary layer with linear amplification range and to design sensitive surface sensors and a fast actuator which generates appropriate high-frequency counterwaves. The main difference compared to previous investigations is the significant increase in flow velocity up to compressible flow conditions. This leads to much larger TS-amplitudes as well as to instable frequencies up to 20 times higher than in low-speed experiments [3, 5]. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 252–259, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Fig. 1. Principle of Active Wave Control (AWC)
2
Experimental Setup
Experimental investigations were conducted in the transonic wind tunnel of the Institute of Aeronautics and Astronautics of the TU Berlin. This continuously working wind tunnel has a nozzle contraction ratio of 48:1 which results in a very low turbulence intensity of T u ≤ 0.15%. The test section features upper and lower adaptive walls (see Fig. 3) for influencing the pressure gradient and thereby fixing boundary layer transition at the desired position regardless of flow velocity. Measurements were carried out up to Mach numbers of Ma = 0.5.
Fig. 2. Experimental setup: wing equipped with AWC-system
Fig. 3. 2-dimensional adaptive test section of transsonic windtunnel
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Fig. 4. Wing model with surface hot wire sensor array
An unswept 2-dimensional wing model with a NACA0004 nose and an overall chord length of 850mm has been developed, see Fig. 2 and 4. Its modular structure, made of aluminium and fibre glass reinforced composite allows the smooth integration of different exchangeable test modules containing the required sensors and actuators. For detection of TS-instabilities several arrays of Surface Hot Wires (SHW, [4]) were applied, see Fig. 4. SHWs are platinum coated tungsten wires with a diameter of 5µm which are flush mounted onto the wing’s surface. They generate negligible interfering surface roughness. Each sensor is clamped across a small cavity of 100µm width. Compared to conventional hot film sensors, less heat is transferred into the wall and therefore an improved signal-to-noise ratio and a cut-off frequency greater than 20kHz allows for detection of very small velocity fluctuations within the boundary layer. In order to properly resolve TS-waves at Ma ≥ 0.3 data was acquired with sampling frequencies of up to 60kHz. Counterwaves were computed by an adaptive filter algorithm running on a fast dSPACETM DS1005 digital signal processor.
3
Investigation of Transitional Flow
Repeatable flow conditions and linear TS-amplification in front of the actuation zone are essential for AWC. Therefore measurements were made with a high spatial resolution in streamwise and spanwise direction at free stream Mach numbers up to Ma=0.4. Location, frequency range and stage of laminarturbulent transition are dependent on Mach number and could be specified. Figure 5(a) shows time traces of 11 streamwise adjacent sensors for Ma=0.4. Typical TS-wave packets are moving downstream between relative chord lengths of 0.24 ≤ x/c ≤ 0.32 with a convection speed of approx. 35% of the local free stream velocity. The boundary layer finally becomes turbulent for x/c ≥ 0.38. The detected overall TS-amplification length of 80mm provides a comfortable spatial range for actuation.
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(a) Time traces downstream at transition stage for Ma=0.4
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(b) Coherence between sensor 5 and sensor 6 (∆x = 15mm)
Fig. 5. Investigation of transitional flow
(a) Power spectra at early TS-amplification stage
(b) Identified frequency range of TSwaves
Fig. 6. TS-frequency range
In Fig. 6(a) the frequency spectra of a transitional sensor signal are displayed up to Ma=0.4. For the example of Ma=0.3 the first characteristic instable TS-mode ranges from 2 - 6kHz. In Figure 6(b) the first mode’s lower and upper instable frequencies are displayed for 0.2 ≤ M a ≤ 0.5 . The instable domain shifts to higher frequencies with increasing flow velocity. With λT S =c/f the mean TS-wavelength for Ma=0.3 is calculated to be 12mm.
4
Actuator Design and Integration
For successful cancellation of TS-waves the actuator has to be able to generate appropriate counterwaves that correspond to TS-waves concerning wavelengths, frequency range and amplitude. A smooth surface membrane actua tor
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Fig. 7. Configuration of membrane actuator for compressible flow
(a) Phase locked membrane deflection for f=4kHz
(b) Amplitude and phase response for noise input signal
Fig. 8. Characteristics of membrane actuator
was designed, because any surface roughness ( slots / obstacles) would not be tolerated by the sensitive boundary layer. A flexible membrane, clamped across a small surface slot, is driven by a plunger coil via a small lifter, see Fig. 7. The oscillating membrane produces a velocity v perpendicular to the wing’s surface. It is converted into a TS-counterwave by the mean flow. According to earlier low speed research [5], an oscillating area of 1/4 of the mean wavelength is most suitable for effective TS-reduction, so a 3mm streamwise slot was deployed. For first experiments at Ma=0.3 the actuator had to cope with frequencies around 4kHz. The moving membrane surface was scanned in the flow direction by a laser vibrometer. Fig. 8(a) shows the membrane deformation shape when excited at 4kHz. The distributions of perturbation velocity v (amplitude) and phase angle between input signal and measured deformation indicate a ‘clean’ membrane oscillation in its first natural mode without interference of high-frequency components. The harmonic surface shape enables a ‘soft’ introduction of the counterwaves into the flow, especially at the membrane edges.
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The performance of the actuator at different frequencies was investigated with a broadband noise input signal, Fig. 8(b). The perturbation velocity obtained at relevant frequencies for Ma=0.3 is around 200mm/s. A short actuator response time is required in order to react rapidly to non-periodic perturbations and to allow the distance between actuator and upstream reference sensor to be minimised. The response time depends mainly on the inertia of the oscillating mechanical system and is calculated by the gradient of phase response, see Fig. 8(b). Although it would be desirable to operate the actuator close to its resonant frequency for amplitude reasons this is unfavourable because of the steep phase gradient and the resulting slow response behaviour. Between 3 and 4kHz a response time of 0.04ms was measured which is nearly five times below the runtime of boundary layer instabilities between the nearest reference sensor and actuator at Ma=0.3. An array of nine reference sensors in a line upstream of the actuator and 11 error sensors behind was integrated into the wing model. The reference sensor next to the actuator (distance 6mm) was preferred because signal correlation decreases with distance. Figure 5(b) indicates good coherence within the TS-frequency range between last reference and first error sensor (distance=12mm) for the case of natural transition. This is a basic requirement for the adaptive filter control algorithm used.
5
Controlling of Artificial TS-Waves and Modifications
As first step for testing the AWC system under high speed flow conditions, 2-dimensional artificial perturbations with variable amplitude and frequency were introduced into the boundary layer upstream of the transition zone. Therefore a second membrane actuator at x/c = 0.14 worked as a perturbation source. A certain minimum amplitude within the TS-frequency range was found to force amplification of these artificial perturbations. Fig. 9 shows the distribution of RMS-values in streamwise direction at Ma=0.25 for a perturbation frequency of 3kHz. For the actuator OFF case, artificial boundary layer instabilities are amplified after the second error sensor and lead to an early transition. With the AWC system operating (actuator ON), perturbations can be reduced significantly, thus instability growth is decreased and transition is delayed by 3-4% of the chord. The first error signal’s RMS value seems to remain unchanged because counter waves need a certain distance for the interaction with the boundary layer. Mach numbers exceeding Ma=0.3 are beyond the means of the present membrane actuator, because amplitudes of TS-instabilities grow further and instable frequencies move beyond 5kHz. An improved piezo-bending type of actuator consists of several spanwise arranged piezo elements, mounted directly at the wing’s surface. Figure 10 illustrates the structure and amplitude response of the piezo actuator. In preliminary tests under flow conditions the
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actuator was used as an efficient perturbation source and will replace the membrane actuator in the future.
Fig. 9. Attenuation of artificial TS-waves (f=3kHz) for Ma=0.25
Fig. 10. Amplitude response of piezo bending actuator
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Conclusion
TS-instabilities were measured in their early amplification stage at Mach numbers up to Ma=0.5. A membrane actuator was developed for generating appropriate counterwaves for Ma=0.3 and tested as an integral part of an active wave control system. Artificially generated perturbations were introduced and reduced significantly for Ma=0.25. For tests at higher flow velocities an improved piezo actuator has been built.
Acknowledgements This paper is mainly based on experimental work that was done within the CAFEDA (Control of Aerodynamic Flows for Environmentally Driven Aircraft) research project, funded by Airbus.
References [1] Baumann, M., Sturzebecher, D., Nitsche, W., 2000: ”Active Control of TSInstabilities on an Unswept Wing” In: Fasel, H. and Saric, W. (Eds.), LaminarTurbulent Transition, IUTAM-Symposium on Laminar-Turbulent Transition, 13.-17. September 1999, Sedona (Arizona, USA), Springer Verlag. [2] Breuer, K., Haritonidis, J., Landahl, M., 1989: ”The control of transient disturbances in a flat plate boundary layer through active wall motion”, In: Physics of Fluids, vol. 1, p. 574-582. [3] Fukunishi, Y., Yokokawa, Y., Sakai, T., Maita, H., Sasaki, K., Izawa, S., 2006: ”Flow Controlling Attemps against a Sound Genarating Cavity Flow and a Transitional Boundary Layer” In: Journal of Japan Society of Fluid Mechanics, 25(2),(2006),pp. 119-126 [4] Sturzebecher, D., Anders S., Nitsche, W., 2001: ”The surface hot wire as a means of measuring mean and fluctuating wall shear stress” In: Experiments in Fluids, Volume 31, Issue 3, pp. 294-301. Springer-Verlag Berlin. [5] Sturzebecher, D., Nitsche,W., 2003: ”Active cancellation of TollmienSchlichting-Instabilities on an unswept wing using multi-channel sensor actuator systems” In: International Journal of Heat and Fluid Flow, Volume 24, Number 4, pp. 572-583. Elsevier Science Ltd.
Evaluation of Initial Amplitudes of Free-Stream Excited Tollmien-Schlichting Waves from Flight-Test Data Arne Seitz DLR-Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7, 38108 Braunschweig, Germany
[email protected]
Summary Flight tests were performed in order to learn more about free-stream excited TollmienSchlichting waves and the role they play in boundary layer transition. Therefore, a multielement hot-film array, placed on the right hand wing of the flying testbed LFU-205, was used to sense the skin friction fluctuations provoked by this type of instability while propagating in the laminar boundary layer. The experimental data acquired were subsequently analyzed with special emphasis on the evaluation of the initial amplitudes of TollmienSchlichting waves. For the streamwise disturbance velocity component their magnitude was determined to be in the order of 10−9 times the boundary layer edge velocity.
1 Introduction Tollmien-Schlichting waves are well known as primary instabilities in transition of boundary layers from laminar to turbulent flow (Tollmien [7], Schlichting [5]). But since almost all experimental investigations on this type of instability have been done in wind-tunnel tests, mainly under conditions of controlled disturbance excitation, very little is known about the technically important case of naturally occurring TollmienSchlichting waves in a wing boundary layer in free-flight. In order to reduce this lack of consolidated findings, carefully prepared in-flight experiments with the flying testbed LFU-205 of DLR were recently performed by Seitz [6]. A multi-element hot-film array, placed on the right hand wing of the flying testbed LFU-205, was used to sense the skin friction fluctuations provoked by TollmienSchlichting waves propagating in the laminar boundary layer. A first analysis of the experimental data revealed that free-stream excited boundary layer disturbances appear as three-dimensional wave packets which are formed by a broadband spectrum of 2d and 3d instabilities. Further investigations concerned the initial amplitudes of Tollmien-Schlichting waves involved in the formation of the wave packets. So far, initial amplitudes of naturally occurring disturbances are an unknown quantity. But if it is the goal to exert more sophisticated transition prediction methods that are capable of treating nonlinear interactions between instabilities, like for example PSE or DNS methods, one needs to have at least an idea of the amplitudes a nonlinear calculation should be started with. The present work is a first step to provide the necessary data. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 260–267, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Initial Amplitudes of Tollmien-Schlichting Waves
Tollmien-Schlichting waves exhibit in their u-disturbance velocity profiles (eigenfunctions respectively) a maximum close to the wall, Fig. 1. Here, amplitude and phase function correspond with the wave ansatz from temporal linear stability theory: u´ = uˆ(z) · exp(ωi t) · exp(α x + β y − ωr t + ϕu (z))
(1)
Using the definition for the wall shear stress, it may be derived from eq. (1) that a velocity disturbance leads to a fluctuating skin friction according to ( dˆ u (( τ´w = µ · exp(ωi t) · exp(α x + β y − ωr t + ϕu (z = 0)) (2) dz (z=0 u/dz)w being the amplitude. It is obvious that Tollmien-Schlichting with τˆw = µ (dˆ instabilities leave traces at the wall which contain all the information about the waves like frequency, wave numbers and amplification. Hot-films are the adequate type of sensor to measure these traces or skin friction fluctuations respectively. Originally introduced by Ludwieg [2] as a device to determine the mean wall shear stress in turbulent flow, soon progress in the development of sensors as well as anemometry systems allowed also for an investigation of unsteady flow phenomena. The principle of hot-film anemometry is based on convective heat transfer from an electrically heated sensing element. Cooling of the element by the flow depends upon the magnitude of the wall shear stress and, hence, a relation exists between the wall shear stress and the electric voltage needed to keep the sensor at a constant temperature. The instantaneous anemometer output voltage e = E0 + e´ may be split up into a mean value E0 which is connected to the mean shear stress τw0 and a fluctuating part e´, which can be related to τ´w via a calibration procedure. The calibration procedure itself as well as an error analysis (on which the accuracy of all quantities subsequently derived from the wall shear stress fluctuations depends) is described in more detail in [6]. As was already mentioned, naturally occurring Tollmien-Schlichting waves appear as wave packets, involving a broadband spectrum of instabilities. Their frequency can easily be determined by a spectral decomposition of the time record from a single hotfilm sensor. In contrast, the spatial periodicity of waves, expressed by wave numbers α and β, can only be identified from measurements if records of multiple sensors distributed across the wing are available. Then, the discrete Fourier decomposition in time and space of records τ´w (t, y) from an equally spaced, spanwise row (x/c = const.) of ns sensors sampled at m discrete time steps yields the amplitude τˆw as a function of Tollmien-Schlichting frequency f = ωr /(2π) and spanwise wave number β, τˆw (fk , βq ) = |Cˆk,q |
(3)
with the Cˆk,q being the complex Fourier coefficients according to Cˆk,q =
2 m(ns − 1)
m−1 s −1)/2 +(n j=0 p=−(ns −1)/2
τ´w (j, p) exp i (−2πfk t − βq y) .
(4)
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Fig. 1. Amplitude and phase distribution of a u-disturbance velocity profile
It should be noted, that a Tollmien-Schlichting wave is well-defined through the specification of two of its parameters like frequency f and spanwise wave number β. The streamwise wave number α then is a dependent variable which is implicitly determined by the dispersion relation ω = ω(α, β). Once the amplitude τˆw (f, β) of a Tollmien-Schlichting wave has been obtained from the hot-film measurements, this result can be utilized to gain the amplitude function uˆ(z) of the streamwise disturbance velocity component: In a first step, the nondimensional, normalized amplitude function u ˆ (z ) of the Tollmien-Schlichting wave with parameters f, β is calculated with a linear stability code. Here the code written by Malik & Orszag [3] was employed. As an input for the stability calculations the correct laminar mean flow has to be provided. This was accomplished by a boundary layer computation (Kaups & Cebeci [1]) based on the measured wing pressure distribution. With the definition of a scaling factor K the amplitude function u ˆ(z) can then be linked to the theoretically determined amplitude function u ˆ (z ) via uˆ(z) = K Ue uˆ (z = z δ1 )
(5)
By differentiating, multiplying with viscosity µ and rearranging eq. (5) the scaling factor K may be evaluated from the present experimental data, i.e. the measured wall shear stress amplitudes, according to: K=
τˆw δ1 1 · . (d uˆ /d z )|z =0 µ Ue
(6)
As was shown in Fig. 1, the uˆ amplitude function exhibits a strong maximum near to the wall. Commonly, this maximum amplitude is taken as a measure for the intensity of a disturbance. With K being known from eq. (6) and taking into account that uˆmax = 1 (due to the normalization of the eigenfunction) one obtains: u ˆmax =K Ue
(7)
While travelling downstream, Tollmien-Schlichting waves are amplified and the total amplification may be expressed by the so called N-factor, which is defined as N =
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ln(A/A0 ), with A/A0 being the ratio of the disturbance amplitude at some position x/c downstream to the initial amplitude at the point of neutral stability. For a TollmienSchlichting wave specified by its frequency and propagation direction, the distribution N (x/c) can be evaluated from the results of linear stability theory. With the disturbance amplitude known at x/c from eq. (7), the initial amplitude can be determined by u ˆmax uˆmax = exp(−N (x/c)) · (8) Ue 0 Ue (x/c)
3 The Experiment Fig. 2 shows the flying testbed LFU-205 of DLR. It is a light aircraft equipped with a laminar glove made from composite material on the right hand wing. The airfoil in the glove region was especially designed to allow for a moderate growth of TollmienSchlichting waves in a constant pressure region or under a slightly adverse pressure gradient. In order to measure the actual pressure distribution, the wing is equipped with 48 orifices in one chordwise row. With an infrared thermovision system placed in the cockpit of the aircraft the location of turbulence onset in the glove region is monitored. Further instrumentation comprises a pitot-tube with total and static pressure ports located on the left wing-tip and a total temperature probe on the lower side of the left hand wing. Data gained from these probes are utilized to compute free stream values of velocity, static pressure, static temperature and density as well as Mach number and Reynolds number.
Fig. 2. Flying testbed LFU-205 of DLR
Although the wing of the LFU-205 is slightly swept forward and tapered (quarter chord sweep −7◦ , taper ratio 0.5), no significant cross-flow can be observed, so the boundary layer may be considered two-dimensional. Therefore, instabilities of the laminar flow will be essentially Tollmien-Schlichting waves. These are measured with an especially designed 69 element hot-film array shown in Fig. 3. Sensor elements are
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Fig. 3. Hot-film array with 69 sensing elements
arranged on the array mainly in two spanwise rows on constant chord positions of x/c = 0.34 and x/c = 0.37. Each row consists of 25 sensors being 4mm apart in spanwise direction. Both rows are connected by a chordwise column of 7 sensors on the centre line of the array. Additional sensors are short coupled (2.5mm distance in chordwise direction) to the elements of both main rows. Each hot film element consists of a 0.2µm thick, 1.5mm wide and 0.1mm deep nickel film connected to 8µm thick copper coated leads. The 0.1mm thick substrate is of polyimide and was bonded to the wing surface. In order to get an installation without forward or backward facing steps, edges of the substrate foil were filled and sanded. For the same reason, the electric leads were routed away from the surface into the interior of the wing before soldered to the wiring. The hot-films were operated in the constant temperature (CTA) mode, with the temperature being held at T = 150◦ C. The fluctuating parts of the sensor signals were sampled simultaneously for one second at a rate of 48 kHz. All test flights were performed in still air within a stable high pressure region. Although not measured in this series of experiments, previously performed flight tests with the LFU-205 under similar weather conditions showed that the free stream turbulence level is not higher than T u = 0.05% (Riedel & Sitzmann [4]).
4 Results of Data Analysis The data aquired at each individual measuring point of a test flight comprise the glove pressure distribution CP (x/c), the free-stream values of V∞ , p∞ , T∞ , an infrared image of the glove region as well as the skin friction fluctuations measured by the hotfilms. As an example, in Fig. 4 10ms long cut outs from the time histories τ´w (t, y) of nine sensor elements located around the centre line (y = 0mm) of the hot-film array at a chord position of x/c = 0.37 are depicted. A first visual inspection of the time histories reveals the principal topology of the disturbance flow. One can clearly discern several coherent structures, which are characterized by a sudden increase of the amplitude over three or four oscillations, a pattern
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Fig. 4. Time histories of signals from nine hot-film sensors located around the centre line of the array (y = 0mm) at a chord position of x/c = 0.37
Fig. 5. Frequency against wave number spectra at x/c = 0.34 and x/c = 0.37, based on data from flight no. 8 / measuring point 22 Table 1. Initial amplitudes of Tollmien-Schlichting-waves flight no. / TS-wave measuring point f [Hz], ψ = 0◦ τˆw [mPa] 8/17 1870 0,26 8/18 1860 0,31 8/19 1810 0,41 8/20 1780 0,43 8/21 1750 0,41 8/22 1700 0,64
(|ˆ u|/Ue )max at x/c = 0.37 0, 83 · 10−5 0, 99 · 10−5 1, 33 · 10−5 1, 42 · 10−5 1, 38 · 10−5 2, 21 · 10−5
(|ˆ u|max /Ue )0 N 8,792 8,928 8,897 9,008 8,783 9,972
1, 26 · 10−9 1, 31 · 10−9 1, 82 · 10−9 1, 73 · 10−9 2, 12 · 10−9 1, 03 · 10−9
that is extended in spanwise direction over several records. This type of amplitude modulation is charcteristic for the occurrence of wave packets. As outlined in section 2, a Fourier decomposition in time and space of the experimental data was performed in
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Experiment Theory
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Fig. 6. Disturbance amplification; comparison between experiment and theory, based on data from flight no. 8 / measuring point 22
order to determine the Tollmien-Schlichting components involved in the formation of these wave packets. Fig. 5 then shows the Tollmien-Schlichting amplitudes plotted as contours over a spectrum of frequencies f and spanwise wave numbers β. Additionally, waves propagating into the direction of the boundary layer edge flow (propagation angle ψ = 0◦ ) can also be identified from this type of chart. The frequency against spanwise wave number spectra at sensor positions x/c = 0.34 and x/c = 0.37 shown in Fig. 5 were evaluated from data measured simultaneously. Consequently, it is possible to derive from these both charts the amplification ∆N = N (x/c = 0.37) − N (x/c = 0.34) of certain Tollmien-Schlichting waves and compare it with the results of a linear stability calculation. This was done for those disturbances propagating with ψ = 0◦ . Amplitude spectra extracted from Fig. 5 for these Tollmien-Schlichting frequencies as well as ∆N (f, ψ = 0◦ ) are shown in Fig. 6. Excellent agreement can be stated when comparing experimentally and theoretically determined amplification exponents. Obviously, the disturbance amplitudes measured were small enough, so that linear stability theory still holds. Hence, it is justified to recalculate the initial amplitudes of instability waves the way described in section 4. The recalculation was then performed for that Tollmien-Schlichting wave of propagation direction ψ = 0◦ which reached, according to the spectrum at x/c = 0.37, the maximum amplitude. In Table 1 the initial amplitudes of Tollmien-Schlichting waves determined from data of six different measuring points that were taken during one test flight are summarized. The values are quite consistent with a mean value of (ˆ umax /Ue )0 = 1.56 · 10−9 and a standard deviation of 0.46 · 10−9 . It should be noted that these amplitudes refer to single Tollmien-Schlichting waves and not to wave packets.
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5 Conclusions Flight tests data were analysed with the objective to determine the initial amplitude of free-stream excited Tollmien-Schlichting waves. Therefore, a multi-element hot-film array was placed on the right hand wing of the flying testbed LFU-205 and used to sense the skin friction fluctuations provoked by disturbances propagating in the laminar boundary layer. It was observed that Tollmien-Schlichting waves form up threedimensional wave packets with a broadband spectrum of instabilities involved. A Fourier decomposition in time and space of the experimental data then yields the wall shear stress amplitudes of Tollmien-Schlichting waves with parameters f and β. It was shown that these wall shear stress amplitudes may be employed to scale the amplitude function of the disturbance velocity in streamwise direction, which is calculated on the basis of linear stability theory. Once the amplitudes of Tollmien-Schlichting waves at a sensor position x/c are known, their initial values at the point of neutral stability can easily be recalculated employing the amplification exponent or N-factor respectively. The analysis of the experimental data from six measuring points revealed that the initial maximum amplitude of free-stream exited Tollmien-Schlichting waves in freeflight in atmosphere is in the order of (ˆ umax /Ue )0 = 1.56 · 10−9 with a standard −9 deviation of 0.46 · 10 .
References [1] K. Kaups, T. Cebeci: ”Compressible laminar boundary layers with suction on swept and tapered wings”. Journal of Aircraft 14, 1977, pp. 661-667 [2] H. Ludwieg: ”Ein Ger¨at zur Messung der Wandschubspannung turbulenter Grenzschichten”. Ing. Archiv 86, 1949, pp. 207-218. [3] M. R. Malik, S. A. Orszag: ”Efficient computation of the stability of three-dimensional compressible boundary layers ”. AIAA-Paper 81-1277, 1981. [4] H. Riedel, M. Sitzmann: ”In-flight investigations of atmospheric turbulence”. Aerospace Science and Technology 86, 1998, pp. 301-319 [5] H. Schlichting: ”Zur Entstehung der Turbulenz bei der Plattenstr¨omung”. Nachr. d. Ges. d. Wiss. zu G¨ottingen, Math.-Phys. Klasse, 1933, pp. 192-208. ¨ [6] A. Seitz: ”Freiflug-Experimente zum Ubergang laminar-turbulent in einer Tragfl¨ugelgrenzschicht”. Dissertation TU-Braunschweig, 2007, published as DLR-Forschungsbericht FB2007-01 ¨ [7] W. Tollmien: Uber die Entstehung der Turbulenz”. Nachr. d. Ges. d. Wiss. zu G¨ottingen, Math.-Phys. Klasse, 1929, pp. 21-44.
The Role of Turbulent Dissipation for Flow Control of Near-Wall Turbulence Bettina Frohnapfel1, Peter Lammers2 , and Jovan Jovanovi´c1 1
LSTM - Institute of Fluid Mechanics, Friedrich-Alexander Universit¨at Erlangen-N¨urnberg, Cauerstr. 4, 91058 Erlangen, Germany
[email protected] http://www.lstm.uni-erlangen.de 2 HLRS - High Performance Computing Center Stuttgart, Nobelstr. 19, 70569 Stuttgart, Germany
Summary One of the major goals of flow control is the reduction of energy consumption in wall bounded flows by minimizing the viscous drag. In the present work it is shown that the turbulent dissipation needs to be minimized in order to obtain energy savings. Analytical considerations lead to the conclusion that this can be achieved by forcing near-wall turbulence to an axisymmetric state. To confirm this finding direct numerical simulations were carried out in which the boundary conditions were such that near-wall fluctuations were forced towards axisymmetry. Additionally, a surface structure that minimizes turbulent dissipation at the wall was designed and tested experimentally. Both, numerical and experimental investigations yield significant drag reduction.
1 Introduction A significant portion of the drag which counteracts the motion of a body through a fluid is generated in the thin viscous region close to the solid boundary where the flow is nearly always turbulent. The viscous contribution to the total drag amounts to about 50% on commercial aircrafts, 90% on underwater vehicles and almost 100% for pipe and channel flows. If we consider current trends in the crude oil prices, worldwide expansion of public transport and initiatives for reduced pollution of the environment, the beneficial outcomes of reducing viscous drag and therefore energy consumption are obvious.
2 Turbulent Dissipation All energy that is used to drive the flow in or around an object is eventually dissipated into heat. In a turbulent flow field the total energy dissipation rate ) is composed of the direct dissipation and the turbulent dissipation: C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 268–275, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
The Role of Turbulent Dissipation for Flow Control of Near-Wall Turbulence
) =ν
∂U j ∂U i + ∂xj ∂xi
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∂U j ∂ui ∂uj ∂uj +ν + . ∂xi ∂xj ∂xi ∂xi
direct dissipation D
(1)
turbulent dissipation
In fig. 1 normalized direct, turbulent and total average dissipation rates extracted from direct numerical simulations of turbulent channel flows [1, 6, 7, 10, 12] are plotted, where averaging and normalization are according to: ν 1 + ) = 4 ) dV. (2) uτ V V 0.16
total dissipation
direct dissipation
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Fig. 1. Contributions of direct and turbulent dissipation to the total dissipation plotted versus the Reynolds number Reτ (left). Distribution of the turbulent dissipation rate versus wall distance normalized with the wall shear velocity and the kinematic viscosity of the flow medium versus distance from the wall for a plane channel flow (right). The upper solid line corresponds to the DNS data at Reτ = 395 by Antonia et al. [1], the lower dotted lines show sketched estimates of + -profiles for decreasing turbulent dissipation at the wall and increasing drag reduction.
At very low Reynolds numbers direct dissipation is the dominating quantity. However, for increasing Reynolds numbers the contribution continuously shifts towards the turbulent dissipation. At Reτ ≈ 590 the contributions are of the same order and at Reτ ≈ 720 the turbulent dissipation constitutes the major part of the total dissipation. For high Reynolds numbers that are of interest in engineering applications no data sets are currently available but the trend suggests that the turbulent dissipation will distinctly outweigh the direct dissipation. Since the average total dissipation balances the pressure drop ∆p/∆L (energy conservation), which in return can be expressed as the work done against the wall shear stress τw (momentum conservation), it holds that Ub ∆p 1 τw AUb ) dV = = , (3) V V ρ ∆L ρV
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where ρ is the fluid density, Ub the bulk flow velocity, A the wetted surface area and V the total volume of fluid. Therefore, it can be concluded that the turbulent dissipation has to be reduced in order to achieve significant drag reduction and energy savings. The distribution of the turbulent dissipation rate versus distance from the wall for a fully developed turbulent channel flow is represented by the upper line in figure 1(right). At the wall it reaches a maximum value. Therefore, a significant gain in drag reduction can be expected when dissipation at the wall is minimized so that the distribution of changes towards the profiles sketched below the solid line.
3
The Limiting State of Turbulence at the Wall
In the near-wall region the instantaneous velocity fluctuations can be expressed by a Taylor series expansion about the wall [9]: ⎫ u1 = a1 x2 + a2 x22 + ... ⎬ u2 = + b2 x22 + ... as x2 → 0, (4) ⎭ u3 = c1 x2 + c2 x22 + ... where the coefficients ai , bi and ci are functions of time and space coordinates x1 and x3 . Since the turbulent dissipation is related to the gradients of the velocity fluctuations, its value at the wall can be evaluated from (4) as: wall = ν a21 + c21 .
Fig. 2. Anisotropy invariant map [8] that bounds all realistic turbulence. The gray errors indicate the structure of the turbulent velocity fluctuations that are found in the limiting states.
The question of how to minimize this value becomes transparent if the problem is analyzed in the functional space which emphasizes the level of anisotropy of the velocity fluctuations, as shown in fig. 2. In close proximity to the wall the normal
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velocity fluctuations (u2 ) decay faster than the streamwise (u1 ) and spanwise (u3 ) fluctuations. Therefore, turbulence in the near-wall region is located on the line that corresponds to two-component turbulence. Its exact location is defined by the value of the second invariant IIa , with the minimum value of IIa = 1/6 at the twocomponent isotropic limit and the maximum of IIa = 2/3 at the one-component limit. Employing existing DNS databases of wall bounded flows it can be shown that the location on the two-component line is linked to the turbulent dissipation at the wall in such a way that wall decreases as IIa increases [5]. We may therefore expect that wall reaches a minimum value as the one-component limit is approached. This limit of highest anisotropy is the point where two-component turbulence and axisymmetric turbulence with the dominating fluctuation on the symmetry axis are simultaneously realized. For velocity fluctuations locally invariant to rotation about the axis aligned in the mean flow direction (x1 ), the derivatives of the velocity fluctuations have the following relationship [4]: (5)
(6)
(7) If the velocity fluctuations very close to the wall given in equ. (4) are forced to fulfill these constraints of local axisymmetry, all coefficients in the series expansion vanish and the turbulent dissipation at the wall attains its minimum value (wall → 0) [5]. It is concluded that energy savings and significant drag reduction can be obtained if near-wall velocity fluctuations are forced towards the one-component state of turbulence. To demonstrate the effect of increasing anisotropy in the near wall region on drag reduction direct numerical simulations of a turbulent channel flow with forced boundary conditions were carried out. A set of nine simulations was performed using a fourth order finite–volume method on a staggered grid [11]. Using a grid with 128 × 128 × 64 points in streamwise, wall-normal and spanwise direction, in one simulation no forcing was applied to the flow at Re = 180. For the other simulations the flow rate was kept constant but an increasing number of points in the viscous sublayer (located at 0.002 ≤ x2 ≤ 0.030 which corresponds to x2 = 0.4, 1.1, 1.8, 2.6, 3.3, 4.0, 4.8, 5.6 in the unforced channel flow) the boundary conditions were modified in such a way to force near-wall turbulence to tend towards an axisymmetric state by imposing spanwise velocity fluctuations (u3 ) to follow the normal fluctuations(u2). The applied forcing yielded decreasing values of Re with a minimum value of Re = 143. The resulting mean flow velocity profile normalized with the respective
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uτ of each flow case is shown in fig. 3(left). It can be seen that U1+ increases significantly as the forcing is applied to more points inside the viscous sublayer which is equivalent to a decrease in uτ and thus a reduced wall shear stress τw . Figure 3(right) shows the corresponding Reynolds stresses normalized with the centerline velocity (which is equivalent for all cases) for better comparison. The forcing that was applied within the viscous sublayer effects a wide range of the flow field resulting in significant reduction of all Reynolds stress components with the peak value of u21 shifting further away from the wall as more points in the viscous sublayer are manipulated. The profiles of u22 and u23 reveal that the stresses are not exactly identical owing to interpolation approximation so that axisymmetry is approached but not exactly fulfilled.
Fig. 3 . Mean flow and Reynolds stresses in a virtual channel flow where an increasing number of points in the viscous sublayer is forced towards an axisymmetric state and therefore towards the one-component limit
Since
and Re are interrelated by: (8)
and since the fluid properties ρ, ν and the channel height 2δ are kept constant, the obtained drag reduction DR is given by: (9) resulting in DR = 8%, 14%, 19%, 24% 28% 31% 34% and 37% respectively. The turbulent dissipation rate versus wall distance is shown in fig. 4. It can clearly be seen that the area under the curve is reduced if an increasing number of points in the viscous sublayer are forced towards an axisymmetric state. The obtained results support the conclusion that drag reduction can be obtained by forcing
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turbulence in the near-wall region towards axisymmetry and thus reducing the turbulent dissipation rate. Indeed numerous existing drag reduction techniques have this mechanism in common: drag reduction is observed if and only if turbulence approaches the one-component state close to the wall [3].
unforced 1 point forced 2 points forced 3 points forced 4 points forced 5 points forced 6 points forced 7 points forced 8 points forced
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Fig. 4 . Turbulent dissipation rate as a function of wall distance . Quantities are normalized with the wall shear velocity of the unforced channel flow and the kinematic viscosity.
4
Drag Reduction with Surface Grooves
The preceding analysis has shown that large drag reduction can be obtained by forcing near-wall turbulence towards an axisymmetric state. To realize this kind of turbulence at the wall a surface structure with grooves aligned in flow direction, as shown in fig. 5, is proposed. Inside the groove the velocity fluctuations in the tangential (x3 ) and normal (x2 ) directions are suppressed due to the side walls and therefore it is expected that turbulence in the groove will tend towards an axisymmetric state. The flat part in between the grooves is needed to avoid singularities which might produce large spikes in the local distribution of the wall shear stress (as for example reported for triangular riblet surfaces by [2]).
x2 x1 x3
h
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Fig. 5 . Surface geometry with grooves aligned in flow direction
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The performance of the grooved surface with respect to drag reduction was tested in an experimental facility in which the pressure drop over a channel test section was measured. Grooves with dimensions of h = 150µm (see fig. 5) were milled into top and bottom walls in the first part of the channel. While the front part of the test section was equipped with a grooved surface, the rear part had smooth channel walls. Before entering the channel test section the flow was tripped some distance upstream which promotes the flow development. The comparison of the pressure drops obtained in each part of the test section can be used to calculate drag reduction DR according to: (10)
0.12
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p [V]
H=35mm H=25mm
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Fig. 6 . On the left a comparison of the pressure drop for a turbulent channel flow with grooved and smooth surfaces respectively is shown. The pressure gradient for the smooth channel shows a steeper slope which implies reduced energy consumption for the grooved surface. The ratio of the two gradients is used to obtain the corresponding drag reduction value DR. On the right all results of the preliminary turbulent drag reduction measurements are shown as a function of the dimensionless groove height h+ for different heights H of the channel test section.
In preliminary experiments performed at low Reynolds numbers drag reduction up to 25% was measured which is significantly higher than any values reported for flow control with surface modifications before. One exemplary measurement result and an overview of the obtained DR values as function of the groove size are shown in fig. 6. It is very interesting to note, that high values of DR are only found for groove sizes which corresponds to one viscous length scale or one-half of the Kolmogorov scale respectively. There is no significant change in pressure drop detected for other non-dimensional groove sizes.
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Conclusion
It is shown that the turbulent dissipation needs to be minimized in order to achieve energy savings in wall bounded flows. Analysis of turbulence in the near-wall region, where the turbulent dissipation reaches its maximum and where flow control techniques can actually be imposed, yields that the turbulent dissipation can be reduced by forcing near-wall turbulence towards the one-component state where axisymmetry and two-component turbulence are simultaneously realized. This finding is confirmed by direct numerical simulations with forced boundary conditions where forcing inside the viscous sublayer (x+ 2 ≤ 5) is sufficient to obtain drag reduction up to 32%. An experimental investigation with a grooved surface structure designed to minimize turbulent dissipation at the wall showed up to 25% drag reduction for grooves with the dimension of one viscous length scale.
Acknowledgements This work received financial support by the Deutsche Forschungsgemeinschaft (Grant Jo 240/5-1).
References [1] R.A. Antonia, M. Teitel, J. Kim, L.W.B. Browne: ”Low-Reynolds-number effects in a fully developed turbulent channel flow”. J. Fluid Mech. 236, 1992, pp. 579-605. [2] K.S. Choi: ”Near-wall structures of a turbulent boundary layer with riblets”. J. Fluid Mech. 208, 1989, pp. 417-458. [3] B. Frohnapfel, P. Lammers, J. Jovanovi´c, F. Durst: ”Interpretation of the mechanism associated with turbulent drag reduction in terms of anisotropy invariants”. J. Fluid Mech., accepted, 2006. [4] W.K. George, H.J. Hussein: ”Locally axisymmetric turbulence”. J. Fluid Mech. 233, 1992, pp. 1-23. [5] J. Jovanovi´c, R. Hillerbrand: ”On peculiar property of the velocity fluctuations in wallbounded flows”. Thermal Science 9, 2005, pp. 3-12. [6] J. Kim, P. Moin, R. Moser: ”Turbulence statistics in a fully developed channel flow at low Reynolds numbers”. J. Fluid Mech. 177, 1987, pp. 133-166. [7] A. Kuroda, N. Kasagi, M. Hirata: ”A direct numerical simulation of the fully developed turbulent channel flow”. Proc. Int. Symp. on Computational Fluid Dynamics, Nagoya, Japan, 1989, pp. 1174-1179. [8] J.L. Lumley: ” Computational modelling of turbulent flows”. Adv. Appl. Mech. 18, 1978, pp. 123-176. [9] A.S. Monin, A.M. Yaglom: ”Statistical Fluid Dynamics”. The MIT Press, Cambridge, Massachusetts, 1999. [10] R.D. Moser, J. Kim, N.N. Mansour: ”Direct numerical simulation of turbulent channel flow up to Reτ = 590”. Phys. Fluids 11, 1999, pp. 943-945. [11] R.W.C.P. Verstappen, A.E.P. Veldman: ”Symmetry-preserving discretization of turbulent flow”. J. Comp. Physics 187, 2003, pp. 343-368. [12] R. Volkert: ”Determination of statistical turbulence quantities for a turbulent channel flow based on direct numerical simulations”. (in German) Ph.D. thesis, University of Erlangen-Nuremberg, 2006.
Radical Farming in Scramjets J. Odam1 and A. Paull2 1
2
Institute of Aerodynamics and Flow Technology, DLR Bunsenstrasse 10, 37073 G¨ottingen, Germany
[email protected] DSTO Hypersonics Branch, mail to: HyShot University of Queensland, Pinjarra Hills 4072, Australia
Summary This paper outlines the theory of radical farming in scramjets and describes the experimental scramjet model that was designed to investigate it. Experiments were conducted at two conditions; a 3 MJ/kg condition corresponding to Mach 7.9 flight at an altitude of 24km and a 4 MJ/kg condition corresponding to Mach 9.1 flight at an altitude of 32km. The results are presented as pressure distributions on the flowpath wall and specific impulse estimates.
1 Introduction Significant scramjet testing has been conducted in the T4 shock tunnel at The University of Queensland since its commissioning in 1987. During this testing it was observed that the location of the combustion-induced pressure rise was coincident with the location of the second impingement on the combustor wall of the reflected shock that originates from the combustor entrance leading edge. To explain this phenomenon, the theory of radical farming was developed. The theoretical and experimental work presented here was conducted to examine this theory.
2 Radical Farming Concept The two-dimensional scramjet model used in the radical farming experiments is illustrated schematically in Fig. l(a). The leading edge shocks are deliberately ingested, producing a system of interacting shock waves and expansion fans reflecting back and forth along the constant-area combustor. This produces regions of elevated pressure and temperature in the flow, which are bounded by the intersecting shock waves at the upstream end and the intersecting expansion fans at the downstream end. The maximum residence time of any streamline in the supersonic flow through such a region is too short to allow for the complete combustion process; however, it is postulated that radicals are formed in this region as the combustion process proceeds and as the flow crosses the expansion fan at the trailing edge, the pressure and temperature differentials are sufficient to freeze the flow, such that the majority of radicals remain in the flow. For this reason, the regions are called radical farms. Such a technique can be exploited either to accelerate the ignition process or where the C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 276–283, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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flow between the radical farms is too cold to sustain combustion, such as in these experiments, to provide the ignition source. In the latter case, the residence time in the radical farm can be too short even to establish ignition but the frozen radicals remain, unreactive, in the flow until they encounter another region of elevated temperature and pressure, where the ignition process may proceed and, if there is sufficiently long accrued residence time at these elevated conditions, the process lead to sustained combustion.
(a)
(b)
Fig. 1. Experimental model: (a) schematic of internal flowpath (not to scale) and (b) mounted in T4
To maximise the effectiveness of radical farming, the fuel and air should be mixed and ready to start reacting in the first radical farm, which by design is at the combustor entrance. This is achieved by injecting fuel from the intake, as indicated in Fig. l(a). It is important to ensure that ignition does not occur on the intake as this would produce a negative t hrust component. It has been demonstrated that in jecting hydrogen through a row of high-pressure portholes spaced evenly across the intake width does not produce combustion on the intake.[l] Additional experiments demonstrated the same results when the intake surface was heated to a temperature of 550 K.[3]
3 Radical Freezing Analysis In order to examine the validity of the frozen flow assumption in the radical farming concept, the changing species concentrations were modelled for an expansion from typical radical farm conditions to typical mean flow conditions over a given expansion fan residence time. The model used a 9-species, 19-reaction scheme that is a modified version from Jachimowski[2]. The reaction scheme is implemented using a sixth-order Runge-Kutta numerical solver and variable time steps. The first order approximation, where the solution is driven by a temperature profile that does not change due to the release of energy from the combustion process, is provided. This estimate is less valid as the temperature decreases towards the ignition limit but nonetheless useful for demonstrating the principals of radical farming. The pressure and temperature in the first radical farm of the 4 MJ/kg condition are estimated as 182kPa and 1185 K respectively. An initially mixed hydrogen-air combination with an equivalence ratio of 0.4 were held at the radical farm conditions for 20µs before being allowed to expand to the post-expansion temperature of
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(a)
(b)
Fig. 2 . Species concentrations for the 4MJ/kg test case: (a) volumetric oncentrations of all species; (b) mass concentrations of hydrogen and oxygen atoms
623 K over a period of 7 µs , representative of the residence time within the trailing edge expansion fan of the centreline combustor flow. The flow is held at the postexpansion conditions for 20 µ s to confirm that the concentrations are no longer changing significantly and then restored to the initial radical farm conditions to confirm the recommencement of the combustion process. Figure 2(a) shows the resulting volumetric species concentrations, clearly illustrating that combustion occurs in the second radical farm interval but not the first. Focussing on the concentrations of the hydrogen and oxygen atoms up until the second radical farm, and presenting them with respect to mass rather than volume, shows that radicals continue to be produced during the expansion process until a threshold temperature is reached. Some recombinationthen occurs but a large portion (85% in this case) of the radicals remain frozen in the flow at the post-expansion conditions, as illustrated in Fig. 2(b) . A repeat of this analysis for the 3 MJ/kg condition showed no sign of combustion. This is in contradiction with the experiments and is a result of the limited validity of the combustion modelling as the conditions approach the ignition limit, as discussed earlier. The radical freezing analysis showed that the assumption of frozen radicals is reasonable and as a result, the accumulated residence time in the radical farms is the primary parameter that determines the occurrence of combustion. In contrast, the mean values of pressure and temperature in the combustor may affect the rate of the combustion process where they are high enough to support combustion but do not dictate whether combustion will occur. This is illustrated experimentally below, where the mean temperature is too low to support combustion.
4 Experimental Work The scramjet model of Fig. l(a) was instrumented with pressure transducers along the centreline of the lower flowpath wall. Some experiments were conducted with glass windows set into the side-plates to allow flow visualisation and the resulting images
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confirmed the expected shock structure withinthe visibleregion of the intake and combustor. A photograph of the windowless model installed in the test section is shown in Fig. 1(a). Table 1. Summary of freestream properties and their relative uncertainties Test Hs Ps Pe Te Condition (MJ/kg) (MPa) (kPa) (K) 1 2.88 39.1 8.02 285 2 4.01 40.4 8.73 411 Uncertainty ±6.5% ±5% ±5% ±9.8%
ρe (kg/m3 ) 0.098 0.074 ±7.2%
Ue Me (m/s) 2282 6.75 2610 6.42 ±3.5% ±1.5%
The freestream properties for each of the two test conditions are summarised in Table 1. At these conditions, the presence of radicals in the freestream flow is insignificant. In addition, the test time is typically greater than 2.5ms, which provides greater than 7 model lengths of test flow. The intake contraction ratio was varied between tests by raising or lowering the entire lower surface of the flowpath. In this way, the combustor height varied from 20 to 32mm and the corresponding contractionratios ranged from 4.1 down to 2.9.
5 Results 5.1 Pressure Distributions Pressure distributions through the entire model length along the centreline of the bottom flowpath surface, are shown in Fig. 3 for the 20 mm and 32 mm combustors (intake compression ratios of 4.1 and 2.9 respectively). The static pressures are normalised with respect to the freestream stagnation pressure (nozzle supply pressure). It is emphasised that the lines between data points only serve the purpose of assisting the viewer to see the pattern of shocks and expansions and can be misleading at specific points, such as the pressure on the end of the intake of Fig. 3 (b). Considering the higher compression ratio case of Fig. 3(a), the intake pressures agree for both the fueloff and fuel-on distributions, diverging just after entering the combustor. The fuel-off distribution shows a short plateau of raised pressure at the combustor entrance, caused by the impingement of the leading edge shock, followed by a further pressure increase due to the impingement of the shock formed at the compression corner on the intake. The pressure then rapidly drops off as the combustor entrance expansion fan reaches the combustor surface. The shocks and expansions proceed to reflect twice more from the combustor surface causing the second and third prominent pressure peaks. By contrast, the fuel-on distribution shows a small oscillation in the pressure at the beginning of the combustor, a first peak that is delayed until almost the location of the second peak in the fuel-off distribution, and then a second, sustained pressure rise that leads to much higher pressures on the thrust surface. A shock appears to impinge on the thrust surface just prior to the final pressure tapping.
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(a)
(b)
Fig. 3. Comparison of normalised pressure distributions at the 4 MJ/kg condition, with and without fuel, for (a) the 20mm combustor (ER=0.4) and (b) the 32mm combustor (ER=0.8)
In the case of the lower compression ratio configuration, the increased flowpath height appears to have led to the merging of the leading edge and compression corner shocks prior to their reaching the combustor surface. The merged shock then impinges approximately 30mm downstream of the leading edge shock impingement location from Fig. 3(a). As a result, the expanded flow region just after the combustor entrance expansion fan is larger for the fuel-off case and extends across the first two combustor pressure tappings. The apparent discrepancy at the end of the intake has already been explained as a product of the straight-line representation between datapoints, in which the pressure drop at the end of the intake is not always displayed due to the resolution of pressure measurements. For this configuration, the shocks and expansions only reflect from the combustor surface twice, resulting in two peaks rather than three. The fuel-on pressure distribution shows little influence from the combustor entrance expansion fan at the beginning of the combustor but otherwise agrees with the general location and magnitude of the first fuel-off peak before increasing to a relatively sustained pressure level. The pressure distributions of Fig. 3 clearly show the shock structure used for radical farming and also indicate the likelihood that combustion is occurring for both fuel-on cases. However, confirmation that the observed pressure rises are due to combustion rather than merely the added mass and momentum of the injected fuel is provided by comparison with data from experiments injecting fuel into a nitrogen, rather than air, freestream. Figure 4 shows such a comparison for each of the test conditions. The static pressures are again normalised with respect to the freestream stagnation pressure although it should be noted that the freestream static pressure to stagnation pressure ratio is approximately 19% lower for nitrogen than for air. In Fig. 4(a), the air and nitrogen pressure distributions for fuel injection are similar until approximately 70mm downstream of the combustor entrance, where the nitrogen distribution drops off sharply while the air distribution remains briefly steady and then starts to rapidly rise. Comparing the nitrogen distribution with the fuel-off air distribution demonstrates that the injection of fuel changes the shock structure
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. out . This is due to the regions of within the , engine appearing to smooth the structure varying specific heat ratio caused by the fuel, which intum lead to regions of varying Mach number. As the shocks pass throughtheseregions,the Mach angles
(a)
(b)
Fig. 4. Comparison of normalised pressure distributions without fuel injection and with fuel injection into both air and nitrogen freestreams for the 20mm combustor at (a) the 4 MJ/kg condition and (b) the 3 MJ/kg condition.
change and the shocks are spread out across the flowfield. Nevertheless, despite the shock structure alteration caused by fuel injection, the pressure rise observed in the fuel-on air distribution is clearly the result of combustion. The fact that combustion is observed within the first 100 mm of the combustor supports the concept advantage of injecting fuel on the intake, allowing pre-mixing of the fuel and air and reducing the required combustor length. Furthermore, the combustion-induced pressure rise occurs in the vicinity of the second shock impingement on the combustor surface. This observation is consistent withnumerous previous experimental observations in T4 and supports the theory that radicals are produced in the region behind the first shock impingement and frozen into the flow to react in the region behind the second shock impingement. Similar observations hold for the lower enthalpy condition shown in Fig. 4(b). The main differences between the two conditions are: the slight delay of approxi mately 20-30 mm in the time until the combustion-induced rise in pressure; and the lower average pressure in the nozzle. 5.2 Specific Impulse Estimates Integration of the measured pressure distributions provided two-dimensional estimates for the intake drag and nozzle thrust. Combining these estimates with skin friction calculations for all internal surfaces, including the sidewalls, led to estimates of the internal flowpath net thrust, from which estimates of the internal flowpath net specific impulse can be made. Experiments were conducted for both test conditions at a range of equivalence ratios from zero to just beyond the point that choking was observed. At the 3 MJ/kg condition, the equivalence ratio at which choking first occurred increased from
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0.33 to 0.68 as the intake contraction ratio was reduced from 4.1 to 2.9. Similarly, at the 4 MJ/kg condition, the choking equivalence ratio increased from 0.41 to 0.88 with decreasing contraction ratio. Figure 5 plots the maximum net specific impulse estimate obtained prior to choking for each intake contraction ratio, at both test conditions. The value is determined using the average thrust obtained over 2-4 experimental runs with equivalence ratios in the proximity of the choking limit. The effect of intake contraction ratio on the dimensions of the radical farms was investigated analytically. Although decreasing contraction ratio leads to a general trend of thinning and lengthening of the radical farms, this effect does not constitute a major change in the radical farm dimensions and the conditions in the radical farm are not dictated by the contraction ratio. This suggests that the combustion process in such a radical farming situation is largely independent of the contraction ratio. In contrast, the combustion process in conventional scramjets is strongly influenced by the fact that the intake contraction ratio determines the precombustion temperature. As expected, the net specific impulse values in Fig. 5 do not display a discernable trend with varying intake contraction ratio. Two significant deviations from the generally constant net specific impulse that is observed for each test condition are noted. In the case of the 3 MJ/kg condition, very low thrust levels were observed at all tests with an intake contraction ratio of 3.2, leading to the low net specific impulse estimate at this point. For the 4 MJ/kg case, the two thrust measurements obtained closest to the choking limit differed significantly, leading to net specific impulse estimates of 263 s and -226 s and therefore averaging out to the low value plotted in Fig. 5.These deviations can be explained as the result of the inherent instability of the choking process and therefore in the conditions near the choking limit. An additional effect may be unfavourable shock structures at particular contraction ratio configurations because the length of the combustor was not tuned to each combustor height. As a result, the shock structure may adversely affect the pressure distribution in the nozzle, such as mismatch in the shock and expansion interactions leading to a large lowpressure region of expanded flow on the thrust surface.
Fig. 5. Net specific impulse estimates of the scramjet internal flowpath as the intake contra ction ratio is varied between 2.9 and 4.1
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6 Conclusion The theory of radical farming was developed to explain the commonly observed phenomenon in T4 of the combustion-induced pressure rise occurring in the vicinity of the second impingement of the leading edge shock on the combustor surface. Preliminary combustion modelling supported the main hypothesis of the theory that radicals are produced in high pressure and temperature regions in the flow and remain frozen in the lower pressure, cooler bulk flow as they pass through expan sions. Experiments further supported the theory by exhibiting combustion at bulk temperatures that are too low to support combustion without a strong shock structure in the scramjet combustor. These experiments also supported the concept of fuel injection on the intake to provide mixed fuel and air to the combustor inlet, reducing the required combustor length. Analysis of net specific impulse estimates for the internal flowpath indicates that in a scramjet engine employing radical farming, the combustion process is largely independent of the intake contraction ratio. This is in contrast to conventional scramjets, where the intake contraction ratio determines the precombustion temperature and therefore has a strong influence of the combustion process. An important point to consider in the employment of radical farming, however, is that the performance of the engine is strongly affected by the combustor height-to-length ratio due to its influence on the shock structure. This must be considered in the engine design.
References [1] [2] [3]
A. D. Gardner. Upstream porthole injection in a 2-D scramjet model. Master’s thesis, The University of Queensland, 2000. C. J. Jachimowski. An analysis of combustion studies in shock expansion tunnels and reflected shock tunnels. Technical Report TP-3224, NASA, 1992. A. Kovachevich, A. Paull, and T. McIntyre. Investigation of an intake injected hot-wall scramjet. In 42nd AIAA Aerospace Sciences Meeting and Exhibit, Nevada, U.S.A., Jan. 2004. AIAA-2004-1037.
Aerothermodynamic Investigation of the Pre-X Configuration in HEG Jan Martinez Schramml and Bodo Reimann2 1
German Aerospace Center (DLR), Institute of Aerodynamic and Flow Technology, Bunsenstraße 10, 37073 Göttingen, Germany
[email protected] 2 Lilienthalplatz 7, 38108 Braunschweig, Germany
[email protected]
Summary During the past years, the Centre National d’Etudes Spatiales (CNES) has initiated several studies concerning the first generation experimental vehicle Pre-X, which is dedicated to mastering glided atmospheric re-entry. CNES has committed at the beginning of 2005 to an important phase with the objective of reaching a preliminary design review by mid 2006. In the framework of phases A2 and B of the Pre-X program the German Aerospace Center (DLR) carried out wind tunnel experiments on a scaled Pre-X vehicle in the High Enthalpy Shock Tunnel Gottingen (HEG) as well as numerical investigations using the DLR TAU code. For a 20° body flap deflection, transitional reattachment was experimentally observed. Additionally, the experiments and the CFD indicated a heating region at the end of the flaps that originated from the impingement of the interference between the bow shock and the shock created by the re-attachment.
1 Introduction The experimental objectives of the French Pre-X program focus on the thermal protection system (TPS) and thermo structures, aerothermodynamic (ATD) phenomena, controlusing aerodynamic flaps, in-flight measurement systems and the system concept end design [3]. Following phase A1 which was performed in 2004, where the objective was to consolidate the aerodynamic shape design, CNES has committedto full development of the Pre-X vehicle. The next phases are led by ASTRIUM- ST as the prime contractor in parallel withthe consolidation studies initiated by ESA in the frame of FLPP (Future Launcher Preparatory Programme). DLR's contribution in the framework of phase A2 of the Pre-X program is the identification of the aerodynamic and aerothermal phenomena on the vehicle. To achieve this, computa- tional fluid dynamics (CFD) simulations using the DLR flow solver TAU, as well as high enthalpy wind tunnel testing in the HEG have been accomplished. The HEG measurement campaign allows cross-checking of the CFD and previous wind tun- nel test results on similar configurations [4]. Due to the large flow speed considered, chemical reactions change the topology of the flowfield: The bow shock is close to the body, therefore shock-shock interactions occur in the vicinity of the deflected flaps. Transition from laminar to turbulent flow is observed in the flap region. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 284–290, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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2 Measurement Technique The HEG is one of the major European hypersonic test facilities. This free piston driven shock tunnel can operate over a wide range of test conditions from low altitude high density Mach 6 flows, through the simulation of Mach 8 flows at approximately 30 km altitude, up to high enthalpy re-entry type conditions. For the experiments discussed in this paper, the tunnel was equipped with a conical hypersonicnozzle with an exit diameter of 0.88 m. Additional information about HEG and previous projects is given in [6] and [5]. The wind tunnel model of the Pre-X configuration has been designed, manufactured and instrumented by DLR. This wind tunnel model has a length of 0.32 m and is instrumented with 30 pressure transducers and 67 thermocouples. Fig. 1 shows the model mounted in the HEG test section. SYMMA
SYMMB
G I H
A C B
Fig. 1. Wind tunnel model in HEG (left) and transducer positions (right). Pressure transducers are colored in blue, thermocouples in red and transducers not considered in black.
On the right hand side of Fig. 1 a schematic view of the positions of transducers on the model surface is shown. The flap system installed on the model allows adjustment of the flap deflection angle. The thermocouples are glued into the flap surface since a threading system could not be installed due to size constraints. Nineteen thermocouples are installed on one flap and ten pressure transducers on the other as shown in Fig. 1. The cuts in the stream-wise direction are labeled FLAPA to FLAPC starting from the outer to the inner sides and FLAPG to FLAP I starting from the inner to the outer sides for each flap. The pressure transducers are of piezoelectric type (KULITE XCEL-093 1.7 bar A). The pressure ports in the model surface have been manufactured to a 0.5 mm diameter. The cavity connecting the ports to the pressure transducers sensitive areahas a volume of 3.004 mm 3 . The thermocouples are fixed by means of a threading system inside the model. The thermocouples are ISA type E (Medtherm Corporation, TCS-061-E). In order to evaluate the heat flux from the measured temperature traces from the thermocouples, the evaluation described in [7] has been used with the discretisation given in [I]. Holographic interferometry has been used for the flow visualisation. With this technique it is possible to measure a line of sight integrated density distribution of the flow field around the model. More
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detailed information about the holographic interferometer at HEG is given in [8] and [9].
3 Numerical Scheme The DLR TAU code is a finite volume EulerlNavier-Stokes solver which can handle structured, unstructured and hybrid meshes and has already been applied to various configurations. The Reynolds averaged Navier-Stokes (RANS) equations are discretized by a finite volume technique using tetrahedrons and prisms. Prismatic
Fig. 2. Numerical results for the flow around the 20 (left) and 10 (right) body flap. Shown are countours of constant Mach number. The black line corresponds to M=1.
Fig. 3. Symmetry plane of the numerical grid. Initial grid (left) and the grid after the fourth adaptation (right). The number of points increases from 1.34M to 2.72M.
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elements were used for the boundary layer regions while tetrahedral elements were used primarily in inviscid flow regions. The AUSMDV second order upwind scheme with MUSCL reconstruction was used for the inviscid fluxes. For time discretization, including local time stepping, a three stage Runge-Kutta as well as an implicit, approximately factorized LU-SGS scheme was implemented. The transition from laminar to turbulent flow is modeled by prescribing of the transition location. For the Pre-X computations with transitional flow, the transition line was set at the windward flap hinge line. This meant that only the flow over the windward sides of the flaps was modeled as turbulent while all other parts, particularly on the leeward side and the vehicle base, were modeled as laminar. In the presented cases, the twoequation k−ω model [l l] was used. To model the thermochemical behavior of the flow, a five species chemical non-equilibrium model was applied [lo]. To generate the grids from the CAD data, the commercial CENTAUR mesh generator was used. The Navier-Stokes mesh had 28 prismatic layers at the wall. Three adaptation steps were performed for the present study. For the first and the second adaptation steps, the number of points was increased about 40%. In the third adaptation step no new points were added, but the available points were redistributed. Fig. 3 shows the symmetry plane of the initial grid and the grid after the adaptation steps.
4 Results The comparison between the experimental and the numerical results of the Pre-X model for HEG condition I is outlined in this section. The nominal free stream quantities for this condition are given in Table 1. The Reynolds number is nondimensionalised with the model length. The angle of attack of the model was 45°. Two cases with flap deflection angles relative to the body of 10° and 20° are discussed. To compare the surface distribution for heat flux and pressure, averaged Table 1. Nominal free stream values for HEG Condition I. Species are in mass fractions. M 8.9
Re 64092
U [g/m3] 1.55
p [Pa] 476
T [K] 901
u [m/s] 5950
1.4
NO N 0.0102 6.00E-07
O 0.2283
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Fig. 4. Comparison of experimental averaged data with the numerical solution for the symmetry plane. Normalized surface heat flux (left) and normalized surface pressure (right).
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data resulting from several runs at the same angle of attack are used for the forebody. The comparison to the surface CFD values is presented in Fig. 4. The CFD result for the surface pressure is well within the standard deviation of the experimental data. The heat flux distribution in the symmetry plane shows good agreement as well, except in the region (260mm>x>220mm). This is the region where the wall curvature of the nose blends into the flat forebody region. There is currently no explanation for the differences between the experimental and the numerical results. A mesh sensitivity study shows that the peak in the heat flux distribution on the nose (x=310mm in Fig. 4) is linked to the grid resolution of the bow shock. The distribution becomes
Fig. 5. Comparison between experiment and CFD for 10◦ flap deflection
Fig. 6. Comparison between experiment and CFD for 20◦ flap deflection
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smoother (the peak becomes smaller) the finer the grid is adapted. The comparison between the numerical and the experimental data on the surface of the flaps is shown in Fig. 5 and in Fig. 6 for flap deflection angles of 10° and 20°, respectively. The normalized heat flux is shown on the upper left of the figures and the normalized pressure is shown on the lower left side. A two dimensional comparison for the normalized heat flux is given on the right hand side of each figure. These two dimensional plots show an interpolation through the thermocouples signals on the model, and care has to be taken when interpreting the interpolation. These distributions are given as an visual orientation aid. The experimental data is shown on the top half of the model, the numerical results on the bottom half. The black rectangles indicate the transducer positions. The boundary on the lower part of the model is the outer boundary of the instrumentation positions. Additionally, stream traces are given to show the location of the separated region within the numerical solution. For the flap deflection case of 10° the comparison for surface pressure and surface heat flux is reasonable. The instrumentation on the flap surface is positioned slightly behind the reattachment point of the separation, thus preventing any comparison to the numerical modeling of the separation itself or any scaling law. However, the heat flux level after reattachment is in good agreement with the numerical data. The comparison for the case with 20° flap deflection in Fig. 6 exhibits apeak heating af- ter reattachment for all cuts. This level could not be reproduced with the assumption of laminar flow for the numerical modeling. The numerical data shown in Fig. 6 is obtained assuming a transitional flow as described in Sec. 3. This transitional case approximately matches the heating level of the experiment, but does not model the subsequent drop in heat flux after the reattachment. The experimental data clearly exhibits a transitional behaviour of the separated flow region. This phenomenon has already been observed during the experimental campaign investigating the X-38 configuration [4]. Even thoughthe free stream Reynolds number (64092) is low, and the forebody measurements for the Pre-X and the X-38 configurations show laminar behaviour, a transitional reattachment on the flap surface was observed. For the Pre-X configuration, this phenomenon was observed for flap angles larger than 10° with an angle of attack of 45°. For the X-38 configuration at 40° angle of attack, transitional reattachment was observed for flap angles larger than 20°. Furthermore the experiments as well as thenumerical data show the interaction between the bow shock of the forebody withthe shock system generated by the separated region. This interaction heats the end region of the flap surface. In Fig. 2 two numerical results for the flow around the flaps with 10° and 20° flap deflection are shown. Here the interaction between the main bow shock and the weak shocks from separation and reattachment can be seen. This interaction impinges on the body flap and heats up the surface for the 20° flap case. The experimental data indicates that this interaction influence is seen just barely measurable at the flap end, while the numerical data shows this interactionheating region further upstream.
5 Conclusion High enthalpy testing of the Pre-X configuration in the HEG has been carried out in the framework of phase A2 of the Pre-X program. The experimental results for a flap
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deflection of 10° are in good agreement with the numerical computation assuming laminar flow. For 20° the heat transfer behind the reattachment region shows better agreement for a transitional computation, but does not model the decreases after reattachment. Similar results were obtained during the X-38 experimental campaign in HEG. For the Pre-X configuration, both the experiment and the CFD indicate an additional heating region at the end of the flaps originating from the impingement of the interference between the bow shock and the shock system created by the separation.
References [1] Cook WJ, Felderman EJ: "Reduction of Data from Thin-Film Heat Transfer Gauges: A Concise Numerical Technique", AIM Journal, 4(3):561-562, 1966. [2] Gerhold T, Friedrich O, Evans J, Galle M: "Calculation of Complex Three-Dimensional Configurations Employing the DLR-TAU-Code", 35th AIM Aerospace Science Meeting and Exhibit, 97-0167, Reno, 1997. [3] Guedron S, Bonnal C, Baiocco P, Moulin J: "Pre-X Project: Short Term Perspectives", 56th International Astronautical Congress of the International Astronautical Federation, The International Academy of Astronautics and the International Institute of Space Law, IAC-05-D2.6.07, Fukuoka, Japan, Oct. 17-21,2005 [4] Hannemann K, Mdnez Schramm, J Bruck S, Longo JMA: "High Enthalpy Testing and CFD Rebuilding of X-38 in HEG, Proc. 23rd International Symposium on Shock Waves, Paper 1552, Fort Worth, USA, 2001. [5] Hannemann K, Schnieder M, Reimann B, Martinez Schramm J: "me Influence and the Delay of Driver Gas Contamination in HEG, 21st AIM Aerodynamic Measurement Technology andGround Testing Conference, 2000-2593, Denver, 2000. [6] Hannemann K: "High Enthalpy mows in the HEG Shock Tunnel: Experiment and Numerical Rebuilding", 41st AIM Aerospace Sciences Meeting and Exhibit, 2003-0978, Reno, 2003. [7] Jones JJ: "Shock-tube heat-transfer measurements on inner surface of a cylinder (simulating a flat plate) for stagnation-temperatures range 4100 to 8300 R, NASA Technical Note D-54, September 1959. [8] Karl S, Mdnez Schramm J, Hannemann K: High Enthalpy Cylinder Flow in HEG: A Basis for CFD Validation, AIM Paper 2003-4252,2003. [9] Mdnez Schramm J, Boutry A, Vital Durand M, Hannemann K: "Time Resolved Holographic Interferometry for Short Duration Hypersonic High Enthalpy Facilities", 24th International Symposium on Shock Waves, Paper 3451, Beijing, 2004. [10] Gupta, RN and Yos, JM and Thompson, RA and Lee, KP: "A review of reaction rates and thermodynamic transport properties for an ll-species air model for chemical and thermal nonequilibrium calculations to 30 000 K, NASA, RP-1232,1990. [11] Wilcox DC: "Turbulence Modeling for CFD, DCW Industries, Inc., 1993.
SHEFEX -- A First Aerodynamic Post- f light Analysis Tarik Barth and Thino Eggers DLR, Institute of Design Aerodynamics, Lilienthalplatz 7, 38108 Braunschweig, Germany
Summary The given paper summarises the status of the post-flight analysis of the SHarp Edge Flight EXperiment (SHEFEX). It focuses on the comparison of numerical and experimental surface pressures as well as the assessment of the free stream vector applying the flight mechanic description of the Digital Miniature Attitude Reference System (DMARS) platform. The results point out that the extraction of angles of attack and sideslip is much more demanding than expected.
1 Introduction Hypersonic systems are complex, difficult to design and expensive to build due to a lack of physical understanding on the involved flow regimes and a lack of data for design. The strategic future tool which will enable a dramatic reduction in the design and development time required for new vehicles is the computational or “virtual” vehicle design and qualification. But this approach is based on mathematical models which require verification and validation to increase their credibility. The improvement of the physical modelling requires good data, acquired in ground facilities and in flight [3].
(a) Forebody
(b) Sensors
Fig . 1 . The SHEFEX Experiment
The objective of the SHEFEX experiment was the investigation of a facetted Thermal Protection System (TPS) concept and the assessment of the potential of sharp edged configurations applying the three point strategy: numerical analysis - ground based C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 291–299, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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facilities - flight experiment. The motivation is neither to perform a re-entry experiment nor to fly at the thermal boundary of modern high temperature materials but to prove in flight that the temperature peaks at the edges of the ceramic-composite panels are lower than those predicted based on a radiation equilibrium hypothesis. The SHEFEX forebody should have as many as possible facetted panels and it should represent as many as possible configuration details of space vehicles, like concave and convex chamfers and a sharp unswept leading edge, see Fig. 1(a). Now, after the flight an enormous amount of scientific data is available and a direct comparison of the flight data with numerical post flight studies and on-ground data is possible.
2 Description of the Experiment The SHEFEX launcher is a two-stage solid propellant sounding rocket system. The launch vehicle consisted of a Brazilian S30 motor as first stage and an Improved Orion motor as second stage. Between the facetted SHEFEX experiment and the second stage were two cylindrical modules which housed the recovery system, the main electronics, the data acquisition devices, the power supply, and the cold gas system (Fig. 2). The vehicle reached an apogee of 211 km. The total flight time was 550 seconds, comprising 45 seconds of experimental time for the atmospheric reentry between 90 km and 14 km. The first atmospheric effects on the acceleration
(a) Launcher
(b) Re-entry configuration
Fig . 2 . SHEFEX launch and re-entry configuration
(a) Re-entry
(b) Pitch and yaw with average
Fig . 3. Re-entry description based on DMARS data
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sensors could be observed at 80 km. Here the pitch and yaw angles (ψ and Θ) started to oscillate and unfortunately the roll rate started to increase, see Fig. 3(a). The vehicle finally achieved a stable flight attitude with a decreasing ellipse-shaped precession around the flight vector, see Fig. 3(b). The flight data enable a detailed flight mechanic description of the complete flight. Data of 59 sensors distributed on the surface of the forebody like pressure transducers, thermocouples, and heat flux sensors are available for an analysis [2].
3
Assessment of the Flow Condition
One requirement for the numerical post-flight analysis is the availability of the free stream condition, namely, the Mach number M∞ , angle of attack α and sideslip angle β. In order to obtain the Mach number or the Reynolds number the 1976 standard atmosphere [1] is applied. The angles of attack and sideslip have to be determined based on the flight vector provided by the DMARS platform. The DMARS flight vector is given in the platform coordinate system. Therefore, e.g. the pitch angle is zero during vertical ascent, 90◦ in horizontal position and so on. The sideslip angle is the lateral deviation from the flight path (Fig. 4). As shown in Fig. 3(a) the re-entry configuration performed a very complex damped coning motion. The pitch and yaw angles oscillate around mean values which are indicated by the dash-dotted lines in the Fig. 3(b). Estimates for the mean values are 164.5◦ for the pitching motion and 3◦ for the yawing motion. Now, α and β are the angles between the longitudinal axis of the vehicle and the flight path, which is not directly known. But, considering Fig. 4, they correspond to the deviation from the mean values for pitch and yaw angle. The roll angle φ may be directly taken from the DMARS data. The flight velocities during the atmospheric descent varied in the region of 1700m/s. The Mach number, see Fig. 5(a), is relatively constant at a value of approx. 5.6 from
Fig . 4 . Definition of pitch, yaw, attack, and sideslip angles
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(a) Mach number, altitude
(b) Reynolds number, velocity
Fig . 5 . Flow conditions during atmospheric re-entry
(a) Extraction of flight data
(b) Pressure sensors
Fig . 6 . Flight analysis
100 km down to 50 km. Then the Mach number increases up to its maximum value 6.2 at 26km. The Reynolds number, see Fig. 5(b), increases up to 10·106 at payload separation. Therefore, a careful comparison of the pressure distributions, temperatures and the heat flux rates from laminar and turbulent Navier-Stokes calculations may allow to detect the altitude where boundary layer transition appears. The actual flow condition and the corresponding experimental values may be obtained applying the procedure given in Fig. 6(a). The post-flight analysis at first required the exact adjustment of the time scales of Radar-, DMARS-, and experimenttime. The Radar data are used to extract the altitude and corresponding time. Applying this time values of pitch-, yaw- and roll angles as well as velocity are obtained from the DMARS data while the measured surface values are contained in the experimental database.
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4 Selected Results of the Post- f light Analysis To compare the numerical with the experimental data Navier-Stokes calculations are performed using the TAU -C ODE. Dependent on the altitude the boundary layer state is assumed to be laminar (> 40 km) or turbulent (< 30 km). Between 30 and 40 km both types are investigated. The wall is always defined to behave radiation adiabatic ( = 0.83). The comparisons of experimental and numerical pressure distributions are done along a line on the lower surface, see Fig. 6(b). Table 1: Transformed angles of attack α and sideslip β
transformed roll angle φ angle of attack α and sideslip β 0◦ 90◦ 180◦ 270◦ 360◦ α α β −α −β α β β α −β −α β
For the numerical calculation the velocity vector is only defined by the angles of attack and sideslip. The roll angle can not be taken into account. Therefore, an orientation (ψ, Θ and φ) and a grid generation of the vehicle have to be carried out for any time step in order to respect the correct flight position. Alternatively, the influence of roll angle may be included using a transformation of the angles of attack and sideslip. On account of the fact that the orientations and grid generations are much time consuming and error-prone the transformation is favoured. Consequently, the transformed values for the angles of attack α and sideslip β which already include the effect of the roll angle are defined by equations (1) and (2). α = cos(φ) · α + sin(φ) · β
(1)
β = cos(φ) · β − sin(φ) · α
(2)
Tab. 1 illustrates the influence of the roll angle on the angles of attack and sideslip. In case that the roll angle is zero, the transformed angle of attack α is identical to the angle of attack α. If now an additional roll angle of 90◦ is introduced at constant angle of attack α is identical to the sideslip angle β. The increase of the roll angle to 180◦ ends to α = −α. A comparison of a result obtained on a rotated vehicle with the one obtained by application of the transformed angles and one additional application using transformed angles prove the procedure (Fig. 7). With respect to save calculation time it is proposed to compute only half of the SHEFEX forebody. That is possible when the influence of the sideslip is marginal and a symmetric flow field may be assumed. For this purpose the influence of the sideslip angle is here investigated. The results point out, that a sideslip angle of less
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Fig . 7. Comparison: rotated grid / transformed angles of attack and sideslip
Fig . 8. Numerical and experimental pressure distributions, influence of the sideslip angle
than 3◦ has only minor influence and the half configuration may be used. In case of larger values the complete forebody has to be considered because the characteristic of the pressure distribution is changed (see Fig. 8 as example). For the numerical calculation of the re-entry the following altitudes are selected: 19.0 km, 27.4 km, 33.8 km, and 55.7 km. The comparison of the pressure distributions for these altitudes are presented in Fig. 9. The agreement is for 19.0 km and 27.4 km reasonably good. Unfortunately, this promising agreement does not hold for the two considered higher altitudes. If the altitude reaches more than 55 km, the influence of the measurement uncertainty (≈ ±50 P a) has to be taken into account. For the altitude of 33.8 km a comparison of laminar and turbulent pressures with the experimental ones suggests that at this altitude the boundary layer may be turbulent. However even for the turbulent flow case large discrepancies in the forward and backward region of the body are observed. Especially in the region of the first
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Fig . 9 . Numerical and experimental pressure distributions
pressure sensor the accuracy has to be much better, because here the flow is almost two dimensional and the pressure value has to be comparable with the oblique shock relations. Here the experimental pressures taken 50 ms later are much closer to the calculation but the agreement is still not promising. The clear influence of flight time on the measured pressure distribution points out that the very fast changes of the flight vector are one obvious problem. Therefore, possible explanations for the large discrepancies between experiment and simulation are the procedure to define the angle of attack and the sensitive adjustment of the experimental time scales (compare [4]). In order to investigate in depth the discrepancies observed at high altitude, two flow conditions (39 km and 55 km) with large offset from the experimental pressures are considered using the oblique shock relations given by equations (3) and (4). M1 2 · sin2 σ − 1 (3) tan ϑ = 2 · cot σ · M1 2 · (κ + cos 2σ) + 2 2 · κ · M1n 2 − (κ − 1) p2 = with M1n 2 = M1 · sin σ p1 κ+1
(4)
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(a) Analysed altitudes
(b) Pressure gradients (39 km)
Fig . 10. Recalculation of the angle of attack based on the oblique shock relations
The Fig. 10(a) shows that for both altitudes the offset is large and the downstream behaviour of the pressure distribution is not well captured for the assumed angles of attack (see section 3). The aim is to establish an analytic angle of attack αshock using the oblique shock relations. In Fig. 10(b) is presented that the flow conditions are not fully two dimensional. Therefore, in a first step the numerical pressure directly at the leading edge is extracted which is obtained using the angle of attack defined in section 3). In oder to include three dimensional effects, a parallel translation of the frontward pressure distribution in a way that which agrees well with the first pressure sensor is being performed in the second step. Finally, a new hypothetic pressure p2 may be extracted at the leading edge which agree with the pressure after the shock. Applying the known geometric deflection angle ϑ and the oblique shock relation within an iteration a new angle of attack may be found. The new angles (αshock = 3.27◦ for 39.1 km and αshock = −5.30◦ for 55.7 km) allow a much better agreement with the measured values. Additional, this procedure shows for 55.7 km that the discrepancies are not caused by the measurement uncertainty, but rather may be traced back to uncertainty on the velocity vector direction.
5 Conclusions An strategy to reduce the flight data of the SHEFEX experiment in order to establish an aerodynamic data basic has been presented and discussed. The results of the
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paper point out that the extraction of angles of attack and sideslip is much more demanding than expected and that relatively straightforward experiments like SHEFEX with an enormous amount of available scientific data and the quite demanding problems of the flight analysis are an essential source to acquire knowledge in the physics of hypersonic flight and the development of more complex vehicles. The presented scheme of work enables to develop the SHEFEX aerodynamic data basic. For further experiments it will be recommended to include an aerodynamic air data system, which allows a direct determination of α and β.
References [1] Digital Dutch: The 1976 Standard Atmosphere. http://digitaldutch.com. [2] G¨ulhan, A.; Requardt, G.; Siebe, F.; Koch, U.; Esser, B.: Instrumentation and In-Flight Data of the SHEFEX Flight Experiment. Proceedings of the 1st International ARA Days on Atmospheric Re-Entry Systems, Missions and Vehicles, Arcachon, France, June 2006 [3] Longo, J.M.A.: Aerothermodynamics A critical review at DLR. Aerosp. Sci. Technol. 7 (2003) 429-438. [4] Turner, J.; H¨orschgen, M.; Stamminger, A.; Turner, P.: SHEFEX Hypersonic Reentry Flight Experiment, Vehicle and Subsystem Design, Flight Performance and Prospects. AIAA Paper 2006-8115, 2006.
Simulation of Magnetohydrodynamic Effects on an Ionised Hypersonic Flow by Using the TAU Code Carmen B¨ottcher1 , Volker Hannemann2, and Heinric hL¨udeke1 1 2
DLR, Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7, 38018 Braunschweig, Germany DLR, Institute of Aerodynamics and Flow Technology, Bunsenstraße 10, 37073 G¨ottingen, Germany
Summary In the present investigation the DLR TAU code is extended to support future experimental investigations of magnetohydrodynamic effects in high temperature hypersonic flows. According to the conditions in the High Enthalpy Shock Tunnel G¨ottingen (HEG) the first steps in enhancing the TAU code are the implementation of a source term formulation of electromagnetic forces and the calculation of the electrical conductivity of air as a gas mixture in chemical non equilibrium. To verify the source term implementation a perfect gas study related to numerical simulations from Poggie and Gaitonde is conducted and shows reasonable agreement. Applied to the experimental conditions the model predicts a noticeable increase of the shock stand off distance.
Nomenclature τ¯ Et q r j
B E σ µ0 m
total stress tensor total energy heat flux term position vector electric current density
1
magnetic induction vector electric field vector electrical conductivity magnetic permeability dipole moment
Introduction
Due to high temperatures between shock and body at hypersonic flights a high level of natural ionisation is generated. About 50 years ago first ideas came up to influence ionised flows with magnetic fields. The theory of the interaction of an electrically conducting fluid with electric and magnetic fields, called magnetohydrodynamics (MHD), was founded. In the present study an extension of the Navier-Stokes equations for the simulation of the influence of magnetic fields on an ionised hypersonic flow will be described. As a tool for the solution of the Reynolds averaged Navier-Stokes equations the DLR TAU code for hypersonic flows [5, 4] is used. It provides a numerical support for HEG experiments as shown in numerous former projects. The code is well validated in terms of hypersonic C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 300–308, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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flows including extensions to calculate real gases with chemical non equilibrium conditions. To simulate MHD effects the code was extended by additional source term formulations of electromagnetic forces. For comparability with results from the literature a perfect gas flow with the assumption of a globally constant electrical conductivity is calculated as a first step. After that a HEG experimental setup is simulated under non equilibrium gas conditions and a local determination of the electrical conductivity. In the hypersonic facility, the High Enthalpy Shock Tunnel G¨ottingen (HEG), the temperatures locally occuring in the experiments are high enough to produce ionised particles. The level of ionisation is sufficient to obtain a measurable influence of magnetic fields on the flow. The DLR is planning corresponding experiments to use the natural (thermal) ionisation in combination with imposed magnetic fields to investigate magnetohydrodynamic effects. The enhanced TAU code with the implementation of magnetohydrodynamic ef fects will be used for the design and support of future MHD experiments in the HEG.
2 Numerical Scheme 2.1 The DLR TAU Code The DLR TAU code [4] is a finite volume solver for the flow field analysis of time dependent, three dimensional viscous and inviscid flows. The spatial discretisation is based on hybrid grids consisting of tetrahedra, prisms, pyramids and hexahedra. Besides the flexibility in grid generation the approach enables local adaptation according to features of the flow field. Further grid handling capabilities are a deformation tool to account for surface changes due to fluid structure coupling and a chimera technique to deal with independent moving parts (i.e. flaps). For time accurate calculations the time derivatives are reformulated such that for each physical time step a steady state problem is solved within a pseudo time. This dual time stepping benefits from the acceleration techniques implemented to reach a steady state solution (multigrid, residual smoothing and local time stepping). The time integration is done via an explicit Runge-Kutta scheme or an implicit LUSGS scheme. Different upwind formulations as well as a central scheme are available. For turbulent flows the modeling varies from several one and two equation models to unsteady Reynolds averaged Navier-Stokes and detached eddy simulations. Several high temperature enhancements (for example: gas mixtures in thermo-chemical non equilibrium) are included to cover hypersonic applications [5]. 2.2 The Source Term Formulation We consider at first the three dimensional compressible Navier-Stokes equations: ∂ρ (1) + ∇ · (ρ U ) = 0 , ∂t ∂ρ U + ∇ · (ρ U ⊗ U + p I) − ∇ · τ¯ = 0 , (2) ∂t ∂ρEt (3) + ∇ · (ρ U Et − τ¯ · U + q) = 0 . ∂t
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Now let a magnetic field be imposed upon the flow. Under the assumption that the imposed field is dominant and the induced magnetic field is negligible the influence of magnetohydrodynamic effects on the flow can be expressed by a source term formulation [7]. The additional terms in the equation system (1-3) are the term j × B in the momentum equation due to the Lorentz force and E · j in the energy equation due to Joule heating and further effects (the resulting equation system is also called the low magnetic Reynolds number model, see Ref. [7]). Assuming that the applied magnetic flux density B does not vary in time Faraday’s law leads to ∂B = 0. ∇×E = − ∂t In the present study we neglect the Hall effect for simplicity [3, 6, 7] as well as for comparison with a test case from the literature [7]. With the assumption of an axisymmetric configuration, a steady flow and j ⊥ U , B, the electric current flows in circumferential direction and the electric field vanishes. Therefore only the momentum equation (2) changes: ∂ρ U + ∇ · (ρ U ⊗ U + p I) − ∇ · τ¯ = j × B . ∂t The electric current density j is obtained from the generalised Ohm’s Law j = σ (E + U × B) = σ (U × B) . Remark: Using the relation
2 B B⊗B , − ∇· µ0 2 µ0 the momentum equation can also be formulated in conservative form, namely ∂ρ U B⊗B + p I − ∇ · τ¯ = 0 , + ∇ · ρU ⊗ U − ∂t µ0 j×B = ∇·
with p = p + B 2 /(2µ0 ). 2.3 The Prescribed Magnetic Field In the modificated TAU code the magnetic induction vector B is calculated pointwise depending on the chosen type of imposed field. In addition to a 3d dipole field also a homogenous field and a 2d dipole field are implemented and can be chosen for simulations. For this study the 3d dipole field is selected. The magnetic field can be described as follows ⎞ ⎛ Bx 3r(r · m) − r2 m µ0 ⎠ ⎝ , B = By = 4π r5 Bz with dipole moment m. In view to the configuration of the planned magnetohydrodynamic HEG experiments and also due to comparability with simulation results in [7] the dipole moment should coincide with the symmetry axis, pointing into negative direction (see Figure 1). Let the x-axis be the symmetry axis. Hence the
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flow m
Fig . 1 . Dipole moment and magnetic field of the imposed 3d dipole
dipole moment can be simplified to m = (m1 , m2 , m3 ) = (m1 , 0, 0) and the magnetic induction vector becomes ⎛ ⎞ m1 (3 x2 −r 2 ) ⎛ ⎞ 5 r Bx ⎟ ⎜ ⎝ By ⎠ = µ0 ⎜ 3 m1 x y ⎟ . 5 ⎠ r 4π ⎝ B z
3 m1 x z r5
The magnitude |B| of the magnetic induction vector in a certain point P = (x0 , y0 , z0 ) is determined by the dipole moment m. Another option is to calculate a suitable dipole moment by choosing a certain magnitude |B| at a certain point P . In magnetohydrodynamics the dipole field is often given by a certain magnitude |B| at the stagnation point. Hence, if the described axisymmetric model is applied and P is located at the stagnation point (xs , ys , zs ) = (xs , 0, 0), the relation between |B| and m reduces to µ0 m1 (3 x2s − r2 ) = µ0 2 m1 . |B| = Bx2 |P + By2 |P + Bz2 |P = 5 3 4π r 4π xs As mentioned before, m shall point in negative x-direction. So we finally get 4π x3s Bx |P . m1 = − µ0 2 2 The magnetic interaction parameter Q = σref Bref Lref /(ρref Uref ) is a measure for the influence of magnetohydrodynamic effects on the flow. For the determination of the reference values see Ref [7]. In section 3.1 identical values for Q are chosen for comparison with the results from Poggie and Gaitonde [7].
2.4 Gas Model and Electrical Conductivity Finally the electrical conductivity σ remains to be determined. The fluid air is modeled at three different levels: as a perfect gas, as a gas mixture in thermo-chemical equilibrium and as a gas mixture in chemical non equilibrium. In the first case the electrical conductivity of the gas, which is necessary for the source term formulation of the magnetic effects, is assumed to be constant throughout the flow field. In thermo-chemical equilibrium the composition of the mixture is a function of only two state variables (i.e. density and internal energy). The expensive calculations of the complete state including the transport properties of the mixture can
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therefore be conducted in advance and stored in curve ts. The basic models for the pre-calculations are the same as the implemented models for the chemical non equilibrium case. Each species of the mixture is described via its partition function or respective curve ts (e.g. Gurvich). In the chemical non equilibrium case the composition of the mixture is determined by a set of mass balance equations (one for each species) including diffusion terms and source terms modeling the chemical reactions. The temperature is calculated from the energy balance, the composition and the species description. Following the approaches given in the thesis of Bottin [1] the transport properties are calculated via collision integrals, which are tted for each pairwise combination of species. The mixture rule from Yos is applied for the viscosity and the translational part of the thermal conductivity. The contribution of the internal modes of energy is taken into account according to Eucken‘s approximation. The thermal conductivity of the electrons and the electrical conductivity are calculated according to Devoto [2] taking two non-vanishing Sonine polynomials into account.
3.1 Perfect G as T est C ase Analogous to [7] a laminar, Mach 5, perfect gas ow over a hemisphere of radius Rb = 0:01m is examined by neglecting any Hall effects. The angle of attack is zero so the calculations can be carried out in an axisymmetric way. A Reynolds number of Re = 1 U1 Rb = 1 = 80000 and a freestream temperature of 100 K are chosen. An axisymmetric 3d dipole with a magnetic eld strength of 5 Tesla at the stagnation point is centered at the origin (see Figure 2). For simpli cation the electrical conductivity is globally set to 1330 S/m to match the given magnetic interaction parameter Q = 6. The magnetic Reynolds number Re σ = U L is approximately 0.02. It has to be pointed out that in [7] is chosen to be zero outside the boundary layer.
0.02
Bx 2 1
Z
0.015
0.01
0 -1
0.005
-2 -3 -4
0 -0.015
-0.01
X
-0.005
Fig . 2 . Two components of the magnetic
0
eld: Bx (coloured), Bz (black isolines)
In the rst simulation Q is xed to zero in order to calculate the ow eld without any imposed magnetic eld. The resulting pressure contours are shown in Figure 3. The shock stand off distance of 16% of the radius Rb is identical to the result from the literature. With a magnetic interaction parameter of Q = 6 the ow eld is
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PRESSURE
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:
:
8
8
Fig . 3 . Pressure contours of a Mach 5, ideal gas flow over a hemisphere without imposed magnetic field
Fig . 4 . Pressure contours of a Mach 5, ideal gas flow over a hemisphere with an imposed axisymmetric 3d dipole, σ const
significantl ychanged b ythe magnetohydrodynamic terms a scan b esee ni nFigur e4 .Compare dt oFigur e3 the shock stand off distanc ei sincreased t o33% of Rb which fits wel lwit hthe literatur evalu eo f30% takin gthe smaller regio nof non vanishin gelectrica lconductivit yint oaccount .The boundar ylayer profiles in Figur e5 and 6 show tha tthe influenc eo fthe magneti cfiel dreduces the stream wis evelocity and the temperature gradien ta tthe wall .Nevertheless the chang eof the qualitative structure o fthe flow i ssmall .Compare dt o[7] the magnetohydrodynamic effects i nthe simulations are marginally stronger. I ti sexpected tha tthis i sdue t othe fact tha tthe electrica lconductivit yi snot restricte dt othe boundary layer a swa sdon ei n[7] .Thi stopic wil lb eexamined i nthe future. Altogethe rthe simulatio nresults are i ngoo dagreement wit h[7].
Q=0 in TAU Q=6 in TAU Q=0 in [5] Q=6 in [5]
0.00012
0.00012 0.0001
wall distance
wall distance
0.0001 8E-05 6E-05 4E-05 2E-05 0
8E-05 Q=0 in TAU Q=6 in TAU Q=0 in [5] Q=6 in [5]
6E-05 4E-05 2E-05
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5
4
6
0
T/T ∞
Fig. 5. Boundary layer profile: temperature, θ = 6.6◦ (red: without MHD effects, blue: with MHD effects)
0
0.02
0.04
0.06
0.08
0.1
0.12
u/u∞
Fig. 6. Boundary layer profile: streamwise velocity, θ = 6.6◦ (red: without MHD effects, blue: with MHD effects)
3.2 Simulations Related to Experimental Setup
I nthe scope o fthe planned magnetohydrodynami cexperiment si nthe HEG a Mac h9 flow ove ra hemisphere-cylinder model i ssimulated .The radiu so fthe
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PRESSURE
PRESSURE
:
:
8
8
Fig. 7. Pressure contours of a Mach 9 flow over a sphere without imposed magnetic field (HEG experimental setup), gas model: chemical non equilibrium
Fig. 8. Pressure contours of a Mach 9 flow over a sphere with imposed 3d dipole (HEG experimental setup), gas model: chemical non equilibrium, σ local variable
spher eamounts t o0.08m .A naxisymmetri c3 ddipol ewit ha magneti cfiel dof 1 Tesl aa tthe stagnatio npoint i sassumed a tthe cente ro fthe hemispher e(Figur e2). Further condition sare a reference density of ρref = ρ∞ = 0.0015 kg/m3 and a freestrea mtemperature of T∞ = 900 K .Due t othe hig hvelocities (about 6 km/ sahead o fthe bow shock )the flow fiel di nthe stagnatio nregio ni sdominated b ynon equilibrium effects .Assumin gchemica lequilibrium leads t otemperatures behin dthe shock which are too low for a significan tionisatio nand respective electrica lconductivity. Calculation swit hthi sassumptio nshow n oinfluenc eo fthe magneti cfiel do nthe flow fiel dand are therefore not include di nthi spaper. Air
ELECTR COND
ELECTR COND
:
:
8
Fig. 9. Electrical conductivity contours without imposed magnetic field
8
Fig. 10. Electrical conductivity contours with imposed 3d dipole
i nchemica lnon equilibrium i sapplied i nthe followin ginvestigation. A sa first ste pa simulatio nwithout magnetohydrodynamic influences i sperformed .The resulting pressur econtour scan b efound i nFigur e7 .Figur e9 show sthe locally calculate delectrica lconductivity. Now the implemented sourc eter meffects o fthe described 3 ddipol ecombine dwit hthe local electrica lconductivit yare take ninto account .Hal leffects are neglected her ebu twil lb einclude di nlater studies .The magneti cReynold snumber Reσ = µ0 σ Uref Lref take sa maximum valu eo f1.17,
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which can still b econsidere da ssmall [7] and justifie sthe application o fthe low magneti cReynold snumbe rmodel .The resulting flow structure i sillustrated i nFigur e8 and 10. Compare dt oFigur e7 again a distinc tincreas eo fthe shock distance i svisible .Als oFigur e1 1and 1 2show tha tthe simulated magnetohydrodynamic effects reduc ethe strea mwis evelocity and the temperature profil enea rthe wall. The qualitative structure o fthe flow i sagain retained.
Q=0 Q=6
0.0012
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wall distance
0.001 0.0008 0.0006 0.0004
0.0008 0.0006 0.0004 0.0002
0.0002 0
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100
200
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Fig. 11. Boundar ylayer profile: tempera - Fig. 12. Boundar ylayer profile: streamwise ture, θ = 6.6◦ (red: without MHD effects, velocity, θ = 6.6◦ (red: without MHD effects, blue: wit hMHDeffects) blue: wit hMHDeffects)
4 Conclusion I nthe presented study a nextensio no fthe DLR TAU cod et osimulat emagnetohydrodynami ceffects i na hypersoni cflow fiel di sinvestigated .The calculation o fa perfect gas flow under influenc eo fa magneti cfiel dshow sgoo dagreement wit hsimulatio nresults fro mthe literatur e[7] .A potential experimental setup i nthe High Enthalpy Shock Tunnel G¨ottinge n(HEG) t oinvestigat emagnetohydrodynami ceffects i sanalyse dassumin gthe air a sa gas mixture i nchemica lnon equilibrium. Applied t othe desig ncondition so fthe experiment sthe model predict sa nincrease o fthe shock stand off distanc ea sexpected .The chang eo fthe shock stand off distanc ewit hand without magneti cfiel di slarg eenoug ht ob ewel lvisualise dwith the optical measurement technique so fthe HEG .The logical nex tsteps wil lb ethe improvement o fthe numerical model b ytakin gthermal non equilibrium and Hall effects int oaccount and eventuall ythe compariso nwit hthe HEG experiments.
References [1] B. Bottin: Aerothermodynamic Model of an Inductively-Coupled Plasma Wind Tunnel. Universite de Liege, VKI, ISBN 2-9600243-0-3, 1999. [2] R.S. Devoto: Simpli_ed Expressions for the Transport Properties of Ionized Monoatomic Gases. Physics of Fluids, No. 10, Issue 10, October 1967. [3] T. Fujino et al.: Numerical Analyses on Flow Control arouns Blunt Body .OREX. by Magnetic Field, AIAA 2003-3760, 2003.
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[4] T. Gerhold et al.: Calculation of Complex three-dimensional con_gurations employing the DLR TAU Code. AIAA-97-0167, 1997. [5] A.Mack, V. Hannemann: Validation of the Unstructured DLR-TAU-Code for Hypersonic Flows. 32nd AIAA Fluid Dynamics Conference, St. Louis (USA), 2002. [6] H.Otsu, A. Matsuda, T. Abe, D. Konigorski: Numerical Validation of the Magnetic Flow Control for Reentry Vehicles, AIAA 2006-3236, June 2006. [7] J. Poggie, D.V. Gaitonde: Computational Studies of Magnetic Control in Hypersonic Flow. AIAA 2001-0196, Reno, January 2001.
Heat Fluxes Inside a Cavity Placed at the Nose of a Projectile Measured in a Shock Tunnel at Mach 4.5 F. Seiler 1, J. Srulijes 1, M. Gimenez Pastor 2, and P. Mangold 2 1
French-German Research Institute of Saint-Louis (ISL) 5 rue du Général Cassagnou, F-68301 Saint-Louis, France 2 Diehl BGT Defence (DBD) Alte Nußdorfer Straße 13, D-88662 Überlingen, Germany
Summary A blunt-nosed supersonic projectile, equipped with a cavity, is a good alternative to reduce high nose heating rates preventing surface melting followed by ablation. For this purpose, tests were done in ISL’s shock tunnel STB. This facility allows reproducing atmospheric flow conditions present during a missile flight at various altitudes. The flow was visualized by shadowgraphs and the heat flux densities were measured with thermocouples at the bottom of various cavity geometries. A numerical simulation was also carried out using the FLUENT code. The comparison between numerical results and measurements is quite satisfactory. An important result of this study is that the deepest cavity has the smallest heat flux.
1
Introduction
For vehicles flying in the earth atmosphere the design objectives must be tailored to the required mission and flight conditions. A commercial aircraft, e.g., is designed for a cost-effective flight. In contrast, economical flight requirements are irrelevant for an armed missile which should be capable of achieving its mission. New concepts on supersonic and hypersonic missiles are worldwide in progress for both, short range air defense guided missiles as well as long range interceptor missiles including re-entry vehicles and hypersonic aircrafts. For high supersonic and hypersonic missiles, aerodynamic heat loads become extremely high. This is a fact that has to be considered during the missile’s development phase. The objective thus is to minimize the thermal stress affecting the structural stability of the missile, in particular its nose and the leading edges of the fins. Beyond Mach 3 missile leading edges are exposed to high heating rates causing high temperatures on the structure. If surface melting and ablation occur, the flight aerodynamics can be perturbed, followed by unacceptable flight trajectory deviations. Additionally, if the head of a high speed missile is equipped with a seeker window for target identification via a radar transmitter, camera or infrared sensor, window blinding or fracturing by extreme aeromechanical and aerothermal loads must be avoided. Another problem is signal refraction passing through the shocked hot gas C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 309–316, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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layer in front of the missile’s seeker head. The preferred shape for these purposes is a hemispherical nose, which becomes critically loaded by thermal stresses at the nose tip. Therefore, it is utmost important to find techniques to reduce the heat at the nose tip of a high-speed missile.
2
Scientific Background
A number of scientific reports on “cavity flows”, in a wider sense, have been published in the past decades. A basic publication concerning the flow in a cavity was made by Hartmann and Troll [9]. They studied the interaction of a supersonic jet flow with a body, which was equipped with a forward-facing cavity. This is well known as the “Hartmann whistle concept” for producing sound of discrete frequencies. More than thirty years later, Burbank and Stallings [1] published that at supersonic Mach numbers the stagnation heat transfer is noticeably lower in a concave nose cone than in a convex geometry. Later, Huebner and Utreja [10] reported on heat fluxes at a nose-cavity arrangement which were found to be less significant at the base of a cavity than at the outer edge. The dynamics of a detached bow shock and acoustic resonance phenomena are investigated by Marquart et al. [11] in a forward-facing cavity at a blunt body configuration. In order to characterize mechanisms to reduce the nose heating loads, Yuceil and Dolling [16] examined the heating phenomenon on blunt bodies equipped with an upstream facing cavity. A number of papers describing in detail the dynamics of a cavity flow, in particular the nose-tip surface heat reduction by a forward-facing cavity at the nose of a supersonically flying vehicle, have been published by Engblom et al. [2-7]. In [6] Engblom and Goldstein published an interesting result on the cooling benefit reached through geometry optimization: “rounding the cavity lip is necessary to reduce local surface heating since the airflow into the cavity directly impinges on the cavity lip”. A recent paper published by Gunes et al. [8] deals with visualizations of the flow around a cavityequipped hemispherical nose fixed in front of a projectile flying at hypervelocities.
Fig. 1. Cavity-nosed missile configurations
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Based on this knowledge, a joint study was initiated by DBD and ISL to determine, for a given missile head geometry [15], how the heating rates at the bottom of a cavity are affected by the cavity depth, which is today strongly relevant for supersonic and hypersonic seeker head configurations. For ISL the main goal is to determine the heat fluxes as they exist in real conditions. Explanations for the mechanisms behind the heat flux behavior will be given later by DBD. For these studies three missile head mock-ups, as shown in Figure 1, were designed and manufactured: one duplicates a blunt missile head (left), the other two are generic cavity-nosed supersonic missile configurations. The middle one in Figure 1 represents a cavity with medium depth configuration, whereas the right one simulates a head design with a significantly deepened hollow space.
3 3.1
Model Configuration
Dimensions
Generic test dome shapes were selected by DBD as shown in the photographs in Figure 1. Their geometries are schematically shown in Figure 2. Test-dome no. 1 resembles a classically designed infrared seeker design with blunt head geometry without cavity. No. 2 and 3 are equipped with a cavity. The model dimensions are given in the drawings in Figure 2. For test-dome no. 2 the cavity depth is 4.7 mm and for no. 3 it is 14.7 mm. The outer cavity edge is rounded, as proposed in [6]. The tapered cavity has a bottom diameter of 55.2 mm. Their length/diameterratios (L/D) are given in Table 1. Table 1 . Test-dome dimensions
test-dome
L [mm]
D [mm]
L/D
2 3
4.7 14.7
55.2 55.2
0.085 0.266
Fig. 2. Test-dome design
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Fig . 4. Measuring locations and instrumented large cavity model
Fig. 3. Thermocouple
3.2
Heat Flux Measurement
For heat flux measurements, thermocouple temperature sensors were used. The thermocouple gauges were bought from the Stoßwellenlabor of the RWTH Aachen University in Germany. Figure 3 shows a photo of the thermocouples used. The thermocouple sensors are made of a Cromel steel rod with 1.9 mm diameter and a centered imbedded Constantan wire. The mirror-polished rod tip of the two materials is grinded with emery to allow thermocouple contact. With the calibration curve delivered by RWTH Aachen, the surface temperature variation 'T can be determined. A solution of the one dimensional heat conduction equation is used to calculate the heat flux q acting from gas to surface structure at the sensor measuring location. The solution is given by Oertel [12] as a function of the measuring time t at the surface at y = 0 as: q( 0 ,t )
UcO d t 'T ( 0 ,t ) ³ dW S dt 0 t W
The variables U, c and O used in this relation are the density, the heat capacity and the heat conduction of the thermocouple Cromel steel material. Figure 4 on the left schematically shows the measuring locations marked by circles. The photo on the right shows shock tube model no. 3 instrumented with thermocouple gauges. Not all thermocouple surfaces can be clearly recognized because of surface damage to the bottom of the cavity caused by shock tube membrane particles’ impact.
4 4.1
Experimental Simulations
Shock Tunnel Description
For this study the ISL high pressure shock tube facility STB (Figure 5 on the right) is used as a shock tunnel and operated in the shock reflection mode with a contoured nozzle. The nozzle exit has a diameter of 220 mm and the throat 50.4 mm in diameter. The length of the driver tube is 3.85 m and the driven tube is 17.75 m long. STB ends with a dump tank of 20 m3. A principle sketch of STB is shown in Figure 6 with the arrangement of driver tube, driven tube, Laval nozzle and dump tank. The shock tube operation and the shock tunnel techniques used have been described in detail by Oertel [13].
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Fig . 5. Shock Tube Laboratory with shock tubes STA and STB
Fig. 6. Arrangement of shock tube STB
4.2
Nozzle Flow
The experiments were performed with a nozzle technically designed for a flow Mach number of M = 4.5. Three atmospheric flow conditions, corresponding to a missile flight at altitudes h = 0, 5 and 20 km, have been tested. In table 2 the experimentally verified nozzle conditions are listed as mean values for the flow data as they are present at the nozzle exit in the shock tunnel experiments performed. The nozzle is mounted at the end of the shock tube and opens into a test section which has a window on each side for visualizing purposes. Figure 7 shows the nozzle arrangement inside the test section with the model placed in front of it as well as a close-up view of the model itself. Table 2 . Flow dimensions
h [km] 0 5 20
Me pe [mbar] 4.52 1027.2 4.58 467.8 4.60 49.5
Te [K] 292.9 234.5 213.3
Ue [kg/m3] 1.18 0.67 0.08
ae [m/s] 348.8 312.1 297.6
Fig. 7. Test-dome in front of the Mach-4.6-Nozzle
ue [m/s] 1578.2 1429.8 1368.9
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Fig . 8 . Without cavity (model no.1), 5-km-condition
Fig. 9. Small cavity (model no. 2), 5-km-condition
Fig . 10 . Large cavity (model no. 3), 0-km-condition
4.3
Experimental Results
Flow visualizations with differential interferometry, as described in [14], have been performed. Figure 8 shows a series of flow pictures, where light intensity variations represent density variations. From left to right, in pictures no. 1 and 2 the nozzle flow develops. Afterwards, pictures no. 3 and 4 show the formation of the stationary bow shock in front of the non-cavity-equipped shock tube model with a flow duration of about 2 ms. A Pitot pressure gauge, seen at the bottom of the pictures, is placed for flow control. A similar flow formation as in the nocavity-case is present for the small-cavity-case, as seen in Figure 9. In the large cavity case stronger flow gradients develop in front of the cavity (Figure 10). As described in section 4.2, the heat fluxes are measured with cavity models no. 1, 2 and 3 and the results are given in Figure 11 for the three flight conditions given in table 2, i.e., h = 0, 5 and 20 km. For case no. 1, without cavity, the heat fluxes increase towards the outer edge of the blunt front section for x o 25 mm, mostly for h = 0 km. The reason for this behavior can be found in the flow acceleration from axis, x = 0, where the velocity equals zero, to x = 25 mm. Boundary layer transition from laminar to turbulent may occur, causing growing heat fluxes. For the cavity models no. 2 and 3 the heat fluxes decrease considerably as the cavity depths increase. One explanation for this behavior is the lessened density near the bottom of the cavity as the cavity depth enlarges, shown by FLUENT calculations. Calculations with the FLUENT code have been performed assuming laminar flow development. An example for the temperature contours is given in Figure 12. The FLUENT results are compared to the experimental results in Figure 13. Deviations present for x o 25 mm, may be caused by boundary layer transition from laminar to turbulent, which is not taken into account in the FLUENT calculations.
Heat Fluxes Inside a Cavity
small cavity 0 km
10
heat flux [MW/m²]
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without cavity 0 km large cavity 0 km
8
small cavity 5 km without cavity 5 km large cavity 5 km
6
small cavity 20 km without cavity 20 km
4
large cavity 20 km
2 0 0
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15
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25
distance from axes x [mm]
Fig. 11. Effects of cavity and flight condition
4
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FLUENT simulation shock tunnel experiment
0
2
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10
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Fig. 12. FLUENT calculated temperature distribution, 0-km-condition
0
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10
15
20
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distance from axes x [mm]
Fig. 13. Comparison of FLUENT calculation and experiment, large cavity, 0-km-condition
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5
Conclusions
This joint study of DBD and ISL deals with a generic missile configuration equipped with a cavity at its front end. Heat fluxes were measured by thermocouples at the bottom of the tapered cavity in the shock tunnel STB of ISL using a Mach 4.5 contoured nozzle. The missile’s on flow was duplicated by this facility as it is in real flight for altitudes h = 0, 5 and 20 km. At the same time, “laminar” FLUENT code calculations are compared with the heat flux results. The agreement is quite good as the cavity boundary layer remains laminar, but larger deviations appear towards the cavity’s outer edge, possibly caused by transition to turbulence.
References [1] [2] [3]
[4] [5] [6] [7]
[8]
[9] [10] [11] [12] [13] [14] [15]
[16]
P.B. Burbank and R.L. Stallings. "Heat transfer and pressure measurements on a flatface cylinder for a Mach number range of 2.49 to 4.44". NASA rep. TMX-221, 1959. W.A. Engblom, B. Yuceil, D.B. Goldstein, D.S. Dolling. "Hypersonic forward-facing cavity flow: an experimental and numerical study". AIAA Paper 95-0293, 1995. W.A. Engblom, B. Yuceil, D.B. Goldstein, D.S. Dolling. "Experimental and numerical study of hypersonic forward-facing cavity flow". Journal of Spacecraft and Rockets, Vol. 33, 1996, pp. 353-359. W.A. Engblom, D.B. Goldstein, D. Ladoon, S. Schneider. "Fluid dynamics of forward-facing cavity flow". AIAA Paper 96-0667, 1996. W.A. Engblom, D.B. Goldstein, "Nose-tip surface heat reduction mechanism". AIAA Paper 96-0357, 1996. W.A. Engblom, D.B. Goldstein. "Nose-tip surface heat reduction mechanism". Journal of Thermophysics and Heat Transfer, Vol. 10, 1996, pp. 598-606. W.A. Engblom, D.B. Goldstein, D. Ladoon, S.P. Schneider. Dolling. "Fluid dynamics of Hypersonic forward-facing cavity flow". Journal of Spacecraft and Rockets, Vol. 34, 1997, pp. 437-444. H. Gunes, I. Fenercioglu, B. Yuceil. "Instantaneous imaging of highly unstable bow shock wave caused by a hypervelocity projectile with a streamwise nose cavity". 11th International Symposium on Flow Visualization, Notre Dame, Indiana, USA, 2004 J. Hartmann and B. Troll. "On a new method for the generation of sound waves". Physcal Review, Vol. 20, 1922, pp.719-727. L.D. Huebner and L.R. Utreja. "Experimental flowfield measurements of a nose cavity configuration". Society of Automotive Engineers, Paper 871880, 1987. E.J.Marquart, J.B. Grubb, L.R. Utreja. “Bo-shock dynamics of a forward-facing nose cavity”, AIAA paper 87-2709, 1987. H. Oertel. "Wärmeübergangsmessungen". In Kurzzeitphysik, Springer Verlag, WienNew York, 1967. H. Oertel. "Stoßrohre", Springer Verlag, Wien-New York, 1966. H. Oertel. "Optische Strömungsmesstechnik", G. Braun Verlag, Karlsruhe, 1989. S. Pataud, B. Sauerwein, A. George, F. Seiler, G. Pastor, F. Boller. "Étude de l'écoulement dans une cavité à la tête d’un missile avec la soufflerie à choc STB à Mach 4,5". ISL-Rep. N 808/2005, 2005. K.B.Yuceil and D.S. Dolling. “Nose cavity effects on blunt body pressure and temperature at Mach 5”. Journal of Thermophysics and Heat Transfer, Vol. 9, 1995.
Experimental Investigation and Numerical Simulation on a Missile Radome at Mach 6 Carl Dankert1 and Hannes Otto2 1
Institute of Aerodynamics and Flow Technology, German Aerospace Center DLR, Bunsenstr. 10, 37073 G¨ottingen , Germany
[email protected] 2 Institute of Aerodynamics and Flow Technology, German Aerospace Center DLR, Lilienthalplatz 7, 38108 Braunschweig, Germany
[email protected]
Summary The aerodynamic loads for a missile radome cruising near ground level are investigated. The investigation consists of experiments at Mach 6 performed in the High Enthalpy Shock Tunnel G¨ottingen (HEG) and CFD calculations using the TAU code of DLR. The first approach duplicated an hypersonic flight at 11 km altitude for some milliseconds with the appropriate Mach number, Re number, flow velocity, pressure and heat load. The experimental results for the heat transfer on the radome fit to the CFD data including the transition from laminar to turbulent.
1 Introduction The development of defence missiles goes towards higher agility and increasing flight velocities. For future systems, Mach numbers of six are envisaged from sea level up to 30 km flight altitudes [2]. Figure 1 shows the major aerodynamic and aerothermodynamic characteristics at Mach 6 and 11 km altitude for such a hypersonic missile. Due to the high kinetic energy during flight, the mechanical loads become high. The heat flux and thermal loads on the surface, especially on the nose of a radome configuration are extremely high. The nose area has to withstand temperatures up to 1800 K and wall pressures of 12 atmospheres during flight. Previous experimental investigations at altitudes of 15 and 21 km and Mach 6 were conducted at the high energy shock tube facility STA of ISL [8] for more slender conical models. In the High Enthalpy Shock Tunnel G¨ottingen (HEG) a new nozzle was designed to create a Mach 6 flow field to test radome models under realistic aerodynamic conditions. A new test section was also built at DLR G¨ottingen. The new contoured Mach 6 nozzle and test conditions produced flow velocities of around 1750 m/s for several milliseconds. The objective was to measure pressure and heat transfer on models like missile radomes and deliver validation data for CFD. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 317–323, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Fig. 1. Aerodynamic conditions at Mach 6 and 11km altitude
2 High Enthalpy Shock Tunnel G¨ottingen The principle of the piston driven wind tunnel HEG is shown in Fig. 2. The compressed air in the buffer drives a piston of 850 kg into the driver gas (nitrogen) [3]. When the main diaphragm breaks, a shock wave passes into the test gas (air) and
Fig. 2. Principle of HEG facility for Mach 6 conditions Table 1. Flow conditions in HEG at Mach 6 Stagnation pressure
390 bar
Stagnation temperature
1600 K
Stagnation enthalpy
1.74 M Jkg −1
Flow velocity
1750 ms−1
Static pressure
0.20 bar
Static temperature
200 K
Mach number
6.2
Core flow diameter
292 mm
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Fig. 3. Distribution of Pitot pressures at nozzle exit plane
is reflected at the end of the shock tube. Simultaneously, a secondary diaphragm at the end of the shock tube bursts and the hot stagnated test gas starts the nozzle flow. The Mach 6 nozzle produces usable test times of about five milliseconds. The corresponding stagnation and flow properties are listed in Table 1. The nozzle was calibrated for these conditions. Figure 3 shows the uniformity of the nozzle flow across the exit plane. The Pitot pressures, measured with a rake, are plotted and compared to CFD calculations. The experimental points show a slightly smaller flow core than the CFD predictions.
3 Experiments at Mach 6 A radome test model of 80 mm diameter and a length of 90 mm was selected. The contour consists of a tangential ogive with a radius of 114.4 mm with spherical nose
Fig. 4. Model for wind tunnel tests, 6 pressure sensors and 7 temperature sensors
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Fig. 5. Schlieren photo showing the flow field structure around the radome model at 5◦ angle of attack
Fig. 6. Measurement of wall temperatures along radome as function of time
tip with a radius of 10 mm. The model was machined of stainless steel and was equipped with six pressure sensors and seven thermocouple sensors. Figure 4 shows a 1/3 axial section through the radome model. The model was fixed in the wind tunnel at angles of attack of 0◦ and 5◦ . Figure 5 gives an impression of the structure of the flow field around the radome model at Mach 6 at 5◦ angle of attack. The shock wave around the model, similar to the one in Fig. 1 can be identified clearly. The shocks at the top of the figure belong to the pressure and heat transfer probes which monitored the undisturbed flow conditions. These probes are located off-axis to avoid interaction with the model on the centreline. The measurements of the wall temperatures along the radome at 0◦ are plotted in Fig. 6 for seven thermocouple sensors. For the data reduction to evaluate heat transfer rates [5], only time values up to 5 ms were selected.
4 Numerical Approach The Centaur software, a commercial package provided by CentaurSoft was used for the generation of the hybrid meshes. In the Navier-Stokes meshes 28 prismatic layers were used to resolve the boundary layer. The initial spacing was adjusted in order to obtain a y + value of approximately unity. The regions of inviscid flow were discretised using tetrahedral cells. To minimize the discretisation errors a mesh convergence study has been conducted. Three meshes with different cell densities in the defined flowfield were generated, the initial grids consist of about 170000 points (coarse mesh), about 310000 points (medium mesh) and about 590000 points (fine mesh). The applied grids were adapted three times to refine regions of large gradients of pressure and density. The meshes were increased about 30 percent per adaptation step. Figure 7a and Fig. 7b show the initial and the final adapted medium meshes in the symmetry plane.
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Fig. 7. Initial medium grid (a) and medium grid after third adaptation (b)
The numerical investigations were performed using the TAU code of DLR, a finite volume Euler- and Navier-Stokes solver, which can handle structured, unstructured and hybrid grids [4]. A validation of the TAU solver for hypersonic flow conditions can be found in [6]. In the present study the AUSMDV second order upwind scheme [1] with MUSCL extrapolation is used. An implicit, approximately factored LU-SGS scheme is used for time discretisation, including the local time stepping. For the transition of laminar to turbulent flow a transition plane has been set at a curvature change between the spherical nose tip and the tangential ogive of the model. As a consequence the nose tip flow is modeled to be laminar and the flow along the tangential ogive is assumed to be turbulent. One- and two-equation turbulence models were tested: Spalart-Allmaras in standard formulation (SA), Spalart-Allmaras with Edwards’ modification (SAE), and the k-ω model. Park’s five species chemical non-equilibrium model [7] was used to model real gas effects of the hypersonic flow.
5
Results
The results for the wall pressures at 5◦ angle of attack are given in Fig. 8. The six pressure measurements are compared to CFD calculations and show good agreement to within the experimental uncertainty. A key problem for the determination of the aerothermodynamic loads is the heat transfer rate. Figure 9 shows the heat transfer normalized by the stagnation value for a radome at Mach 6.2 and 5◦ angle of attack corresponding to a flight altitude of 11 km, a flow velocity of 1750 m/s and a Reynolds number of 44·106 . The experimental data fit very well with the CFD calculations. The numerical results support the experimental observation that the flow changes from laminar to turbulent near the alteration in curvature of the model surface. In the numerical simulation the transition was fixed at the location assumed in the experiment. The standard two-equation k-ω model described by Wilcox [9]
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Fig. 8. Wall pressure along radome at 5°, CFD and experiment
Fig. 9. Normalized heat transfer along radome at 5°, experiment and CFD with medium mesh
has been selected since it provides the best correlation with the experimental data. The transition from laminar to turbulent increases the heat transfer by a factor of three. The fineness of the CFD grid has only a small influence on the heat transfer results (see Fig. 10).
Fig. 10. Normalized heat transfer along radome at 5°, CFD on different meshes
6 Conclusion A new nozzle and test section were built to create the flow field for a missile flying at Mach 6.1 at 11 km altitude. Flow velocity, Mach number and Reynolds number were fully duplicated with the new nozzle for HEG. The measurement time (approximately 5 ms) is sufficient to obtain wall pressures and heat transfer rates on a radome model at moderate angles of attack (0◦ and 5◦ ). The numerical results
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confirm the experimental evidence of laminar-turbulent transition at the change in curvature between the spherical nose tip and the tangential ogive (see Fig. 9). Furthermore, the k-ω turbulence model has proven to be the model which fits best to the experimental results for this type of flow.
References [1] B. Bottin: ”Aerothermodynamic Model of an Inductively-Coupled Plasma Wind Tunnel”, Dissertation, Universit´e de Li`ege, 1999 [2] H.K. Ciezki et al.: ”The DLR Research and Technology Project ”Hochagiler Flugk¨orper” (Missile of High Agility)”. Proceedings of the Innovative Missile Systems Symposium, RTO-MP-AVT-135, Paper 17, May 15-19 2006, Amsterdam, NL, pp 17-1 17-17. [3] G. Eitelberg, T.J. McIntyre, W.H. Beck:” The High Enthalpy Shock Tunnel in G¨ottingen”, AIAA paper, 92-3942, 1992. AIAA 17th Aerospace Ground Testing Conference. [4] T. Gerhold, O. Friedrichs, J. Evans, M. Galle: ”Calculation of Complex ThreeDimensional Configurations Employing the DLR-Tau-Code”, AIAA 97-0167, 1997 [5] G. Koppenwallner: ”Fundamentals of Hypersonics: Aerodynamics and Heat Transfer”. In: Short Course Notes entitled Hypersonic Aerodynamics, Rhode Saint Genese, Belgium, 1984. Von Karman Institute for Fluid Dynamics. [6] A. Mack, V. Hannemann: ”Validation of the unstructured DLR-TAU-Code for Hypersonic Flows”, AIAA 2002-3111, 2002 [7] C. Park: ”Review of Chemical-Kinetic Problems of Future NASA Missions”, I: Earth Entries. Journal of Thermophysics and Heat Transfer, No. 3, 385-398, 1993. [8] F. Seiler, J. Srulijes, M. Havermann, P. Hennig, P. Gleich:“Heat transfer measurements at the nose of a high speed Mach 6 missile.”, RTO AVT-135/RSY-020 Symposium on Innovative Missile Systems, 15-19 May 2006, Amsterdam, The Naval Barracs. [9] D.C. Wilcox: ”Turbulence Modeling for CFD”, DCW Industries, Inc., 1993.
Control of Flow Separation in Adverse Pressure Gradients by Means of Crosswise Grooved Surfaces for Turbo Machine Applications W. Hage, R. Meyer1, and C.O. Paschereit2 1
DLR (German Aerospace Center), Institute of Propulsion Technology, Dept. of Engine Acoustics, Müller-Breslau-Str. 8, D-10623 Berlin, Germany
[email protected] 2 Technical University of Berlin, Institute of Fluid Mechanics and Engineering Acoustics, Müller-Breslau-Str. 8, D-10623 Berlin, Germany
Summary The total pressure losses in a compressor stage are strongly affected by sec-ondary flow effects such as corner separation. Experiments in a highly loaded com-pressor cascade were performed to minimize the losses using a grooved surface on the side walls under cross flow conditions. This geometry hampers the secondary flow in the cascade in such a way, that the corner separation is less pronounced. The total pressure losses are reduced by 9.8% and the flow turning angle is increased. The experiments were accomplished at Reynolds numbers up to Re = 0 56 · 10 (based on 40 mm chord of the vane profile) and Mach numbers up to M = 0.67. To understand the flow physics of the crosswise grooves, additional experiments at lower Reynolds numbers were performed.
1 Introduction The losses in highly loaded axial compressors are dominated by flow separations at the walls of the casing and the hub as well as on the blades. The strong adverse pressure gradient combined with a relatively large boundary layer-thickness at these sidewalls inevitably leads to flow separation. For smooth walls, there are established values of the boundary layer (momentum loss) thickness and of the pressure gradient, up to which the flow remains attached [4]. For surfaces with crosswise grooves however, an appreciably larger pressure gradient can be maintained without flow separation. In diffusers [3] and external body flows [6],[14],[11], the effectiveness of crosswise grooves was already demonstrated. Although such crosswise grooves increase the losses compared to that of a smooth surface with attached flow, the envelope in which the flow remains attached is significantly extended. Thus, a given pressure rise can be realized on a considerably reduced distance without substantial loss increase. Therefore, surfaces with crosswise grooves were investigated with respect to their applicability for flow control in axial compressors. The realization of a pressure increase with moderate losses resulting from the crosswise grooves will lead to a considerable improvement of the compressor stage efficiency of gas turbines. The C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 324–331, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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effectiveness of these crosswise grooves on the sidewalls of a compressor cascade was tested experimentally in a test facility with Mach-and Reynolds number similarity to the flow conditions in a middle stage of the compressor in an axial turbo machine.
2 Experiments with Simplified Diffuser Geometries In order to obtain better insight into the flow physics of the grooved geometries, two sets of experiments in simplified diffuser geometries were made. (a) Experiments in a conventional wind tunnel with a two dimensional Borda-Carnot diffuser: The aim of this investigation was to find an optimized groove geometry in order to minimize the reattachment length of the separated flow. Measurement of the static pressure on the diffuser walls and in between the grooves provided information about the efficiency of the different geometries. Reynolds number of this experiment was Re = 0.2 · 106, based on the conditions at the inlet diameter of the diffuser. The data from the wind tunnel experiments showed clearly that the existence of the grooves in between the blades is of great importance for the effect of pressure recovery. The blades are separating areas of different static pressure in the groove from the higher static pressure in the next groove downstream. Thus, a pressure recovery over a shorter length is possible. If the grooves between the ribs are filled
Fig. 1. Pressure recovery as measured in different wind tunnel diffuser configurations. 6 Reynolds number Re= 0. 2. 10 .
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Fig. 2. Velocity data and streamlines from PIV measurements in oil channel experiments, 4
Reynolds number Re = 1. 6·10 , left: reference configuration; right: configuration with grooves.
with a solid body as shown in Figure 1, the effect of a better pressure recovery is less pronounced. If the tips of the blades are covered with a cowling, the pressure is less pronounced. If the tips of the blades are covered with a cowling, the pressure recovery is of the same magnitude as for a diffuser with an opening angle of 35◦ . The experiments showed that the isolation of areas of different pressure levels in the grooves leads to a shorter reattachment length of the separated flow in the diffuser. (b) Experiments in an axially symmetrical Borda-Carnot diffuser in an oil channel: The experimental setup of the oil channel of the ILR TU Berlin is described in Brunn & Nitsche [1]. The Reynolds number of this experiment was Re = 1.6·104 , based on the conditions at the inlet diameter of the diffuser. In order to obtain deeper insight in the flow between the ribs, PIV measurements were carried out. Figure 2 shows a well pronounced vortex system in the grooves. In the grooves, stationary vortices are found, which are co-rotating with the mean flow velocity. This vortex system causes an expansion of the flow field. Thus, a more divergent behaviour of the streamlines is observed, which is equivalent to a retardation of the flow. This leads to a higher pressure increase with a shorter reattachment length of the separated flow.
3 Experiments in a High Speed Compressor Cascade The experimental investigation of grooved structures on the sidewalls of a compressor cascade (as shown in Figure 3) was carried out in a high-speed wind tunnel of DLR Institute of Propulsion Technology, Dept. of Turbine Acoustics in Berlin. The wind tunnel has a rectangular cross section of 40 mm width and 90 mm height. Flow velocities up to M = 0.7 with Re = 0.6 · 106 (based on 40 mm chord of the NACA 65 K48 profile) can be obtained. The total pressure losses in the cascade were measured with a wake rake (26 Pitot-probes to measure the total pressure distribution of the outflow and 4 Conrad-probes). To detect the region of separated flow on the vane, a flow visualisation technique was used. The main geometrical parameters and design flow conditions are given in Table 1. Boundary layer suction at all four channel walls was possible and could be adjusted independently (Figure 3). The suction at the upper and lower walls allowed the adjustment of the static pressure over the channel height.
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Table 1. Geometrical and design flow conditions M1 = 0.67 β1 = 132◦ βS = 112.5◦
c = 40 mm t/c = 0.55 h/c = 1
Fig. 3. (a) Test section with mounted cascade; (b) Rake with 26 total pressure probes and a rake with four Conrad probes combined with a static pressure probe
(a)
(b)
(c)
Fig. 4. (a) pressure distribution [7]; SL = separation line; cp = lines of constant static pressure; (b),(c) flow visualisation of secondary flow on the sidewall of a two dimensional compressor cascade with NACA 65 K48 stator vanes. Mach number M = 0.67, Reynolds number Re = 0.56 · 106
Thereby, a homogeneous inflow according to an ”infinite blade cascade” was achieved. A more detailed description of the cascade test facil-ity, including a specification of the measurement accuracy, is given by Hergt et al. [5]. To describe the losses in the axial compressor cascade, the local total pressure loss coefficient [2],[12] is used which is defined in equation (1) and equation (2) with a mass flow averaging.
ζ(u, z) =
pt1 − pt2 (u, z) q1
(1)
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ζm (u, z) =
m ˙ local · ζ(u, z) m ˙ total
(2)
Based on the local total pressure loss coefficient distribution it is possible to calculate the loss distribution ζ(z) in the vane height direction by integrating all losses over one pitch at each z-Position. The total pressure loss coefficient ζT of a cascade passage is computed by integration of the total pressure loss coefficient distribution ζ(z) in the vane height direction, which is composed of three loss components, as shown in equation (3):
ζT = ζBL + ζP + ζSP
(3)
These three pressure loss components are (i) the side wall boundary layer losses ζBL , (ii) the profile losses ζP caused by the friction of the flow around the vane profile (with infinite aspect ratio) and (iii) the losses due to corner separation ζSP caused by the interference between the wall and the vane boundary layer and the high positive pressure gradient in flow direction. 3.1 The Mechanism of Grooved Structure on the Sidewall of a Compressor Cascade The flow in an axial compressor stator stage (as shown in the schematic views in Figure 5) is complex and three dimensional. Therefore, a better understanding of the secondary flow effects is necessary to improve the efficiency of turbomachines. The corner separation is caused by the interference between the wall and vane boundary layers and the high positive pressure gradient in the flow direction. The grooved structure on the sidewall hampers that peculiar secondary flow, which feeds the corner separation with additional fluid. If the secondary flow from the pressure to the suction side of the vanes is less pronounced, the corner separation is also prominent. 3.2 Description and Effectiveness of the Groove Geometry on the Sidewalls of the Compressor Cascade The grooves are formed by inserting thin steel sheet strips with a thickness of 0.05 mm into the passage of the cascade. The sheet metal is inserted through the slits shown in Figure 5 and the depth can be varied arbitrarily. The slit pattern follows the streamline distribution of the flow in the middle of the passage at 50% of the vane height. The distance between two slits and the length of the slits are 5% and 50% of the vane chord length respectively, starting at the trailing edges of the vanes. A loss reduction with grooves on the sidewalls is only achieved if the geometry is designed properly. Eight grooves each with a height of 1 mm , on each side of the cascade, shows a reduction of the total pressure coefficient of 2%. If the grooves are all of an equal height of 4 mm and although some are protruding out of the secondary flow region, they decrease the pressure loss by up to 2.6%. Thus, it became clear, that
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equal groove height is not necessarily optimal, but rather decreasing groove height with increasing distance from the suction side of the vane, like shown in Figure 6. With this optimized configuration, the groove height was adjusted to the extension of the area, where a higher loss due to corner separation and secondary flow was observed. With this configuration a loss reduction of 9.8% was found, see Figure 7. Table 2. Nomenclature h M p pt q Re t u
[m] [−] [P a] [P a] [P a] [−] [m] [m]
vane height Mach number static pressure total pressure dynamic pressure Reynolds number pitch circumferential coordinate
z β1 β2 βS ζ λ ρ
vane height coordinate [m] inflow angle [◦ ] outflow angle [◦ ] stagger angle [◦ ] total pressure loss coefficient [−] aspect ratio [−] [kg/m3 ] density
Fig. 5. (a) Flow phenomena in a cascade passage according to Kang [8]; (b) secondary flow in a cascade passage; (c) Geometry of grooves on the sidewalls
(a)
(b) 6
Fig. 6. (a) Total pressure loss data of reference configuration, M = 0.67, Re= 0.56·10 ; (b) Schematic view of the configuration of optimized grooves
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(a)
(b)
(c) Fig. 7. (a) Total pressure losses of the cascade with optimized grooves; (b) Difference of total pressure loss data of reference and grooved configuration. The green colour indicates a lower total pressure loss. A loss reduction of 9.8% is obtained. (c) The total pressure losses and outflow angle ȕ2 plotted over the height of the stator vane. Mach number M = 0.67, Reynolds number Re = 0.56 · 106 .
4 Conclusion The use of grooved sidewalls shows an effective suppression of secondary flow on the sidewalls of the compressor cascade. A total loss reduction of 9% and of an increased flow turning angle of up to 2◦ in vicinity of the sidewalls was found. This corresponds to an efficiency increase of 1.3% in a comparable compressor stage. Further work will examine the behaviour of the grooves in off design conditions of the cascade.
Acknowledgements This research is funded by the German National Science Foundation (DFG) under the umbrella of the Special Research Activity (Sonderforschungsbereich SFB 557, ”Beeinflussung komplexer turbulenter Scherstr o¨ mungen”) at the DLR Berlin and is gratefully acknowledged.
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References [1] Brunn, A. & Nitsche, W.: ”Separation control by periodic excitation in a turbulent axissymmetric diffuser flow”, in: Journal of Turbulence 4-2003-003, IOP Publish-ing Ltd., April 2003 [2] Cumpsty, N.A.(2004): ”Compressor Aerodynamics”, Krieger Publishing Company [3] Eck, B. (1966): ”Technische Str¨omungslehre (hier: Untersuchungen zu diskontinuierlichen Oberfl¨achen mit hohen Druckgradienten”, von B. Regenscheit, p. 249), 7. Auflage, Springer-Verlag, Berlin, Heidelberg, New York [4] Fernholz, H. (1966): ”Eine grenzschichttheoretische Untersuchung optimaler Unterschalldiffusoren”, Ingenieur-Archiv, 35.Band, 3.Heft, pp 192-201 [5] Hergt, A., Meyer, R., Engel, K.(2006): ”Experimental Investigation of flow control in Compressor Cascades”, Proc. of GT2006 ASME Turbo Expo 2006: Power for Land, Sea and Air May 8-11, 2006, Barcelona, Spain [6] [6] Howard, F.G. & Goodman, L. (1985): ”Axissymmetric bluff-body drag reduction through geometrical modification”, AIAA, J. Aircraft, Vol. 22, No. 6, June 1985, pp 516522 [7] H¨ubner, J. (1996): ”Experimentelle und theoretische Untersuchungen der wesentlichen Einflussfaktoren auf die Spalt-und Sekund¨arstr¨omungen in Verdichtergittern”, PhDThesis, Universit¨at der Bundeswehr, M¨unchen [8] Kang S. & Hirsch, C. (1991): ”Three Dimensional Flow in a Linear Compressor Cascade at Design Conditions”, ASME 91-GT-114 [9] Meyer, R., Hage, W., Paschereit, C.O. (2006): ”Flow control with crosswise grooves in diffuser flows”, Proc. EFMC6 KTH Euromech Fluid Mechanic Conference 6, Stockholm, Sweden, June 26-30 [10] Migay,V.K. (1962): ”The aerodynamic effectiveness of discontinuous surfaces”, Inzhenero-Fisicheskiy Zhurnal Vol.5, No.4, pp 20-24 [11] Mair, W.A. (1965): ”The effect of rear-mounted disc on the drag of a blunt based body of revolution” Aeronautical Quarterly 16, Seite 350-360 [12] Scholz, N. (1965): ”Aerodynamik der Schaufelgitter”, Verlag Braun [13] Stark, U. & Bross, S. (1996): ”Endwall boundary layer separations and loss mechanisms in two compressor cascades of different stagger angle”, AGARD-CP 571, Paper 1 [14] Viswanath, P.R. (1995): Flow management techniques for base and afterbody drag reduction”, Progress in Aerospace Sciences 32, pp 79-129
Wave Drag Reduction Approach for Lattice Wings at High Speeds E. Schülein and D. Guyot*
*
DLR, Institut für Aerodynamik und Strömungstechnik, Bunsenstr. 10, 37073 Göttingen, Germany TU Berlin, Institut für Strömungsmechanik und Technische Akustik, Müller-Breslau-Str. 8, 10623 Berlin, Germany
Summary The investigations of the aerodynamic performance of new locally swept lattice wings include numerical simulations as well as wind tunnel measurements. The investigations were performed at free-stream Mach numbers from 2 to 6 for angles of attack varied from 0 to 10 degrees. The performance of the lattice wings was assessed on the basis of the zero-lift drag and lift-to-drag ratio. The numerical and experimental results show that the novel design of the lattice wings has distinct advantages in comparison to the conventional unswept configurations. Compared to conventional lattice wings the maximum benefit e.g. of the zero-lift total drag for the investigated locally swept lattice wings is of the order of 30% - 40%.
1 Introduction Lattice wings (also called grid fins) are compact constructions of several aerodynamic surfaces (members), which are held together by a common framework and build a versatile aerodynamic device. This device can be used as aerodynamic stabilizer, lifting or control element. Each member usually represents a very thin high-aspect-ratio rectangular wing of constant chord. In the last two decades, lattice wings have been the object of intensive investigations in the scientific community working on missile technology, [1]-[5]. At that time several research activities have been conducted also at the German Aerospace Center DLR. Some aspects of these activities are described in [6] and [7]. The reasons for the interest in lattice wing devices are their unique aerodynamic and structural characteristics. To their main advantages belong: 1) the high aerodynamic effectiveness at low weight and volume; 2) well adjustable aerodynamic characteristics for wide ranges of Mach numbers and deflection angles; 3) enhanced yaw stability at high incidence angles and improved roll stability; 4) small hinge moments with minimal shift of the centre of pressure. The biggest drawbacks of these control elements are their relatively high drag levels at given lift characteristics as well as the weak stability at transonic speeds. The reduction of the lattice wing’s drag is still very important and would mean a clear improvement of their performance. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 332–339, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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2 Proposed Wave Drag Reduction Approach The usual wave drag reduction approaches for conventional lattice wings are the minimization of leading edge bluntness and total surface thickness [1, 2], both typical for common nose wave drag reduction measures. The request for very thin surface elements with leading edges as sharp as possible lead to very high manufacturing costs for drag optimized lattice wings. Furthermore, the enormous thermal and mechanical loads on the sharp leading edges associated with higher supersonic speeds restrict the possibilities to use such strips of thin metal sheets as construction elements of grid fins. One of the characteristics of conventional grid fins is the unswept rectangularshaped plan form of its basic internal members. The advantages of swept leading edges for planar wings in contrast to unswept ones are well known. Their aerodynamic wave drag due to thickness at higher supersonic speeds is much smaller than that of a comparable rectangular wing. Another important advantage of the planar swept wing is a clear improvement of their stability at transonic speeds.
ÅÅÅÅÅ a.
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According to ideas presented in [8] the effective wave drag reduction for a grid fin can be reached if the leading edges of each internal member are locally swept. The plan-form of all members has to be similar to a row of delta wings located sideby-side. Figure 1 presents, beside the conventional lattice wing, the two most simple grid fin configurations with locally swept leading and trailing edges (LSE) that differ only in the position of the tooth relatively to the grid cell. The view direction agrees here approximately with the flow direction. Some other configurations could be created by varying the main geometrical parameters: the relative size of each periodical “tooth”, its number and position within the grid cell, as well as the local sweep angle. To the authors knowledge there are no investigations on similar locally swept grid fins up to now. This kind of lattice wings is expected to be more favorable for the use at high speeds than the conventional configurations.
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The object of these investigations isÅ a parametric study of locally swept wing Å configurations regarding their performance at high speeds. The preliminary Å investigations include numerical simulations of a supersonic flow around the Å conventional and locally swept planar and lattice wings of infinite span, which were performed within the framework of a diploma thesis [9], as well as wind tunnel experiments on corresponding realistic finite span models [10]. The examination of the planar wings additionally to the lattice controls shows the potential and the influence of each geometrical parameter that characterizes the LSE form – sweep angle as well as the relative tooth size in span and chord direction. These results are described in detail in [10]. The present paper includes some results for the locally swept lattice wings (LSLW) in comparison to the conventional unswept elements. The results show the effects of free-stream Mach number, angle of attack, local sweep angle and relative tooth position within the grid cell.
3 Geometrical Parameters of the Locally Swept Lattice Wings The conventional grid fin can generally be defined by a number of geometrical parameters such as span s, height h, chord c, member thickness t and cell spacing. For locally swept lattice wings at least two new additional parameters are needed for the description of the grid geometry [10]: the relative tooth position for crossing members and the number of teeth nt inside of each grid cell. The most simple configurations of locally swept grid fins (Fig.1b, c), have a single tooth per cell (nt = 1) and two different arrangements of teeth at the intersection of crossing members. The first one in Figure 1b presents a valley-type configuration and the second – a peak-type (Fig.1c). The logic applied for this definition is explained in Figure 2, where two regular types of intersection for crossing locally swept planes are presented in detail. The flow direction in Figure 2 is meant to come from top to bottom. For the peak-type both crossing teeth have a common peak between four valleys (Fig.2a), and for the valley-type correspondingly a common valley between four peaks (Fig.2b). These two types of member intersection should be sufficient for all symmetrical cases with integer numbers of teeth in each grid cell. Figure 3 sketches the single locally swep t grid fin member with a hexagonal cross section. Apart from the half-angles of the leading (LE) and trailing (TE) edge sharpness ßnLE and ßnTE, measured in the planes normal to the edges, their geometry is characterized by the sweep angle M, tooth length ct and nominal wing
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chord cnom. The relative tooth length [ marks the relation of the tooth length to the wing’s chord ([ = ct /cnom) and characterizes the global plan-form of the locally swept member. All these parameters together describe the geometry of a LSLW. A-A
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Accordingly to the definition given in [10] the common name for the investigated LSLWs should include some main parameters, which allows a sufficient description of the geometry in the shortest form: LSLWtype_En_M_[. The type means the kind of plane intersection: “p” for peak- and “v” for valley-type. For example, the name LSLWv2_17°_55°_0.714 (Fig.1b) defines a double valley-type wing with En = 17°, M = 55° and [ = 0.714. The definition double marks in this case that both edges (leading and trailing edge) are locally swept. The designation of a conventional rectangular lattice wing is defined simple as RLWE.
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The numerical simulations have been performed with the TAU-code, an internal development of the German Aerospace Centre (DLR), which solves threedimensional Navier-Stokes equations using a finite volume approach and can handle structured, unstructured and hybrid meshes. Three different model configurations, the reference unswept wing and both main types of novel lattice wings (peak- and valley-type), were investigated numerically. The first important simplification of the numerical models was the neglect of the outer framework influence, so that only cut-out segments from grid fins of infinite height h and span s were investigated. And the second one was the free-slip condition for the model walls. So the CFD results show wave drag of infinite grid fins, not accounting for the friction drag. These simplifications were made for fast and effective numerical calculations of numerous investigated wing configurations and flow conditions. These conditions are sufficient to prove the effect of the local leading edge sweep on the wave drag reduction without the relinquishment of the important flow phenomena.
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Figure 4 shows the effect of the leading edge sweep on the zero-lift wave drag (left) and on the zero-lift wave drag reduction (right) for the investigated Mach number range. To quantify the improvement of the performance of LSLW in comparison to the well known conventional lattice wings, the relative profit e.g. in the wave drag is presented here as: Relative wave drag reduction cWD LSLWtype_E _ M_ȟ cWD RLWE 100% . cWD RLWE
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It is very impressive that both locally swept wings clearly show a more favorable total drag in the investigated Mach number range. The profit in the wave drag lies between 32% and 42%. Furthermore, the results show that according to zero-lift wave drag the wing of the peak-type is somewhat better than other LSLW. This effect has been analyzed in more detail with the inspection of the surface pressure and Mach number distributions in [10].
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At incidence angles the improvement in the lift-to-wave-drag ratio relative to the ones of the unswept lattice wing (Fig.5) grows with Mach number. This improvement is determined similarly to the drag reduction relation used above. At D = 5° the profit reaches values between 25% and 37%. Different types of locally swept wings show different characteristics. Furthermore, in opposite to the impression one gets from the zero-lift behavior, the wing of the valley-type is suddenly more effective than one of the peak-type. This result is not unambiguously correlated with the described observations at D = 0° and therefore is somewhat unexpected. Responsible for this reversion in the effect of the favorable wing configuration is the enormous increase of the lift coefficient for the valley-type of the LSLW. So at incidence of 5 degrees this wing shows about 8% higher lift coefficients than the rival. At D = 10° this gain increases to ~9%.
5 Experimental Results The force measurements were conducted in the Ludwieg Tube Facility RWG. Unfortunately, for technical reasons the wind tunnel models ordered with the same geometrical parameters than the grid configurations investigated numerically were manufactured with unacceptable accuracy concerning the sharpness of the edges. Therefore, in these preliminary tests only accepted models were investigated experimentally (Fig.1). The models tested include a reference unswept lattice wing RLW17° (a) and two locally swept configurations: LSLWv2_17°_55°_0.714 (b) and LSLWp2_17°_55°_0.714 (c). Different from the numerical investigations, the half-angle of the sharpness En was here 17°, and not only the leading but also the trailing edges of the wind tunnel models were locally swept. These differences allow only qualitative comparisons of numerical and experimental results. As expected, the experimental results show also more favorable total drag for both LSLW-configurations and confirm the expected trends in the Mach number (Fig.6). On the one hand, the effect in total drag is certainly weaker than in pure wave drag predicted numerically. On the other hand the effect was reinforced due the higher values of the angle En. The results obtained show after all a reduction of
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the total drag between 32% and 38%. The presented experimental data show that the advantages of the peak-type configuration versus the rival are just as predicted. The gain in the total drag lies between 3% and 5%. It is possible that this effect is stronger in the measurements than in the CFD results due to the “double-tooth” design of the wind tunnel models as mentioned above.
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Fig. 7. Effect obtained experimentally at Į = 5° (unswept wing: RLW17° , valley 2-type: LSLWv2_17°_55°_0.714 and peak2-type: LSLWp2_17°_55°_0.714)
The lifting performance of the locally swept lattice wings is presented in Figure 7 on the example of results obtained at D = 5°. Apart from mainly lower absolute values, compared with the calculated results from the lift-to-wave-drag ratio, the experimental results provide a full confirmation of the numerically predicted effects. So, the “valley2”-type wing shows the highest values of the lift-to-drag ratio in comparison to the unswept wing. The values in the profit for the “valley2”-type lie between 16% and 21% and for the “peak2”-type wing between 10% and 15%. Consequently, the advantages of the locally swept lattice wings have been confirmed in the present work not only numerically but also experimentally.
6 Conclusions Å Å The
investigation of locally swept lattice wing configurations shows an essential Å improvement of the high speed performances for these novel control elements. Å The most important results can be summarized as follows:
The profit gained for the investigated realistic locally swept lattice wing configurations in comparison to the conventional ones in the zero-lift total drag amounts up to 38% and in the lift-to-drag ratio up to over 20%. The lowest zero-lift wave drag is shown by the lattice wing construction with the peak-type of locally swept lattice wings (LSLWp and LSLWp2). On the contrary, better lift-to-drag performance at wing incidence is shown by the valley-type of LSLW (LSLWv and LSLWv2).
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The effect of the LSE increases with the free stream Mach number, the local sweep angle, and the relative thickness of the members and the bluntness of their leading edges. It decreases with the incidence angle and for bigger relative tooth-sizes. Å
References
[1] S.M. Belotzerkovsky et al., “Reschetchatye Kryl’ya”, Moscow, Mashinostroeniye, 1985 (in Russian); see also: S.M. Belotzerkovsky et al., “Wings with internal Framework,” Machine Translation, FTD-ID (RS) T-1289-89, Foreign Technology Div., 1987. [2] W.D. Washington and M.S. Miller, “An Experimental Investigation of Grid Fin Drag Reduction Techniques,” AIAA Paper 93-0035, Jan. 1993. [3] G.M. Simpson and A.J. Sadler, “Lattice Controls: A Comparison with Conventional, Planar Fins”, Missile Aerodynamics, Meeting Proceedings RTO-MP-5, Paper 9, 1998, pp. pp. 9.1–9.11. [4] W.D. Washington and M.S. Miller “Experimental Investigations of Grid Fin Aerodynamics: A Synopsis of Nine Wind Tunnel and Three Flight Tests”,Missile Aerodynamics, Meeting Proceedings RTO-MP-5, Paper 10, 1998, pp. 10.1–10.13. [5] R.W. Kretzschmar and J.E. Burkhalter, “Aerodynamic Prediction Methodology for Grid Fins”, Missile Aerodynamics, Meeting Proceedings RTO-MP-5, Paper 11, 1998, pp. 11.1–11.11. [6] Ph. Reynier and E. Schülein, „Incorporation of an Actuator Disc for Lattice Wing Modeling in an Unstructured Navier-Stokes Solver,”New Results in numerical and experimental fluid mechanics IV, edited by Ch.Breitsamter et al., Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Volume 87, Springer, Berlin 2004, pp.132-139. [7] Ph. Reynier, J.-M. Longo and E. Schülein, „Simulation of Missiles with Grid Fins Using an Actuator Disk,” Journal of Spacecraft and Rockets , Vol.43, No.1, Jan.-Feb.2006, pp. 84-91. [8] E. Schülein, “Wing for an Aircraft or Spacecraft”, US Patent No. 7 114 685 B1 [9] D. Guyot „Experimentelle und numerische Untersuchung zum Einfluss der Lamellengeometrie auf die Leistung eines Hochgeschwindigkeitsgitterflügels,“Diploma thesis, TU Berlin, February 2005 (in German). [10] E. Schülein and D. Guyot, “Novel High-Performance Grid Fins for Missile Control at High Speeds: Preliminary Numerical and Experimental Investigations”,Innovative Missile Systems, Meeting Proceedings RTO-MP-AVT-135, Paper 35, 2006, pp. 35-1 – 35-26, RTO, Available from: http://www.rto.nato.int/abstracts.asp.
Forces and Velocity Measurements in Ship Propulsion Systems Jo˜ao Pˆego1 , Hermann Lienhart2, Franz Durst2 , and Kay Tigges3 1
2
Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Lehrstuhl f¨ur Str¨omungsmechanik (LSTM), Universit¨at Erlangen-N¨urnberg, Cauerstraße 4, 91058 Erlangen, Germany 3 Siemens AG, ATD IS 3 Postfach 10 56 09, 20038 Hamburg, Germany
Summary Pod drives are modern outboard ship propulsion systems with a motor encapsulated in a watertight pod. The motor’s shaft is connected directly to one or two propellers. The whole unit hangs from the stern of the ship and rotates azimuthally, thus providing thrust and steering. In the past decade pod drive propulsion has developed to be a reliable and serious alternative to conventional ship propulsion, especially for cruiser liners [1,2]. The overall efficiency of the system is determinant for the economical success of the vessel and for this reason it is necessary to improve the efficiency of the propulsor recurring to non-conventional solutions. Force and phase-resolved LDA measurements were performed for inline co-rotating and contra-rotating propeller pod drive models. The measurements permitted to characterize these ship propulsion systems in terms of their hydrodynamic performance and lead to their efficiency optimization. The most efficient co- and contra-rotating propellers were subjected to detailed study of the swirling wake flow by means of laser Doppler anemometry.
1 Introduction Advancements in ship propulsion systems to increase their efficiency have reached a state of research and development that requires the combined employment of analytical, numerical and experimental techniques. For these reasons an extended test facility was set up inside and around the water tunnel of LSTM-Erlangen consisting of torque and force measuring techniques as well as a laser-Doppler system to study not only integral forces but also the mean and turbulence quantities in the wake flow of ship propellers. Based on a design by Siemens AG, a set of 24 pod drive models was studied [3] using the ship propulsion research facility at LSTM [4]. This research aimed to improve the efficiency of the propulsor, for which the produced thrust and delivered torque were measured with the water tunnel balance. Pod drive models with tandem in line coand contra-rotating propellers were used for the experiments. The measurements have C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 340–347, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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shown the influence of the pod geometry on the propulsor overall efficiency and also proved that contra-rotating propellers provide higher efficiency. This increase of efficiency can be either used to increase the speed of the ship without increasing fuel consumption or to reduce the consumption of fuel without reducing the ship’s speed. The present article discusses the results of the force and phase-resolved LDA measurements performed for two of the pod drive models, these being the most efficient coand a contra-rotating propeller models (CoRP and ContraRP).
2 Test Rig and Measurement Technique 2.1 Water Tunnel The experiments were performed in the closed loop water tunnel at LSTM-Erlangen, which is shown in Fig. 1. The test section has a rectangular shape with a width of 800 mm and a height of 1000 mm in a longitudinal extension of 8800 mm. Large windows of glass build the side, bottom and top walls of the test section, thus providing unrestricted optical access to the interior. Continuous control of the flow rate was achieved by setting the pitch angle of the pump’s impeller blades.
Fig. 1. Pod drive model and balance in the water tunnel
2.2 Ship Propulsion Models The ship propulsion models tested were two pod drive models with co- and contrarotating propellers. Fig. 2 presents the most relevant components of the models, showing also the balance used for the force measurements to determine the pod drive models efficiency. A light barrier inside the balance delivered a TTL signal marking the beginning of every new turn of the motor. This signal was used to measure the propeller’s frequency and also for the phase-resolved LDA measurements. The propellers had a diameter of 250 mm and were designed to accomplish the criteria of optimum efficiency, while avoiding cavitation. The procedure used for the design of the propellers, pod and water tunnel balance has been previously reported ([4], [3]) and should be referred to for detailed information. Apart from the rear propeller rotation sense, both pod drive models were designed with the same criteria. For the given propellers diameter the blockage ratio was 6% of the water tunnel test section.
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1 – DC motor 2 – Force gauge 3 – Torque sensor 4 – Toothed belt 5 – Front propeller 6 – Rear propeller 7 – Strut 8 – Balance Fig. 2. Pod drive model and water tunnel balance
2.3 LDA System and Signal Processing For this set of measurements a two-component Aerometrics Phase Doppler Particle Analyser (PDPA) was adapted to operate as LDA. The probe created a measuring volume of about 100 µm diameter at a distance of 665 mm from the front lens in water. This set-up allowed simultaneous two-component velocity measurements in backscatter mode. The LDA signals were then processed by two Dantec 57N20 BSA enhanced units, operating in synchronous mode. The complete experimental set-up used during the LDA measurements is presented in the diagram of Fig. 3.
Fig. 3. LDA measurements setup showing pod drive model in the water tunnel
3 Discussion 3.1 Open Water Efficiency Fig. 4 shows the characteristic curves of the two pod drive models, which had the same distance between propeller ratio, LP rop /D1 = 1.248, used for the propeller design. Here LP rop is the distance between propeller planes and D1 is the front propeller diameter. For co-rotating propellers only synchronous propellers were considered, because this is the only meaningful practical solution. If the propellers weren’t synchronous
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a gear would have to be built, thus having the complexity of contra-rotation without its benefits. For contra-rotating propellers the highest efficiency was obtained for n2 /n1 = 0.9, where n is the propeller frequency and indexes 1 and 2 refer to the front and rear propeller. For these reasons these models were chosen for comparison. The graphs show the values of the thrust coefficient, KT = ρDT4 n2 , torque coefficient KQi =
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Here, T is the produced thrust, Qi the delivered propeller advance ratio J = torque, Va is the propeller advance speed and ρ is the fluid density. The Reynolds number, Re0.7R1 , based on the chord length of the front propeller blade section at radius (r/R1 = 0.7) and the section advance speed, VS , was about 380 000. R1 = D1 /2 is the front propeller radius and the section advance speed at radius r/R1 = 0.7 was computed according to VS = Va2 + (πn1 · 0.7D1 )2 . Measurements were taken for a front propeller frequency of revolution of 7.5 Hz. The results presented here are average values of 10000 samples, which correspond to the following relative statistical errors: 0.5% for the thrust, T , 0.3 % for the torque, Qi , and 0.3% for the advance speed, Va . Accounting for error propagation in measurements, the expected maximum relative statistical errors for the quantities presented in the graphics are: KT – 0.5%, KQ – 0.3% and ηO – 0.37%. The figures show very distinctive patterns for the CoRP and the ContraRP models. While for the co-rotating propeller model the load distribution between front and rear propeller (see the KQ1 and KQ2 curves in figure 4(a)) was rather unbalanced, this was not the case in the CoRP model. It is also possible to conclude that the ContraRP was less loaded (lower values of KT , KQ1 and KQ2 ). Yet, the most striking difference occurs with the efficiency curve of the two models. The CoRP model presents a curve, which is flat near the highest efficiency at J = 0.98 and reaching lower values. The ContraRP model shows a narrower curve with an efficiency always superior to the CoRP model in the whole range of advance ratio and with the maximum efficiency located at J = 0.85. The increase of efficiency from one model to the other is about 6%, which is fairly good.
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3.2 Mean Flow Fig. 6 and 7 show the distribution of the mean axial and mean tangential velocity components in the wake of the CoRP and the ContraRP pod drive models, measured using the phase-resolved LDA technique. Two axial positions were considered, corresponding to x/D1 = 1 and 5, where x is the distance from the rear propeller plane Each line in the radial distribution representation corresponds to an angular position in the bladefixed frame of reference. To avoid a too dense graphic representation of the data, only 6 of the 60 measured angular positions are plotted. Their location in the blade-fixed frame of reference is shown in Fig. 5 where the line styles identify the lines of figures Figs. 6 and 7. All velocity components presented were normalized by the propeller advance speed, Va , which in the experiment corresponded to the inflow velocity of the water tunnel.
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The idea of using contra-rotating propellers to improve the efficiency of the ship propulsor is triggered by the fact that the rear propeller can be used to regain some of the energy lost to the swirl in the slipstream of the front propeller. Contra-rotating propellers thus should produce higher thrust and weaker swirl in the wake flow. Comparing the results of the mean axial velocity at one diameter distance from the rear propeller plane it is clear how in the contra-rotating propeller pod drive model the axial momentum reaches higher values, while presenting a more uniform distribution. Near the axis, the mean axial velocity shows a strongly marked depression, which is present in both the CoRP and the ContraRP pod drive models. While for the CoRP the velocity drops to almost half of the propeller advance speed, Va , for the ContraRP a model the velocity never falls below U Va = 1. The low values of axial momentum near the propeller axis evidence the velocity defect due to the influence of the propeller hub, which is not producing thrust. In this region there is nearly no dependence of the phase, θ, that is, the velocity here is independent of the propeller position. At a radial position of about 80% of the propeller radius a strong dependence of the phase is evidenced by the changing profiles. In both cases the tip vortex is made visible by a solid body rotation pattern in line θ = 80◦ at position r/R1 ≈ 0.8 (Figs. 6(a) and 6(b)). In the far wake the angle dependency of the flow is much attenuated, which is recognized by the overlapping of the six curves in Figs. 6(c) and 6(d). This results in a
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weak correlation between the rear propeller angular position and the flow field in the far wake. In both the co- and the contra-rotating propellers models the effect of distance in the axial momentum profile is to make it more uniform and to flatten the depression in the axis. Yet, the mean axial velocity distribution is much different in both cases. The CoRP model shows a distribution which is characterized by the hub velocity defect at inner radii and an area of momentum increase in the interval 0.2 < r/R1 < 1.0, which coincides with the area of thrust production. The distribution of mean axial velocity in the case of the ContraRP model is much more homogeneous at this axial position and the hub velocity defect is imperceptible. The distributions of mean tangential velocity component (see Fig. 7) are rather interesting because they differ very much when comparing the co- and the contra-rotating propeller models. The former presents a mean tangential velocity distribution that has the shape of a concentrated vortex, which extends until r/R1 = 0.22, asymptotically decaying with increasing radius (see Fig. 7(a)). Near the propeller plane the swirl intensity is high and the mean tangential velocity component reaches values of the same order of magnitude as the advance speed, Va . The swirl intensity decays fast with the distance from the propeller and is about half of the previous value at position x/D1 = 5 (see Fig. 7(c)). A far different distribution results for the contra-rotating propeller model (Figs. 7(b) and 7(d)). In this case the swirl intensity is much smaller than before. Apart from a restricted area near the axis and tips, and only for the very near wake
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(x/D1 = 1), the mean tangential velocity vanishes everywhere. This means that the rear propeller in the ContraRP model was able to eliminate completely the swirl in the slipstream of the front propeller. Though this was indeed the aim of the design, it is nevertheless surprising how efficient this pod drive model behaved.
4 Conclusions and Remarks A new experimental facility ([4], [3]) for detailed investigation of force and flow measurements of ship propulsion systems has been introduced. This facility has been built in the water tunnel at LSTM-Erlangen. It is equipped with a water tunnel balance for force/momentum measurements and a drive unit for single and tandem propellers. For the detailed research of ship propulsion systems a laser Doppler anemometer was used. Phase-resolved LDA measurements in the slipstream of two pod drive models (with tandem co- and contra-rotating propellers) were presented for two measuring stations, one in the near and the other in the far wake. The radial distribution of the mean axial and tangential velocity components were discussed. It has been shown that the contrarotating propeller pod drive model produces a more homogeneous wake, with higher increase in axial momentum and almost no swirl component. Compared with the corotating propeller pod drive model it produces more thrust, while losing less energy in swirl production.
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Acknowledgements The authors gratefully acknowledge the support from the Fundac¸a˜ o para a Ciˆencia e Tecnologia, Portugal, through grant SFRH / BD / 3394 / 2000 to finance the stay of Jo˜ao Pedro Pˆego at LSTM-Erlangen and from SIEMENS AG to carry out the development work described in this paper.
References [1] F. A. Mewis: “Pod drives - pros and cons”. HANSA - Schiffahrt - Schiffbau - Hafen, 2001, 138, 25-30. [2] J. Friesch: “Investigations of podded drives in a large cavitation tunnel”. In PRADS 2001, Shangai, China, September 2001. [3] J. P. Pˆego: “Advanced Fluid Mechanics Studies of Ship Propulsion Systems”. PhD thesis, Technische Fakult¨at Friedrich-Alexander Universit¨at Erlangen-N¨urnberg, 2007. [4] J. Pˆego, H. Lienhart, F. Durst, and O. Badran: “Construction of a test facility for the research of ship propulsion systems”. Emirates Journal for Engineering Research, 2005, 10(2):1–8.
Numerical Simulation of the Flow Around a Finite Cylinder with Ground Plate in Comparison to Experimental Measurements Octavian Frederich1 , Erik Wassen1 , Frank Thiele1 , Mario Jensch2 , Martin Brede2 , Frank H¨uttmann2, and Alfred Leder2 1
Berlin University of Technology, Institute of Fluid Mechanics and Engineering Acoustics, M¨uller-Breslau-Str. 8, 10623 Berlin, Germany
[email protected] 2 University Rostock, Chair of Fluid Mechanics, Albert-Einstein-Str. 2, 18051 Rostock, Germany
Summary Simulations and experiments were performed to capture the spatio-temporal flow field around a finite circular cylinder mounted on a ground plate. In order to provide a combined database and testcase for future simulations and experiments, the flow is investigated using state-of-the-art techniques with a high resolution in time and space, namely Large-Eddy Simulation and Detached-Eddy Simulation for the numerics and time-resolved PIV as well as LDA for the measurements. The predicted time-averaged and unsteady flow field from simulations corroborate well the experiments, giving new insights into the complex turbulent separated flow behind a quite simple geometry.
1 Introduction Unlike the flow around an infinite circular cylinder, the spatio-temporal flow field around a wall-mounted finite circular cylinder is characterised by complex structures and includes different flow phenomena like horseshoe vortex, separation, turbulent wake, transition, etc. [1], which interact with each other. On this account such a flow configuration at Reynolds number ReD = 200 000 has been chosen in the scope of projects funded by the German Research Foundation (DFG) as a basis for all investigations in the research program “Imaging Measuring Methods for Flow Analysis”. Concomitant to the experiments using time-resolved PIV (TR-PIV) and LDA, simulations with high resolution were to be performed. Therefore, a Direct Numerical Simulation (DNS) would be desirable, but due to the present Reynolds number only Large-Eddy Simulation (LES) and the hybrid approach Detached-Eddy Simulation (DES) can be considered for the description of the spatio-temporal transport processes. These methods are able to resolve the dominant, energy-containing structures in the separated turbulent flow with an affordable high resolution. The results from numerical simulations and data obtained experimentally for the flow field around the finite cylinder is intended to provide a combined database useful for validation and development of future modelling or measuring methods. On the way C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 348–355, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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to this objective, the analysis of a complex spatial flow field by imaging measuring methods is to be supported and the capabilities of numerical flow prediction and the influence of the model is to be investigated. Following the description of the investigated configuration in section 2, an overview of the numerical method and parameters for the simulations is given in section 3. The simulation results and the comparison to experimental measurements are presented in section 4.
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As an example for a complex three-dimensional and unsteady flow configuration a wallmounted finite circular cylinder (figure 1) is chosen. The present Reynolds number of ReD = 200 000 is based on the cylinder diameter D = 0.12 m, the inflow velocity U∞ = 26 m/s and the kinematic viscosity ν of air. The cylinder with an aspect ratio of L/D = 2 is mounted with its center 1.5D downstream of the beginning of a flat plate in order to minimise the influence of the plate’s boundary layer. The plate has a rounded leading edge and is 1.3 m long. At the smooth transition from the rounded to the flat part of the plate a trip wire is included to fix the laminar-turbulent transition. The described cylinder configuration is investigated experimentally in an open wind tunnel of the G¨ottinger type at the University Rostock using LDA and time-resolved PIV currently. The turbulence level of the inflow was measured with Tu = 0.5 %. The coordinate system used with x in streamwise, y in lateral and −z as cylinder axis is shown in figure 1. Reynolds-number Turbulence level
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hanging nodes was generated, whereby the transition on the plate is realised by a geometrically modelled wire. The spatial resolution at the wall was estimated such that the wall coordinate of the first grid point form the wall is below y + = yuτ /ν = 1, and the first five grid points are below y + = 5. The results show a maximum wall coordinate of ymax = 0.57 . A slip condition is applied to the upper and lateral boundaries, and a convective outflow condition to the downstream boundary. No-slip conditions are applied to all physical walls. A spatially variable velocity inlet profile obtained from an additional simulation with an extended upstream region is used, accounting for the blockage of the cylinder. The temporal discretisation is based on the required time step of ∆t = 0.005 UD∞ obtained from initial simulations. Using this time step, a Large-Eddy Simulation with the standard Smagorinsky subgrid-scale model, Cs = 0.1, and a Detached-Eddy Simulation with LLR k-ω background model [5] were carried out. The investigations were performed using the code ELAN developed at the Berlin University of Technology. This numerical procedure uses a conservative finite-volume discretisation based on general curvilinear coordinates of the Navier-Stokes equations for incompressible and compressible flows. The spatial discretisation of the diffusive and convective terms is realised using central differencing, and a backward difference quotient of second order accuracy is used for the temporal derivative [6]. The code is able to handle all common simulation approaches like U-RANS, DES, LES as well as Direct Numerical Simulation. The code has been parallelised in order to be used on today’s most powerful massively-parallel computer architectures.
4
Results
The numerical simulations in the present work were carried out in close collaboration with experiments conducted at the Chair of Fluid Mechanics at University Rostock. In figure 4 a comparison is shown between numerical and experimental results – obtained by LES resp. LDA – for profiles of the time-averaged velocity components at different positions in two planes aligned orthogonally in the wake. The overall agreement of
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separation recirculation vortex side tip vortex
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the results is very good, especially taking into account the highly three-dimensional turbulence around this configuration. In order to analyse the unsteady flow pattern first the time-averaged flow has been evaluated extensively as a reference. In figure 2 a and b the unsteady and time-averaged vortex structures predicted by LES are illustrated using the vortex criterion λ2 [3]. While the instantaneous snapshot reveals the unsteadiness of the flow, the time-averaged evaluation allows for the detection of the vortex shedding regime. The identification of the individual topological structures (figure 3) and their quantification is based on a detailed analysis of the time-averaged flow field obtained by LES. Due to the good agreement of the experiments [4] and LES the extracted time-averaged vortex shedding regime is appropriate for both. Starting with the classification of the vortical structures at the inflow, the boundary layer on the plate stagnates in front of the cylinder, constituting a horseshoe vortex. At the leading edge of the cylinder top the oblique separation occurs and the flow reattaches close to the leading edge, surrounding a very complex recirculation region on the top. The flow on the cylinder shell separates close to 80o , starting to generate a typical alternate vortex shedding. This vortex street is disturbed massively by the flow over the free end, and their combination leads to a recirculation arch vortex with tornado-like regions downstream of the cylinder. Due to this recirculation region the main flow reattaches to the plate (in the symmetry plane close to x/D = 2.9). Near this reattachment region trailing vortices are generated by the lateral velocity components of the sideways vortex shedding and the flow down to the plate. The analysis reveals that for this small cylinder aspect ratio no coherent longitudinal vortices exist in the whole wake as stated in most investigations on finite cylinders. Instead the two side tip vortices separating from the cylinder top are merged into the recirculation region, and the trailing vortices are generated after the reattachment. The recirculation regions in the wake force surrounding fluid towards the symmetry plane and the plate. This induces a large-scale vortical
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motion resembling longitudinal vortices, as indicated by the λ2 -criterion in figure 2 b between the recirculation region and the trailing vortices. However, while the agreement between experiments and LES is very good, the DES predicts a spatially smaller wake region. By expecting a laminar separation of the boundary layer on the cylinder shell with transition following closely, the problems of the DES arise from the production of eddy viscosity by the RANS model in the attached boundary layer. This problem of the modelling approach itself leads to a particular turbulent boundary layer, which stays attached too long and reduces the wake size in lateral and streamwise direction. To overcome this wrong behaviour, a transition
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model has to be used or the production term of the k-equation has to be deactivated locally which is nontrivial due to the interaction of several boundary layers. A quantitative comparison of numerical and experimental results is shown in figure 5 for time-averaged velocity components and RMS values using an exemplary line in the cylinder wake. The agreement of the numerical values obtained by LES and the LDA measurements are very good. The newly available TR-PIV measurements [2] are still influenced by methodic errors in the PIV evaluation and interpolation to a coarse grid, but agree also well with LES and LDA results. The DES results predict the right
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Table 1. Strouhal numbers for the first and second dominant frequency f1 and f2 compared between LDA [4], TR-PIV [2] and numerical results St for f1 St for f2 LDA PIV LES DES
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tendency but are affected by the wrong boundary layer treatment on the cylinder shell. Especially, the extremely good congruence of the RMS values of LES and LDA make clear how well the numerical prediction and the experimental measurements of the unsteady flow match. The spectral analysis of local velocity and pressure signals as well as of global force coefficients expose a dominant frequency corresponding to the Strouhal num= 0.16 found in numerical and experimental data. In addition, a second ber of St = Uf D ∞ frequency can be recognised in the wake with St = 0.20 (table 1). This is typical for the flow around infinite cylinders and confirms that there are relicts of the typical van K´arm´an vortex formation. Both frequencies will be used to synchronise the unsteady flow data of the simulation and experiments and to perform a phase-averaging of the whole flow field.
5 Conclusion Simulations with high resolution in time and space were performed for the turbulent flow around a wall-mounted finite cylinder. The comparison to concomitant experiments show a very good agreement of LES and measurements with respect to the time-averaged flow, statistical moments and dominant frequencies. It was shown that the vortical flow around the finite circular cylinder is a superposition of several vortex structures. This result is also predicted by the DES but with reduced wake size due to the modelling approach in the attached boundary layer. The ongoing work and further research will be concentrated on the analysis and understanding of the unsteady flow pattern. Therefore, phase-averaging, POD, structure tracing and uncertainty visualisations will be applied to numerical and experimental results.
Acknowledgements The work presented is supported by the German Research Foundation (DFG) within the scope of the research project ”Imaging Measuring Methods for Flow Analysis” ( Bildgebende Messverfahren f¨ur die Str¨omungsanalyse”). All simulations were performed on the IBM pSeries 690 Supercomputer of the North German Cooperation for High-Performance Computing (HLRN). Both organisations we thank cordially for their support. ,,
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References [1] J.H. Agui, J. Andreopoulus: “Experimental investigation of a three-dimensional boundarylayer flow in the vicinity of an upright-wall mounted cylinder”. ASME J. Fluids Engineering, 114(4):566-576, 1992. [2] M. Jensch, M. Brede, F. Richter, A. Leder: “Verwendung des Time-Resolved StereoPIV Messsystems zur Ermittlung zeitaufgel¨oster Geschwindigkeitsfelder im Nachlauf eines Kreiszylinders”. In: D. Dopheide, H. M¨uller, V. Strunck, B. Ruck, A. Leder (eds.), “Lasermethoden in der Str¨omungsmesstechnik - 14. Fachtagung der GALA e.V. 2006”, PTB Braunschweig, pp. 39.1-39.8, 2006. [3] J. Jeong, F. Hussain: “On the identification of a vortex”. J. Fluid Mech. 285:69-94, 1995. [4] F. Richter: “Experimentelle Untersuchungen zur Charakterisierung der Str¨omungs- und Turbulenzstrukturen im Nachlauf eines Kreiszylinderstumpfes unter Ber¨ucksichtigung der Zentrifugalbeschleunigung”. Dissertation, University Rostock, Monsenstein und Wannerdat, M¨unster, 2005. [5] T. Rung, F. Thiele: “Computational modelling of complex boundary-layer flows”. In Proc. of 9th Int. Symp on Transport Phenomenain Thermal-Fluid Engineering, 1996. [6] L. Xue: “Development of an efficient parallel solution algorithm for the three-dimensional simulation of complex turbulent flows”. Dissertation, TU Berlin, 1998.
Numerical Simulation of Aerodynamic Problems with the SSG/LRR-ω Reynolds Stress Turbulence Model Using the Unstructured TAU Code Bernhard Eisfeld Institute of Aerodynamics and Flow Technology, German Aerospace Center (DLR), Lilienthalplatz 7, 38108 Braunschweig, Germany
[email protected]
Summary The SSG/LRR-ω differential Reynolds stress model has been implemented into DLR’s unstructured flow solver TAU. The model is a blend between the Speziale-Sarkar-Gatski model (SSG) in the far field and the Launder-Reece-Rodi model (LRR) near walls, combined with Menter’s baseline ω-equation for the length scale. The implementation has been checked for the transonic flow around the RAE 2822 airfoil, comparing with the results obtained with DLR’s structured flow solver FLOWer. In the following the model is applied to the L1T2 three-element airfoil at high incidence, demonstrating the model’s applicability to 2D high-lift problems. Furthermore results are shown for the transonic flow around the two wings of the Third AIAA Drag Prediction Workshop and the ONERA M6 wing.
1 Introduction Virtually any technical flow is turbulent, so that the turbulence modelling is decisive for the accuracy of the predictions. In the field of aircraft aerodynamics there is a particularly high demand for precise predictions of the drag, being composed of friction drag, induced drag and wave drag. Consequently, boundary layers, free vortices and shock locations must be accurately predicted. Furthermore the simulation of high-lift flows, especially the break down of lift, requires the correct prediction of flow separation. Ideally a turbulence model should be able to cope with all of these phenomena, but yet this has been found very difficult to achieve. Simulations of technical flow problems are usually based on the Reynolds averaged Navier-Stokes (RANS) equations, and most turbulence models used in practice still rely on the Boussinesq hypothesis, i. e. it is assumed that the Reynolds stress tensor be aligned with the mean strain tensor. The proportionality coefficient is called eddy viscosity and is added to the molecular viscosity of the fluid. In aircraft aerodynamics the Spalart-Allmaras model [10] and the k-ω models of Wilcox [13] and Menter [8] are probably the most popular eddy viscosity models. They involve one or two additional transport equations, respectively, for which efficient numerical solution techniques have been developed [5]. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 356–363, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Despite their merits for predicting attached and to some extent even separated flows, eddy viscosity models lack from the simplification of the Boussinesq hypothesis. Some improvement has been achieved by Explicit Algebraic Reynolds Stress Models (EARSM), like the one by Wallin and Johansson [12], which allow the Reynolds stresses not to be aligned with the mean strain components. However, these models use the same transport equations as eddy viscosity models, only replacing the Bousinesq hypothesis. Therefore they inherit the associated deficiencies. The highest level of turbulence modelling based on the RANS equations is consituted by so-called differential Reynolds stress models, providing an individual transport equation for each component of the Reynolds stress tensor and some length scale supplying variable. Within the EU project FLOMANIA, DLR supported by F. Menter formulated and implemented the SSG/LRR-ω model in the structured flow solver FLOWer and validated it for a variety of test cases [6]. In the following, promising results were obtained on structured grids for various aeronautical flows [4,3]. Therefore the implementation was transfered to DLR’s unstructured solver TAU, being more flexible with respect to the application to complex flows. The current paper shows first results, obtained with the DLR TAU code, using the SSG/LRR-ω model.
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The Speziale-Sarkar-Gatski (SSG) Reynolds stress model [11] has been developed for homogeneous turbulence, i. e. for flows in the absence of walls. Its length scale is supplied by a standard transport equation for the dissipation rate . In contrast Wilcox [14] has shown, that the Launder-Reece-Rodi (LRR) Reynolds stress model [7] is capable of treating the near wall turbulence, when using a transport equation for the specific dissipation rate ω and omitting the so-called wall-reflection terms. As a matter of fact both models for the pressure-strain term can be written in a unified way as 1 Πij = − C1 + C1∗ P (k) bij + C2 bik bkj − bmn bmn δij 3 2 + C3 − C3∗ bmn bmn kSij + C4 k bik Sjk + bjk Sik − bmn Smn δij 3 −C5 k (bik Wkj − Wik bkj ) , (1) where is the dissipation rate, P (k) = −Rij ∂Ui /∂xj represents the production of kinetic turbulence energy k, bij = Rij /(2k) − δ/3 is the anisotropy tensor of the Reynolds stress tensor Rij , and Sij and Wij are the symmeric and the antisymmetric part of the velocity gradient tensor ∂Ui /∂xj , respectively. The SSG and LRR model differ only in the values of the coefficients Ci and Cj∗ , so that the blending formula (LRR)
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gradually changes the model from LRR near walls (F1 = 1) to SSG further away (F1 = 0). This is the same technique as used with Menter’s BSL ω-equation [8], where the coefficients are blended in such a way that they correspond with an -equation near walls and an ω-equation further away. Thus, using Menter’s blending function F1 and his BSL ω-equation, yields a model, changing consistently from SSG+ (in ω-notation) in the far field to LRR+ω near walls. This model has been termed SSG/LRR-ω model and validated, e. g. in the EU-project FLOMANIA [6].
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The DLR TAU code has been applied to a couple of test cases, using the SSG/LRR-ω Reynolds stress model. The test cases have been selected for verification of the implementation by comparison with FLOWer results and for demonstrating the applicabilty of the model on unstructured grids in two and three dimensions. 3.1 RAE 2822 Airfoil The RAE 2822 airfoil is a typical test case for transonic flow. For the conditions referred to as case 9 (Mach number Ma = 0.73, Reynolds number Re = 6.5 · 106 , incidence α = 2.8o ) the occurring shock does not induce separation, while for the conditions referred to as case 10 (Mach number Ma = 0.75, Reynolds number Re = 6.2 · 106 , incidence α = 2.8o ) it does [1]. Although there may be some doubts on the reliability of the experimental data, the predicted shock positions of test cases 9 and 10 are rather sensitive with respect to the choice of the turbulence model. Therefore they are very well suited to verify the RSM implementation in TAU with respect to FLOWer. Figure 1 shows the pressure distributions, obtained for case 9 and case 10 on a very fine structured grid of 736 × 176 cells, guaranteeing a fairly grid independent solution. RAE 2822 Case 9
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Fig. 1. RAE 2822. Pressure distributions for case 9 (Ma = 0.73, Re = 6.5 · 106 , α = 2.8o ) and case 10 (Ma = 0.75, Re = 6.2 · 106 , α = 2.8o ).
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As one can see, the TAU and FLOWer results for the SSG/LRR-ω model are virtually identical. Only in the immediate vicinity of the shock there occur very small differences, which may be due to some differences in the baseline numerical procedures between a structured and an unstructured code. For comparison the result with the Wilcox k-ω model is plotted in the same figures, in order to compare with the differences occuring when changing the turbulence model. 3.2 L1T2 Three-Element Airfoil The L1T2 three-element airfoil has been studied within the British National High Lift Programme and serves as a validation test case for high lift flows [9]. The experiments have been carried out at a Mach number of Ma = 0.197, a Reynolds number of Re = 3.52·106, based on the chord length without deployed high lift devices, and an incidence of α = 20.18o. The hybrid computational grid consists of 108556 points, where the boundary layers are resolved with 26 to 30 structured layers. Except small regions in the vicinity of the
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suction peaks of slat and main wing the distance of the wall nearest grid point in wall units, y + = d1 uτ /ν, is below 1.5, where d1 is the distance, uτ is the friction velocity, and ν is the kinematic viscosity of the fluid. The grid is the finest of a series of four grids used for a grid convergence study. Fig. 2 shows the pressure distributions on slat, main wing and flap. As one can see, the predictions with the SSG/LRR-ω model agree quite well with the measurements. However, there is virtually no difference of the results compared to simpler models like the Menter SST. 3.3 AIAA Drag Prediction Workshop Wings For the Third AIAA Drag Prediction Workshop two different wings for transonic flow have been designed particularly for evaluating drag differences [15]. The wings have been designed as a blind test for the participants, so that there are no experiments available. The flow conditions have been defined by a Mach number of Ma = 0.76 and a Reynolds number of Re = 5·106 based on the mean aerodynamic chord. The incidence varies between α = −1.0o and α = +3.0o . The computations have been carried out on grids with more than 5 · 106 nodes for each wing. These were the medium grids of a grid refinement study at α = 0.5o , involving also a coarse mesh with approximately 2 · 106 grid nodes and a fine mesh with approximately 10 · 106 grid nodes for each wing. AIAA DPW Wing 1
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Fig. 3. AIAA Drag Prediction Workshop. Lift vs. incidence for wing 1 and wing 2.
Fig. 3 shows the lift vs. the incidence as they are predicted with the SSG/LRR-ω and the Menter SST model. As one can see, at low angle of attack, both models predict virtually the same CL at a given α, indicating that the flow is attached. When increasing the angle of attack, the SST model yields a smaller maximum lift at lower incidence compared to the SSG/LRR-ω model, indicating earlier separation. Unfortunately it is impossible to judge upon the models’ accuracy without any experimental data. Nevertheless the test case shows the applicability of Reynolds stress models at least to simple three-dimensional aeronautical configurations.
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3.4 ONERA M6 Wing The ONERA M6 wing is a low aspect ratio tapered wing, which is considered a standard test case for three-dimensional transonic flow [1]. Two flow conditions have been chosen, one where the flow remains attached (Mach number Ma = 0.84, Reynolds number Re = 11.72 · 106 , incidence α = 3.06o) and one, where the flow is separated on the outboard part of the wing (Ma = 0.84, Re = 11.71 · 106 , α = 6.06o). η = 0.65
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Fig. 4. ONERA M6 wing, α = 3.06o . Pressure distributions at 65% and 90% span. η = 0.65
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Fig. 5. ONERA M6 wing, α = 6.06o . Pressure distributions at 65% and 90% span.
The computations have been carried out on a grid, consisting of 4.5 · 106 nodes, which is considered fine enough for obtaining fairly grid independent solutions. For both flow conditions the wall distance of wall nearest grid points, d1 , corresponds to a value of y1+ < 1 on the entire wing. Fig. 4 shows the pressure distributions in two sections at 65% and at 90% span at the lower incidence of α = 3.06o. As has been noted earlier [2], there is not much difference between the results obtained with different models. Nevertheless at 90% span
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some slight deviation from the experiment is observed with the SST model, indicating a small shock induced separation, which is not present with the SSG/LRR-ω model. At the higher incidence of α = 6.06o it was impossible to achieve a converged solution, using the Menter SST model. Apparently this is due to a much too large separation predicted on almost the entire top of the wing. In contrast, the SSG/LRR-ω model yields a converged solution where the separation is restricted to the outboard part of the wing. As one can see in Fig. 5, the pressure distribution at 65% span agrees quite well with the experiments, whereas some deviation is observed at 90% span, indicating that the separation is still not captured perfectly. Note, that the wiggles in the pressure distributions are no numerical artefacts, but stem from the complex separation pattern.
4 Conclusion The SSG/LRR-ω Reynolds stress turbulence model has been implemented into DLR’s unstructured flow solver TAU. For the RAE 2822 airfoil the results obtained on a structured grid agree very well with the predictions by DLR’s structured flow solver FLOWer. The applicability of the model to high-lift problems, at least to twodimensional flow, is demonstrated for the L1T2 three-element airfoil. Computations for the transonic flow around the two wings of the Third AIAA Drag Prediction Workshop and the ONERA M6 wing show, that the SSG/LRR-ω model is also well applicable to three-dimensional flows. Furthermore it appears that the SSG/LRR-ω model is able to predict reasonably well the rather difficult flow around the ONERA M6 wing at an incidence of α = 6.060 , where shock induced separation occurs. In contrast the Menter SST predicts a massively separated flow field for this case, so that no converged solution could have been obtained. Generally there have not been observed any particular numerical difficulties with respect to convergence or stability, when using a Reynolds stress model. Nevertheless the experience with FLOWer shows, that such type of model may be more sensitive to the grid quality than simpler eddy viscosity models. Concerning the computing times, an increase by a factor of 1.64 has been observed for the ONERA M6 wing, comparing the SSG/LRR-ω model with the Menter SST model. Unfortunately no information on the memory requirements has been available for the computations presented. Anyway, transfering the experience with the FLOWer code, an increase by a factor of 2 should be expected, comparing a Reynolds stress model with an eddy viscosity model. In the future it is planned to increase the complexity of computed test cases, in order to explore the range of applicability of the SSG/LRR-ω model in the DLR TAU code. These studies should involve generic aircraft, delta wing and 3D high-lift configurations.
Acknowledgements The author wants to thank Florian Menter for supporting him in formulating the SSG/LRR-ω model within the framework of the FLOMANIA project. The author wants to thank Peter Eliasson and Stefan Wallin from the Swedish Defence Research Agency FOI for hosting him for a three months stay, during which the model implementation
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has been completed and parts of the computations have been carried out. Furthermore the author is indebted to his colleagues Stefan Melber-Wilkending, Jochen Wild and Olaf Brodersen for providing grids for the computations.
References [1] J. Barche (Ed.): “Experimental Data Base for Computer Program Assessment”, AGARDReport AGARD-AR-138, 1979 [2] B. Eisfeld: “Numerical simulation of aerodynamic problems with a Reynolds stress model”. In: H.-J. Rath, C. Holze, H.-J. Heinemannm R. Henke, H. H¨onlinger (Eds.): New Results in Numerical and Experimental Fluid Mechanics V, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Vol. 92, pp. 413-419, 2006 [3] B. Eisfeld, “Computation of Complex Compressible Aerodynamic Flows with a Reynolds Stress Turbulence Model”. In: G. Lube, G. Rapin (Eds.): International Conference on Boundary and Interior Layers, BAIL 2006, G¨ottingen, 2006 [4] B. Eisfeld, O. Brodersen: “Advanced Turbulence Modelling and Stress Analysis for the DLR-F6 Configuration”. AIAA-Paper 2005-4727, 2005 [5] J. K. Fassbender: “Improved Robustness for Numerical Simulation of Turbulent Flows around Civil Transport Aircraft at Flight Reynolds Numbers”. Dissertation, Technische Universit¨at Braunschweig, DLR Forschungsbericht, DLR–FB 2003–09, 2004 [6] W. Haase, B. Aupoix, U. Bunge, D. Schwamborn (Eds.): “FLOMANIA – A European Initiative on Flow Physics Modelling”. Notes on Numerical Fluid Mechanics, Vol. 94, Springer, 2006 [7] Launder, B. E., Reece, G. J., Rodi, W., Progress in the development of a Reynolds-stress turbulence closure, Journal of Fluid Mechanics, 68, 1975, pp. 537-566 [8] F. R. Menter: “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications”. AIAA Journal 32, 1994, pp. 1598-1605 [9] Moir, I. R. M., Measurements on a two-dimensional aerofoil with high-lift devices, AGARD-Report AGARD-AR-303, Vol. II, Chapter A2, 1994 [10] P. R. Spalart, S. R. Allmaras: “A One-Equation Turbulence Model for Aerodynamic Flows”. AIAA-Paper, 92-439, 1992 [11] C. G. Speziale, S., Sarkar, T. B. Gatski: “Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach”. J. Fluid Mech. 227, 1991, pp. 245-272 [12] S. Wallin, A. V. Johansson: “An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows”. J. Fluid Mech. 403, 2000, pp. 89-132 [13] D. C. Wilcox: “Reassessment of the Scale Determining Equation for Advanced Turbulence Models”. AIAA Journal 26, 1988, pp. 1299-1310 [14] D. C. Wilcox: “Turbulence Modeling for CFD”. DCW Industries, La Ca˜nada, USA, 2nd ed., 1998 [15] Third AIAA CFD Drag Prediction Workshop, San Francisco, June 3-6, 2006 URL: http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpw/ (Status: November 14, 2006)
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DNS of Compressible Inert and Infinitely Fast Reacting Mixing Layers Inga Mahle, J¨orn Sesterhenn, and Rainer Friedrich Fachgebiet Str¨omungsmechanik, TU M¨unchen, Boltzmannstr. 15, 85748 Garching, Germany
[email protected]
Summary This paper deals with the effects of heat release and compressibility on temporally evolving, turbulent mixing layers. Direct Numerical Simulations (DNS) of such layers at two different convective Mach numbers are performed with and without combustion which allows to study the effects of heat release and compressibility separately and combined. It is shown that both, compressibility and heat release, dampen the turbulence activity and lead to a reduced growth of the mixing layer. Alterations in pressure fluctuations are a main reason for the changes. The effects of compressibility are not as strong in the reacting mixing layer as in the inert one. A significant difference between the inert and the reacting mixing layers at high convective Mach number is that entropic density fluctuations prevail over the acoustic ones when reaction takes place while both contribute to nearly equal parts in the non-reacting compressible flow.
1 Introduction When the convective Mach number increases, there is a large reduction of the thickness growth rate and of the turbulent intensities in a mixing layer. This has important consequences for applications such as scramjet engines. Vreman et al. [1] found out that the decrease in pressure fluctuations led to reduced pressure strain terms and is responsible for the changes in growth rate of a compressible mixing layer. The present work aims at confirming and complementing these and other investigations on the effects of compressibility. Additionally, not only compressibility effects are taken into account, but also heat release effects due to combustion. To gain an understanding of both, the separate effects and the interaction of compressibility and heat release effects, DNS data of inert and infinitely fast reacting mixing layers at different convective Mach numbers are collected and analyzed. The paper is organized as follows: The first section gives an overview over the generic configuration of the mixing layer and the four different test cases in particular. In the next section, the combustion model and the integrated transport equations are described shortly and the most important features of the numerical method are C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 372–380, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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given. Then, in the main part of the paper, a selection of results is presented. The gained insight leads to an outlook on the way to be pursued in the future and to the conclusions drawn in the last section.
2 Mixing Layer Configurations The test cases correspond to plane mixing layers which develop between two streams of gas that have different velocities. The layers are studied numerically in a reference frame moving with the average speed of the two streams and under the parallel-flow assumption which makes them temporally evolving mixing layers. Figure 1 shows the initial configuration. All test cases are 3D with x and x denoting the homogeneous, periodic streamwise and spanwise directions and x denoting the transverse direction, which is the direction of the mean shear. For the inert cases (denoted by inert-X), the upper stream (index 1) is pure oxygen and the lower stream (index 2) pure nitrogen. At the beginning of the simulations, temperature and pressure are constant. This results in an initially nearly constant density across the mixing layer. The upper stream of the infinitely fast reacting mixing layers (denoted by inf-X) contains air and the lower stream a mixture of hydrogen and nitrogen with a stoichiometric mixture fraction of 0:3. The inert and reacting mixing layers are simulated at two convective Mach numbers Mc = u= (c + c ), respectively: at 0:15 (inert-0:15 and inf-0:15), which is quasiincompressible, and at 0:7 (inert-0:7 and inf-0:7). The velocity difference between the two streams is denoted by u, and c and c are their sonic speeds. The inert flows are initialized by a hyperbolic-tangent profile for the mean streamwise velocity and density. In order to accelerate the transition to turbulence, broadband perturbations in the velocity components are used. The Reynolds number at the beginning of the computations Reω, = u ω, = is 640. It is based on the initial vorticity thickness ω, , on the density = ( + ) =2 and on the viscosity = ( + ) =2. The reacting test cases are initialized by appropriately scaled, fully turbulent velocity fields taken from the inert test cases and a hyperbolic-tangent mixture fraction profile. Grid and domain sizes are given in table 1. The grid-spacing is constant in all directions, except for inert-0:15 where a 3% grid stretching is used within the inner third of the computational domain.
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Table 1 . Simulation parameters: Box dimensions in streamwise, spanwise and transverse directions, L1 , L2 and L3 , with N1 , N2 and N3 grid points, respectively L1 /δ!;0 L2 /δ!;0 L3 /δ!;0 N1 N2 N3 inert-0.15 129.38 32.25 92.23 512 128 512 inert-0.7, inf-0.15, inf-0.7 86.25 21.5 43.0 768 192 432
3
Combustion Model, Equations and Numerics
The reaction of hydrogen and oxygen is simplified by a one-step irreversible global reaction. This implies that fuel and oxidizer cannot be simultaneously present at the same place. In addition, a common Schmidt number is used for all species. These assumptions allow to relate the species mass fraction to a single passive scalar, the mixture fraction z by the piecewise linear Burke-Schumann relations [2] [3]. Therefore, only the transport equation for the mixture fraction, a passive scalar without a source term, must be integrated, and the species mass fractions can be computed from the resulting mixture fraction distribution. In addition to the continuity and the momentum equations, the transport equations of the mixture fraction in the reacting cases and the species mass fractions in the inert cases, the energy equation, in the form of a pressure or temperature equation, is integrated in order to retain a fully compressible formulation. The equations are implemented in the characteristic formulation of Sesterhenn [4] which facilitates the treatment of the non-reflecting boundaries in the sheardirection. The integration in time is performed with a third-order low-storage RungeKutta scheme [5] whereby the spatial derivatives are discretized by using sixth-order compact central schemes [6]. The primitive variables are filtered every 20th time step to prevent spurious accumulation of energy in the highest wavenumbers using a sixth-order compact filter [6].
4
Results and Their Analysis
4.1 Instantaneous Scalar Fields Figure 3 shows instantaneous, fully turbulent scalar fields: the oxygen mass fraction for the inert mixing layers and the mixture fraction, which is directly related to the oxygen and all other mass fractions, for the reacting mixing layers. In addition, the isoline of the respective scalar at 0:3 is shown. In case of the mixture fraction, this is the stoichiometric value, and the curve gives the position of the flame. Comparing the scalar distributions and isolines of the reacting test cases with those of the inert test cases, the smoothening effect of the flow structures due to heat release is striking. It is apparently dampening the turbulent fluctuations. While the present increase in Mach number seems to have less impact on the reacting mixing layers and is probably obscured by the much stronger consequences of the heat
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release, one can see that for the inert test cases the isoline becomes less wrinkled with increasing Mc . Hence, the dominant wrinkling length scale increases. 4.2 Mixing Layer Growth The stabilizing and turbulence reducing effect of compressibility and heat release can also be observed from the temporal growth of the mixing layer. Figure 2 shows the momentum thickness Z 1 1 1 u − hu if dx h i (1) θ = 4 u 1 as a function of the non-dimensional time ω = t u= ω . Here and in the following, quantities with brackets, like h i, denote Reynolds averaged quantities, and quantities with an additional index f , like hu if denote Favre averaged quantities. Compressibility and heat release reduce the growth of the mixing layer. After some time, all mixing layers have a nearly constant growth rate which gives evidence of the fact that a self-similar state is reached [1] [3]. All profiles in the following sections are averaged spatially over the homogeneous directions and temporally over the self-similar state. 4.3 Mean Profiles While heat release and compressibility have only little effect on the profile of the streamwise velocity (not shown), the density profile (Fig. 4a) is heavily influenced, in particular by the heat release which leads to a strong reduction within the mixing layer. Furthermore, the density profiles of the reacting mixing layers are broadened by thermal expansion compared to the ones of the inert mixing layers. At high M c , the mixing layer heats up by dissipation which also results in a reduction of the density. This effect is clearly visible for the inert test cases, but is overshadowed by the strong consequences of heat release for the reacting test cases.
18 16 14
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Fig. 2 . Momentum thickness,+: inert-0.15,
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4.4 Reynolds Stresses and Their Transport Equations In Fig. 2, it has been shown that the momentum thickness growth rate of the mixing layer is reduced by compressibility and heat release. To clarify the reason, we simplify the exact expression for the growth rate by neglecting mean viscous effects [1] and obtain Z 1 @hu if _θ = − 1 h iR dx ; (2) u @x 1 which contains the Reynolds shear stress h iR = h u00 u00 i. As mentioned before, the velocity profile hardly changes between the test cases. On the contrary, the profile of the Reynolds shear stress (Fig. 4b) is affected by compressibility and heat release. Therefore, this quantity is responsible for the reduced growth rate of the momentum thickness. Similar observations, namely a reduction by compressibility and heat release, can be made for the other (non-zero) Reynolds stresses h iRij = h u00i u00j i and the turbulent kinetic energy, k = 0:5Rii (not shown). This is in agreement with the smoothening effect of compressibility and heat release that has been noticed from the instantaneous scalar fields in Sect. 4.1. To further clarify the mechanisms that lead to a reduction of the Reynolds stresses, we investigate the most important terms in their transport equations. For the streamwise Reynolds stress h iR , these are the production rate, the pressure strain rate and the dissipation rate. Since the latter one is not very much influenced by compressibility and heat release, we focus here on the production (Fig. 5a) @hu if P = −2R (3) @x and the pressure strain rate (Fig. 5b)
= 2hp0
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(c)
(d)
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Fig. 3 . Instantaneous scalar fields: (a) inert-0.15, (b) inert-0.7, (c) inf-0.15, (d) inf-0.7, the black curve is the isoline at 0.3. The scalar shown is the oxygen mass fraction for the inert cases and the mixture fraction for the reacting ones.
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Both quantities are reduced by compressibility and heat release. The key-role of the pressure strain rate reduction can also be observed for the other components of the Reynolds stresses. It becomes clear when taking into account that the pressure strain rates are responsible for the redistribution of energy from the streamwise component of the Reynolds stresses to the transverse and spanwise components. 4.5 Thermodynamic Fluctuations The reduction of the pressure strain terms is closely related to alterations of the fluctuating pressure field (Fig. 6a). RMS values of other thermodynamic quantities are shown in Figs. 6b to 7b: The reduction of the molecular weight fluctuations with compressibility for the inert test cases can be explained by the smaller mixing efficiency due to less turbulence and the strong increase for the reacting test cases by the presence of hydrogen. Besides an increase of the temperature fluctuations with increasing Mach number due to dissipative heating, the main increase of this quantity is clearly caused by heat release. For the reacting test cases, the temperature is strongly correlated with the density which becomes evident when comparing the shape of the corresponding RMS profiles. The density fluctuations can be split up (b)
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into an acoustic part, which is related to the pressure fluctuations, and an entropic part, which is connected to temperature and molecular weight fluctuations [7]: 0
=
0ac
+
0en
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;
= p0ac =h ci ;
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This split-up, which is shown in Fig. 8 for both test cases at Mc = 0:7, gives some insight into the nature of the density fluctuations: It is clearly visible that while both, the entropic and the acoustic part, contribute to the density fluctuations in the inert case, the entropic part plays the dominant role in the reacting case. This gives important implications for future research: Instead of computing the pressure fluctuations from a convective wave equation, as it had been done by Pantano & Sarkar [3] for inert mixing layers at high Mc because of strong acoustic effects, the pressure fluctuations of the reacting mixing layer at high Mc , should be correctly obtained from a Poisson equation. Such an equation results when taking the divergence of the momentum equations, inserting the continuity equation and subtracting the average of the resulting Poisson equation. Neglecting higher order terms as well as the second temporal derivative of the pressure fluctuations gives the Poisson equation for p 0 . Knowing the Green function of this equation could then lead to a further detailed understanding of the effects of heat release on compressible mixing layers. (b)
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Fig. 8 . Split-up of density fluctuations: +: total RMS value, RMS value of ρ0en , inert-0.7 (a) and inf-0.7 (b)
: RMS value of ρ0ac , ∗ :
4.6 Conclusions DNS of inert and reacting, temporally evolving mixing layers at two different convective Mach numbers have been performed to investigate the effects of heat release due to combustion and compressibility. It is found that both, compressibility and heat release, act in the same direction by reducing the turbulent fluctuations, the growth of the mixing layer and the turbulent mixing which has important implications for practical applications like scramjet engines the efficiency of which depends on a proper mixing of oxidizer and fuel. The damping of the turbulence activity is caused in both cases by a reduced production of turbulent fluctuations and a reduced distribution of turbulent kinetic energy from the streamwise component of the Reynolds stresses to the other diagonal stresses. The underlying mechanism is that changes in the fluctuating pressure field occur that lead to reduced pressure strain rates. An important difference between the thermodynamic fluctuations of the inert and reacting mixing layer at high convective Mach number is that for the latter, most density fluctuations come from the so-called entropic mode and that the acoustic mode is nearly negligible. This implies that one way to further investigate the behaviour of the fluctuating pressure is to solve its Poisson equation with the help of a Green function.
References [1] A.W. Vreman, N.D. Sandham, K.H. Luo: ”Compressible mixing layer growth rate and turbulence characteristics”. J. Fluid Mech. 320, 1996, 235-258. [2] S.P. Burke, T.E.W. Schumann: ”Diffusion flames”. Industrial and Engineering Chemistry 20, 1928, 998-1005. [3] P. Pantano, S. Sarkar: ”A study of compressibility effects in the high-speed turbulent shear layer using direct simulation”. J. Fluid Mech. 451, 2002, 329-371.
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[4] J. Sesterhenn: ”A characteristic-type formulation of the Navier-Stokes equations for high order upwind schemes”. Computers & Fluids 30, 2000, 37-67. [5] J. Williamson: ”Low-storage Runge-Kutta schemes ”, J. Comput. Phys. 35, 1980, 48-56. [6] S.K. Lele: ”Compact finite difference schemes with spectral-like resolution”, J. Comput. Phys. 103, 1992, 16-42. [7] S. Sarkar: ”The high-speed turbulent shear layer”. In: ”Variable Density Fluid Turbulence, Fluid Mechanics and its Applications”, Eds: P. Chassaing, R.A. Antonia, F. Anselmet, L. Joly, S. Sarkar, Kluwer, 2002, 235-259.
Developement of a Numerical Procedure for Direct Simulations of Turbulent Convection in a Closed Rectangular Cell Matthias Kaczorowski*, Andrei Shishkin, Olga Shishkina, and Claus Wagner German Aerospace Center (DLR) – Institute of Aerodynamics and Flow Technology, Bunsenstr. 10, 37073 G¨ottingen, Germany ∗
[email protected]
Summary A Seperation of Variables (SoV) scheme has been developed in order to obtain a direct solution to the Poisson problem arising from Chorin’s projection method within rectangular enclosures. The algorithm is compared to the previously used FFT with periodic boundaries and methods to improve the performace of the required transformations are presented. Finally, first results of thermal convection obtained in a closed box geometry are compared to results obtained in a periodic rectangular cell.
1
Introduction
In recent years great efforts have been made to investigate Rayleigh-B´enard convection, where fluid between horizontal walls is heated from below and cooled from above. As far as numerical studies are concerned most simulations assume either periodic boundary conditions in horizontal directions or deal with cylindrical geometries. However, it was found by Daya and Ecke [2] that the shape of the container has a significant impact on the turbulent flow properties. In order to improve the understanding of the mechanisms of turbulent thermal convection it is therefore essential to be capable of conducting numerical studies in both cylindircal and rectangular geometries. Since Shishkina and Wagner [5] have studied thermal convection in cylindrical containers by means of direct numerical simulations, it is a straight forward conclusion to perform similar investigations in a rectangular geometry. Therefore the existing programme of Shishkina and Wagner [6] was rewritten in Cartesian coordinates. In the case of periodic boundaries the Poisson problem arising from Chorin’s [1] pressure correction method can be decoupled by means of a Fast Fourier Transform (FFT), so that a set of linearly independent equations can be solved. When solid walls are employed rather than periodic boundaries the near wall region needs to be sufficiently resolved. A Discrete Cosine Transform (DCT) which satisfies the zero gradient boundary conditions might be used instead, but the required equidistant C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 381–388, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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mesh will be extensively large in order to resolve the boundary layers, and hence waste computational resources. Therefore a new pressure correction algorithm based on the method of Separation of Variables (SoV) is implemented in order to efficiently solve the incompressible Navier-Stokes equations. Thus sufficiently small grid spacing can be used in the vicinity of the walls whereas larger cell sizes can be realised in the centre of the domain. In the following an outline of the mathematical background of this algorithm is provided together with possibilities to improve the performance of the required transformations, and hence the overall programme. Tests of both codes with the SoV and the FFT decoupling are carried out in order to determine advantages and disadvantages of both algorithms. Finally, first results of Rayleigh-B´enard convection within a rectangular container are presented.
2 2.1
Mathematical Background
Splitting Scheme
In order to simulate thermal convection the conservation equations of mass, momentum and energy need to be solved. Following the summation convention the non-dimensional Navier-Stokes equations can be written in the following form . P r ∂ 2 ui ∂p ∂ui ∂ui + θδ1i (1) + − = −uj Γ 3 Ra ∂x2j ∂xi ∂xj ∂t
where ui are the velocity components in i-direction, θ and p represent the temperature and pressure respectively and δij is the Kronecker symbol. In this particular case gravitational forces are acting in the x1 -direction, i.e. the vertical direction. The non-dimensional constants P r = ν/κ, Ra = αgH 3 ∆T /(νκ) and Γ = W/H are defined by the kinematic viscosity ν, the thermal diffusivity κ, the thermal expansion coefficient α, the temperature difference between top and bottom walls ∆T and the height H and width W of the fluid layer. Eq. (1) can be solved using Chorin’s [1] pressure correction method splitting the Navier-Stokes equations into transport equations neglecting the pressure term yielding an estimated velocity field u∗i u∗i − uni n+1/2 ∂ui = −uj ∂xj ∆t
.
n+1/2
+
P r ∂ 2 uni + θn δ1i Γ 3 Ra ∂x2j
(2)
and the Poisson equation for the pressure where the source term Q is the divergence of the velocity field u∗i ; the superscript n denotes the time level. ( ∂u∗j ∂pn+1 (( ∂ 2 (pn+1 ∆t) =0 (3) , =Q= ∂n (wall ∂xj ∂x2j
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From u∗i and the solution of the Poisson equation pn+1 ∆t the divergence-free velocity field of the time step n + 1 can be calculated explicitly from = u∗i − un+1 i
2.2
∂pn+1 ∆t ∂xi
(4)
The Poisson Problem
The discretisation of the Poisson equation (3) on a non-equidistant but Cartesian grid with Neumann boundary conditions at the walls reads Dx1 ψ(I,J,K) + Dx2 ψ(I,J,K) + Dx3 ψ(I,J,K) = Q(I,J,K)
(5)
with ψ = pn+1 ∆t and ⎧ 1 ψ(2) −ψ(1) ⎪ ;Ni =1 ∆xi(1) · ∆xui ⎪ ⎪ i(1) ⎪ ⎨ ψ −ψ ψ −ψ (Ni−1) (Ni+1) (Ni ) 1 i) − (N∆x ui ui Dxi ψ(Ni ) = ∆xi(Ni ) · ;1
i(Ni,max−1)
where ∆xi and ∆xui i denote the cell sizes in i-direction of the p and the staggered ui cells respectively and Ni represents the grid point in that direction. Obviously eq. (5) cannot be solved efficiently. In order to achieve a direct solution of the problem it is therefore necessary to obtain a set of linearly independent equations which can be achieved by representing ψ and Q in terms of the eigenvectors vlx2 = (vxl 2 (1) , vxl 2 (2) , . . . , vxl 2 (Jmax ) )T and eigenvalues λlx2 , l = 1, . . . , Jmax of the geometry matrix Dx2 . Hence, their linearly independent ˆ are obtained, which are related to ψ and Q by eq.’s (7) and components ψˆ and Q (8). Q(I,J,K) =
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(7)
l vxl 2 (J) ψˆ(I,K)
(8)
l=1
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Decoupling eq. (5) in y-direction leads to the following set of Jmax linearly independent systems that can be solved efficiently by means of a five point matrix solver. J max l=1
$
J max % ˆl Q Dx1 (I) + λlx2 + Dx3 (K) ψˆ(I,K)l = (I,K)
(9)
l=1
The back transform of the decoupled solution to the Poisson equation ψˆ can be perˆ the following formed using eq. (8). In order to obtain the decoupled source term Q transformation can be used
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ˆl Q (I,K) =
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||vlx2 ||
Q(I,J,K)
(10)
where ||vlx2 || =
J max
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(11)
J=1
2.3
Speed-Up of the Transformation
For the decomposition of the source term into its linearly independent components 2 by eq. (10) N = O(Jmax ) operations are required which is a significant increase compared to the commonly used FFT algorithms. However, it is possible to improve the performance of the transformation by exploiting the fact that the grid is symmetric with respect to the center. Thus, the source term Q can be decomposed into its symmetric and antisymmetric parts Q+ and Q− given by 1 (Q(I,J,K) + Q(I,Jmax +1−J,K) ); 2 1 = (Q(I,J,K) − Q(I,Jmax +1−J,K) ); 2
Q+ (I,J,K) =
Q− (I,J,K)
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1≤J ≤
(12)
(13)
which in turn are transformed into their linearly independent components through eq. (10) using the symmetric and antisymmetric eigenvectors of Dx2 respectively 2 which requires only Nspd−up = Jmax /2 − Jmax /2 operations. Taking into account that there are additional summations for the decomposition of the source term 2 2 + 4Jmax rather than N = 2Jmax − Jmax operations are required. Nspd−up = Jmax + − ˆ ˆ The solution ψ and ψ can then be back transformed by eq. (8) again using symmetric and antisymmetric eigenvectors of Dx2 respectively, so that the solution to eq. (3) is obtained by + − + ψ(I,J,K) ; ψ(I,J,K) = ψ(I,J,K)
+ − ψ(I,Jmax +1−J,K) = ψ(I,J,K) ; − ψ(I,J,K)
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(14)
(15)
Additionally to the speed-up of the transformation the decomposition into symmetric and antisymmetric parts entails two other advantages. Numerical inaccuracies resulting from the calculation of the eigenvalues and eigenvectors can be avoided and the storage for the eigenvectors vlx2 is also reduced by a factor of two.
3
Results
Prior to the first numerical simulation the implementations of transform and back transform have been verified. In the case of equidistant grid spacing the scaled eigenvalues Λl are identical with a cosine transform, since
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l−1 Λlx2 = λlx2 / ∆x2(J=1) ∆xv2(J=1) = 2 cos 2π Jmax
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(16)
Transform and back transform and the matrix solver have shown to work reliably for both equidistant and non-equidistant grids. Table 1 compares the corrected velocity fields obtained with cyclic and Neumann boundary conditions. The results indicate that in both cases the corrected velocity field is divergence free within machine accuracy. Table 1 . Divergence of the corrected velocity field on a 32 × 32 × 32 grid using the FFT with cyclic and the SoV with Neumann boundaries in the direction of transformation
FFT
SoV
∼ 10−19 (∇ui )max ∼ 10 |∇ui | ∼ 10−17 ∼ 10−16 ∇ui ∼ 10−26 ∼ 10−28 −21
3.1
Performance Tests
The subsequent figures show the performance of the programme with different pressure correction algorithms. All pressure correction algorithms discussed in the following rely on the 2-D cyclic reduction method by Swarztrauber and Sweet [7, 8], so that the different performances result from their respective transformation. However, the matrix solver is almost as efficient as an FFT, and therefore not critical as far as the computational efficiency is concerned. It can be seen from Fig. 1 (left) that solving eq. (3) consumes about 40% of the computational time when an FFT is used. Using the separation of variables scheme the computational effort for the solution of the Poisson equation increases significantly. The optimised algorithm performs noticeably better than the standard transformation, Nonetheless, in terms of efficiency it cannot compete with a conventional FFT. However, it has to be taken into account that due to the equidistant grid spacing required for a DCT significantly more grid points are needed. Fig. 1 (right) illustrates the overall performance of an FFT-like transformation when k × Ni grid points would be used in order to resolve the near wall region. It can be seen that increasing the number of grid points, i.e. increasing k, quickly consumes the advantages of the fast transformation, not to speak of the enormous increase in storage requirements. When large numbers of grid points are required the parabolic behaviour of the transformation becomes a problem since the pressure correction algorithm tends to determine the required computational time as can be seen from Fig. 1 (left). Driscoll et al. [3] have developed a Fast Polynomial Transform (FPT) that is applicable to
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Fig . 1 . Left: Contribution of the Poisson solver (eq. (3)) to the overall performance of the programme. Right: Computing time per time step as a function of the grid points Ni in transformation direction; the FFT is plotted for k × N i points.
any set of orthogonal polynomials and represents a generalisation of the FFT. The FPT requires NFPT (Ni ) ≤ 18Ni log22 (Ni ) + 3Ni log2 (Ni )
(17)
operations. Figure 2 indicates that it is only reasonable to apply this method when more than about 103 grid points are used in transformation direction. 3.2
Thermal Convection in a Rectangular Cell
Direct numerical simulations of thermal convection have been conducted in a long rectangular cell. Flow fields have been simulated for Rayleigh numbers up to Ra = 3.5 × 106 with 96 × 400 × 96 grid points. Fig. 3 illustrates that the Nu − Ra scaling 20 15
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Fig . 3 . Nusselt number as a function of Rayleigh number for the periodic (•) and the closed () rectangular cell; scaling Nu ∼ Ra0.284 (—)
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Fig . 4 . Isothermals of Rayleigh-B e´ nard con vection in a rectangular cell with periodic (left) and Neumann boundaries (right) in longitud inal direction; Ra = 3.5 × 10 5 (top) and Ra = 3.5 × 10 6 (bottom), white (θ = +0 .5) to black (θ = −0.5)
matches the effective scaling of the 3-D regime obtained in a periodic rectangular cell for Rayleigh numbers up to 6.0 × 108 : Nu ∼ Ra0.284 . This is also very close to Grossmann and Lohse’s [4] theoretical prediction of an effective exponent of β ≈ 0.29 for 1 P r 7. However, in the case of the periodic cell Nusselt numbers are significantly smaller. This is due to the fact that the transition to a 3-D flow field occurs earlier in the closed box geometry. The isothermals of two different Rayleigh numbers for the enclosure and the periodic cell shown in Fig. 4 illustrate this behaviour.
4
Conclusion
A new pressure correction algorithm has been developed for Neumann boundary conditions in all coordinate directions, in order to simulate thermal convection in an enclosure. Comparison of performance tests with an FFT algorithm using k × Ni grid points has shown that the drawback of a relatively slow transformation with O(Ni2 ) operations is easily compensated, since the number of grid points can be significantly reduced by concentrating them in the regions of interest, i.e. in the vicinity of the walls. However, it could also be seen that for relatively small numbers of grid points the SoV performes better than the FPT which should only be employed for very large problems, since it scales with O(Ni log2 (Ni )). For less than approximately 103 grid points in the direction of transformation the improved SoV should be used.
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First results of thermal convection obtained with the new pressure correction algorithm have been presented and compared to results obtained in an aspect ratio unity rectangular container with periodic boundaries in longitudinal direction. It can be concluded that the additional lateral wall has a non- negligible impact on the flow field, and thus the heat transfer between top and bottom walls. However the scaling Nu ∼ Raβ is similar in both cases, which underlines that the new pressure correction algorithm works reliably. It is now possible to obtain a direct solution to the Poisson problem within rectangular enclosures, so that the simulations do not have to rely on iterative approaches. The properties of turbulence of these flow fields will be compared to velocity fields of cylindrical or periodic domains. Hence, the influence of the geometry and the bounding walls can be analysed.
Acknowledgments The authors would like to thank the Helmholtz Association of German Research Centers (HGF) for funding this work through the project Virtual Institute - Thermal Convection.
References [1] A. J. Chorin. Numerical solution of the Navier-Stokes equations. Mathematics of Computations, 22:745 – 762, 1968. [2] Z. A. Daya and R. E. Ecke. Does turbulent convection feel the shape of the container? Phys. Rev. Lett., 87(18):184501, 2001. [3] J. Driscoll, J. D. M. Healy, and D. Rockmore. Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs. SIAM J. Comput., 26:1066 – 1099, 1997. [4] S. Grossmann and D. Lohse. Fluctuations in turbulent Rayleigh-B´enard convection: The role of plumes. Phys. Fluids, 16(12):4462 – 4472, Dezember 2004. [5] O. Shishkina and C. Wagner. Analysis of thermal dissipation in turbulent RayleighB´enard convection. J. Fluid Mech., 546:51 – 60, 2006. [6] O. Shishkina and C. Wagner. A fourth order finite volume scheme for turbulent flow simulations in cylindrical domains. Computers and Fluids, 2006. [7] P. N. Swarztrauber. A direct method for the discrete solution of separable elliptic equations. SIAM J. Numer. Anal., 11:1136 – 1150, 1974. [8] P. N. Swarztrauber and R. Sweet. Efficient Fortran subprograms for the solution of elliptic equations. Technical Report IA-109, National Center for Atmospheric Research, 1975.
Computation of Supersonic Base Flow Using Detached Eddy Simulation V.K. Togiti and H. Ludek ¨ e e DLR, Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7, D-38108 Braunschweig, Germany
[email protected]
Summary In the present study, Detached Eddy Simulation is applied to the supersonic axisymmetric base flow. Computations are done on coarse and fine unstructured grids to examine the influence of grid refinement on flow predictions. The final results are compared with available experimental data of Herrin and Dutton, which includes boundary layer velocity profile, base pressure, centerline velocity profile, Mach and turbulent kinetic energy distributions. The influence of grid refinement is observed and the reasons are discussed.
1
Introduction
Turbulent base flow is a complex type of flow, which usually occurs in the wake of missiles, rockets, projectiles and re-entry vehicles. The aerodynamic characteristics of these vehicles are significantly influenced by the base flow field. Therefore, an accurate estimation and understanding of the effects in this region will enable possibilities to improve the aerodynamic stability and controllability of the vehicle. Schematic features of the supersonic base flow at M ∞= 2 46 are depicted in Fig. 1. The large turning angle at the transition of the contour from the wetted surface to the base causes separation and a region of reverse flow as visible in the wake. The separation bubble in Fig. 1 is in fact a vortex ring around the centerline. The point along the axis of symmetry where the streamwise velocity is zero is considered to be a saddle point of the flow field. In this region the flow is forced to turn along the axis of symmetry causing a re-compression shock to be formed. In recent years, this configuration has been studied using Detached Eddy Simulation (DES) on structured and unstructured grids by numerous scientists [1]&[2]. Their results have shown good agreement with experiments. Quantitative deviation of second order statistics, which are resolved Reynolds stresses and turbulent kinetic energy, with experiments was thought to be due to insufficient grid resolution, which is needed to resolve shear layer rollup. The same configuration, unsteady 3D supersonic baseflow, was studied by Frank Simon et. al. [3] on structured grids to find the influence of numerical parameters, related to hybrid RANS/LES methods, and sensitivity of DES constant on flow predictions. Several zonal DES and other RANS/LES computations of the base flow have been performed on two high C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 389–396, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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Fig. 1. Schematic of mean axisymmehic base fbw
density grids. It has been stated that capturing of the weak three-dimensional instabilities of highly compressible mixing layer could be possible with high grid resolution in the azimuthal direction for 3D computations. The aim of the current work is to investigate the base flow using DES and to examine the influence of grid resolution on flow predictions especially in the shear layer behind the base.
2 Numerical Algorithm For the current investigation the DLR-TAU-code was used, which is an unstructured compressible flow solver. The TAU-code is a second order finite-volume flow solver for the Euler and Navier-Stokes equations in integral form. Different numerical schemes like cell-centered for sub- and transonic and AUSMDV for super- and hypersonic flow conditions are implemented. Second-order accuracy for the upwind scheme is obtained by the MUSCL extrapolation, in order to allow the capturing of strong shocks and contact discontinuities. A three-stage RungeKutta as well as an implicit LUSGS scheme are available to advance the solutions in time for steady flow fields. For acceleration of the convergence local time stepping, implicit residual smoothing and full multigrid are optional. For fast and accurate transient flow simulations a dual time stepping scheme, following Jameson, is implemented, which is an implicit algorithm and not restricted to the choice of the smallest timestep in the flow field. To overcome this limit the time derivative in the Navier-Stokes equations is discretized by a second order backward difference, resulting in a non-linear equation system which converges towards the subsequent timestep by using an inner pseudo-time. Within this inner loop all the mentioned acceleration techniques such as local time stepping and residual smoothing are applicable. With this approach an acceleration of time accurate calculations of second up to third order is possible. Several one- and two equation turbulence models are available for steady and unsteady simulations.
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In the presented work, the latest version of DES, called delayed DES (DDES), is applied. Details can be found in [5] and [6]. The filter width used in the formulation is the cube root of cell volume instead of maximum of grid spacing in the three spatial directions. 2.1 Test Conditions The configuration of this study is based on the experimental work done by Herrin and Dutton [4]. In the experiment the axisymmetric cylinder has a base radius of 31 75 mm. The free stream Mach number is set to 2 46. The unit Reynolds number per meter is 45 × 106 and the free stream pressure and temperature are 31415 P a and 145 K, respectively. 2.2 Computational Grids Two unstructured grids are used in the current study to examine the influence of grid resolution on flow predictions and shear layer instabilities. Both grids are generated by the Centaur grid generator and have prismatic cells in the wall regions and tetrahedron cells in the remaining domain. The coarse grid has about 900000 nodes. To obtain a finer resolution this grid is refined in the base and wake region. The fine grid has 6 million nodes. The value of the first y + for both grids calculated at the end of the cylinder is less than 0.5 for the Spalart-Allmaras model. Both, coarse and fine grid are shown in Figs. 2 and 3.
Fig . 2 . Coarse grid
Fig. 3 . Fine grid
3 Results In the computations, the flow is initially set to free stream conditions and then advanced in time with a physical time step of 5 × 10−6 s for 3000 steps. Later
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statistical sampling is done over15 ms in physical time and finally the time averaged results are compared to the experiments. The incoming turbulent boundary layer needed to be resolved properly as the turbulent content in the free shear layer comes from the boundary layer that separates at the base corner. A comparison of the experimental and computed data of the boundary layer velocity profiles on the cylinder surface 1mm behind the base edge is shown in Fig 4. The profiles show that the boundary layer is properly resolved on both the coarse and the fine grid. The mean pressure along the base is an important variable for an industrial application, since it contributes strongly to the total drag. The profiles obtained on different grids and the experimental data are shown in Fig 5. The base pressure on the coarse grid is fairly matching with the
%XPERIMENTAL #OARSE &INE
%XPERIMENT #OARSE &INE
#P
9 MM
U 5 d
Fig . 4 . Boundary layer velocity distribution lmm behind the base edge
Y2
Fig. 5. Pressure along the base
%XPERIMENT #OARSE &INE
U 5 d
X2
Fi g . - . Streamwise velocity distributiom along the wake axis
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experiments. But, on fine grid it has been over predicted by 30%. The centerline velocity profiles are shown in Fig. 6. The coarse and the fine grid have predicted the peak reverse velocity correctly but the location of the peak velocity moved further downstream on the fine grid. The length of the recirculation is slightly over predicted on the coarse grid though the base pressure agrees well with the experiments. On the fine grid it is almost two times to the one in the experiments. Mach contours, obtained on different grids, are compared with experimental data and are shown in Figs. 7 and 8. Mach contours on the coarse grid and of the experimental data look alike. On the fine grid, Mach distribution differs from the experiment. The region of the gradient of local Mach number along the shear layer is well predicted on the coarse grid compared to the experimental data. However on the fine grid, because of low turbulent eddy viscosity, which is directly related to the grid size, the region has become thinner. Non-dimensional streamwise turbulent intensity is shown for the both grids in Figs. 9 and 10. The peak values are observed near the free stagnation point in the experiment. In Figs. 11 and 12, contours of non-dimensional resolved turbulent kinetic energy, TKE, are shown. On the coarse grid, TKE is under predicted in the shear layer, which is because of turbulent viscosity convected from the separated boundary layer. However, on the fine grid the complete shear layer is underpredicted. The reason could be the overprediction of the base pressure. Moreover the shear layer rollup was not captured by the fine grid.
4 Conclusion In the current study, DES was applied to supersonic base flow and the influence of grid resolution on flow predictions was studied. The predictions on the coarse grid were in good agreement with the experiments except the second order statistics, which are resolved Reynolds stress tensor and turbulent kinetic energy. This is a results of the high turbulent eddy viscosity in the shear layer. On the fine grid, a larger recirculation is observed which is due to inability to predict the free shear layer growth correctly. This shear layer growth depends on the development of flow instabilities. In the specified simulated case, instabilites develop after an unrealistic long transition range in time and space. This phenomena caused the larger recirculation, the high pressure near the base and the low turbulent kinetic enegy in the free shear layer.
References [1] J.R. Forsythe, K.A. Hoffmann, K.D. Squires: Detached-Eddy Simulation with compressibility Corrections Applied to a Supersonic Axisymmetric Base Flow, AIAA 02-0586, 2002. [2] H. L¨udeke, A. Filimon, Time Accurate Simulation of Turbulent Nozzle Flow by the DLR TAU-code. AG STAB, 2004.
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[3] Frank Simon, Sebastien Deck, Philippe Guillen, Pierre Sagaut: RANS-LES Simulations of Supersonic Base FLow. 44th AIAA Aerospace Sciences Meeting and Exhibit, 9-12 January 2006. [4] J.L. Herrin, J.C. Dutton: Supersonic Base Flow Experiments in the Near Wake of a Cylindrical Afterbody, AIAA Journal, Vol 32, No. 1, January 1994. [5] P.R. Spalart: Young-Person’s Guide to Detached-Eddy Simulation Grids, NASA/CR-2001-211032, 2001. [6] Andrey K. Travin, Mikhail L. Shur, Philippe R. Spalart and Mikhail Kh. Strelets: Improvement of Delayed Detached-eddy simulation for LES with wall modelling, ECCOMAS CDF 2006.
Large-Eddy Simulation of Tundish Flows Using Preconditioning Nouri A. Alkishriwi, M. Meinke, and W. Schr¨oder Aerodynamisches Institut, RWTH Aachen, W¨ullnerstraße zw. 5 und 7, D-52062 Aachen, Germany
[email protected] www.aia.rwth-aachen.de
Summary A large-eddy simulation using preconditioning is described and applied to investigate continuous tundish flow. The NAVIER–S TOKES equations for compressible flow are approximately solved using an implicit dual-time stepping scheme combined with low Mach number preconditioning and a multigrid acceleration technique. The impact of a turbo-stopper on jet spreading, jet impingement on the wall, wall jets, and steel quality is investigated. The numerical analysis shows the turbostopper flow to generate a more homogeneous tundish flow and as such a higher steel quality can be expected.
1 Introduction Large-eddy simulation (LES) has become as a viable and powerful tool to investigate the physics of unsteady compressible and incompressible turbulent flows. However, LES of low Mach number flows based on compressible solvers has not adequately been discussed in the literature. At low Mach numbers the performance of LES codes developed for the conservation equations of compressible fluids deteriorates in terms of speed and accuracy due to the presence of two different time scales associated with acoustic and convective waves. In many subsonic turbulent flows low Mach number regions exist, which require large integration times until a fully developed flow is established. Different methods have been proposed to solve such mixed flow problems by modifying the existing compressible flow solvers. One of the most popular approaches is to use low Mach number preconditioning methods for compressible codes [1]. The basic idea of this approach is to modify the time marching behavior of the system of equations without altering the steady state solution. This is, however, only useful when the steady state solution is sought. A straightforward extension of the preconditioning approach to unsteady flow problems is achieved when it is combined with a dual-time stepping technique. A highly efficient large-eddy simulation method is presented based on an implicit dual-time stepping scheme combined with preconditioning and multigrid. This method has been validated by well-known case studies in Alkishriwi et al.[2],[3],[4], C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 397–404, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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[5]. This preconditioned large-eddy simulation method is used to simulate the incompressible flow field in a continuous casting tundish. The main function of a continuous casting tundish is to act as distributor of molten steel from the ladle to the moulds. The tundish plays a critical role in determining the final steel quality. In continuous casting of steel, the tundish enables to remove nonmetallic inclusions from molten steel and to regulate the flow from individual ladles to the mold. There is two types of flow conditions in a tundish. Steady-state casting where the mass flow rate through the shroud msh is equal to the mass flow rate through the submerged entry nozzle into the mold mSEN , and transient casting in which the mass of steel in a tundish varies in time during the filling or draining stages. The motion of the liquid steel is generated by jets into the tundish and continuously casting mold. The flow regime is mostly turbulent, but some turbulence attenuation can occur far from the inlet. The characteristics of the flow in a tundish include jet spreading, jet impingement on the wall, wall jets, and an important decrease of turbulence intensity in the core region of the tundish far from the jet, Gardin et al. [7] and Alkishriwi et al. [5]. In previous studies, a large amount of research has been carried out to understand the physics of the flow in a tundish mainly through numerical simulations based on the Reynolds-averaged NAVIER–S TOKES (RANS) equations plus an appropriate turbulence model. To gain more knowledge about the time-dependent turbulence process, which cannot be achieved via RANS solutions, large-eddy simulations of the tundish flow field are performed in this study.
2 Governing Equations The governing equations are the unsteady three-dimensional compressible NAVIERS TOKES equations written in generalized coordinates ξi , i = 1, 2, 3 ∂Q ∂(Fci − Fvi ) =0 , + ∂t ∂ξi
(1)
where the quantity Q represents the vector of the conservative variables and Fci , Fvi are inviscid and viscous flux vectors, respectively. As mentioned before, preconditioning is required to provide an efficient and accurate method of solution of the steady NAVIER–S TOKES equations for compressible flow at low Mach numbers. Moreover, when unsteady flows are considered, a dual-time stepping technique for time accurate solutions is used. In this approach, the solution at the next physical time step is determined as a steady-state problem to which preconditioning, local time stepping, and multigrid are applied. Introducing of a pseudo time τ in (1), the unsteady two-dimensional governing equations with preconditioning read Γ−1 where R represents
R=
∂Q ∂Q + + R = 0, ∂τ ∂t
∂Ec ∂Fc ∂Ev ∂Fv + − − ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2
(2) (3)
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and Γ−1 is the preconditioning matrix, which is to be defined such that the new eigenvalues of the preconditioned system of the equations are of similar magnitude. The preconditioning technique from Turkel [1] has been implemented. It is clear that only the pseudo time terms in (2) are altered by the preconditioning, while the physical time and space derivatives retain their original form. Convergence of the pseudo time within each physical time step is necessary for accurate unsteady solutions. This means, the acceleration techniques such as local time stepping and multigrid can be immediately utilized to speed up the convergence within each physical time step to obtain an accurate solution for unsteady flows. The derivatives with respect to the physical time t are discretized using a three-point backward difference scheme that results in an implicit scheme, which is second-order accurate in time ∂Q = RHS ∂τ RHS = −Γ
(4)
3Qn+1 − 4Qn + Qn−1 + R(Qn+1 ) . 2∆t
3
Numerical Procedure
The governing equations are the NAVIER–S TOKES equations filtered by a low-pass filter of width ∆, which corresponds to the local average in each cell volume. The monotone integrated large-eddy simulations (MILES) approach is used to implicitly model the small scale motions through the numerical scheme. Since the turbulent flow is characterized by strong interactions between various scales of motion, schemes with a large amount of artificial dissipation significantly degrade the level of energy distribution governed by the small-scale structures and therefore distort the physical representation of the dynamics of small as well as large eddies. Using a mixed central-upwind AUSM (Advective Upstream Splitting Method) scheme with low numerical dissipation does remedy this problem. The approximation of the convective terms of the conservation equations is based on a modified second-order accurate AUSM scheme using a centered 5-point low dissipation stencil (Meinke et al. (2002)) to compute the pressure derivative in the convective fluxes. The viscous stresses are discretized to second-order accuracy using central differences, i.e., the overall spatial approximation is second-order accurate. In the dual-time stepping technique the solution at the next physical time step is determined as a steady-state problem which is tackled by preconditioning, local time stepping, and multigrid. A second-order 5-stage Runge-Kutta method is used to propagate the solution from time level n to n+1. The Runge-Kutta coeffi4 9 12 24 6 , 24 , 24 , 24 , 24 ) are optimized for maximum stability of a centrally cients αl = ( 24 discretized scheme. The physical time derivative is discretized by a backward difference formula of second-order accuracy. The method is formulated for multi-block structured curvilinear grids and implemented on vector and parallel computers.
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4
Tundish Flow
In the following the numerical setup of the tundish flow, the computational domain, and the boundary conditions are briefly described. Figure 1 shows the geometry of the tundish with and without turbo-stopper and its main dimensions. Furthermore, the notation of the boundaries is given.
dsh shroud dsr
z
stopper rod sen
x Lsh Lsen
z
γ
H
y
dsen B
L
Fig . 1 . Description of the main parameters of the tundish geometry and integration domain. Without turbo-stopper (left); with turbo-stopper (right) . Table 1. Physical parameters of the tundish flow Parameter tundish length L tundish width B tundish height H inclination of side walls diameter of the shroud dsh diameter of the SEN dSEN diameter of the stopper rod dsr hydraulic diameter dhyd Re based on the jet diameter
value 1.847 m 0.459 m 0.471 m 7o 0.04 m 0.041 m 0.0747 m 0.6911 m 25000
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The minimum grid spacing used for the pipe flow and the round jet at ReD = + = 1.75. In the circum25000 near the wall region in the radial direction is ∆rmin ferential direction, the grid spacing various linearly with r. It reaches a minimum value of 12 (∆r∆θ)min = 8.1 near the centerline of the shroud of the tundish and the pipe and a maximum value 12 (D∆θ)max = 25.8 in terms of viscous wall units at the shroud wall. In the streamwise direction a spacing of ∆z + = 14.7 is used with a relatively small grid anisotropy in the center region of the pipes. In the remaining part of the tundish, the maximum grid spacing in the x, y, and z-direction is ∆x+ = 50, ∆y + = 40, and ∆z + = 30, respectively. The Mach number based on the centerline velocity of the jet is set to M a = 0.02 and the physical time step ∆t = 0.01. The numerical simulations consist of two simultaneously performed computations. On the one hand, the pipe flow is calculated to provide time-dependent inflow data for the jet into the tundish and on the other hand, the flow field within the tundish is computed. The geometrical values and the flow parameters of the tundish are given in Table 1. The computational domain is discretized by 12 million grid points, 5 million of which are located in the jet domain to resolve the essential turbulent structures. Since the jet possesses the major impact on the flow characteristics in the tundish, it is a must to determine in great detail the interaction between the jet and the tundish flow. The boundary conditions consist of no-slip conditions on solid walls. At the free surface, the normal velocity components and the normal derivatives of all remaining variables are set zero. An LES of the impinging jet requires a prescription of the instantaneous flow variables at the inlet section of the jet. To determine those values a slicing technique based on a simultaneously conducted LES of a fully developed turbulent pipe flow is used.
5 R esults In the following, the LES of the tundish flow with and without turbo-stopper is discussed. The numerical results of both problems are compared. We now turn to analyze the no-turbo-stopper case. The typical structure of the velocity field in the tundish is shown in Fig. 2. The figure shows the computed instantaneous flow pattern in the longitudinal cross sections at y/B1 = 0 and y/B1 = −0.5, i.e., in the geometrical symmetry plane and the near side-wall plane within the tundish, respectively. It is clear that the turbulent flow in the tundish contains a wide range of length scales. Large eddies with a size comparable to the diameter of the shroud occur together with very small eddies. The figure evidences the jet spreading, jet impingement on the wall, and wall jets. Furthermore, it is evidenced that a pronounced entrainment occurs just downstream of the jet exit. The jet has a small spreading rate, which is why it reaches the bottom of the tundish at high velocities, spreads in all directions, and generates an upward flow along the side walls of the tundish such that a large vortex between the jet and the near side walls occurs in Fig. 2. This behavior is additionally evidenced in Figs. 4 and 6. In these figures the turbulent flow and the vortical structures in a tundish are evidenced by the time averaged turbulent kinetic energy and the λ2 criterion. The pronounced
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Fig. 2 . LES velocity field of the tundish flow without turbo-stopper at y/B1 = 0, i.e., symmetry plane, (top); and at y/B1 = −0.5, i.e., side-wall, (bottom)
Fig . 3 . LES velocity field of the tundish flow with turbo-stopper at y/B1 = 0, i.e., symmetry plane, (top); and at y/B1 = −0.5, i.e., side-wall, (bottom)
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Fig. 4 . Time averaged turbulent kinetric energy distribution of the tundish flow without turbo-stopper at y/B1 = 0, i.e., symmetry plane
Fig . 5 . Time averaged turbulent kinetric energy distribution of the tundish flow with turbo-stopper at y/B1 = 0, i.e., symmetry plane
Fig . 6 . λ2 -contours and color-coded magnitude of velocity for the instantaneous flow field at x = Lsh of the tundish flow without (left) and with (right) turbo-stopper
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vertical flow structure near the shroud is emphasized. Such a flow pattern leads to nonmetallic inclusions, which are to be avoided. The comparison with the results of the with-turbo-stopper case in Figs. 3, 5, and 6 shows that the tundish flow can be controlled by a turbo-stopper, i.e., the flow pattern can be optimized. Since the shroud jet is redirected back on itself, turbulence is generated, and kinetic energy is dissipated within the transient shear layers and vortex systems. Thereby, in the remaining tundish a more homogeneous flow is produced. Regions of high absolute turbulence are only limited to the direct inlet area of the tundish.
6 Conclusion An efficient large-eddy simulation method for nearly incompressible flows has been applied to continuous tundish flows. The method uses an implicit time accurate dualtime stepping scheme in conjunction with low Mach number preconditioning and multigrid acceleration. The flow field in a tundish with and without turbo-stopper is analyzed to determine the impact of such a device on the steel quality. The findings evidence many intricate flow details and indicate a more homogeneous flow pattern when the turbo-stopper is used.
References [1] E. Turkel: ”Preconditioning techniques in computational fluid dynamics”. Annual Review of Fluid Mechanics, 31, 1999, pp. 385-416. [2] N. Alkishriwi, M. Meinke, W. Schr¨oder: ”Efficient Large-eddy simulation of low Mach number flow”. 92, 14th -DGLR Symposium der STAB, Nov. 16-18, 2004, Bremen, Germany. [3] N. Alkishriwi, M. Meinke, W. Schr¨oder: ”Preconditioned large-eddy simulations for tundish flows”. Proc. of turbulence and shear flow phenomena TSFP-4, Williamsburg, June 27-29, 2005. [4] N. Alkishriwi, M. Meinke, W. Schr¨oder: ”A Large-eddy simulation method for low Mach number flows using preconditioning and multigrid”. Computers and Fluids, 35/10, 2006, pp. 1126-1136. [5] N. Alkishriwi, M. Meinke, W. Schrder, A. Braun, H. Pfeifer: ”Large-Eddy Simulations and particle-image velocimetry measurements of tundish flow”. J. of Steel research, 77 No.8, 2006, pp. 565-575. [6] M. Meinke, W. Schr¨oder, E. Krause, T. Rister: ”A comparison of second- and sixth-order methods for large-eddy simulations”. Computers and Fluids, 31, 2002, pp. 695-718. [7] P. Gardin, M. Brunet, J.F. Domgin, K. Pericleous: ”An Experimental and numerical CFD study of turbulence in a tundish container”. Second International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia, 6-8 December.
Numerical Investigations of Effusion Cooling in Hypersonic Boundary-Layer Flow Jens Linn and Markus J. Kloker Institut f¨ur Aerodynamik und Gasdynamik, Universit¨at Stuttgart, Pfaffenwaldring 21, D-70550 Stuttgart, Germany
[email protected]
Summary Direct numerical simulations (DNS) are carried out to investigate the effect of effusion cooling by blowing through spanwise slits and discrete holes in a laminar flat-plate boundary layer at various Mach numbers. The comparison with experimental data for a Mach-2.67 boundary layer with a cool wall and a spanwise slit shows good agreement. For an adiabatic Mach-6 boundary layer, the focus of the study is the effectiveness of various effusion-cooling configurations (slits and holes), the generated vortices and shear-layer systems. In terms of cooling effectiveness slits are preferable over (a few) holes.
1 Introduction For aerospace or hypersonic cruise vehicles, the state of the boundary layer is of great importance, because for turbulent boundary layers, the thermal loads and skin friction are higher than in laminar boundary layers. Therefore, knowledge of cooling features and laminar-turbulent transition is necessary for the design and the thermal protection system (TPS). Different strategies are used to reduce the thermal loads of hypervelocity vehicles, e.g. radiation, transpiration, effusion cooling or ablation. The subject of this paper is the effusion or film cooling, where cold air is blown into the boundary layer through spanwise slits or rows of holes. This implies a change in the laminar stability properties of the boundary layer.
2
Numerical Method and Boundary Conditions
The numerical method is based on the complete 3-d unsteady compressible NavierStokes equations in conservative formulation (ρ, ρu, ρv, ρw and ρe). The equations are solved in a rectangular integration domain on the flat plate, which does not contain any shock wave induced by the leading edge. In streamwise (x-) and wall-normal (y-) direction, the discretisation is realized by splitted compact finite differences of 6th order [7]. In the spanwise (z-) direction, the flow is assumed to be periodic, thus a Fourier spectral representation is employed. The time integration is done with a classical 4th -order Runge-Kutta method. A detailed description of the discretisation and the algorithm is reported in [1]. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 405–412, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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All length scales are nondimensionalised with respect to a reference length L = Reference values for velocity, density, temperature, viscosity and conductivity are their freestream values at the inflow (indicated by subscript ∞). The pressure is 2 normalised by ρ∞ u∞ , where the superscript denotes the dimensional quantities. With u L = 105 . The air these definitions, the global Reynolds number is defined as Re = ∞ ν∞ is considered as a non-reacting calorically perfect gas [3,9]. At the inflow boundary (x = x0 ), profiles from boundary-layer theory are fixed for all flow variables. For the boundary conditions at the outflow (x = xN ), all equations are solved neglecting the second x-derivative terms. At the freestream boundary (y = yM ), the gradient of the flow variables is set to zero along spatial characteristics [3]. The steady blowing of cold air through holes at the wall with a radius rc (figure 1) is modelled by prescribing a wall-normal momentum distribution ν∞ Re . u ∞
(ρv) = (ρv)c,max · c(r),
(1)
where (ρv)c,max is the maximum wall-normal momentum (maximum blowing ratio). The wall temperature distribution over the hole is prescribed by Tc = Tw · 1 − c(r) + Tc,core · c(r), (2) where Tc,core is the core temperature of the cold air and Tw the local wall temperature at the edge of the hole. The distribution function c(r) is a polynomial of 5th order, which has been already used in [10] for suction and blowing to generate disturbances at the wall. Both the gradient and curvature are zero at r = 0 and r = rc : c(r) = 1 − 6 ·
r rc
5 + 15 ·
2 2 r = (x − xc ) + (z − zc )
r rc
4
,
− 10 ·
r rc
3 ,
(3)
0 ≤ r ≤ rc ,
where xc and zc stand for the center coordinates of the hole. Outside the hole, (ρv) is zero at the wall, Tw is the local adiabatic wall temperature ((∂T /∂y)|w = 0) or has a constant value (isothermal wall), and the pressure gradient in wall-normal direction at the wall and in the holes is zero. For steady blowing through a spanwise slit of width bc = 2 · rc , the distribution function c(r) in equation (3) is independent from the zcoordinate, i.e. z = zc . Grid refinements studies showed no dependence of the grid spacing (∆x, ∆y, ∆z, ∆t) on the steady state solutions discussed below.
3
Results
3.1 Comparison with Experiments at Mach 2.67 In this section, we compare our simulation results with the measurements of Heufer and Olivier [4,5] at the Shock Wave Laboratory Aachen (SWL). They investigated
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Fig. 1. (ρv)-distribution at the wall for one row of holes
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Fig. 2. Comparison of cooling effectiveness ηc from simulation and experiment for an effusion cooled boundary layer at Mach 2.67 for two blowing rates (blue - (ρv)c,max = 0.065; green - (ρv)c,max = 0.043)
Fig. 3. Visualisation of the temperature field with streamlines in a longitudinal cut for the effusion-cooled boundary layer at Mach 2.67 ((ρv)c,max = 0.065). Isolines of the u-velocity for u = 0 (dashed line) and u = 0.99 (dashed dotted line). ∆x = 0.253·10−2 and ∆y = 0.6·10−3 .
an isothermal laminar boundary layer on a wedge with a deflection angle of 30◦ and a given post-shock Mach number 2.67, with M = 7 and T0 = 1368K. Cold air is blown through one spanwise slit. The given post-shock freestream temperature is = 564K(≈ 0.45 Trec ), the pressure p∞ = 0.1489bar (⇒ L = 24.57mm), and T∞ ). This wall temperature means the wall temperature is TW = 293K = const(≈ 14 Trec that the wall is strongly cooled. Investigations based on the Linear Stability Theory (LST) have shown (see, e.g.[8]), that wall cooling stabilises 1st -mode (vorticity) disturbances and destabilises the 2nd -mode (acoustic) disturbances that do not exist at Mach numbers lower than approximately 3.5. In addition, the basic boundary layer investigated is subcritical (Rx,crit Rex (xN )) due to the strong wall cooling.
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Figure 2 shows the comparison of the experimental [4,5] and the simulation data for two blowing ratios, (ρv)c,max = 0.043 and (ρv)c,max = 0.065, with Tc,core = 293K(= Tw ). The slit width is bc = 0.5mm (≈ 0.57 δc , δ - boundary-layer thickness), and the cooling effectiveness is defined by ηc = 1 − q˙c /q˙ref , where q˙c = ϑ · ∂T /∂y|w is the heat flux into the wall with effusion cooling and q˙ref is the heat flux without effusion cooling. For both blowing rates the cooling effectiveness is higher in the simulations. In the low-blowing case, there is a heat flux into the slit and cold-gas reservoir in (that was realized the experiment resulting in a higher cooling gas temperature Tc,core but not quantitatively measured) thus lowering the cooling effectiveness ηc . Note that no experimental data of the boundary-layer evolution are available and thus the local thickness parameters and Reynolds numbers may differ. Higher cooling effectiveness was also found by tentative numerical simulations at SWL. A longitudinal cut of the temperature field with streamlines is shown in figure 3 for the high blowing rate. In front of the slit is a reversed-flow region near the wall with a clockwise rotating vortex with its center marked by the dot. For both blowing rates, no instability regions were found using LST despite a separation region exists in front of the slit. The basic cooling by the cool wall is so strong that it stabilizes even the blowing. Here effusion cooling is applied in a case where the flow is already strongly cooled by a cool wall, thus this case is unrealistic. A simple transfer of the results to cases with significantly different wall temperature gradients is not possible. 3.2 Comparison of Effusion-Cooling Configurations at Mach 6 In this section we investigate an adiabatic boundary layer at an edge Mach number 6 in which cold air is blown through spanwise slits and rows of holes. The freestream = 89K(≈ 17 Trec ) and the pressure is p∞ = 0.0038bar(⇒ L = temperature is T∞ 36.28mm), matching the flow parameters of experiments in the hypersonic wind tunnel H2K of DLR-K¨oln [2]. Table 1 summarizes the parameters. Slits were used in cases a, b and holes in cases c, d. The integrally injected massflow and the cooling gas temperature = 293K(≈ 12 Trec ) is in all cases the same. In case c the two rows of holes are Tc,core aligned in contrast to case d where the rows are sz /2-staggered. The first spanwise slit or row of holes is positioned at xc = 2.205. Table 1. Parameters of the slits and holes configurations for cases at Me = 6 case slits row of (ρv)c,max hole diameter streamwise spanwise rows holes or slit width d spacing sx spacing sz z-offset a 2 0.0284 0.058 ≈ 0.6 δ 0.1378 ≈ 1.4 δ b 1 0.0284 0.087 0.1378 c 2 0.15 0.058 0.1378 0.1378 d 2 0.15 0.058 0.1378 0.1378 sz /2 x0 = 0.225, xN = 7.33, yM = 0.54 ≈ 4 δ at x = xN , S P r = 0.71, κ = 1.4, µ (T ) = µ (T∞ ) · T 3/2 · T1+T +TS −5 kg 110.4 with S = T and µ∞ (T∞ = 280K) = 1.735 · 10 ms
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Fig. 4. Wall temperature for steady blowing into an adiabatic flat-plate boundary layer at Mach 6 through two spanwise slits (a), one slit (b), two aligned rows of holes (c) and two sz /2-staggered rows of holes (d)
Fig. 5. Visualisation of the temperature field and isolines of the u-velocity in the crosscut at x = 2.5 for the aligned rows (a) and for the sz /2-staggered rows (b)
The resulting wall temperature is shown in figure 4 for all four configurations. Cases a and b look very similar and show a significant lower wall temperature than the two
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Fig. 6. Visualisation of vortical structures via λ2 -isosurface (λ2 = −15.0) for aligned rows of holes (a) and for staggered rows of holes (b). The arrows indicate the rotation sense.
other cases c and d. In the ,,aligned” case c, the wall temperature is only slightly reduced and strongly varies in the z-direction. In the ,,staggered” case d, the wall temperature is lower than in case c, and does not vary as strong in the z-direction. The temperature and u-distribution in the crosscut is shown in figure 5. The aligned rows blow more cold gas from the wall into the boundary layer and show stronger ∂u ∂z -gradients than the staggered rows. The reason why the slits are more efficient is that the blowing surface is much larger than with the holes, translating into a lower wall-normal velocity in the slits. Thereby the cold gas keeps closer to the wall. The vortical structures of the hole configurations are visualised via the λ2 -criterion [6] in figure 6. From the holes, counter-rotating vortex pairs (CVPs) emerge which are along the jet trajectory and have such a rotation sense that gas is transported away from the wall in the streamwise hole center line. Furthermore exists a toroidal neck vortex at each hole edge. It has a counter-clockwise rotation sense upstream the hole, in contrast to the considered slit case. A horseshoe vortex is not observed in the simulations due to the low blowing ratio. In studies of jets in crossflow (JICF), where typically a horseshoe vortex (with a rotation sense opposite to the CVP) is found, (ρv)c,max ≈ O(1) and
Numerical Investigations of Effusion Cooling in Hypersonic Boundary-Layer Flow
Fig. 7. Stability diagram (from LST) for an adiabatic, effusion-cooled Mach-6 boundary layer (case a) for a (2nd -mode) 2-d wave (fre f ). Dashed red line is the quency: ω = 2πL u∞ αi = 0 curve of the adiabatic Mach-6 boundary layer without blowing, displaying also 1st mode instability for x > 4.7.
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x Fig. 8. N-factors (= − x0 αi dx) of case a with (blue lines) and without blowing (red lines with circles) for various frequencies ω (see also figure 7)
d > δ. In the aligned case (figure 6a), the second row enhances the CVPs from the first row and the vortices lay wall-parallel. In contrast, figure 6b shows the CVPs from the second row pushed downwards, keeping the cold gas at the wall. However, the CVPs of both cases decay downstream. For a Mach-6 boundary layer, the strongest amplified disturbance mode is the 2nd mode as a 2-d wave (γ = 0). Figure 7 shows the stability diagram for case a. The dash-dotted lines mark the streamwise positions of the slit centers. The 1st mode is completely stabilised for γ = 0 because of the cooled wall and is not present anymore in the considered streamwise region. The 2nd mode is shifted to lower frequencies in the region of the slits, and the maximum amplification rate αi,max is about twice as large as in the case without blowing. A small additional instability region is developed at the slits over the 2nd mode, which seems a higher viscous mode. Figure 8 shows the N-factors for case a with and without blowing. For the frequency ω = 12.5 the N-factor is approximately four times higher at the end of the considered streamwise domain than without blowing. Primary LST uses the assumption that the spanwise baseflow gradients are zero. This assumption is no longer valid in the cases c and d with the holes and therefore primary LST can not be used to predict the instability of these cases. For cases c and d, unsteady background disturbances will be introduced in the DNS to investigate the instability properties of the full 3-d case. Enhanced laminar instability can compromise the cooling effect.
4 Conclusions The comparison between numerical and experimental results for effusion cooling of a laminar, basically already cooled isothermal Mach-2.67 boundary layer through slits
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shows good agreement. The simulation predicts a slightly higher cooling effectiveness which may be caused by differing boundary-layer parameters that are not available from the experiment. The presented results for effusion-cooling configurations with slits and holes of an adiabatic Mach-6 boundary layer shows that slits are better than (a few) holes. The slit blowing velocity is, at same mass flow, smaller than that of (a few) holes. Aligned rows of holes induce a strong variation of the wall temperature and less cooling effectiveness compared to staggered rows of holes. A counter-rotating vortex pair (CVP) is generated at each hole, decaying downstream. In the staggered case, the decay is stronger than in the aligned case. The analysis with the linear stability theory shows that the maximum amplification rate of 2nd -mode disturbances grows and the amplified frequency band is shifted to lower frequencies. Defined 3-d instability investigations will be performed by unsteady DNS.
Acknowledgments This research has been partially sponsored by the Helmholtz-Gemeinschaft (HGF) within the German joint project RESPACE - Key Technologies for Reusable Space Systems.
References [1] Babucke, A., Linn, J., Kloker M., Rist, U.: ”Direct numerical simulation of shear flow phenomena on parallel vector computers”, In High Performance Computing on Vector Systems (ed. M. Resch & el.) Proc. High Performance Computing Center Stuttgart, pp. 229-247, Springer (2006). [2] Bierbach, M.: ”Untersuchungen zur aktiven K¨uhlung der Grenzschichtstr¨omung an einem Plattenmodell”, Diplomarbeit, Technische Universit¨at Darmstadt (2005). [3] Eißer, W.: ”Numerische Untersuchungen zum laminar-turbulent Str¨omungsumschlag in ¨ Uberschallgrenzschichten”, Dissertation, Universit¨at Stuttgart (1998). [4] Heufer, K.A., Olivier, H.: ”Film cooling for hypersonic flow conditions”, Proc. 5th European Workshop on Thermal Protection Systems and Hot Structures 2006, ESA. [5] Heufer, K.A., Olivier, H.: ”Film Cooling of an Inclined Flat Plate in Hypersonic Flow”, AIAA 2006-8067. [6] Jeong, J., Hussain, F.: ”On the identification of a vortex”, J. Fluid Mech., vol. 285, pp. 69-94 (1995). [7] Kloker, M.: ”A robust high-resolution split-type compact FD-scheme for spatial direct numerical simulation of boundary-layer transition”, Applied Scientific Research, vol. 59, pp. 353-377 (1998). [8] Malik, M.R.: ”Prediction and control of transition in supersonic and hypersonic boundary layers”, AIAA-J., vol. 27, pp. 1487-1493 (1989). [9] Thumm, A.: ”Numerische Untersuchung zum laminar-turbulenten Str¨omungsumschlag in transsonischen Grenzschichtstr¨omungen”, Dissertation, Universit¨at Stuttgart (1991). [10] Stemmer C., Kloker M.: ”Interference of wave trains with varying phase relations in a decelerated 2-d boundary layer”, In Recent Results in Laminar-Turbulent Transition (ed. S. Wagner, M. Kloker, U. Rist), NNFM, vol. 86, pp. 91-110, Springer (2003).
Numerical Investigation of Flow-Induced Noise Generation at the Nozzle End of Jet Engines Andreas Babucke, Markus Kloker, and Ulrich Rist Institut f¨ur Aerodynamik und Gasdynamik, Universit¨at Stuttgart Pfaffenwaldring 21, D-70550 Stuttgart, Germany
[email protected] http://www.iag.uni-stuttgart.de
Summary Sound generation downstream the nozzle end of a subsonic laminar jet has been investigated using two-dimensional direct numerical simulations (DNS). The nozzle end is modeled by a finite flat plate with Mach numbers of MaI = 0.8 above and MaII = 0.2 below the splitter plate. Behind the nozzle end, a combination of a wake and mixing layer develops. Due to the high amplification rates, disturbances saturate before a pure mixing layer occurs. Non-linear generation mechanisms produce higher harmonic disturbances in the upper boundary layer, resulting in an only quasi periodic solution. The main acoustic sources correspond to the positions of vortex pairing. Broadband noise is emitted instead of tonal noise, known from the pure mixing layer without a splitter plate [4].
1
Introduction
Noise reduction is an important issue for a wide range of technical problems but the mechanisms of flow-induced sound are not yet understood properly. The current investigation focuses on jet noise as it is a major noise source of aircrafts. Aeroacoustic simulations are a relatively new field in computational fluid dynamics demanding high requirements in computational performance and the accuracy of the computational scheme itself. On the one hand, a high resolution is needed to compute the noise sources accurately, on the other hand, it demands a large computational domain to obtain the relevant portions of the acoustic far-field. As the acoustic amplitudes are small compared to the flow-field disturbances, boundary conditions have to be chosen carefully, in order not to spoil the acoustic field with reflections. Previous investigations of jet noise have been focusing on either pure mixing layers [3, 4] or low Reynolds number jets [5], where a S-shaped velocity profile is used at the inflow. In this two-dimensional DNS, we include the nozzle end, modeled by a thin flat plate with two different free-stream velocities on top and below. This allows to investigate the influence of the wake formed by the two boundary layers. Additionally, the splitter plate gives the possibility to model real actuators for future noise reduction instead of unrealistic volume forcing. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 413–420, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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2 2.1
Computational Configuration
Numerical MPPethod
The two-dimensional simulation is a first step and has been performed using the NS3D code [2]. It solves the unsteady compressible Navier-Stokes equations. The code is written in conservative formulation, solving for density ρ, the momentum densities ρu, ρv and the total energy per volume E. The velocity components are 2 normalized by the inflow velocity U ∞ , the pressure with ρ · U ∞ and all other quantities by their inflow values of the upper boundary layer (see figure 1), marked with the subscript ∞ . Length scales are made dimensionless with a reference length L and the time with L/U ∞ , where the overbar denotes dimensional quantities. The viscosity is modeled using the Sutherland law with a reference viscosity µ(T ∞ = 280K) = 1.735 · 10−5 kg/(ms). Since we can assume a weak temperature dependence, the Prandtl number P r = 0.71 and the ratio of specific heats κ = 1.4 are taken constant. In streamwise (x) and normal (y) direction, the flow-field is discretized by 6th order compact finite differences. Alternating up- and downwind-biased differences [7] are applied to convective terms for de-aliasing. Second derivatives are computed directly instead of applying the first derivative twice. This leads to better resolved viscous terms and improves the stability of the code [1]. Grid transformation in the x-y plane is implemented by mapping the physical grid on an equidistant computational ξ-η grid. The equations are integrated in time using the standard 4th -order Runge-Kutta scheme. The domain decomposition in the x-y plane is not only used for parallelization, it also allows to define neighbours or specific boundary conditions for each domain. Therefore, the nozzle end can be included easily without any special treatment of corner points. At the free stream boundaries, a one-dimensional characteristic boundary condition [6] is used. An additional damping zone forces the flow variables smoothly to a steady state solution. Prescribing amplitude and phase distributions from linear stability theory allows to introduce defined disturbances at the subsonic inflow with characteristic boundary conditions [6]. The outflow is the most crucial part as one has to avoid large structures passing the boundary and contaminating the acoustic field. Therefore, a combination of grid stretching and spatial low-pass filtering is applied in the sponge region. Disturbances become increasingly badly resolved as they propagate through the sponge region. As the spatial filter depends on the step size in x-direction, perturbations are smoothly dissipated before they reach the outflow boundary. This procedure shows very low reflections and has been already applied by Colonius et al. [4]. 2.2
FPPlow PPaPrameters
For the current investigation, an isothermal subsonic jet with the Mach numbers MaI = 0.8 for the upper and MaII = 0.2 for the lower stream has been selected. As both temperatures are equal (T1 = T2 = 280K), the ratio of the streamwise
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velocities is UI /UII = 4. This large factor leads to strong instabilities behind the nozzle end, so a moderate number of grid points in x-direction is sufficient to simulate the aeroacoustic source. The Reynolds number Re = ρ∞ U1 δ1,I /µ∞ = 1000 is based on the displacement thickness δ1,I of the upper stream at the inflow. With δ1,I (x0 ) = 1, length scales are normalized with the displacement thickness of the fast stream at the inflow. The boundary layer of the lower stream corresponds to the same origin of the flat plate. The cartesian grid is decomposed into eight subdomains, four in streamwise and two in normal direction. Each subdomian contains 650 x 425 points in x- and y-direction. The mesh is uniform in streamwise direction with a step size of ∆x = 0.15 up to the sponge region, where the grid is highly stretched. In normal direction, the finest step size is ∆y = 0.15 in the middle of the domain with a continuous stretching up to a spacing of ∆y = 1.06. The origin of the coordinate system (x = 0, y = 0) is located at the end of the nozzle. The nozzle end is modeled by a finite thin flat plate with a thickness of one ∆y. Due to the vanishing thickness of the nozzle end, an isothermal boundary condition at the wall has been chosen. The temperature of the plate is T wall = 296K, being the mean value of the adiabatic wall temperatures of the two streams. The initial condition along the flat plate is obtained from similarity solutions of the boundary-layer equations. Further downstream, the full boundary-layer equations are integrated downstream, providing a flow-field sufficient for an initial condition and linear stability theory. The resulting streamwise velocity profiles of the initial condition are shown in figures 1 and 2. Behind the nozzle end, the flow field keeps its wake-like shape for a long range. As high amplification rates occur here, the flow is already unsteady before a pure mixing layer has developed. This means that the pure mixing layer investigated earlier [1, 3, 4] has to be considered as a rather theoretical approach.
Fig . 1 . Profile of the streamwise velocity u for the upper and lower boundary layer at the inflow
Fig . 2 . Downstream evolution of the streamwise velocity profile behind the nozzle end
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Linear Stability Theory
Spatial linear stability theory (LST) is used both to introduce disturbances at the inflow, as well as for further analysis of the flow-field. As the flow is highly unsteady behind the nozzle end and enforcing an artificial steady state does not work properly, we use the initial condition derived from the boundary-layer equations to compute eigenvalues and eigenfunctions. The computation is based on a 4th -order matrix solver combined with Wielandt iteration. According to figure 3, a fundamental angular frequency of ω = 0.0688 was chosen for the upper boundary layer. The amplification keeps almost constant in downstream direction. As the two boundary layers emerge from the same position, the lower boundary layer is stable up to the nozzle end. Behind the edge of the splitter plate, amplification rates 50 times higher than in the upper boundary layer occur due to the inflection points of the streamwise-velocity profile. Maximum amplification in the mixing layer takes place for a frequency of roughly three to four times of the fundamental frequency of the boundary layer as illustrated in figure 4.
Fig . 3 . Amplification rates of the upper (fast) boundary layer given by linear stability theory for various x−positions
4
Fig . 4 . Amplification rates for various x−positions behind the splitter plate predicted by linear stability theory
Numerical Results
The flow is forced at the inflow of the upper boundary layer with a single disturbance of the fundamental frequency ω0 and an amplitude of umax = 5 · 10−3 , using the corresponding eigenfunction from linear stability theory. As the flow field is only quasi periodic in time, a time interval of 64 fundamental periods with a sampling rate of 4000 timesteps is used for analysis. To ensure that no initial disturbances falsify the results, a total number of 556000 timesteps, corresponding to 139 periods of the fundamental frequency, has been computed. As the maxima of u-disturbances in the boundary layer are located near the wall, analysis of amplitudes is mainly based on the maximum of the streamwise velocity along y. Thus only a small data strip in y-direction with high temporal resolution is required for postprocessing. The spatial growth rates αi are compared with linear stability theory in figures 5 and 6 for the upper boundary layer and the following mixing layer, respectively. For the latter,
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Fig . 5 . Amplification rates of streamwise velocity u in the upper boundary layer, based on maximum amplitudes along y. Symbols denote results from linear stability theory.
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Fig . 6 . Amplification rates of norm velocity v behind the nozzle end, based on maximum amplitudes along y. Symbols denote results from linear stability theory.
Fig . 7 . Maximum amplitude of the streamwise velocity fluctuation u in the upper boundary layer (x < 0) and behind the splitter plate (x ≥ 0). The fundamental frequency and its five higher harmonics are plotted in bold and labeled separately. The non-integer higher harmonics are colored in dark and bright blue, dark and bright green, yellow and grey, respectively. Each color corresponds to a frequency range of 1 · ω0 . (h, 0) means frequency h · ω0 with zero spanwise wavenumber.
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the normal velocity v is chosen as it is mainly associated with vorticity and does not contain upstream propagating sound. The maximum amplitudes of the streamwise velocity u are shown in figure 7 where the fundamental frequency and its first five higher harmonics are highlighted. Note that along the splitter plate (x ≤ 0), the values of the upper boundary layer are used. In the upper boundary layer, the fundamental disturbance (1, 0) behaves according to linear stability theory. The amplification rates of the first two higher harmonics (2, 0) and (3, 0) approach the one of the fundamental frequency, indicating that they are generated by the fundamental disturbance. Near the end of the splitter plate (−20 < x < 10), the growth rates differ from linear stability results due to the discontinuity in geometry at x = 0. The amplitudes of the streamwise velocity u shown in figure 7 reveal the generation of higher harmonics in the upper boundary layer, leading to a wider disturbance spectrum for the subsequent mixing layer. Behind the splitter plate, the first three higher harmonics are now strongly amplified and grow according to linear stability theory (figure 6). Due to increased amplitudes of several frequencies, also originally stable modes grow now caused by non-linear interaction of the amplified disturbances. With many disturbances having large amplitudes, only a quasi periodic solution can be reached. Saturation of the second higher harmonic (3, 0) takes place at x ≈ 60. In figure 8, we can see that this is the position where the mixing layer rolls up and the first vortex is generated. Further downstream, the lower frequencies (1, 0) and (2, 0) dominate the flow field, corresponding to subsequent vortex pairing. At x = 150, two out of three eddies pair and the resulting vortex merges with the remaining eddy at x = 220. Thereby, it is only quasi-deterministic whether the single eddy pairs with the precedent or subsequent bigger vortex. → The resulting emitted sound is visualized by the dilatation field ∇− u together with the spanwise vorticity in figure 9. Although the sound generation is less clear than for a pure mixing layer [4], the dilatation field allows to determine roughly two acoustic sources. Their positions correspond to the locations of vortex pairing mentioned above. By placing an observer in the acoustic field (x = 195, y = −121.8), the emitted sound can be evaluated more precisely. Looking at the time dependent
Fig . 8 . Snapshot of the spanwise vorticity behind the nozzle end (located at x = 0). Contour levels range from −0.53 to 0.12.
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Fig . 9. Snapshot of the far-field sound showing the dilatation ∇u in a range of ±3·10−4 . The two boundary layers and the evolving mixing layer are illustrated by spanwise vorticity with the same contour levels as in figure 8. The position of the observer is marked by a white cross.
Fig . 10 . Time-dependent signal of the acoustic pressure fluctuations in the far-field at the observer’s position (x = 195, y = −121.8)
Fig . 11 . Amplitude spectrum of the acoustic pressure relative to the fundamental frequency at the observer’s position (x = 195, y = −121.8). Analysis is based on the time frame shown in figure 10.
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pressure signal in figure 10, almost random fluctuations can be observed. The spectrum of the acoustic pressure in figure 11 shows a broadband noise with decaying amplitudes for higher frequencies. Despite the full spectrum, peaks at higher harmonics of the fundamental frequency (ω/ω0 = 2, 3, 4, 6) can be observed.
5
Conclusion
The nozzle end of an isothermal jet with Mach numbers MaI = 0.8 and MaII = 0.2 has been simulated using two-dimensional DNS. As a model for the nozzle end of a jet, the mixing of two co-floating streams downstream of a splitter plate is used. The baseflow obtained from the boundary-layer equations shows a combination of wake and mixing layer behind the nozzle end. The flow becomes highly unsteady already before a pure mixing layer develops. As the disturbances saturate, roll up of the mixing layer and, further downstream, vortex pairing occurs. Due to large amplitudes, non-linear mechanisms produce a wide spectrum of disturbances, leading to a less deterministic flow-field as it is the case for the pure mixing layer. This results in a broadband noise emission, still showing peaks at the higher harmonics in the pressure spectrum. Further investigations will include three-dimensional simulations and the development of actuators for noise reduction.
Acknowledgements The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for its financial support within the the subproject SP5 in the research group FOR-508 ”Noise Generation in Turbulent Flows”. Supercomputing time was provided by the H¨ochstleistungsrechenzentrum Stuttgart (HLRS) within the projects ”Lamtur” and ”Teraflop Workbench”.
References [1] A. Babucke, M. J. Kloker, and U. Rist. DNS of a plane mixing layer for the investigation of sound generation mechanisms. to appear in Computers and Fluids, 2006. [2] A. Babucke, J. Linn, M. J. Kloker, and U. Rist. Direct numerical simulation of shear flow phenomena on parallel vector computers. In High performance computing on vector systems: Proceedings of the High Performance Computing Center Stuttgart 2005. Springer Verlag Berlin, 2006. [3] C. Bogey, C. Bailly, and D. Juve. Numerical simulation of sound generated by vortex pairing in a mixing layer. AIAA J., 38(12):2210–2218, 2000. [4] T. Colonius, S. K. Lele, and P. Moin. Sound generation in a mixing layer. J. Fluid Mech., 330:375–409, 1997. [5] J. B. Freund. Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9. J. Fluid Mech., 438:277–305, 2001. [6] M. B. Giles. Nonreflecting boundary conditions for Euler equation calculations. AIAA J., 28(12):2050–2058, 1990. [7] M. J. Kloker. A robust high-resolution split-type compact FD scheme for spatial DNS of boundary-layer transition. Appl. Sci. Res., 59:353–377, 1998.
Numerical Analysis of Sound Generating Mechanisms of a High-Lift Device Daniel K¨onig1 , Wolfgang Schr¨oder, and Matthias Meinke Institute of Aerodynamics, RWTH Aachen University, W¨ullnerstraße zw. 5 u. 7, 52062 Aachen, Germany
[email protected] http://www.aia.rwth-aachen.de
Summary In this study the flow field and the acoustic field over a high-lift configuration consisting of a deployed slat and a main wing is numerically analyzed. The flow parameters are M a = 0.16, Re = 1.4 ∗ 106 , αSlat = 23o and αgeo = 13o . The flow data, which are computed via a large-eddy simulation, provide the distributions being plugged in the source terms of, e.g. the acoustic perturbation equations (APE) or the Ffowcs-Williams and Hawkings (FWH) equation, to compute the acoustic near field. The analysis shows the interaction of the shear layer of the slat trailing edge and the slat gap flow to generate higher vorticity than the main airfoil trailing edge shear layer. Thus, the slat gap is the dominant noise region.
1 Introduction The progress in reducing the emitted sound by jet engines of commercial aircraft in recent years has shifted the focus of interest to airframe noise. Especially during landing, when the engines run in idle mode, airframe noise is as important as jet noise. Experimental investigations [14] have shown that the generated noise of the flow over deployed slats is a dominant part to the overall sound pressure level radiated by aircraft. The reduction of this noise requires a detailed knowlege of the sound generating mechanisms. For this purpose, the underlying turbulent flow field must be investigated since it is the development of turbulent wall-bounded and free shear layers that governs the noise generation. The following sections describe the large-eddy simulation of the flow field over an airfoil consisting of a slat and a main wing. The results are the input for further aeroacoustic investigations which are done by the acoustic perturbation equations [6] and a Ffowcs-Williams and Hawkings formulation [12].
2 Numerical Methods and Computational Setup 2.1 Large-Eddy Simulation A monotone-integrated large-eddy simulation (MILES) [3],[13] approach is used to to solve the three-dimensional compressible Navier-Stokes equations. The blockstructured solver is based on a vertex centered finite-volume technique, which uses C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 421–429, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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a modified second-order accurate AUSM discretization to approximate the inviscid fluxes [11]. The viscous terms are discretized by a central scheme of secondorder accuracy. The temporal integration from time level n to n + 1 is done by an explicit five-step Runge-Kutta method, whereas the coefficients are optimized for maximum stability. For a detailed description of the flow solver the reader is referred to Meinke et al. [13]. At low Mach number flow a preconditioning method in conjunction with a dual-time stepping scheme can be used [2]. Furthermore, a multigrid scheme is implemented to accelerate the convergence of the dual-time stepping procedure. 2.2 Acoustic Simulations All acoustic analyses are performed using a two-dimensional approach. That is, the spanwise extent of the computational domain of the large-eddy simulation can be limited since especially at low Mach number flows the turbulent length scales are significantly smaller than the acoustic length scales and as such the noise source can be considered compact. This two-dimensional acoustic analysis tends to result in somewhat overpredicted sound pressure levels which are corrected following the method described by Ewert et al. in [7]. Acoustic Perturbation Equations The first method to determine the acoustic field is based on the acoustic perturbation equations (APE) derived for compressible fluids, i.e., the APE-4 form introduced by Ewert and Schr¨oder [6] are used. This system of equations is part of the DLR PIANO code which is applied in this investigation. The spatial derivatives are approximated by the 7-point dispersion-relation preserving finite-difference scheme of Tam and Webb [17]. The time integration is done by an alternating 5-6 stage lowdispersion low-dissipation Runge-Kutta scheme [8]. To avoid spurious oscillations the solution is filterd every 2nd iteration step using a 6th-order explicit commutative filter [16], [18]. For the outer boundaries radiation boundary conditions by Tam and Webb [17] are applied. Ffowcs-Williams Hawkings Formulation The second approach is based on an extension of the general theory of Lighthill that contains not only acoustic sources but also convective and refraction effects in the source terms. The solver for this integral equation of Ffowcs-Williams and Hawkings (FWH) [19] is written in the frequency domain as discussed in [12]. The formulation in the frequency domain avoids the requirement of an infinite number of time levels, which are needed by the 2-d Greens function in the time domain. The quadropole term in the FWH equation is neglected due to its M a8 behavior in comparison to the M a6 proportionality of the surface source term.
∗ i
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extremely influence the acoustic analysis, a sponge layer following Israeli et al. [9] is imposed. On the walls an adiabatic no-slip boundary condition is applied. In the spanwise direction periodic boundary conditions are used. The computation is performed for a free-stream Mach number of M a = 0.16 at an angle of attack of 13o . The Reynolds number based on the clean chord length of the main wing is 1.4 million. The high-lift configuration and these parameters correspond to data of an experimental setup in the DLR Braunschweig wind tunnel (AWB), in which measurements of the flow and acoustic field have been performed within the national project FREQUENZ.
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Fig . 5 . Grid resolution near the wall: Slat cove
The acoustic mesh for the APE solution has a total number of 1.8 million points, which are distributed over 24 blocks. The grid spacing is chosen to resolve a maximum frequency of 8 kHz in the whole domain.
3 Results and Discussion 3.1 Flow Simulation The initial conditions were obtained from a two-dimensional RANS simulation and the subsequent large-eddy simulation has been run for about 5 non-dimensional time units based on the freestream velocity and the clean chord length to ensure a fully developed turbulent flow field. Subsequently, samples were recorded for the statistical analysis and also to compute the aeroacoustic source terms. The time interval was chosen to be approximatly 0.0015 time units covering an overall time of approx. 6 non-dimensional time units resulting in about 4000 data sets using 7 Terabyte of disk space.
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,%3 2!.3 %XP DATA
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Fig . 6 . Comparison of the cp coefficient Fig . 7. Streamlines and Mach number distribetween LES, RANS [5] and experimental bution of the time and spanwise averaged flow field data [10]
The quality of the results can be assessed by the mesh resolution. Hence, first of all the correct mesh resolution near the wall is determined and depicted in Figs. 2 to 5. It is evident that the flat plate approximation yields satisfactory results concerning the grid resolution. However, due to the accelerated and decelerated flow on the suction and pressure side, respectively, the grid resolution departs somewhat from the approximated values. In the slat cove region the resolution reaches everywhere the required values for LES of wall bounded flows (∆y + ≈ 2, ∆x+ ≈ 100 ∆z + ≈ 20)[1]. In Fig. 6 the pressure coefficient cp computed by the LES solution is compared with RANS data [5] and experimental distributions. The measurements were carried out in an anechoic wind tunnel with an open test section to capture the flow and the acoustic field. These experiments are compared to numerical solutions which mimic
Fig . 8 . λ2 contours in the slat cove region at M a = 0.16, Re = 1.6 · 106 and α = 13o
Fig . 9 . λ2 contours in the slat cove region at M a = 0.16, Re = 1.6 · 106 and α = 13o
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uniform freestream conditions. Therefore, even with the correction of the geometric angle of attack of 23o in the measurments to about 13o in the numerical solution no perfect match between the experimental and numerical data can be obtained. The Mach number distribution and some selected streamlines of the time and spanwise averaged flow field are depicted in Fig. 7. Apart from the two stagnation points an aera of low Mach number flow is visible which belongs to a recirculation domain in the slat cove region. This recirculation zone is bounded by a shear layer emanating from the slat cusp and reattaching close to the trailing edge of the slat. The peak Mach number occurs just downstream of the slat gap.
Fig . 10 . λ2 contours in the slat gap area
Fig . 11 . Time and spanwise averaged tur` ´ bulent kinetic energy k = 21 u2 + v 2 + w2 in the slat cove region
The turbulent vortex structures are shown by λ2 contours in Figs. 8 to 10. The grey scales mapped onto the contours indicate the local velocity magnitude. The shear layer between the recirculation area and the flow passing through the slat gap develops large vortical structures near the reattachment point. Most of these structures are convected through the slat gap while some vortices are trapped in the recirculation area and are moved upstream to the cusp. This behavior is in agreement with the findings of Choudhari et al. [4]. Furthermore, like the investigations in [4] the analysis of the unsteady data indicates a fluctuation of the reattachment point. On the suction side of the slat, shortly downstream of the leading edge, the generation of the vortical structures in Fig. 8 visualizes the transition of the boundary layer. This turbulent boundary layer passes over the slat trailing edge and interactes with the vortical structures convected through the slat gap. Fig. 10 illustrates some more pronounced vortices being generated in the reattachment region and whose axes are aligned with the streamwise direction. They are some kind of G¨ortler vortices created by the concave deflection of the flow. distribution%of the time and spanwise averaged turbulent kinetic energy k = $ The 1 2 + v 2 + w2 is shown in Fig. 11. In agreement with [4] the peak value ocu 2 curs in the vicinity of the reattachment point. This corresponds to the strong vortical structures in this area evidenced in Fig. 9.
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Fig . 12 . Streamwise component of the purturbed lamb vector
Fig . 13 . Perturbed Lamb vector. Top: streamwise component.Bottom: normal component.
Fig. 14 . Pressure contours based on the LES solution
Fig . 15 . Pressure contours based on the LES/APE solution
3.2 Acoustic Analysis Figs. 12 and 13 show the distribution of the acoustic sources by means of the components of the perturbed Lamb vector (ω × u) . The strongest acoustic sources are due to the normal component of the Lamb vector. The peak value occurs on the suction side downstream of the slat trailing edge whereas somewhat smaller values are determined near the main wing trailing edge.
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In Figs. 14, 15 and 16 snapshots of the pressure fluctuations based on the LES, the APE and the FWH solutions are shown. The interaction between the trailing-edge noise of the main wing and that of the slat is clearest in the APE solution. A closer look reveals the slat source to be dominant compared to the main airfoil trailing edge. It is evident that the LES mesh is not capable to resolve the high frequency waves in the near far field. The quality of the FWH results may suffer from neglecting the quadrupole terms.
Fig . 16 . Pressure contours based on the LES/FWH solution
Fig . 17 . Directivities for a circle with R = 1.5 based on the APE solution
The directivities of the slat gap noise source and the main airfoil trailing edge source are shown in Fig. 17 on a circle with the radius R = 1.5 centered near the trailing edge of the slat. The following geometric source definitions were used. The slat source covers the part from the leading edge of the slat through 40% chord of the main wing. The remaining part belongs to the main wing trailing edge source. An embedded boundary formulation is used to ensure that no artificial noise is generated [15]. It is evident that the sources located near the slat cause a stronger contribution to the total sound field than the main wing trailing edge sources. This behavior corresponds to the distribution of the Lamb vector.
4
Conclusion
The application of a hybrid flow/acoustic simulation approach to the flow over a high-lift configuration and its generated sound field has been presented. The LES data are plugged in either the APE or the FWH formulation. The analysis shows the interaction of the shear layer of the slat trailing edge and slat gap flow to generate higher vorticity than the main airfoil trailing edge shear layer. Thus, the slat gap is the dominant noise source region. The results of the large-eddy simulation are in good agreement with data from the literature. The acoustic analyse shows the correlation between the areas of high vorticity, especially somewhat downstream of the slat trailing edge and the main wing trailing edge, and the emitted sound.
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Acknowledgment This work is done in the framework of the national project FREQUENZ. The authors thank Elmar Gr¨oschel for his assistance in running the PIANO code.
References [1] [2]
[3] [4]
[5] [6] [7] [8]
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
LESFOIL: Large Eddy Simulation of Flow Around a High Lift Airfoil, chapter Contribution by ONERA. Springer, 2003. N. Alkishriwi, W. Schr¨oder, and M. Meinke. A large-eddy simulation method for low mach number flows using preconditioning and multigrid. Computers and Fluids, 35(10):1126–1136, 2006. J. P. Boris, F. F. Grinstein, E. S. Oran, and R. L. Kolbe. New insights into large eddy simulation. Fluid Dynamics Research, 10:199–228, 1992. M. M. Choudhari and M. R. Khorrami. Slat cove unsteadiness: Effect of 3d flow structures. In 44st AIAA Aerospace Sciences Meeting and Exhibit. AIAA Paper 2006-0211, 2006. M. Elmnefi. Private communication. Institute of Aerodynamics, RWTH Aachen University, 2006. R. Ewert and W. Schr¨oder. Acoustic perturbation equations based on flow decomposition via source filtering. Journal of Computational Physics, 188:365–398, 2003. R. Ewert, Q. Zhang, W. Schr¨oder, and J. Delfs. Computation of trailing edge noise of a 3d lifting airfoil in turbulent subsonic flow. AIAA Paper 2003-3114, 2003. F. Q. Hu, M. Y. Hussaini, and J. L. Manthey. Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics. Journal of Computational Physics, 124(1):177–191, 1996. M. Israeli and S. A. Orszag. Approximation of radiation boundary conditions. Journal of Computational Physics, 41:115–135, 1981. A. Kolb. Private communication. FREQUENZ, 2006. M. S. Liou and C. J. Steffen Jr. A new flux splitting scheme. J. Comput. Phys., 107:23– 39, 1993. D. P. Lockard. An efficient, two-dimensional implementation of the ffowcs williams and hawkings equation. Journal of Sound and Vibration, 229(4):897–911, 2000. M. Meinke, W. Schr¨oder, E. Krause, and T. Rister. A comparison of second- and sixthorder methods for large-eddy simulations. Computers and Fluids, 31:695–718, 2002. M. Pott-Pollenske, J. Alvarez-Gonzalez, and W. Dobrzynski. Effect of slat gap on farfield radiated noise and correlation with local flow characteristics. AIAA. W. Schr¨oder and R. Ewert. LES-CAA Coupling. In LES for Acoustics. Cambridge University Press, 2005. J. S. Shang. High-order compact-difference schemes for time dependent maxwell equations. Journal of Computational Physics, 153:312–333, 1999. C. K. W. Tam and J. C. Webb. Dispersion-relation-preserving finite difference schemes for computational acoustics. Journal of Computational Physics, 107(2):262–281, 1993. O. V. Vasilyev, T. S. Lund, and P. Moin. A general class of commutative filters for les in complex geometries. Journal of Computational Physics, 146:82–104, 1998. J. E. F. Williams and D. L. Hawkings. Sound generation by turbulence and surfaces in arbitraty motion. Philosophical Transactions of the Royal Society London, A342:264– 321, 1969.
On the Design of Silent Trailing-Edges M. Herr Deutsches Zentrum für Luft- und Raumfahrt e.V . (DLR ), Institut für Aerodynamik und Strömungstechnik , Lilienthalplatz 7 , D-3810 8 Braunschweig, Germany
[email protected]
Summary Trailing-edge noise data from an experimental study on a flat-plate and on a twodimensional NACA 0012-like airfoil are presented. Within extensive measurements in the Aeroacoustic Wind Tunnel Braunschweig various flow-permeable (comb-type) edge modifications were tested with respect to their noise reduction capability. Among numerous design parameters a narrow slit width was identified as major requirement for a low-noise trailing-edge design whereas flexibility of the comb material was found to be not imperative for such edge noise reduction means. Because both boundary-layer tripping and trailing-edge thickness show significant influence on trailing-edge noise, a detailed description of these effects is provided to enable an accurate interpretation of corresponding noise test data.
1 Introduction Trailing-edge (TE) noise reduction is subject of intense efforts due to its practical relevance for a wide range of applications (such as the high-lift systems and turbomachinery or wind-turbine blades). TE noise is directly related to the turbulence produced in the boundary layer of an airfoil, [7] and [9]. When vortices pass the TE, sound is emitted due to the abrupt change in boundary condition. Numerous theoretical and experimental studies have been conducted in the past with the aim to gain TE noise prediction capability, the latter being a necessary requirement for the development of noise reduction means for full-scale applications. But up to now, there is still the need for a more detailed description. The current work extends the parametric noise-reduction study of [8], in which several flow-permeable brush-type TEs, [11], were successfully tested at realistic Reynolds numbers. With respect to a prospective full-scale application additional test parameters (such as stiffness or spacing of the brush fibres and angle of attack) were included into the test program. Apart from measurements on a flatplate configuration, also measurements on a NACA 0012-like airfoil representing a more realistic geometry were performed. As an alternative low-noise TE design (instead of TE brushes) also a comb-type slit TE, [4], was tested. The latter design was chosen based on the outcome of recent numerical calculations in terms of the aerodynamic performance of different (stiff) slat TE modifications, [14]. This study showed that the aerodynamic properties can be fully preserved by the comb-type slit TE, whereas application of brush-type TE extensions (composed of C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 430–437, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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cylindrical fibres, exhibiting a step between solid geometry and brush root) would require a readjustment of the slat gap and overlap settings to retain maximum lift.
2 Experimental Setup and Test Conditions Directional microphone and hot-wire measurements were conducted in the openjet anechoic test section of DLR’s Aeroacoustic Wind Tunnel Braunschweig (AWB). Two basically different symmetric 2D airfoils were tested: (i) a vertically mounted flat-plate airfoil with a variable chord length between 0.8 m and 2.0 m and (ii) a 0.4 m chord NACA 0012-like airfoil which was placed horizontally between two sideplates attached to the rectangular 0.8 m by 1.2 m AWB nozzle exit, Figure 1 (top). The geometry of the latter airfoil corresponds to an original NACA 0012 section for the first 95 % of chord length, but comprises a modified TE region by reducing the original TE thickness (h = 1.008 mm) to h = 0.15 mm. Thus, a slightly increased TE taper was obtained (exact coordinates on request). A set of exchangeable TE sections (starting at 80 % of the chord length) was manufactured, Figure 1 (bottom); covering increased solid TE thicknesses (realized by cutting the modified NACA 0012 geometry so that the chord length was reduced) and different flow-permeable edge-modifications. The exchangeable solid slit TE was realized for the modified NACA 0012 configuration (h = 0.15 mm) with slits of 45 mm in length, a mean width of 0.1 mm and a slit spacing of 0.4 mm. A more detailed description of the flat-plate experimental arrangement is provided in [8]. For the assessment of TE noise spectra a directional microphone system (elliptic mirror) was used. In [5] the underlying calibration of the specific mirror resolution and gain is presented. The integral boundary layer parameters, such as the displacement thickness G* at the TE, were measured by hot wires for the flat plate only. Since boundary layer measurements were not available for the NACA 0012-like airfoil these values had to be approximated by XFOIL calculations, [6]. Chord-based Reynolds numbers ranged from 2.1 x 106 to 7.9 x 106 for the flat-plate configurations and from 1.1 x 106 to 1.6 x 106 for the NACA 0012-like airfoil. Within a parametric study the effects of flow speed u (40 m/s to 60 m/s), geometric angle of attack D (0 deg to 13.9 deg, corresponding to an aerodynamic angle of attack D* of 8.7 deg, as derived from pressure distributions available for the solid NACA 0012-like airfoil), boundarylayer tripping (60 deg-zigzag strips of different height at different positions) and TE thickness h (0.15 mm to 1 mm) on TE noise were determined. Brush design parameters included fibre diameter (0.3 mm to 0.5 mm), fibre length (5 mm to 100 mm), fibre yaw orientation (0 deg vs. 10 deg), slit width s (< 0.1 mm to 0.3 mm) and flexibility of the fibre material (polypropylene vs. steel). This study was focussed on broadband turbulent boundary-layer TE noise and quasi-tonal bluntness noise, related to von Kármán-type vortex-shedding from the TE. Therefore, to prevent the known mechanism of laminar boundary-layer instability noise, [12], the boundary layers were symmetrically tripped on both the pressure and the suction sides in most of the test cases. For the flat-plate model data as presented in this paper transition was forced by a backward facing step (of 0.1 mm in height) at a fixed position 55 mm downstream of the leading edge, [8].
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For the modified NACA 0012 mainly the results for a 0.4 mm thick zigzag trip (12 mm in length, leading edge at 9.25 % of the full chord length) are shown. This comparably thick tripping was chosen based on diagnostic stethoscope measurements and infrared thermography which verified the effectiveness of the tripping devices. Without tripping these preparatory measurements showed that at a 0 deg angle of attack and an inflow turbulence level of Tux = 0.4 % natural transition leads to a fully developed turbulent boundary layer between 45 % (uf = 60 m/s) and 51 % (uf = 40 m/s) of the uncut chord length. Slit TE
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Fig . 1 . Flat-plate model (top left) and quasi NACA 0012 model (top right) in the AWB. Schematic view of the different solid reference and brush TE geometries (bottom).
3 Results 3.1 Effect of Boundary-Layer Tripping on TE Noise Figure 2 demonstrates the effect of boundary-layer tripping in comparison with the case of free transition at a 0 deg angle of attack. For zero incidence it is known that no laminar boundary-layer instability noise is present, [12]. Since transition occurs more upstream for the tripped configurations, the turbulent boundary-layer TE noise maximum shifts towards lower frequencies, Figure 2 (left). This effect can also be noted when comparing the corresponding spectra of [3]. A respective frequency shift and moreover, a significant decrease in peak level can be observed for the quasi-tonal component, related to blunt TE vortexshedding, an effect which is also documented in [2]. Similar trends are shown for different tripping configurations: moving forced transition further downstream has the effect that low-frequency contributions (corresponding to larger turbulence scales) decrease and high-frequency contributions increase. Consequently, a
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Strouhal-scaling of the TE noise spectra based on G* as proposed by Brooks, [3], leads to a collapse of at least the turbulent boundary-layer TE noise peak frequency and of the low and high frequency level roll-off, Figure 2 (right). This frequency scaling behaviour may (among other measurement-related reasons) explain discrepancies when comparing the test data from different research groups, such as the recently reported differences between NACA 0012 data from Oerlemans, [13], and Brooks and co-workers, [3]. In both studies TE geometry, chord length (228.6 mm), flow velocity and incidence were identical, but entirely different tripping devices and wind tunnel turbulence levels (< 0.4 %, [3], vs. 0.5 %, [13]) were present. Therefore, it seems not exactly defined (i) where transition really takes place and (ii) whether the tripping devices cause additional disturbances to the turbulent boundary-layer flow. An artificial thickening of the boundary-layer due to the “heavy” tripping (being much “heavier” than in the current study) was stated in [3] while no G* data was provided in [13]. Assuming a smaller G* for this dataset (e. g. by usage of XFOIL, as it was done in the present study) could explain the observed frequency shift in measured TE noise spectra. 3.2 Effect of Solid Trailing-Edge Thickness on TE Noise In the relevant semi-empirical prediction approaches (which are mostly based on [3]) bluntness noise related level peaks are considered to be simply superimposed on the broadband turbulent boundary-layer TE noise of a “sharp” airfoil. The latter was interpreted as the remaining continuous spectrum after subtraction of the quasi-tonal bluntness contribution. However, from the results of the current investigations this assumption can not be confirmed. According to Figures 3 and 4 rather a spectral redistribution of sound intensity is observed: For blunt TEs, due to the presence of additional turbulence production at the TE, those frequency bands containing the vortex-shedding frequency dominate TE noise spectra, whereas a significant roll-off in level at the adjacent higher frequencies is observed. This is obvious from both the flat-plate and the quasi NACA 0012 data, Figures 3 and 4. Note in this context the significant level difference in the higher frequency range when noise data obtained for the thin 0.15 mm TE are compared
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against those for the thicker 1 mm TE. This trend can also be established from a more recent study of Hutcheson and Brooks on a (cambered) NACA 63-215 airfoil, where results from small-aperture directional-array (SADA) measurements on a 0.13 mm TE and a 3.3 mm TE are compared, [10]. In this study a “spectral hump near 10 kHz” was determined in TE noise spectra for the sharp TE, and the spectral high-frequency roll-off in level was obviously steeper for the blunt TE case. In summary, these results may show the benefit of using focussing measurement techniques which are less susceptible to contamination from extraneous noise sources and coherence loss effects than e.g. the coherent output power (COP) method being applied in [2] and [3]. With respect to an accurate validation of the noise reduction potential of TE modifications these effects have to be kept in mind when choosing a particular reference configuration. The presence of bluntness noise is primarily related to the thickness ratio h/G*. The relevant literature suggests that “for h/G* < 0.3 (and certainly for < 0.05)” a TE is considered “sharp”, i.e. the condition for no vortex shedding from the edge is satisfied, [1]. The assumption of h/G* < 0.3 (which was mainly based on the NACA 0012 data of [2]) was verified by the current quasi NACA 0012 data if one discounts the above-stated differences in energy distribution. However, as already emphasized in [8], the results as obtained for the flatplate airfoil prove that this vortex-shedding condition is not universally valid, Figure 3. As known from Blake, [1], vortex-shedding amplitudes are highly dependent on TE geometry which has to be taken into account in TE noise prediction schemes. Due to the fact that dominating level maxima are still present for h/G* << 0.1 a more conservative estimate of h/G* < 0.05 for a “sharp TE” is supported by the present data. 3.3 Effect of Flow-Permeable Edge-Modifications Some basic effects of brush TE-extensions on TE noise spectra have been presented in [8] for the flat-plate airfoil. Accordingly, TE-brush devices provide a
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significant broadband noise reduction potential. Figures 5 to 7 show corresponding results for brush-devices at the quasi NACA 0012 model, thus documenting a broadband noise reduction of about 4 to 10 dB for the best configuration tested. This was a flexible brush, composed of a single layer of polypropylene fibres (0.4 mm in diameter and 35 mm in length) with a spacing of s < 0.1 mm. It should be noted that for this configuration measured 1/3-octave band sound pressure levels exceeded the wind-tunnel self noise by only 3 dB which underlines the effectiveness of this noise reduction measure. In Figure 5 1/3-octave band TE noise spectra are presented for three selected TE modifications compared with the solid TEs of different thickness. Note that, due to the above-mentioned effect of bluntness, the achieved noise reduction in the mid to high frequency range (fmG*/uf > 0.1) depends on the choice of the respective solid reference TE. For clarity only the sharp TE (h = 0.15 mm) will be considered in the following, Figures 6 to 8. In Figure 6 the effect of flow velocity on TE noise spectra is shown for the same three TE modifications as displayed in Figure 5 (compared with the sharp reference TE). Herein, data reduction follows the theoretical (velocity) 5 dependence, [7, 9], and is based on G* as a characteristic length scale of the boundary-layer flow, [3]. As has been theoretically derived, [7], low-impedance TEs are assumed to show the same velocity dependence of TE noise spectra as rigid ones. According to Brooks et al., [3], the maximum of turbulent boundary-layer trailing-edge noise is expected in the Strouhal number range from fmG*/ uf = 0.06 to 0.1 which is in good agreement with the current reference data. In this Strouhal range the TE modifications are highly effective. Normalized data are shown to coalesce within 3 dB. However, a better fit of normalized spectra for all TE modifications is obtained by using a higher velocity exponent of about 6. The normalized reference data show a mismatch of TE noise spectral peaks which was already discussed in [3] and [8].
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With respect to the influence of different brush characteristics, it was found that the flexibility of the brush material is favourable, but not essential to achieve a noise reduction. In fact, as shown in Figures 5 and 6, also stiff metal fibres
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(needles with the same dimensions as the polypropylene fibres) showed a similar noise reduction capability. Due to difficulties in the manufacturing of brush devices (e.g. reproducible fibre spacing) it is not finally proven whether or not these results reflect a systematic beneficial effect of fibre flexibility. From all parameter variations in the current study, the mean fibre spacing was identified as the most important design parameter. According to Figure 7 an almost zero-spacing of the fibres (s < 0.1 mm) revealed the best results. This supports the working hypothesis in [8], assuming the noise reduction to be the consequence of a viscous damping of turbulent flow pressure amplitudes in the brush area. However, the exact relationship between flow parameters and the slit width is difficult to identify due to the limitations in the manufacturing accuracy. The actual spacing explains why in Figure 5 the slit TE (s | 0.1 mm) achieved significantly less noise reduction than the TE extensions (s < 0.1 mm). To support the statement, that the slit TE could be further optimized by adapting the slit width, TE brush extensions with a comparable coarser spacing were tested, Figure 7. For s | 0.1 mm the results for the slit TE are in between those as obtained for the flexible and the stiff TE-extensions, respectively. In addition to theses findings the following trends could be identified: (i) a minimum fibre length l | 15 mm is required to achieve a measurable noise reduction (ii) fibres aligned in flow direction are more favourable than (yaw) inclined ones, (iii) thick fibres are more effective compared to thin ones, [8], (iv) a growing angle of attack diminishes the overall noise reduction because solid (reference) TE noise emission decreases in the mid-frequency range. Figure 8 displays TE noise spectra for different angles of attack D. A significant difference between the brush TE-extension and the slit TE is obvious. The latter represents, at a lower level though, the same angle-of-attack dependence as the solid reference TE, whereas the brush TE-noise spectrum remains nearly unaffected by changes in angle of attack.
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4 Conclusions The results of an experimental study on trailing-edge noise revealed that boundary-layer tripping and design details of the edge geometry have a significant effect on trailing-edge sound emission. Some discrepancies between the available trailing-edge noise data (presented here and from different research groups) indicate the general need for a more detailed consideration of specific test conditions in terms of their respective effects on noise emission. Several flowpermeable edge-modifications (brushes and a slit trailing edge) were found to effectively reduce trailing-edge noise for both a flat-plate and a NACA 0012-like airfoil at different flow velocities and angles of attack. The relevant design criteria for such devices were derived from parametric tests. Accordingly, a narrow slit width (< 0.1 mm) is imperative for a low-noise trailing-edge design whereas material flexibility is considered favourable, but not a necessary requirement.
References [1] W. K. Blake. “Mechanics of Flow-Induced Sound and Vibration, Complex FlowStructure Interactions”. In Applied Mathematics and Mechanics, Volume 17-II, Academic Press, Inc., Orlando, 1986, pp. 756-767. [2] T. F. Brooks and T. H. Hodgson. “Trailing Edge Noise Prediction From Measured Surface Pressures”. In Journal of Sound & Vibration, Volume 78, No. 1, 1981, pp. 69-117. [3] T. F. Brooks, D. S. Pope and M. A. Marcolini. “Airfoil Self-Noise and Prediction”. NASA Reference Publication 1218, 1989. [4] J. Delfs. “Aerodynamisches Bauteil mit einer geschlitzten Hinter- oder Seitenkante“. German Patent Application DE 10 2006 049 616.7, 2006. [5] W. Dobrzynski, K. Nagakura, B. Gehlhar and A. Buschbaum. “Airframe Noise Studies on Wings with Deployed High-Lift Devices”. AIAA Paper 98-2337, 1998. [6] M. Drela. “An Analysis and Design System for Low Reynolds Number Airfoils”, Conference Paper, Conference on Low Reynolds Number Airfoil Aerodynamics, University of Notre Dame, France, 1989. [7] J. E. Ffowcs-Williams and L. H. Hall. “Aerodynamic Sound Generation by Turbulent Flow in the Vicinity of a Scattering Half-Plane”. In Journal of Fluid Mechanics, Volume 40, No. 4, 1970, pp. 657-670. [8] M. Herr and W. Dobrzynski, "Experimental Investigations in Low-Noise TrailingEdge Design”. In AIAA-Journal, Volume 43, No. 6, 2005, pp. 1167-1175. [9] M. S. Howe. “Acoustics of Fluid-Structure Interactions”. Cambridge University Press, 2000. [10] F. V. Hutcheson and T. F. Brooks. “Effects of Angle of Attack and Velocity on Trailing Edge Noise”. AIAA Paper 2004-1031, 2004. [11] K. Mau and W. Dobrzynski, "Anordnung zur Minderung des aerodynamischen Lärms an einem Vorflügel eines Verkehrsflugzeuges“, European Patent EP 1314642 B1, 2005. [12] E. C. Nash, M. V. Lowson and A. Mc Alpine. “Boundary-Layer Instability Noise on Aerofoils”. In Journal of Fluid Mechanics, Volume 382, 1999, pp. 27-61. [13] S. Oerlemans. “Wind Tunnel Aeroacoustic Tests of Six Airfoils for Use on Small Wind Turbines”. Report of the National Renewable Energy Laboratory NREL/SR500-35339, 2004. [14] J. Ortmann and J. Wild. “Effect of Acoustic Slat Modifications on Aerodynamic Properties of High-Lift Systems”, AIAA Paper 2006-3842, 2006.
Investigation of Flow Induced Sound Radiated by a Forward Facing Step C. Hahn2 , S. Becker1 , I. Ali1 , M. Escobar2, and M. Kaltenbacher2 1
Institute of Fluid Mechanics (LSTM), Friedrich-Alexander-University Erlangen-N¨urnberg, D-91058 Erlangen, Germany
[email protected] http://www.lstm.uni-erlangen.de/aeroakustik/ 2 Department of Sensor Technology (LSE), Friedrich-Alexander-University Erlangen-N¨urnberg, D-91052 Erlangen, Germany
Summary In the present paper we investigate the aeroacoustic characteristics of the flow over a forward facing step. The purpose is to identify relevant regions which are responsible for the noise generation. Correlation measurements are carried out for the pressure fluctuations on the wall, velocity characteristics of the flow and the recorded far-field acoustic pressures. The application of the correlation techniques provides a useful tool in the characterisation of the flow-induced sound sources. For further understanding, flow and acoustic experiments are conducted using different Reynolds numbers and radii of curvature.
1 Introduction The perturbation of the flow over the exterior surface of airplanes by small steps plays an important part in the generation of aeroacoustic noise. Steps on the exterior surface appear at skin lap joints, window gaskets and other flow perturbing obstacles. Due to manufacturing imperfections step heights observed on aircraft surfaces due to skin-joint mismatches and glassing contours are in the range of 0.8 mm to 5 mm [1], [2]. Therefore, the precise understanding of the correlation between fluid and acoustic quantities for this kind of geometry is of relevance. The physics behind the noise generation process is quite complicated and still not fully understood for this simplified geometry. The strategy of our investigations is to combine the numerical and experimental tools at LSTM and LSE at the University of Erlangen-N¨urnberg. This paper is focused on the experimental part of the investigations. For the experiments in a low noise wind tunnel environment we make use of correlation techniques and a microphone array to identify relevant regions which are responsible for the noise generation. Correlation analysis is applied to the wall pressure fluctuations and velocity characteristics of the flow in respect to the recorded far-field sound pressure. C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 438–445, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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2.1 Test Setup Measurements are carried out in an aeroacoustic wind tunnel which is integrated into an anechoic chamber. The chamber has a lower cut-off frequency of 300 Hz. The anechoic environment allows for the measurement of the directional pattern of the radiated sound. The closed return wind tunnel has an open test section with a nominal cross section of 200 mm x 260 mm and yields a maximum exit velocity of 50 m/s. For the experiments described, a larger nozzle with a cross section of 250 mm x 330 mm is used as only velocities ranging from 10 m/s to 30 m/s are investigated. The free stream turbulence ratio is found to be 0.15% for the respective velocities. The distribution of the velocity in a plane perpendicular to the flow direction is homogeneous. The deviations of the streamwise velocity component relative to the core velocity are below 0.5%. 2.2 Measurement Techniques Different measurements have been taken using flows with Reynolds numbers (based on step height) ranging from 8000 to 24000. The step height H is always 12 mm. Investigations at Re=8000 are carried out with a laminar boundary layer while at Re=24000 a tripping tape is used to generate a turbulent boundary layer at the inflow. The boundary layer profile is measured 0.2 m (= 16.6 H) upstream of the step. A displacement thickness of 1.24 mm and a momentum thickness of 0.52 mm is observed for a flow velocity of 10 m/s. At flow velocity of 30 m/s a displacement thickness of 0.61 mm and a momentum thickness of 0.49 mm is obtained. Several measurements techniques are being used including the following: – Hot Wire Measurements: to characterise the velocity distribution in the boundary layer and to measure the velocity fluctuations in the shear layer behind the forward facing step. Hereby a constant temperature anemometer (Dantec Type 55M01) equipped with a single-wire probe is used. The probe is calibrated with the help of a prandtl tube prior to the measurements. – Stationary Pressure Measurements: yielding information about the distribution of the static pressure on the surface of the plate. The data are used for validation of numerical calculations. – Instationary Pressure Measurements: are used to analyse the pressure fluctuations near the forward facing step. The pressure fluctuations are later on correlated with the sound measurements in the far-field. The local wall pressures are recorded with flush mounted Kulite pressure transducers XCQ-093. – Microphone sound field measurements: are undertaken with 1/2” free field condenser microphones (B&K 4189) in order to record the flow induced sound field. – Correlation measurements: simultaneously recording the radiated sound in the farfield and the local wall pressures allows for the indirect localization of the radiated noise. In order to guarantee simultaneity a respective data acquisition card is used (NI 4472).
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РMicrophone Array based Source Localization: in order to directly localize the generated noise a microphone array consisting of 32 simultaneously recorded microphones is used. The 1/4Ӭ omnidirectional electret microphones (Sennheiser KE 4211-2) are provided in this setup with a developed preamplifier yielding a sensitivity of 60 mV/Pa which can be further increased up to 30 V/Pa via additional amplifiers. Microphone selfnoise is 27 dB(A). 2.3 Test Model and Geometries The general measurement setup is shown in Figure 1. A flat plate (length 1 m in streamwise direction, width 0.66 m) is flush mounted to the nozzle. At a distance of 37 cm downstream of the nozzle a step is attached to the plate. Three different geometries of the upper edge of the step are investigated: A sharp edge, a rounded edge with a radius equal to half the step height and a fully rounded edge (radius equal to step height). The varied geometries are depicted in Figure 2.
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Fig. 3. Flow visualization of a forward facing step with sharp edge
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3.1 Radiated Sound Field The flow induced noise field of the forward facing step is recorded at a distance of 1 m at an angle of 45◦ and 90◦ relative to the plate with microphones placed outside the flow (see Figure 4). For each velocity the acoustic measurement is carried out twice. At first the step is removed and only the background noise stemming from tunnel and plate is recorded. Then the step is mounted on the plate and the measurement is repeated. The difference in the respective recorded spectra shows the contribution of the step alone. The results displayed in Figure 5 illustrate the broadband nature of the radiated noise of the step. Hereby the cross spectrum between the simultaneously recorded signals at an
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Sound pressure spectra − sharp edged step Cross spectra of microphones at 45° and 90° 60 Plate U =10 m/s (laminar) 0
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angle of 45◦ and 90◦ is evaluated in order to suppress uncorrelated noise between these two microphones and thereby yielding a higher ratio between the signal of interest and background noise. 3.2 Variation of Geometry Investigation of the sound radiated by steps with rounded edges using only a single microphone leads to no further insight, because the background noise level of the tunnel is too high compared to the noise level of the modified steps. Therefore no difference is visible in the frequency spectrum whether only a flat plate is examined or one with a step on top. Using correlation analysis, a spatial separation of the various noise sources can be accomplished. For this purpose a microphone array consisting of 32 microphones is used. The microphones are evenly distributed on a circle with a diameter of 0.9 m at a distance of 2 m (see Figure 6). The microphones are flush mounted in a wooden plate with sidelength 1 m. Thereby uncontrolled sound reflections between the microphones are prevented. Furthermore, the sound pressure observed by individual microphones is doubled. It has to be noted that the distance between some of the microphones to the edge of the plate is comparable to the wavelength of the lowest frequency used for evaluation thereby leading to incomplete reflection of the sound wave yielding inaccurate source strength levels. However, the main focus of this investigation is not to obtain exact source strength levels but to examine the effect of geometry variations on the radiated sound which can still be observed in this setup. The signals are recorded simultaneously with a sample rate of 96 kHz. Afterwards a beamforming approach in the time domain is used including the deletion of the diagonal elements of the microphone cross correlation matrix. The signals are evaluated in a frequency range of 2 kHz-10 kHz, which represents the range that shows major noise contributions of the step. No amiet-correction is applied, as the Mach number used is
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Fig. 7. Localization of flow induced sound radiated by sharp edged step via microphone array
Fig. 8. Flow induced sound radiated by step with rounded edge (R=H/2)
Fig. 9. Flow induced sound radiated by step with rounded edge (R=H)
small (Ma = 0.09 in this case) and the array is placed far away from the sources so that only a narrow cone of view is used [3]. In Figure 7 the sharp edged step is clearly visible. Due to limited spatial resolution of the array and the width of its main lobe, the exact location of the generated noise cannot be pinpointed and the question whether the noise is created in front of the step or downstream of it cannot be answered by this measurement. Steps with rounded edges generate less noise as can be seen in Figure 8 and Figure 9. There is only a little amount of noise generated and additional noise sources of the wind tunnel itself become visible. The noise coming from the nozzle on the left side and a detached eddy on the right side at the end of the flat plate can be discerned. Peak levels of the acoustic source strength are reduced by over 10 dB for the rounded edge with R=H/2. Additional increase of the curvature radius of the step (R=H) further reduces peak levels by 2 dB in the region of the step.
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Investigation of Noise Generating Region
4.1 Instationary Pressure Field and Spatial Correlation The instationary wall pressures are recorded at several positions in front of the step and downstream of it. Figure 10 displays the local pressure spectra measured along a line parallel to the main flow direction containing the center of the plate. The broad band nature of the pressure fluctuations is again clearly visible. Wall pressures in front of the step (at -10 mm and at -5 mm) are lower than those downstream of it. For the case shown (Re=24000) the spectra drop off around 500 Hz until they can no longer be discriminated from the self-noise of the transducers.
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The coherence between wall pressures in streamwise direction is high when only transducers belonging to the same fluid region are concerned. There is no coherence found between transducers positioned in front of the step and others positioned downstream of it. Of special interest is the distribution of the coherence in lateral direction. Figure 12 displays the coherence spectra for transducer positions located 10 mm in front of the step when their lateral distance is varied from 5 mm to 30 mm. As expected the coherence is reduced when the distance between the sensors is increased. However, a significant amount of coherence is always present in the frequency range of 2 kHz to 10 kHz which represents the dominant range of the radiated flow induced noise (at Re=24000). In contrast, the coherence in lateral direction downstream of the step displays a different behaviour (see Figure 13). Here, only for a distance of 5 mm between the sensors a significant level of coherence can be found for frequencies greater than 100 Hz. At increased distances the coherence in lateral direction quickly drops off. No coherence is found in the frequency range of 2 kHz to 10 kHz. I.e. wall pressures measured in front of the step possess greater coherence in lateral direction than wall pressures recorded at locations downstream of the step.
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Coherence betwen wall pressures in lateral direction sharp edged step (H=12 mm), U0=30 m/s
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4.2 Correlation and Coherence Between the Pressure and Sound Field Using coherence analysis [4], [5] in order to indirectly localize the origin of the radiated sound yields only for positions in front of the step significant levels of coherence between local wall pressure and far-field sound. Figure 11 displays the coherence between a far-field microphone at an angle of 90 and wall pressures in front of the step as well as positions downstream of it. Furthermore, the coherence found is in the relevant range of 2 kHz to 10 kHz. 4.3 Velocity Field The basic flow field is depicted in Figure 14. Mean flow and standard deviation are shown for a free stream velocity of 10 m/s. The flow separation at the upper edge of
Fig. 14. Velocity distribution of a sharp edged step at U0 = 10 m/s
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the step is visible. Additionally, a vortex region is visible in front of the step. Inserting smoke into the test section and illuminating it via laser light sheet technique, the vortex at the base of the step can be visualized (see Figure 3). Therewith it can be observed that the vortex size is reduced in case steps with rounded edges are used.
5 Conclusion Experimental investigations of the flow induced noise from a forward facing step are presented. The sound generated by the forward facing step is broadband in nature radiating significant amounts of noise in the frequency range of 2 kHz to 10 kHz (at Re=24000). Extensive investigations are done by using the correlation technique. The objective is to get a better understanding of the mechanics of flow induced sound concerning the forward facing step. The results show that only in the closed region before and around the sharp edge one can find a connection between velocity distribution and local wall pressure to the far-field sound pressure. This region contains a large twodimensional vortex structure. Behind the step the flow is three dimensional. The spatial correlation of wall pressures in lateral direction to the mean flow is less correlated. The turbulent flow pattern undergoes transition into an isotropic structure. Correlation measurements will be supplemented by correlating local velocity fluctuations with radiated sound. For this purpose investigations using a hotwire anemometer with a three wire probe are under way. A characterisation of the flow field with a 3d Laser-Doppler-Anemometer is planned as well. Furthermore, numerical calculations based on similar flow conditions are currently being carried out.
Acknowledgements The results presented here were obtained in a project financed by BFS (Bayerische Forschungsstiftung) which is gratefully appreciated.
References [1] B. Efimtsov, N. Kozlov, S. Kravchenko: ”Wall pressure-fluctuation spectra at small forwardfacing steps”. AIAA 99-1964, 1999. [2] B. Efimtsov, A. Golubev, S. Rizzi, A. Andersson, E. Andrianov: ”Influence of small steps on wall pressure fluctuation spectra measured on a TU-144LL flying laboratory”. AIAA 20022605, 2002. [3] P. Sijtsma, S. Oerlemans, H. Holthusen: ”Location of rotating sources by phased array measurements”. NLR-TP-2001-135, National Aeroaspace Laboratory NLR, 2001. [4] J. Bendat, A. Piersol: ”Engineering Applications of Correlation and Spectral Analysis”. John Wiley & Sons Inc, 1980. [5] Y. Guo, M. Joshi, P. Bent, K. Yamamoto: ”Surface pressure fluctuations on aircraft flaps and their correlation with far-field noise”. J. Fluid. Mech., Vol. 415, pp. 175-202, 2000.
Application of PIV Techniques for Rotor Blade Tip Vortex Characterization H. Richard 1 , B.G. Van der Wall 2 , M. Raffel 1, and M. Thimm 1 1
DLR, Institute of Aerodynamics and Flow Technology - Technical Flows Bunsenstraße 10, D-37073 Göttingen, Germany 2 DLR, Institute of Flight Systems - Helicopter Lilienthalplatz 7, D-38108 Braunschweig, Germany
Summary To investigate the development of blade tip vortices under different rotor conditions like thrust and rotational speed, both 2C and 3C PIV measurements were performed on a 40% Mach scaled Bo105 model rotor in hover condition. The vortices were traced from the trailing edge of the blade up to half a revolution behind the blade with azimuth steps between 1° to 10° and different spatial resolutions. In addition, a sequence of three-component measurements was performed just after the vortex creation at finer azimuth steps of 0.056° to generate a 3D volumetric data set of the blade tip vortex. The influence of the PIV image analysis parameters on the vortex parameters derived - in particular sampling window size and window overlap - has been investigated. The measurements presented are part of the HOTIS (HOver TIp vortex Structure) project.
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Introduction
Within the last decade a large number of experimental investigations were performed in order to better understand and to model the development of rotor blade tip vortices. Most of these studies were done in hovering condition, because the flow field is azimuthally axisymmetric under this condition, the vortices are convected below the rotor plane and are isolated in early stage. While earlier velocity measurements were obtained using intrusive techniques such as hot-wires, more recent flow measurements rely exclusively on optical techniques like described in [1], mainly LDV and PIV. Within the HOTIS project velocity field measurements were conducted using two (2C) and three components (3C) PIV on a four-bladed rotor in hover condition in ground effect inside the rotor preparation hall at DLR Braunschweig. In order to investigate the aging process of the blade tip vortex and the influence of the rotor parameters on the vortex characteristics measurements were made at different vortex ages: Ȍv, from 3.6° to 152.6° with respect to quarter chord line (=1° to 150° behind the trailing edge) for several rotor parameters: rotation speed (200, 540 and 1041rpm) and thrust (from 0N to 3500N). The blade tip vortex was measured with different spatial resolutions: low and very high spatial resolution in case of 2C-PIV measurements and with high spatial resolution for 3C-PIV. The spatial resolution, C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 446–453, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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defined by the field of view and the size of the interrogation window, is an important parameter when looking for vortex properties such as maximum swirl velocity or core radius. In addition to these measurements, 3C measurements were performed for vortex ages between 6.036° and 9.634° using very fine age increments of 0.056° to generate an averaged volumetrically resolved velocity data set of the vortex. 160 instantaneous PIV images were recorded for every condition.
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The rotor is a 40%, dynamically and Mach scaled Bo105 main rotor with a radius of R=2m. It is composed by four hingeless blades that have a precone of 2.5° at the hub with chord length of c=0.121m, -8°/R linear twist and a solidity of ı=0.077. The model was installed in the center of the 12m*12m*8m rotor testing hall of DLR Braunschweig. The rotor was operated in ground effect – the hub center located 2.87m above the ground – at different rotation speeds of 200, 540 and 1041rpm, corresponding to tip Mach numbers of MH=0.122, 0.329 and 0.633, and with different thrust coefficients varying from CT/ı=0 to 0.072. Due to the closed hall, recirculation existed and generated an inherently unsteady flow field. 2.2
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The illumination source of the PIV setup consisted of Nd:YAG laser and light sheet forming optics which were bolted to the ground below the rotor. The light sheet was oriented vertically upward and parallel to the trailing edge of the rotor blade when the blade was at 0° angle and had a waist thickness of 1-2mm at the measurement plane and a width of around 30cm. According to the blade geometry, the azimuth of the trailing edge is 2.6° behind the quarter chord line at the blade tip. Three CCD cameras were used: one for 2C PIV and for recording the position of the blade tip and the two other cameras in stereo arrangement for 3C PIV. They were mounted on a support structure which was bolted to the wall. The complete PIV setup was described in details in [2].
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The aim of these measurements was to better understand the development of the blade tip vortex. The velocity vector fields are used to extract vortex characteristics. Prior to the processing of the images, the influence of the PIV interrogation window size and overlap on these characteristics was investigated. Sampling window size: One of the most important PIV parameters is the size of the interrogation or sampling window. In table 1 vortex characteristics in term of core radii: rc, maximum swirl velocity: Vșc, tangential velocity: V0 and vorticity: Ȧy0 in the vortex centre, extracted from averaged results (Ȍv=7.6°, MH=0.633 and
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CT/ı=0.036) obtained using different cross-correlation window sizes (WS) expressed also in term of measurement length, Lm, with a constant window overlap of 6px are listed. Table 1. Comparison of vortex characteristics extracted from 3C averaged result processed with different interrogation window sizes (Ȍv=7.6°, MH=0.633, CT/ı=0.036) rc/c
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The decrease of window size results in an increase of maximum swirl velocity, axial velocity and a decrease of vortex radius. The size of the window should be as small as possible but it is limited by the experimental parameters like seeding distribution, in-plane and out-of-plane loss of pair, or image background noise. Most of these parameters can be optimized by adapting the pulse separation time, light sheet width or light sheet thickness. Nevertheless the parameter which still limits the decrease of the window size is the particle density inside the core. Indeed vortex in hover condition are characterized by a low seeding density inside the core due to the centrifugal force which drives the seeding particles outwards from the vortex center. In order to decrease this void, smaller particles can be used but it has to be kept in mind that a decrease of particle diameter requires a quadratic increase of laser energy in order to keep the amount of light scattered by the particles constant. The use of too small window size results in outliers in the vortex core which should be interpolated and can not be used for analysis like the 16*16px window size presented in the table 1, where most of the vector inside the core were interpolated as shown in Fig. 1. As it can be seen the number of outliers in the vortex core is increasing as the window size is decreasing from the vortex center towards the outer part of the vortex following the particle density. In this paper, all the PIV images were processed using 24x24px windows because it allows over the complete vortex age to keep a low number of outliers. Sampling window overlap: The second parameter study was done on the sampling window overlap value. This value defines the interrogation interval (sample points) and is very often set by default to 50% of the window size. Numerical investigations of the effect of correlation window overlap on vortex characteristics (maximum swirl velocity and core radius) were performed in [1] using a Vatistas model. The same investigation was reproduced using an instantaneous PIV image (Ȍv=7.6°, MH=0.328 and CT/ı=0.036).
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In a second step the image was processed using window sizes of 48, 64, 96, and 128px size with overlap between 2px and the window size value. The maximum swirl velocity and core radius were computed by averaging circularly the tangential velocity profile versus r (two dimensional analysis), the radial distance from the vortex center. This radial averaging method is known to be more robust [3]. The curves obtained are presented on Fig. 3. The effect of the window size is still noticeable but the curves are smoother and the oscillations which were observed in Fig. 3 are nearly completely damped. This investigation shows that the overlap parameter can play an important role when looking for vortex characteristics and - in order to avoid random effect - an overlap as large as possible should be used in order to avoid these sampling artifacts. 3.2 Conditional Average and Post-processing In order to retain vortex characteristics, e.g. core radius or maximum swirl velocity, simple (ensemble) average of the 160 instantaneous vector fields can not be used because of vortex wander. The scatter of vortex position in space observed for young vortex is mainly due to the non periodicity of the blade tip position as shown on Fig. 4. The radial and vertical scatter increases linearly with age till the next blade passage. In the present article, the conditional average is done after locating the vortex centers using wavelet detection method as described in [4]. Once the centers are detected, they are shifted to a common point before performing the average.
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Sequences of 3C measurements were performed with vortex age increments of about ¨Ȍ=0.056° from Ȍv=6.036° to 9.634° for different rotor conditions. These small increments allow reconstructing a 3D volume of the vortex due to the azimuthally axisymmetric behavior of the flow field. The azimuthal space is similar to the in-plane resolution within each image and thus the flow gradients are of the same accuracy as those computed from the single flow field itself. The resulting volume allows the computation of the two missing components of the vorticity vector, based on out-of-plane derivatives of u, v and w.
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4 4.1
Results and Discussion
Aging Process
The development of swirl velocity and core radius extracted from the averaged vector field are presented on Fig. 6 for MH=0.328 and 0.633 and a CT/ı=0.036. As shown, the swirl velocity is decreasing with age from 38% of the blade tip speed at Ȍv=3.6° to 9% at Ȍv=152.6° for MH=0.328 and from 35% of the blade tip speed to 9% for MH=0.633. In parallel the core radii increase from 2.1% to 3.8% of the blade chord which is in good agreement with other measurement performed on rotor operating in hover [5]. The non-dimensional vortex parameters like swirl velocity and core radius seem independent of the tip speed. 4.2
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Fig. 7 show the variation of vortex characteristics as function of rotor loading for Ȍv=7.6° and MH=0.328 and 0.633. The swirl velocity is increasing with the
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rotor loading and like the aging process seems to be independent from the tip speed and the core radius seems to be independent of the thrust at least for low thrust values. For CT/ı=0, we can observe a negative swirl velocity which is due to the linear twist of the rotor blade which requires negative loading at the tip for zero thrust to counter balance the lift in the inboard area.
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In this paper, the HOTIS (HOver TIp vortex Structure) test performed in the rotor testing hall of DLR Braunschweig using 2C and 3C-PIV was described. This test was performed with a 40% Mach scaled model rotor of the Bo105 in hover condition at different RPM and thrust. The blade tip vortices were traced from their creation up to half a revolution with small age increments. A parametric study of the two main PIV parameters - the correlation window size and the overlap - was done. It shows that the overlap parameter should be as large as possible in order to avoid random effects and to improve the accuracy of vortex characteristics which can be extracted from the result. Preliminary results show that in order to resolve properly the vortex characteristics and particularly the core radius of young vortices, higher resolution than the one used for the stereo measurement is mandatory like the one used for the 2C very high resolution.
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A three dimensional reconstruction of the blade tip vortex has been applied for vortex ages between Ȍv=6.036° and 9.634° ° using a very fine age increment step of ǻȌv=0.056°. This volume reconstruction allows computing all the differential quantities which are not possible to obtain with normal 3C-PIV. A first postprocessing and analysis shows that the vortex characteristics - swirl velocity, core radius and axial velocity - seem to be independent of the blade tip speed at least in the range measured during the test (0.122 MH 0.633).
References [1] van der Wall BG, Richard H (2005) Analysis methodology for 3C PIV data. Proc: 31st European Rotorcraft Forum, Florence, Italy. [2] Richard H., Bosbach J, Henning A, Raffel M, Willert C, van der Wall BG (2006) 2C and 3C-PIV measurements on a rotor in hover condition. 13th Int. Symp., Appl. of Laser Techniques To Fluid Mechanics, Lisbon, Portugal. [3] Burley CL, Brooks TF, van der Wall BG, Richard H, Raffel M, Beaumier P, Delrieux Y, Lim JW, Yu YH, Tung C, Pengel K, Mercker E (2002) Rotor wake vortex definition and validation from 3-C PIV HART II study. Proc: 28 th European Rotorcraft Forum, Bristol, England. [4] Schram C, Riethmuller ML (2001) Vortex ring evolution in an impulsively started jet using digital image particle velocimetry and continuous wavelet analysis . Measurement Science and Technology vol. 12, pp 1413-1421. [5] Duraisamy K, Ramasamy M, Baeder JD, Leishman JG (2006) Computational/ Experimental Study of Hovering Rotor Tip Vortex Formation. Proc. 62 nd Annual Forum of the American Helicopter Society, Phoenix, USA.
Automatic Differentiation of an Entire Design Chain for Aerodynamic Shape Optimization Nicolas R. Gauger1,2 , Andrea Walther3 , Carsten Moldenhauer3 , and Markus Widhalm1 1
German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7, 38108 Braunschweig, Germany
[email protected] 2 Humboldt University Berlin, Department of Mathematics, Unter den Linden 6, 10099 Berlin, Germany 3 Technische Universit¨ at Dresden, Institute of Scientific Computing, 01062 Dresden, Germany
Summary Detailed numerical shape optimization will play a strategic role for future aircraft design. It offers the possibility of designing or improving aircraft components with respect to a pre-specified figure of merit. Optimization methods based on exact derivatives of the cost function with respect to the design variables still suffer from the high computational costs if many design variables are used. However, these gradients can be efficiently obtained by the solution of so-called adjoint flow equations. Furthermore, it is possible to derive these adjoint solvers in an automated fashion by the use of so-called automatic differentiation (AD) tools. In the present paper the efficient and automated differentiation of an entire design chain, including the flow solver, is presented. As test case for AD-generated adjoint sensitivity calculations an inviscid RAE2822 airfoil is chosen under transonic flight conditions for drag reduction.
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Introduction
In aerodynamic shape optimization, the task of computing sensitivities is essential for the application of gradient-based optimization strategies. Gradient computations for a given cost function I(P ) for a design vector P out of a defined design space can generally be done with several methods. There are at least three ways to approximate or compute these sensitivities. The first one is the finite difference method (FD) which approximates the components of the gradient by the difference quotients ∂I I(P + h ∗ ei ) − I(P ) (P ) ≈ ∂pi h
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where n is the number of design parameters, ei the ith unit vector, and h is the scalar step size. As can be seen in the approximation (1), the cost function C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 454–461, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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I(.) has to be calculated once at point P and further n times at (P + h ∗ ei ) for 1 ≤ i ≤ n. This results in (n + 1) cost function evaluations. Hence, the computational effort for the gradient approximation using finite differences is proportional to the number of design variables. Therefore, problems with this method occur if the computation of the cost function is extremely expensive or if there are many design variables. Another problem may occur if the step size h is not accurately chosen. This is based on the truncated Taylor expansion I(P + h ∗ ei ) = I(P ) +
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If h is chosen too large, the first and higher order terms on the right side of equation (2) would have a large impact on the quality of the approximation in (1). This means on the one hand, that h has to be chosen relatively small. On the other hand, if h is chosen too small, the division in the approximation term (1) is very error intensive. This would result in numerical noise and the loss of numerical stability. Therefore the step length h has to be manually tuned with respect to the cost function, parameterization and used geometry. The second alternative to provide gradient information is the continuous adjoint approach. This method has been first used in aerodynamic shape optimization by Jameson in his works on potential equations [5] and later also for the Euler equations [7] as well as Navier-Stokes [6]. Its main advantage over the finite differences is a significant speed increase because the corresponding numerical effort is independent of the number of design variables. However, the implementation of the continuous adjoint approach may be time-consuming and error-prone. The third method is automatic differentiation (AD) which is a comparatively new field of mathematical sciences. This technique is based on the observation that various elemental operations (like +, −, ∗) build up the cost function as their concatenation. Therefore, applying the chain rule to this concatenation results in a automated differentiation of the cost function (after having dealt with possible inconsistencies and other problems which are beyond the scope of this paper, see [1] for detailed information). Depending on the starting point of the differentiation process—either at the beginning or at the end of the respective chain of concatenations—one distinguishes between the forward mode and the reverse mode of AD. Using the reverse mode of AD, gradients can be computed very accurately at a computational cost that is independent of the number of design variables. This is the reason why this method also is called a discrete adjoint method.
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At DLR there are few efforts in AD computations. One attempt is the differentiation of the DLR TAUij code (see Sec. 3) by ADOL-C [2], which is validated in the present work. In the following sections a complete optimization chain will be presented. Its differentiated code will be validated against finite differences. The test case for the validation of the differentiated design chain is the transonic RAE2822 airfoil as an inviscid 2D test case.
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In aerodynamic shape optimization, a geometry is either given by a parameterization or can be changed by a parameterized deformation. This means in detail that based on the given parameters a shape can be build up or deformed by a design vector. Furthermore, the obtained shape has some aerodynamic properties like the drag coefficient or its pressure distribution. Therefore, the task of the aerodynamic shape optimization is to optimize the design vector and its dependent shape by minimizing some aerodynamic cost function. When optimizing, there must be a chain of function evaluations to calculate the cost function value at a given parameterization. This can be done by deforming a static initial shape or surface mesh and its dependent computational grid based on the parameterization and afterwards evaluating the cost function (see Fig. 1).
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The basic idea for deforming the surface of an airfoil is to evaluate functions and add their values to the upper and lower side of the surface. Therefore every design parameter is used to scale a specific function which is afterwards added to the shape. The result is a surface deformation which maintains the airfoil thickness. This maintenance of the airfoil thickness is needed to prevent the optimization to converge to a flat plate.
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In this paper, two possible kinds of functions are considered for the deformation. First, Hicks-Henne functions defined by ha,b : [0, 1] → [0, 1] : ha,b (x) = (sin(πx
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For the optimizations presented in Sec. 4, we used the four following tools covering the four steps of the design chain: The surface deformation is performed by the tool defgeo. This tool has been implemented in order to compute deformations based on Hicks-Henne as well as cosine functions. To compute the difference vectors of the original to the transformed shape geometry, i.e., the deformed surface x obtained by defgeo, another program named difgeo had to be implemented. The grid deformation within our optimization chain is done by a tool named meshdefo. This tool uses a public domain linear equations solver to compute the above mentioned coefficients of the interpolation. DLR’s flow solver TAUij is used to compute the flow around the deformed airfoil. TAUij is a quasi 2D version of TAUijk [3], which again is based on a cell centered developer version of
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the DLR TAU code [10]. TAUij solves the quasi 2D Euler equations. For the spatial discretization the MAPS+ [8] scheme is used. To achieve second order accuracy, gradients are used to reconstruct the values of variables at the cell faces. A slip wall and a far field boundary condition are applied. For time integration a Runge-Kutta scheme is used. To accelerate the convergence, local time stepping, explicit residual smoothing and a multigrid method are used. The code TAUij is written in C and comprises approximately 6000 lines of code distributed over several files. The chain to compute the cost function value is visualized in Fig. 1. This entire optimization chain has been differentiated (see [9] for details) using the AD-tool ADOL-C [2] which operates in reverse mode based on operator overloading. The differentiated chain can be written as ∂drag ∂drag ∂m ∂dx ∂x = · · · . ∂P ∂m ∂dx ∂x ∂P Note that the first term on the right side corresponds to the differentiation of TAUij, the second term to the differentiation of meshdefo, the third term to the differentiation of difgeo and the last term to the differentiation of defgeo. Since difgeo computes only the differences dx = x−xs and xs is a static initial surface, its Jacobian ∂dx ∂x becomes the unit matrix. Therefore, we obtain ∂drag ∂m ∂x ∂drag = · · . ∂P ∂m ∂dx ∂P
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As test case for the validation and application of AD-generated adjoint sensitivity calculations an inviscid RAE2822 airfoil is chosen with a Mach number of 0.73 and an angle of attack of 2◦ . The drag coefficient for this test case has been optimized with both parameterizations, Hicks-Henne and cosine function parameterizations (see Sec. 2). In both optimizations 20 design parameters have been used. The computational grid has 161x33 grid points. First, a validation framework based on finite difference for the AD sensitivities was generated by tuning the step size h as described in Sec. 1. For this purpose, the difference quotients of both parameterizations have been calculated for varying step sizes with respect to all n = 20 parameters. As can be seen in Fig. 2, a good choice for the step size h is 10(−3) for both parameterizations.
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Fig. 2. Difference quotients of Hicks-Henne (left) and cosine (right) functions parameterization for parameters 6, 10 and 15 and varying stepsizes
Fig. 3. Optimization history of FD and AD for Hicks-Henne (left) and cosine (right) functions parameterization
The optimization history based on a gradient approximation by finite differences is shown in Fig. 3 for both parameterizations. In case of the HicksHenne functions parameterization the optimization converges after nine gradient computations which are marked by a filled out square. The optimization with cosine functions converges after 13 gradient computations. The pressure distribution for both optimizations is illustrated in Fig. 4. The optimal geometries can be found in Fig. 5. As one can see, the strong shock of the original baseline geometry nearly vanished in both cases and one ends up with more than 50 % decrease in drag. In contrast to the optimum obtained for the Hicks-Henne functions parametrization, the optimum with cosine functions parameterization shows little oscillations in the pressure distribution which is based on the fact, that cosine functions only have local impacts on the surface deformation whereas Hicks-Henne functions have global impacts. Fig. 6 also shows comparisons between the FD and AD gradients for the original RAE2822 airfoil with Hicks-Henne and cosine functions parameterization. This validates the AD approach and shows also that the chosen step size for the FD gradient is accurate enough.
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As with the finite differences method the optimizations have also been done with the automatic differentiation approach. The optimization histories, the pressure distributions and the surface geometries of the optimum obtained with AD are also visualized in Figs. 3, 4 and 5. This clearly validates the automatic differentiation approach.
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Within the previous sections a complete optimization chain has been presented and differentiated using automatic differentiation. The AD-generated adjoint sensitivity calculations have been validated against finite differences. Furthermore a comparison between two possible parameterizations, HicksHenne and cosine functions parameterization, has been done. The method of automatic differentiation is a powerful tool for calculating accurate gradients. Although it is not widely spread in the present it surely will become more popular in the future and also has high potential for larger applications.
References [1]
[2]
[3]
[4]
[5] [6]
[7] [8] [9]
[10]
A. Griewank: Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation. Frontiers in Appl. Math. 19, Society for Industrial and Applied Mathematics, Philadelphia, 2000. A. Griewank, D. Juedes, J. Utke: ADOL-C: A Package for the Automatic Differentiation of Algorithms Written in C/C++. ACM Trans. Math. Softw., 22:131–167, 1996. R. Heinrich: Implementation and Usage of Structured Algorithms within an Unstructured CFD-Code. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Vol. 92, 2006. M.H.L. Hounjet, B.B. Prananta, R. Zwaan: A Thin Layer Navier Stokes Solver and its Application for Aeroelastic Analysis of an Airfoil in Transonic Flow. Proceedings International Forum on Aeroelasticity and Structural Dynamics, Manchester, June 1995, pp. 1.1-1.9, 1995. A. Jameson: Aerodynamic design via control theory. Journal of Scientific Computing, 3:233-260, 1988. A. Jameson, L. Martinelli, N.A. Pierce: Optimum aerodynamic design using the Navier-Stokes equations. Theoret. Comput. Fluid Dynamics, 10:213-237, Springer, 1998. A. Jameson, J. Reuther: Control theory based on airfoil design using the Euler equations. AIAA Proceedings 94-4272-CP, 1994. C.-C. Rossow: A flux splitting scheme for compressible and incompressible flows. Journal of Computational Physics, 164:104–122, 2000. S. Schlenkrich, A. Walther, N.R. Gauger, R. Heinrich: Differentiating Fixed Point Iterations with ADOL-C: Gradient Calculation for Fluid Dynamics. Proceedings of High Performance Scientific Computing, Hanoi, March 6-10, 2006. M. Widhalm, C.-C. Rossow: Improvement of upwind schemes with the least square method in the DLR TAU code. Notes on Numerical Fluid Mechanics, Vol. 87, 2004.
Computational Study of Mean Flow and Turbulence Structure in Inflow System of a Swirl Combustor 2 ˇ c1 , Suad Jakirli´c1 , Dalibor Cavar ˇ Sanjin Sari´ , Bj¨orn Kniesner 1 , Paul Altenh¨ofer 1, and Cameron Tropea 1 1
2
Fachgebiet Str¨omungslehre und Aerodynamik, Technische Universit¨at Darmstadt Petersenstr. 30, 64287 Darmstadt, Germany Fluid Mechanics Secton, Dept. of Mech. Engineering, Technical University of Denmark Nils Koppels Alee 403, 2800 Kgs. Lyngby, Denmark
Summary Flow structure in the annular section of the inlet system of a tuboannular swirl combustor with respect to the swirl intensity influence was investigated computationally complementary to the recent experimental study by Palm et al. [1]. In addition to the non-swirling case, two different swirling configurations corresponding to the swirl numbers S = 0.6 and 1.0 were considered. The simulations were performed by using Large Eddy Simulation (LES) method and a two-layer model scheme hybridizing a near-wall k − ε RANS (Reynolds-averaged Navier Stokes) model covering the wall layer and LES method in the outer layer employing Smagorinsky model. Special attention was devoted to the position of the interface. An in-depth analysis of the mean velocity and turbulence fields reveals an increasingly asymmetric axial velocity profile in the annular pipe and an appropriatelly shaped profile of the Reynolds stress components corresponding to the enhanced turbulence production in the outer part of the concentric annulus. The present study also aimes at generation of reliable swirling inflow data for future LES of the flow in the combustor flue.
1 Introduction The flow in the annular region of the combustor inlet section representing primary air is usually swirled contributing significantly to the mixing intensification with the non-swirling inner jet (representing fuel) in the near field (primary zone) of a combustion chamber. This mixing process influences to a large extent the efficiency of the entire combustion process. A free, swirl-induced recirculation zone enhances therefore the mixing and stabilizes the flame. Due to complex geometry of the actual swirl generators, it is desirable to computationally generate proper inflow conditions for Large Eddy Simulation of the swirling flow in the combustor flue, instead of accounting for the vanes and movable blocks for swirl intensity adjustment. Pierce and Moin (1998) proposed such a method based on the computation of the so-called equilibrium swirling flow assuming the fully-developed conditions. This method implies introduction of an additional force in terms of a fictituous pressure gradient in the circumferential directions. While this method does not necessarily provide C. Tropea et al. (Eds.): New Res. in Num. and Exp. Fluid Mech. VI, NNFM 96, pp. 462–470, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com
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the compatibility of the circumferential and axial velocity fields - a consequence of the fully-developed flow assumption (axial velocity profile largely retains its symmetric shape), an additional forcing of the axial momentum has to be introduced. In order to do it, one needs an a priori known, e.g. from experiment, mean axial velocity, which serves as a target profile. In addition to the simulations of entire inlet system including swirl generator and annular section (complementary to the original swirl generator configuration, several geometrical simplifications of the inlet plenum and the swirl-intensity-determining movable blocks were considered), this forcing method for inflow generation was also tested in the present study. The reference database arose from the recent experimental investigation by Palm et al. (2006) considering the flow structure in the annular section of inlet system of a swirl combustor model with respect to the mass flow rate and swirl intensity influence. The combustor model including the swirl generator is depicted in Fig. 1. As the experimental flow conditions indicate a strong departure from a fully developed annular flow, a due attention has to be payed when generating inflow boundary conditions. In the present work the series of simulations performed using LES and hybrid LESRANS (HLR) approaches provides a valuable assessment of possible influence of the inflow boundary conditions on the predictions of mean flow and turbulence in the combustor flue.
Fig. 1. Schematic of the swirl combustor, Dm,i = 36mm, Dm,o = 40mm, Df = 200mm, Dc = 100mm, δ = 7.5o , Lm = 720, Lf = 1200mm, lc = 120mm
2 Computational Method All computations were performed with an in-house computer code FASTEST based on a finite volume numerical method for solving both 3-D filtered and RANS equations on block-structured, body-fitted, non-orthogonal meshes. A cell centered (collocated) variable arrangement and Cartesian vector and tensor components are used. The well-known SIMPLE algorithm was applied for coupling the velocity and pressure fields. The convective transport of all variables was discretized by a secondorder central differencing scheme (CDS), whose stability was enhanced through the
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so-called deferred correction approach (the fraction of the CDS scheme in this flux blending method was 100% for all equations except for the k and ε - equation in the RANS domain in the HLR-framework). Time discretization was accomplished applying the 2 nd order (implicit) Crank-Nicolson method. The subgrid scales were modeled with either the standard or the dynamic Smagorinsky model. In the hybrid LES-RANS method, the low-Reynolds number k − ε model due to Launder and Sharma (1974) was applied in the near-wall region. In the first step the well-known subgrid-scale model due to Smagorinsky is used in the core flow. The model coupling is realized via turbulent viscosity, representing an approach which enables the solution to be obtained by using one system of equations. Depending on the flow zone, the turbulent viscosity is either computed from the RANS formulation: µt = " Cµ fµ k 2 /ε
(1)
or from the Smagorinsky model: ( ( µt = " Cs2 ∆2 (S (
(2)
where the Smagorinsky constant C s takes the value of 0.1, ∆ being the filter width and S the strain rate. The interface values for k and ε representing actually the boundary conditions for the corresponding equations in the RANS region are obtained by estimating the subgrid-scale kinetic energy and dissipation: kSGS =
C 2 ∆2 |S|2 νt |S| = s ; εSGS = νt |S|2 = Cs2 ∆2 |S|3 0.3 0.3
(3)
Such a procedure provides the continuity in k and ε profiles across the interface. Consequently, a fairly smooth transition of the turbulent viscosity µ t is ensured in accordance to the equations (1), (2) and (3). The interface position is one of the most important issues in the hybrid LES/RANS framework. In the present model, the interface position can vary in accordance to the control parameter which represents the ratio (fraction) of modeled to the total turbulent kinetic energy in the LES region, averaged over all grid cells at the interface belonging to the LES domain: k ∗ = kmod /(kmod + kres ). Its typical value is about 20%. The resulting interface position in the present hybrid simulation is at y + ≈ 200. More details on this recently proposed hybrid approach can be found in Jakirli´c et al. (2006). Table 1 summarizes the simulations performed with respect to the methods used and cases considered. Some relevant computational details are also given. Swirling flow in a concentric annulus pertinent to the inlet section of a swirl combustor was computed first in order to examine this less expensive means of inflow generation. Boundary conditions correspond to the experiment of Palm et al. [1] (Rec ≈ 50500 based on bulk velocity and hydraulic diamater (D c − Dm,o )). These computations assumed fully developed flow conditions, whereby the (equilibrium) swirling motion was created by introducing a fictitious pressure gradient into the momentum equation governing the circumferential velocity in line with the method proposed by Pierce and Moin (1998). The magnitude of the pressure gradient (with
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Table 1 . Summary of the simulations Computation LES DSM
Model
Dynamic Smagorinsky LES DSM Dynamic Smagorinsky LES Standard Smagorinsky LESc Standard Smagorinsky HLRc Hybrid LES-RANS
Nr × NΘ Configuration Swirl Ngrid (see Figs. 1, 2 and 3) intensities 49 × 128 only annulus 0,0.6,1.0 401, 000 49 × 128 Fig. 3-left 0,0.6,1.0 665, 000 only simplified swirler 49 × 128 Fig. 3-right 0,0.6,1.0 665, 000 simplified swirler + combustor 40 × 192 Fig. 2-right 1.0 787, 000 swirler + combustor 40 × 192 Fig. 2-right 1.0 787, 000 swirler + combustor
remark streamwise periodic convective outlet combustor included combustor included combustor included
constant value over the cross-section) was iteratively adjusted until the computed U and W velocity fields satisfied the prescribed swirl intensity S. As mentioned in the introductory section, additional forcing introduced into the equation governing the axial momentum had to be introduced as well. This method was successfully applied by Pierce and Moin [2] for generating the swirling inflow for LES of a model combustor investigated experimentally by Roback and Johnson (1983). The solution domain with the length of L x = 2.67π(Rc − Rm,o ) was meshed by Cartesian grid with Nx × Nr × NΘ = 64 × 49 × 128 cells. The maximum value of the CFL number was 0.85. The original experimental swirl generator system was accounted for in the work of Grundmann (2005) and Palm et al. (2006), Fig. 2-left (not considered here). In order to partially avoid the complex geometry of the swirler two geometrical simplifications were followed in the present study: one excluding the annular-like inflow plenum (Fig. 2-right; this configuration was simulated by the LESc and HLR methods ) and second excluding the channels between movable blocks
Fig. 2. The experimental swirl generator system including the concentric annular section (left, only half of it shown, the grid used by Grundmann [4]), a simplified swirl generator including guide vanes (right, only a quarter of it shown)
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Fig. 3 . A simplified swirl generator (left) including the combustor flue (right, a grid slice in the x-y plane)
(i.e. guide vanes), Fig. 3. The conventional LES using Smagorinsky model and its dynamic variant (DSM) were used when simulating the latter geometry. Different simulations were performed accounting only for the swirl generator and annular pipe with outlet boundary located at its end, that is at the flue inlet (see Fig. 1), but also accounting for the entire combustor (flue). Preserving the same resolution used for the afore-mentioned concentric annulus computations, the grid consisted of Nx × Nr × NΘ = 106 × 49 × 128 cells. The experimental mass flow rate was imposed at the inlet. In the case of geometries displayed in Fig. 3 the tangential (azimuthal) velocity component was adjusted in time until the desired swirl intensity was obtained. Separate precursor LES of a fully developed non-swirling (central) pipe flow corresponding to the experiment (Re m ≈ 23500) had to be conducted to generate the pipe inflow data when accounting for the combustor flue. The solution domain consisting of the flue, simplified swirler and central pipe was discretized by about 4.6 Mio cells (Fig. 3-right). All three cases corresponding to the swirl numbers of 0.0, 0.6 and 1.0 were simulated. Additional LES (LESc) and hybrid LES-RANS simulations employing a somewhat coarser (in streamwise and radial directions) grid (comprising about 3.3 Mio cells with the flue included) of the combustor configuration characterized by the strongest swirl S = 1.0 were conducted. Though simplified, this swirler configuration includes guide vanes as shown in Fig. 2-right.
3
Results
Fig. 4 displays the contours of the mean axial velocity field for the cases with S = 0 (left) and S = 1.0 (right) in the radial and 90 o -turned part of the inlet section and the annular pipe indicating very complex flow pattern deviating substantially from the equilibrium (fully-developed) conditions. Zones with flow reversal (blue areas) are situated behind the movable blocks (flow past a bluff body). Strong acceleration of the flow occurs when the flow in the radial part of the swirler transforms into the axial motion within the annular pipe (red areas). In the case with strong swirl (Fig. 4-right), a swirl-induced recirculation zone bounded at the inner wall (thin blue area) was detected corresponding to the region with strongest acceleration of the circumferential flow.
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Fig. 4. Contours of mean axial velocity in the x − y plane obtained by LES: S = 0 (left) and S = 1.0 (right) 0.02
1.6
Exp. (S=0) LES DSM (annulus-forcing) LES DSM (simplified swirler) 0.01 LES (combustor+swirler)
0.015
1.2 2
1 uv/Ub
U/Ub - axial velocity
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0.005 0 -0.005
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0.2 0 -0.2 0
Fig. 5.
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0.8
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LES predictions, case S=0: mean axial velocity and mean shear stress u’v’
The LES predictions for the non-swirling case are displayed in Fig. 5. Mean axial velocity and shear stress u v obtained by LES of the simplified swirler agree closer with the experimental results when compared to the case which assumes fully developed annular flow. Inclusion of the combustor flue in the computational domain seems to improve the prediction of the shear stress u v (Fig. 5-right). However, the additional LES of the simplified swirler employing the standard Smagorinsky model (not shown here) yields very similar results indicating the influence of the SGS modeling. Predictions of the mean axial and tangential (azimuthal) velocity components for the case S = 0.6 also demonstrate superiority of the simulations which include the swirl generator system (Fig. 6). Mean axial and tangential velocity profiles obtained by LES of the similar annular configuration with a slightly smaller swirl intensity (S = 0.41, PM - Pierce and Moin [2]) are displayed as reference as well. The results corresponding to the case featuring maximum swirl intensity S=1.0 are presented in Fig. 7. Overall, LES and hybrid LES–RANS yield comparable results. However, it can be observed that LES DSM run with the simplified swirler returns not as accurate results as in the previous two cases. It appears that the dynamic Smagorinsky model exibits a higher sensitivity to the grid resolution as a consequence of the strong swirl intensity. On the other side, the DSM of the
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468 1.6
W/Ub - tangential velocity
U/Ub - axial velocity
Exp. (S=0.6) LES DSM (annulus-forcing) LES DSM (simplified swirler) LES (combustor+swirler) LES-PM (S=0.4)
2
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Exp. (S=0.6) LES DSM (annulus-forcing) LES DSM (simplified swirler) LES (combustor+swirler) LES-PM (S=0.4)
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Fig. 6. LES predictions, case S=0.6: mean axial and mean circumferential velocities
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2 W/Ub - tangential velocity
U/Ub - axial velocity
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Exp. (S=1.0) LES DSM (annulus-forcing) LES DSM (simplified swirler) LES (combustor+swirler) LESc (combustor+swirler) HLRc (combustor+swirler)
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1 Exp. (S=1.0) LES DSM (annulus-forcing) LES DSM (simplified swirler) LES (combustor+swirler) LESc (combustor+swirler) HLRc (combustor+swirler)
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Exp. (S=1.0) LES DSM (annulus-forcing) LES DSM (simplified swirler) LES (combustor+swirler) LESc (combustor+swirler) HLRc (combustor+swirler)
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Exp. (S=1.0) LES DSM (annulus-forcing) LES DSM (simplified swirler) LES (combustor+swirler) LESc (combustor+swirler) HLRc (combustor+swirler)
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e)
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f)
Fig. 7. LES predictions, case S=1.0: a) mean axial velocity, b) mean circumferential velocity, c) streamwise turbulence intensity, d) radial turbulence intensity, e) circumferential turbulence intensity, f) mean shear stress u’v’
Computational Study of Mean Flow and Turbulence Structure x/Df =0.2
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4
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5 6 u’u’/Uf
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8
9
c)
Fig. 8. LES predictions of the streamwise turbulence intensity just downstream of the combustor inlet: a) S = 0, b) S = 0.6, c) S = 1.0
annular flow with the forcing performs much better at S=1.0. This is supported by Fig. 8c displaying more accurate predictions of streamwise turbulence intensity just downstream of the combustor inlet at x = 0.2D f . Based on the results displayed in Fig. 8 one can deduce that inclusion of the swirl generator system in the computation yields an improvement of results which, however, weakens with the increase in swirl intensity.
4 Conclusions LES and hybrid LES-RANS simulations of flow in the swirl generation system of a tuboannular combustor type were performed for different swirling intensities. In addition to the original swirl-generator configurations, different simplification levels of the inflow system geometry were considered. Also considered, in the framework of the separate computations, is the flow in the concentric annulus assuming fullydeveloped flow conditions applying appropriate forcing on both axial and angular momentum. The aim of this study was also to enlighten the flow structure in such a complex geometry influenced strongly by the streamline curvature due the imposed swirl and the 90o turned bend-like form of the annular section. The resulting mean velocity and profiles of the turbulent quantities follow closely the experimental data in all cases considered. Especially good agreement was documented when the configuration accounted for the flue was considered. In such a way the adverse pressure gradient effects originating from the sudden expansion of the inflow system into the flue positioned 40mm downstream of the measurements cross-section could be properly captured.
Acknowledgments The financial support of the Deutsche Forschungsgemeinschaft through the grants SFB568 ”Flow and Combustion in Future Gas Turbine Combustion Chambers” (TPC3, B.K.) and the research group on ”LES of complex flows” (FOR 507/1, JA ˇ is gratefully acknowledged. 941/7-1) (S. S.)
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References ˇ c, S., Jakirli´c , and Tropea, C. (2006): Ex[1] Palm, R., Grundmann, S., Weism¨uller, Sari´ perimental characterization and modelling of inflow conditions for a gas turbine swirl combustor. Int. J. Heat and Fluid Flow, Vol. 27, pp. 924-936 [2] Pierce, C.D. and Moin, P. (1998): Method for Generating Equilibrium Swirling Inow Conditions. AIAA Journal, Vol. 36(7), pp. 1325-1327 ˇ c, S and Hanjali´c, K.: Merging near-wall RANS models [3] Jakirli´c , S., Kniesner, B., Sari´ with LES for separating and reattaching flows. Proc. of FEDSM2006 ASME Joint U.S. European Fluids Engineering Summer Meeting, Miami, FL, USA, July 17-20. [4] Grundmann, S. (2003): Numerische Untersuchung drahlbehafteter Str¨omungen in einem realit¨atsnahen Drahlbrennermodell. Diploma thesis, Darmstadt University of Technology (presented at the DGLR Kongress, 2004).
Author Index
Ali, I. 438 Alkishriwi, N.A. 397 Alrutz, T. 186 Altenhöfer, P. 462
Hannemann, V. 300 Heinrich, R. 62 Henke, R. 44 Henningson, D. 235 Herr, M. 430 Herrmann, U. 137 Himisch, J. 52, 62 Hoefener, L. 244 Hörnschemeyer, R. 44 Horstmann, K.H. 62 Hüttmann, F. 348
Babucke, A. 413 Barth, T. 291 Becker, S. 438 Böttcher, C. 300 Brede, M. 348 Brezillon, J. 194 ýavar, D.
462
Jakirliü, S. 462 Jensch, M. 348 Jovanoviü, J. 268
Dankert, C. 317 Dettmar, H. 219 Domhardt, J. 96 Durst, F. 340 Dwight, R.P. 194 Egami, Y. Eggers, T. Eisfeld, B. Engert, M. Escobar, M.
Kähler, C.J. 88 Kaltenbacher, M. 438 Karczorowski, M. 381 Kauertz, S. 44 Klein, C. 227 Klinge, F. 104 Kloker, M.J. 405, 413 Knauss, H. 121 Kniesner, B. 462 Koch, S. 9 König, B. 80 König, D. 421 Krämer, E. 121, 364 Krämer, E. 80 Krimmelbein, N. 210 Krumbein, A. 210
227 291 356 252 438
Fey, U. 227 Frederich, O. 348 Friedrich, R. 372 Frohnapfel, B. 268 Gaisbauer, U. 121 Gardner, A.D. 170 Gassner, G. 154 Gauger, N.R. 454 Gerhold, T. 162 Gimenez Pastor, M. 309 Grote, A. 26 Guyot, D. 332
Lammers, P. 268 Leder, A. 348 Leuckert, J. 96 Lienhart, H. 340 Liersch, C. 62 Linn, J. 405 Lörcher, F. 154 Lübon, C. 145 Lüdeke, H. 300, 389 Lutz, T. 80, 364
Hage, W. 324 Hahn, C. 438 Hannemann, K. 129 [
472
Author Index
Mahle, I. 372 Mangold, P. 309 Mannini, C. 36 Martinez Schramm, J. 129, 284 Marxen, O. 235 Meinke, M. 397, 421 Melber-Wilkending, S. 52, 113 Mertol, B.A. 1 Meyer, R. 324 Moldenhauer, C. 454 Munz, C.-D. 154 Nagel, B. 62 Neumann, J. 162 Neuwerth, G. 44 Nitsche, W. 96, 244, 252 Odam, J. 276 Orlt, M. 186 Ortmanns, J. 88 Otto, H. 317 Pallek, D. 104 Paschereit, C.O. 324 Pätzhold, M. 80 Pätzold, A. 252 Paull, A. 276 Pêgo, J. 340 Peltzer, I. 96 Pfingsten, K.-C. 71 Radespiel, R. 26, 71, 88 Raffel, M. 446 Raichle, A. 52 Reimann, B. 284 Richard, H. 446 Richter, K. 9, 170 Rist, U. 413 Rixen, C. 44
Robinson, M. 129 Rödiger, T. 121 Rosemann, H. 9, 170 Rossow, C.-C. 202 Šariü, S. 462 Schewe, G. 36 Schmid, S. 364 Schröder, W. 17, 397, 421 Schülein, E. 332 Schütte, A. 186 Schwarz, T. 178 Seiler, F. 309 Seitz, A. 260 Sesterhenn, J. 372 Shishkin, A. 381 Shishkina, O. 381 Soda, A. 36 Srulijes, J. 309 Streit, Th. 62 Thiele, F. 348 Thimm, M. 446 Tigges, K. 340 Togiti, V.K. 389 Tropea, C. 462 Van der Wall, B.G. Voß, R. 36
446
Wagner, C. 381 Wagner, S. 145 Walther, A. 454 Wassen, E. 348 Wichmann, G. 113 Widhalm, M. 186, 454 Wild, J. 219 Zurheide, F.
17
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