Summary of Flow Modulation and Fluid-Structure Interaction Findings: Results of the Collaborative Research Center Sfb 401 at the Rwth Aachen University, Aachen, Germany, 1997-2008

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109

Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM)

Editors W. Schröder/Aachen B.J. Boersma/Delft K. Fujii/Kanagawa W. Haase/München M.A. Leschziner/London J. Periaux/Paris S. Pirozzoli/Rome A. Rizzi/Stockholm B. Roux/Marseille Y. Shokin/Novosibirsk

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Summary of Flow Modulation and Fluid-Structure Interaction Findings Results of the Collaborative Research Center SFB 401 at the RWTH Aachen University, Aachen, Germany, 1997-2008 Wolfgang Schröder (Editor)

ABC

Prof. Dr.-Ing. Wolfgang Schröder Chair of Fluid Mechanics and Institute of Aerodynamics RWTH Aachen University Wüllnerstraße 5a 52062 Aachen Germany E-mail: [email protected]

ISBN 978-3-642-04087-0

e-ISBN 978-3-642-04088-7

DOI 10.1007/978-3-642-04088-7 Notes on Numerical Fluid Mechanics and Multidisciplinary Design

ISSN 1612-2909

Library of Congress Control Number: 2010928119 c 2010

Springer-Verlag Berlin Heidelberg

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. 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. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 543210 springer.com

NNFM Editor Addresses

Prof. Dr. Wolfgang Schröder (General Editor) RWTH Aachen Lehrstuhl für Strömungslehre und Aerodynamisches Institut Wüllnerstr. 5a 52062 Aachen Germany E-mail: [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: [email protected] Dr. Werner Haase Höhenkirchener Str. 19d D-85662 Hohenbrunn Germany E-mail: [email protected] Prof. Dr. Ernst Heinrich Hirschel (Former General Editor) Herzog-Heinrich-Weg 6 D-85604 Zorneding Germany E-mail: [email protected] Prof. Dr. Ir. Bendiks Jan Boersma Chair of Energytechnology Delft University of Technology Leeghwaterstraat 44 2628 CA Delft The Netherlands 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. Sergio Pirozzoli Università di Roma “La Sapienza” Dipartimento di Meccanica e Aeronautica Via Eudossiana 18 00184, Roma, Italy E-mail: [email protected] Prof. Dr. Jacques Periaux 38, Boulevard de Reuilly F-75012 Paris 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]

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Preface

In this issue of Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM) the results of the collaborative research center SFB 401 Flow Modulation and Fluid-Structure Interaction at Airplane Wings at the Rheinisch-Westf¨alische Technische Hochschule (RWTH) Aachen University are reported. The funding was provided by the Deutsche Forschungsgemeinschaft (DFG). The research was performed from 1997 through 2008 and on the average consisted of more than 14 subprojects per year. Approximately 110 scientists from universities of the Austria, Belgium, France, Great Britain, Italy, Japan, Netherlands, Russia, South Korea, Sweden, Switzerland, United States, and international research organizations such as DLR, NASA, NLR, ONERA were invited. The distinct scientists from all over the world gave seminars on topics related to the research ﬁelds tackled in the collaborative research center SFB 401. Some of them stayed for just a few days, others were hosted for a longer time to intensify the joint research. Besides the scientiﬁc value the Flow Modulation and Fluid-Structure Interaction at Airplane Wings program possesses a pronounced educational merit. This becomes evident by the fact that 35 doctoral theses, 80 diploma theses, and 117 study theses were stimulated by the research program of the SFB 401 and ﬁnished before 2010. The authors of this issue of NNFM acknowledge the valuable support from all guest scientists and everybody scientiﬁcally involved in the SFB 401. They are especially grateful to the reviewers of SFB 401 Hans-J¨ urgen Christ (University of Siegen), Franz Durst (University of Erlangen-N¨ urnberg), Gerhard Dziuk (University of Freiburg), Leonhard Fottner (Universit¨ at der Bundeswehr), Rainer Friedrich (TU M¨ unchen), Wolfgang Geißler (DLR), Michael Griebel (University of Bonn), Roger Grundmann (TU Dresden), Ernst Heinrich Hirschel (MBB/DASA), Reinhard Hilbig (Airbus Deutschland GmbH), Peter Horst (TU Braunschweig), Jean Hourmouziadis (TU Berlin), Dietrich Hummel (TU Braunschweig), Andreas Kablitz (University of Cologne), Horst Kossira (TU Braunschweig), Edwin Kreuzer (TU Hamburg-Harburg), Di-

VIII

Preface

etmar Kr¨ oner (University of Freiburg), Bernd-Helmut Kr¨ oplin (University of Stuttgart), Hans J. Markowitsch (Bielefeld University), Wolfgang Nitsche (TU Berlin), Frank Obermeier (TU Freiberg), Ekkehard Ramm (University of Stuttgart), Rolf Rannacher (Heidelberg University), Klaus Starke (University of Freiburg), Martin Stutzmann (TU M¨ unchen), Cameron Douglas Tropea (TU Darmstadt), Stefan Turek (TU Dortmund), Bernhard Wagner (Dornier GmbH), Siegfried Wagner (University of Stuttgart), Gerald Warnecke (University of Magdeburg), Knut Wilhelm (TU Berlin), Christoph Zenger (TU M¨ unchen). Some of them accompanied the SFB 401 from the very beginning in 1997 to the very end in 2008. They deﬁnitely ensured the scientiﬁc quality in the various projects. It is the desire of the authors to express their deep gratitude to Drs. Dieter Funk and Thomas M¨ unker from the Deutsche Forschungsgemeinschaft for their valuable suggestions and the competent guidance whenever funding or organizational questions occurred. Finally, the editor expresses his thanks to Dr. Frank Zurheide who did the technical editing work and last but not least to Springer Verlag. Without their commitment this edition in the series Notes on Numerical Fluid Mechanics and Multidisciplinary Design wouldn’t have been published. Aachen, December 2009

Wolfgang Schr¨ oder

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV Vortex Sheets of Aircraft in Takeoﬀ and Landing . . . . . . . . . . . . Robert Sch¨ oll, Rolf Henke, and G¨ unther Neuwerth An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows about Elastic Conﬁgurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gero Schieﬀer, Saurya Ray, Frank Dieter Bramkamp, Marek Behr, Josef Ballmann Timestep Control for Weakly Instationary Flows . . . . . . . . . . . . Christina Steiner, Sebastian Noelle Adaptive Multiscale Methods for Flow Problems: Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wolfgang Dahmen, Nune Hovhannisyan, Siegfried M¨ uller

1

25

53

77

Interaction of Wing-Tip Vortices and Jets in the Extended Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Frank T. Zurheide, Guido Huppertz, Ehab Fares, Matthias Meinke, Wolfgang Schr¨ oder Experimental and Numerical Investigation of Unsteady Transonic Airfoil Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Atef Alshabu, Viktor Hermes, Igor Klioutchnikov, Herbert Olivier Enabling Technologies for Robust High-Performance Simulations in Computational Fluid Dynamics . . . . . . . . . . . . . . . 153 Christian H. Bischof, H. Martin B¨ ucker, Arno Rasch

X

Contents

Inﬂuencing Aircraft Wing Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 181 R. H¨ ornschemeyer, G. Neuwerth, R. Henke Development of a Modular Method for Computational Aero-structural Analysis of Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . 205 Lars Reimer, Carsten Braun, Georg Wellmer, Marek Behr, Josef Ballmann A Uniﬁed Approach to the Modeling of Airplane Wings and Numerical Grid Generation Using B-Spline Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Karl-Heinz Brakhage, Wolfgang Dahmen, Philipp Lamby Parallel and Adaptive Methods for Fluid-Structure-Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Josef Ballmann, Marek Behr, Kolja Brix, Wolfgang Dahmen, Christoph Hohn, Ralf Massjung, Sorana Melian, Siegfried M¨ uller, Gero Schieﬀer Iterative Solvers for Discretized Stationary Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Bernhard Pollul, Arnold Reusken Unsteady Transonic Fluid – Structure –Interaction at the BAC 3-11 High Aspect Ratio Swept Wing . . . . . . . . . . . . . . . . . . . 325 P.C. Steimle, W. Schr¨ oder, M. Klaas Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 J.A. Kengmogne Tchakam, H.-G. Reimerdes Aero-structural Dynamics Experiments at High Reynolds Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Josef Ballmann, Athanasios Dafnis, Arndt Baars, Alexander Boucke, Karl-Heinz Brakhage, Carsten Braun, Christian Buxel, Bae-Hong Chen, Christian Dickopp, Manuel K¨ ampchen, Helge Korsch, Herbert Olivier, Saurya Ray, Lars Reimer, Hans-G¨ unther Reimerdes Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

List of Contributors

Atef Alshabu Shock Wave Laboratory of RWTH University, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected] Arndt Baars Lehr- und Forschungsgebiet f¨ ur Mechanik, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany

Christian H. Bischof Institute for Scientiﬁc Computing, RWTH Aachen University, Seﬀenter Weg 23, 52056 Aachen, Germany [email protected] Alexander Boucke Lehr- und Forschungsgebiet f¨ ur Mechanik, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany Karl-Heinz Brakhage Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected]

Josef Ballmann Lehr- und Forschungsgebiet f¨ ur Mechanik, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany Frank Dieter Bramkamp [email protected] Lehr- und Forschungsgebiet f¨ ur Mechanik, RWTH Aachen University, Schinkelstraße 2, Marek Behr 52062 Aachen, Germany CATS, Chair for Computational Analysis of Technical Systems, Carsten Braun RWTH Aachen University, Lehr- und Forschungsgebiet f¨ ur Schinkelstraße 2, 52062 Aachen, Mechanik, RWTH Aachen Germany University, Schinkelstraße 2, [email protected] 52062 Aachen, Germany

XII

Kolja Brix Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected]

List of Contributors

Ehab Fares Institute of Aerodynamics, RWTH Aachen University, W¨ ullnerstraße 5a, 52062 Aachen, Germany

Christian Buxel ilb, Institut f¨ ur Leichtbau, RWTH Aachen University, W¨ ullnerstraße 7, 52062 Aachen Germany

Rolf Henke Institute of Aeronautics and Astronautics, RWTH Aachen University, W¨ ullnerstraße 7, 52062 Aachen, Germany [email protected]

Martin B¨ ucker Institute for Scientiﬁc Computing, RWTH Aachen University, Seﬀenter Weg 23, 52056 Aachen, Germany [email protected]

Viktor Hermes Shock Wave Laboratory of RWTH University, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected]

Athanasios Dafnis ilb, Institut f¨ ur Leichtbau, RWTH Aachen University, W¨ ullnerstraße 7, 52062 Aachen Germany [email protected] Bae-Hong Chen CATS, Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany Wolfgang Dahmen Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected] Christian Dickopp Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany

Christoph Hohn Lehr- und Forschungsgebiet f¨ ur Mechanik, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany [email protected] Nune Hovhannisyan Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected] Ralf H¨ ornschemeyer Institute of Aeronautics and Astronautics, RWTH Aachen University, W¨ ullnerstraße 7, 52062 Aachen, Germany [email protected] Guido Huppertz Institute of Aeronautics and Astronautics, RWTH Aachen University, W¨ ullnerstraße 5a, 52062 Aachen, Germany

List of Contributors

Jules A. Kengmogne Tchakam Department of Aerospace Structures, RWTH Aachen University, W¨ ullnerstraße 7, 52062 Aachen, Germany [email protected]

XIII

RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected]

Siegfried M¨ uller Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen Michael Klaas University, Templergraben 55, Institute of Aerodynamics, RWTH Aachen University, W¨ ullnerstraße 5a, 52056 Aachen, Germany [email protected] 52062 Aachen, Germany [email protected] G¨ unther Neuwerth Igor Klioutchnikov Institute of Aeronautics and Shock Wave Laboratory of RWTH Astronautics, RWTH Aachen University, RWTH Aachen University, W¨ ullnerstraße 7, University, Templergraben 55, 52062 Aachen, Germany 52056 Aachen, Germany [email protected] [email protected] Sebastian Noelle Helge Korsch Institute for Geometry and Practical ilb, Institut f¨ ur Leichtbau, RWTH Mathematics IGPM, RWTH Aachen Aachen University, W¨ ullnerstraße 7, University, Templergraben 55, 52062 Aachen Germany 52056 Aachen, Germany [email protected] Manuel K¨ ampchen ilb, Institut f¨ ur Leichtbau, RWTH Herbert Olivier Aachen University, W¨ ullnerstraße 7, Shock Wave Laboratory of RWTH 52062 Aachen Germany University, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany Philipp Lamby [email protected] University of South Carolina, Industrial Mathematics Institute, Columbia SC, 29208, USA Bernhard Pollul [email protected] Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen University, Templergraben 55, Matthias Meinke 52056 Aachen, Germany Institute of Aerodynamics, RWTH Aachen University, W¨ ullnerstraße 5a, [email protected] 52062 Aachen, Germany [email protected] Arno Rasch Institute for Scientiﬁc Computing, RWTH Aachen University, Seﬀenter Sorana Melian Weg 23, 52056 Aachen, Germany Institute for Geometry and [email protected] Practical Mathematics IGPM,

XIV

Saurya Ray Lehr- und Forschungsgebiet f¨ ur Mechanik, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany Lars Reimer CATS, Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany [email protected] Hans-G¨ unther Reimerdes Department of Aerospace Structures, RWTH Aachen University, W¨ ullnerstraße 7, 52062 Aachen, Germany [email protected] Arnold Reusken Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected] Gero Schieﬀer CATS, Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany [email protected]

List of Contributors

Wolfgang Schr¨ oder Institute of Aerodynamics, RWTH Aachen University, W¨ ullnerstraße 5a, 52062 Aachen, Germany [email protected] Robert Sch¨ oll Institute of Aeronautics and Astronautics, RWTH Aachen University, W¨ ullnerstraße 7, 52062 Aachen, Germany [email protected] Per Christian Steimle Institute of Aerodynamics, RWTH Aachen University, W¨ ullnerstraße 5a, 52062 Aachen, Germany [email protected] Christina Steiner Institute for Geometry and Practical Mathematics IGPM, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany [email protected] Georg Wellmer CATS, Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany [email protected] Frank Thomas Zurheide Institute of Aerodynamics, RWTH Aachen University, W¨ ullnerstraße 5a, 52062 Aachen, Germany [email protected]

Introduction

The Collaborative Research Center SFB 401: Flow Modulation and FluidStructure Interaction at Airplane Wings investigates numerically and experimentally fundamental problems of very high capacity aircraft having large elastic wings. The ﬂow regimes are deﬁned by cruise ﬂight and high-lift conditions. This issue summarizes the ﬁndings of the 12-year research program at RWTH Aachen University which was funded by the Deutsche Forschungsgemeinschaft (DFG) from 1997 through 2008. The research program covered the following three main topics of large transport aircraft: (i) Model ﬂow, wakes, and vortices of airplanes in highlift-conﬁguration, (ii) Numerical tools for large scale adaptive ﬂow simulation based on multiscale analysis and a parametric mapping concept for grid generation, and (iii) Validated computational design tools based on direct aeroelastic simulation with reduced structural models. In the various contributions, the fundamentals and methods for ﬂow modulation, the physical and mathematical modeling, the mathematical wellposedness, and the numerical solution are elaborated on 14 projects which were led by several institutes of RWTH Aachen University. To be more precise, the research team consisted of the Institute of Aircraft Design and Aeronautics (ILR), Institute of Aerodynamics (AIA), Institute of Lightweight Structures (IfL), Institute of Flight Dynamics (DF), Institute of Mathematics and Numerical Mathematics (IGPM), Institute of Scientiﬁc Computing (SC), and Institute of Mechanics (LFM). Any research ﬁeld considered in the Collaborative Research Center SFB 401 is tackled via numerical and experimental analyses. A supercritical proﬁle based on the three-element high-lift airfoil system BAC3-11/30/21 which is described in AGARD-AR 303 served as reference conﬁguration. Due to the generous funding by DFG it was possible to cover the technologically relevant Mach number and Reynolds number regimes by using various experimental facilities such as water tunnels, wind tunnels, and shock tubes. Considering the structure of the Collaborative Research Center SFB 401 the following scientiﬁc objectives were pursued. The detailed understanding of

XVI

Introduction

the essential ﬂow phenomena in the wake of aircraft wings was to be improved by experimental and numerical investigations. Using this knowledge novel methods were to be developed to enhance the decay rate of wake vortices of aircraft especially at take-oﬀ and landing. Furthermore, a fundamental analysis of the ﬂuid-structure interaction phenomenon was to be performed especially in the transonic state to design physically based numerical methods and mathematical models to predict the impact of the interaction between ﬂuid mechanics and structure mechanics on the overall aerodynamics at highlift and cruise ﬂight conditions. The methods being derived in the SFB 401 are expected to increase the eﬃciency and the safety in developing and operating the next generation aircraft. In the following a concise summary of the scientiﬁc results of the SFB 401 will be given. It goes without saying that a more elaborate discussion of the various ﬁndings can be found in the contributions of the projects where the scientiﬁc details are presented. As to the wake ﬂow the measurements showed two vortex pairs to be necessary to generate an inherently unstable vortex system. The excitement of these instabilities via rudder oscillations accelerates the vortex decay and as such reduces the induced rolling moment on the trailing aircraft. These wake ﬂow analyses were completed by pressure-sensitive paint measurements on the surface of the wing to understand the ﬂow physics in the near-surface region and by particle-image velocimetry measurements which were performed on the one hand, to capture the ﬂow structure in the very near ﬁeld downstream of the wing and on the other hand, to provide inﬂow data to numerically investigate the wake ﬂow further downstream, i.e., in the near far-ﬁeld. Longwave instabilities could be determined by solutions of the three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations. To spatially and temporally capture the development of short-wave and long-wave instabilities of the aforementioned two-vortex system large-eddy simulations were performed. This direct cooperation between experimental and numerical approaches was especially successful in the analysis of the interaction between engine jet and tip vortices at high-lift and cruise-ﬂight conditions. For instance, it was shown that the outer ﬂap-tip vortex is clearly inﬂuenced by the engine jet, i.e., the distributions of the azimuthal and the axial velocity and the vorticity are modiﬁed resulting in a diﬀerent trajectory of the vortex. However, the tangential velocity distribution of the wing-tip vortex was not evidently changed by the engine jet. In the context of the RANS investigations the method of automatic diﬀerentiation was used to determine the sensitivity of a newly developed one-equation turbulence model on the constants occurring in the formulation. The deviation of the approximate solution of the local shearstress distribution from the analytical solution deﬁned the cost function to determine the sensitivity. The investigations on the dynamics of ﬂuid-structure interactions (FSI) focused on the transonic ﬂow regime. Steady and unsteady measurements were performed to experimentally analyze the interaction of the boundary

Introduction

XVII

layer, shock wave, and trailing-edge separation which showed a pronounced nonlinear behavior. At forced pitch oscillations about the axis normal to the freestream direction, shock and separation oscillations were observed the phase location related to the wing motion of which varied depending on Mach number, angle of attack, amplitude, and reduced frequency. This behavior can be considered a mechanism to generate transonic limit-cycle oscillations. To reach even in the transonic regime freestream Reynolds numbers being comparable to technologically relevant values experiments were performed for an oscillating elastic wing in the European Transonic Wind Tunnel (ETW). These measurements were deﬁned in the projects High Reynolds Number Aero-Structural Dynamics (HIRENASD) and the transfer program Aero-Structural-Dynamics Methods for Airplane Design (ASDMAD). The experimental FSI-data of the various projects was used to validate novel numerical schemes for aeroelastic simulations. These methods are based on the multi-domain decomposition concept. The coupling of the ﬁnite-volume ﬂow solver and the ﬁnite-element structure solver is based on the transfer of load and shift data between the various approaches by satisfying the conservation of energy. The good agreement between the experimental and the numerical ﬁndings showed the Navier-Stokes-based-aeroelastic method to accurately describe the response of an elastic wing in subsonic and transonic ﬂows. The measurements at high Reynolds numbers under cruise-ﬂight conditions possessed an unsteady interaction of high-frequency pressure waves with the outer ﬂow and the boundary layer. On the upper surface of the wing low amplitude pressure oscillations ranging from 1 kHz to 2 kHz were observed. The results match the data determined in the NASA Ames High Reynolds Number Facility. The development of the adaptive ﬂow solver of the three-dimensional Navier-stokes equation QUADFLOW can be considered successful. Steady and unsteady ﬂows can be computed on moving meshes. The implicit algorithm was parallelized for computers with distributed memory. The processor load was dynamically balanced by decomposing, distributing, and gathering various blocks deﬁning the mesh. The nonlinear systems of equations resulting from the implicit discretization scheme were linearized through the Jacobi matrices of a ﬁrst-order discretization. The investigations on preconditioning methods ensued the point-Gauss-Seidel and point-block-ILU preconditioners to yield the best results. Moreover, various theoretically based a-posteriori error estimators were implemented to adaptively control the time step. The error was decomposed into a spatial and a temporal part such that a time step control for unsteady ﬂow problems could be developed. Besides the grid generation the multiscale approach was successfully used to develop a scale adaptive solver. The grid generator takes advantage of B-spline representations to create independent from the spatial resolution high quality composed meshes. Among other features, various approximation and smoothing algorithms are used, for instance, to adaptively deform the meshes.

XVIII

Introduction

Overall, the intensive cooperation of engineers and mathematicians resulted in the highly eﬃcient QUADFLOW software. It has to be mentioned that QUADFLOW cannot only be applied to ﬂuid-structure interaction problems at cruise-ﬂight or high-lift conditions. The method can be used to simulate any problem which can be described by the fundamental equations of ﬂuid and solid mechanics. Considering the investigations on trailing vortices and ﬂuid-structure interaction also experimental methods were successfully developed which allow a detailed and fundamental analysis of mechanisms characterized by interacting phenomena. As to the dynamics of wake vortices the signiﬁcance of high frequency perturbations on the stability behavior was evidenced while regarding the shock-boundary-layer interaction the impact of trailing-edge generated perturbations on the shock strength and hence, on the separation location was clariﬁed. In the following, the results of the various projects of the Collaborative Research Center SFB 401 Flow Modulation and Fluid-Structure Interaction at Airplane Wings are discussed at length. Each contribution is meant on the one hand, to show the development over the 12-year funding period and on the other hand, to emphasize the major ﬁnal ﬁndings.

Vortex Sheets of Aircraft in Takeoff and Landing Robert Schöll, Rolf Henke, and Günther Neuwerth

Abstract. In the present paper the development of vortex wake starting from the vortex sheet at the trailing edge of a transport aircraft wing up to the far field over 60 spans downstream is investigated. Different configurations of a half model were investigated in wind and water tunnels as well as in a towing tank by hot-wire anemometry and particle image velocimetry. In addition to an understanding of the development of the wake, means for the attenuation of the impact on following aircraft were investigated. A fin was installed on the suction side of the wing to investigate the impact on the vortex system downstream. The possibility of exploiting short-wave cooperative instabilities to accelerate vortex decay were investigated as well. This included active excitation of instabilities via ailerons oscillated in antiphase.

Nomenclature Symbols C coefﬁcient D drag L lift Re Reynolds number S wing area b span c mean aerodynamic chord l rolling moment m, n indices t time Robert Schöll · Rolf Henke · Günther Neuwerth Institute of Aeronautics and Astronautics, RWTH Aachen University e-mail: [email protected] W. Schröder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 1–23. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

2

Symbols (continued) u, v, w velocity components x, y, z coordinates Γ circulation α angle of attack δ aileron deﬂection ε ﬁn angle ζ , η perturbations η slat or ﬂap deﬂection ξ meandering amplitude

R. Schöll, R. Henke, and G. Neuwerth

Sub- and superscripts c core f follower if inner ﬂap of outer ﬂap s slat Abbrevations DST Development Centre for Ship Technology and Transport Systems ILR Institute of Aeronautics and Astronautics FAA Federal Aviation Administration PIV Particle Image Velocimetry

1 Introduction The problem posed by vortex wakes has been under scientific scrutiny for a very long time. High aspect ratio wings such as those used on transport aircraft produce compact, long-lived wake vortices. These are proportional in strength to the lift and hence to the weight of the generating aircraft. To prevent accidents due to vortex wakes, pilots have to meet separation standards such as those of the FAA [1]. Especially during takeoff and landing these standards are an impediment to the growth of air traffic. Kraft conducted flight investigations into vortex wakes as early as 1955 [2]. Research activity peaked in the United States with the advent of new, heavy transport aircraft in the 1960s and 1970s [3]. Rossow [4] and Gerz et al. [5] give an overview of the efforts to understand and to attenuate vortex wakes up until the beginning of the current decade. Until the 1970s the main approach towards wake attenuation was to increase vortex core size, thereby decreasing tangential velocity and the impact on encountering aircraft. Several devices for altering the wake are tested, and fins are described as particularly effective [4]. The influence of a fin on time-averaged and instantanous wakes in the extended near field is discussed in the Results section. A major breakthrough in wake vortex research occurred with the discovery of the sinusoidal instability by Crow [6]. He found that wake vortex systems are not primarily destroyed by diffusive processes, but by a cooperative instability between the counter-rotating vortices, which finally leads to a vortex ring state. More recent research identifies instabilities in systems of two trailing vortex pairs [7, 8]. The growth rates of these instabilities are considerably higher than those of the Crow instability, leading to a more rapid decay of the vortex system. Ortega et al. show that these theoretical results can be substantiated in experiments [9]. Haverkamp et al. use active excitation by means of oscillating control surfaces to shift the onset of instabilities to lower distances behind the wakegenerating wing [10].

Vortex Sheets of Aircraft in Takeoff and Landing

3

The problem with the aforementioned experiments is that the Reynolds numbers are rather low and the vortices generated by the models have large diameters with regard to the vortex spacing. In addition, to get high growth rates of the unstable modes, counter-rotating vortices of considerable strength are produced. Conventional wings are highly unlikely to generate such optimal vortex systems due to constraints such as geometry of the high-lift system and maximum wing root bending moment. Experiments conducted in towing tanks such as those by Veldhuis et al. [11] and Albano et al. [12] use realistic configurations, but no specific effort to actively manipulate the vortex wake is undertaken. The Results section shows results of the wing configuration used in this project in a towing tank test campaign for more than 60 spans downstream. The impact of the wake modification on the vortex-generating wing is also discussed.

2 Experimental Facilities The core of the work of the present paper was experimental in nature. The different models used were all half wings without a fuselage. Near field experiments were conducted in a wind tunnel with an open test section. The extended near field wake was investigated in a circulating water tunnel. Finally, the far field experiments were conducted in a towing tank facility.

2.1 Wind Tunnel The closed circuit wind tunnel of the Institute of Aeronautics and Astronautics (ILR) had an open test section with a 1.5 m diameter and a length of 3 m. A photograph of the test section is shown in figure 2. The half models were mounted upright on a flat plate with leading edge suction. To reduce the influence of turbulence induced by mixing with the ambient air, the base plate was extended towards the diffuser. The model support allowed the inclusion of a 6 component strain gauge balance. In the velocity regime used for the experiments, wind tunnel turbulence was below 3 %.

2.2 Water Tunnel The circulating water tunnel of ILR was chosen for the size of its test section, which allowed models of the same span as in the wind tunnel. The closed test section had a rectangular cross section 1.5 m wide, 1.0 m high, and 6.5 m long. The tunnel walls were equipped with glass panes, allowing access for optical measurement methods, as can be seen in figure 3a. Additional windows were inserted at the very end of the test section to ease measurements at this point. The model was mounted on the tunnel ceiling near the nozzle. As in the wind tunnel, a strain gauge balance could be fitted for force measurements. The average turbulence outside the boundary layer was 2 %.

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R. Schöll, R. Henke, and G. Neuwerth

2.3 Towing Tank The towing tank experiments were conducted in the Small Deep Water Tank of the Development Centre for Ship Technology and Transport Systems (DST). Overall length was 70 m, but for the current experiments only 38 m were used. The width was 3 m and the water level was adjusted to 2.6 m. For the downstream distances investigated, end effects did not adversely affect the measurements. Surface waves caused by the displacement of the model sting were damped with floating mats covering the tank 10 m upstream and downstream of the measurement station. In between measurements the water was allowed to settle, so that residual velocities were less than 0.5 mm/s. Figure 1 depicts the installation of the measurement equipment in the test section, for a top view compare fig. 3b. A big watertight tank with the open top above the water surface was constructed to house a set of digital cameras. A box with a glass bottom was installed to allow for an undisturbed entry of a laser light sheet into the water. Finally, the model support had to span the whole width of the tank.

Fig. 1 Cross section of the towing tank with half model and setup for 2C-PIV measurements

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3 Measurement Instrumentation For the wake measurements two distinct techniques were employed: hot-wire anemometry for the wind tunnel measurements and particle image velocimetry (PIV) for the water tunnel measurements. Force measurements were carried out with the help of a 6 component half model strain gauge balance accurate to within 1 % for lift forces.

3.1 Hot-Wire Anemometry The setup for hot-wire anemometry is illustrated in figure 2. The probe was a three wire probe to measure all three velocity components. The effective wire length was 1 mm. The sampling rate was 1 kHz and the measurement interval 1 s. To capture the complete wake, a coarse measurement mesh with a spacing of 5 mm or 7.5 mm was used. To better capture the vortex properties, a finer mesh with a spacing of 2 mm was used at the location of the vortices.

Fig. 2 Wind tunnel test stand with hot-wire probe

3.2 Particle Image Velocimetry The general setup for 3C-PIV is illustrated in figure 3a. A pulsed laser illuminated planes perpendicular to the mean flow. Digital cameras on both sides of the tunnel captured the images of polyamide 12 particles, which were used as tracers. Prismatic water-filled boxes between the cameras and the glass panes allowed a nearly refraction-free recording of the flow. The measurement resolution was 5 mm with a 50 % overlap for the interrogation windows. The measurement frequency was 7 Hz, and flowfields could be investigated in a time-averaged and time-resolved manner. In order to capture the complete wake, the cameras had to be set up in different vertical positions at one downstream measurement station. For the towing tank experiments, a 2C-PIV setup was used. A stack of three cameras was mounted in a tank with an open top above the water surface. The tank was placed at a distance of 2.55 m, and the optical axis was at an angle of 19° with

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R. Schöll, R. Henke, and G. Neuwerth

(a) Water tunnel test section with half model and 3C-PIV setup for wake measurements

(b) Top view of the 2C-PIV setup at the towing tank, the Model is shown at x/b = 0

Fig. 3 PIV setups at different experimental facilities

respect to the model trajectory. The cameras had overlapping fields of view and, combined, gave a viewport of 0.55 m by 1.55 m. The viewport was laterally placed such that the vortices could be tracked from rollup up to the end of the measurement. The lower edge was at 0.8 m above the floor of the tank, effectively ending the measurement before the ground effect became dominant. Figure 3b illustrates the setup. To ensure sufficient light intensity, two double-cavity Nd:YAG lasers with 120 mJ pulse energy each were coupled. In order to generate a light sheet not influenced by ripples on the water surface, a box with a glass bottom was installed. Seeding was the same as in the water tunnel experiments. A 0.65 m wide rake with 10 nozzles was used to distribute the seeding evenly in the measurement plane. The measurement resolution was 5 mm in both dimensions and the window overlap roughly 50 %. The measurement frequency was set to 2 Hz, and 155 images were recorded in sequence.

4 Model The half-wing used was a model featuring some of the attributes of a transport aircraft wing. Figure 4 shows a sketch of the model. The airfoil section was the BAC 3-11/RES/30/21 [13]. The thickness to chord ratio was 15 % at the wing root, thinning out to 11 % at the kink and remaining at that value out to the tip. The wing had no twist and no dihedral. The mean aerodynamic chord c was 0.178m and the area S was 0.207 m2 . The model could be set up in a clean configuration for cruise conditions as well as a landing or takeoff configuration with the high-lift devices deflected. Different sets of slats and single-slotted flaps were available: slats of 0° and 25° deflection as well as flaps of 0°, 10°, and 20°. The inner and outer flap were completely independent to enable differential flap settings. For experiments aimed at triggering instabilities in the wake vortex system, a pair of ailerons was added. Both ailerons had the same planform area, so that changes

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Vortex Sheets of Aircraft in Takeoff and Landing

Fig. 4 Layout of the half model, flaps are shown fully deployed (ηif = ηof = 20°). The slashdotted lines mark the plan view of the clean configuration. Dimensions are given in mm

in overall lift were minimized when the ailerons were actuated with the same amplitude in antiphase. A shaft/hollow-shaft system was integrated into the wing to drive the ailerons. The system was designed to keep the outer shape of the wing unchanged, and no part extended more than 1 mm outside the former shape. All parts were flush to prevent flow detachment and keep wake data comparable with pre-modification results.

5 Results In the course of twelve years of research many different results concerning the vortex wakes of wings were obtained, some of which are reported in the following sections. Results obtained from experiments with less complex models, such as rectangular wings with and without flaps are omitted. All of the experiments shown were conducted with a swept trapezoidal wing with a high lift system. For the most recent experiments, a set of independently actuated ailerons was added to the model. The maximum distance to the wake generator was increased in order to satisfy the demand for statements about the wake at realistic separations.

5.1 Near Field To get an understanding of the wake structure in the near field, a number of different high-lift system configurations were investigated at α = 4° and 8°. The model

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R. Schöll, R. Henke, and G. Neuwerth

was configured in a clean configuration (flaps and slats retracted) and high-lift configurations (slats always 25°). Flaps were set at 10° and 20°, and there was also a differential flap setting with an inboard loading. Table 1 gives an overview of the configurations and the respective lift coefficients at an angle of attack of 8°. Table 1 Configurations of the high-lift system and respective CL , α was set to 8° for all cases slat εs = 0° εs = 25° εs = 25° εs = 25° flap εif = εof = 0° εif = εof = 10° εif = εof = 20° εif = 20°, εof = 10° CL

0.65

0.89

1.34

1.10

Oil streak pictures showed that there was only minimal flow separation in the wake of the slat and flap supports. Smoke injection showed that the trailing vortices were compact and did not burst. Hot-wire anemometry was used to derive time-averaged velocity fields in planes perpendicular to the mean flow. Figure 5 shows a visualization of the axial vorticity calculated from the in-plane velocities. The values were non-dimensionalized by multiplication with c/u∞ . It can be seen that there was a vortex sheet in the wingtip plane, which had two starting points for vortex rollup. These were located at the wingtip and at the tip of the outer flap. Further downstream, these developed into two distinct vortices incorporating more and more circulation from the original vortex sheet. It could also be observed that the trailing vortices started into an orbital movement about one another, caused by mutually induced velocities. Using detailed measurements with a higher resolution around the vortex locations, vortex parameters such as core radius, maximum tangential velocity and circulation were derived. The core radius rc was defined as the radius where the maximum tangential velocity was recorded. Figure 6a shows the wingtip vortex’s core radius development for different settings of the high-lift system and increasing downstream distance. For α = 4° the core radius was highly dependent upon the flap setting. Clean configuration lead to a continously compact vortex, while deployment of the slats led to an increase in radius because of the additional turbulence in the boundary layer. Increasing the flap deflections led to an increased induced angle of attack near the wingtip, which in turn led to a more compact trailing vortex. For α = 8° the relative influence of the flap setting was greatly reduced, all measurements were closely spaced. In the outboard flap vortex, the same tendencies could be observed: wider spacing for α = 4° and smaller radii for greater flap deflection anlges. There, the differential flap setting was generally situated between 10° and 20° flaps. It could also be observed that the relative increase in radius between x/b = 0.0 and 0.8 was greater, especially for α = 4°. This could be explained by the greater diffusion due to more turbulence in the outer flap vortex coupled with lower relative circulation. Vortex core radii and tangential velocities were not sufficient information to estimate the impact of the vortex system on encountering aircraft. Since induced

Vortex Sheets of Aircraft in Takeoff and Landing

9

-0.4 -0.2

z/b

0 0 0.2

-0.1 -0.2

wing tip vortex 0

outboard flap vortex

0.2

y/b

0.4

b x/

0.4 0.6

ωc/u∞

0.8 -2

1

6

12

18

24

30

Fig. 5 Time-averaged axial vorticity distribution in the wake of the half model, both flaps were extended 20°. Re = 4 × 105 and α = 8°

rolling motion was deemed most dangerous for following aircraft, especially near the ground, an induced rolling moment coefficient was defined as the ratio of rolling moment to the product of dynamic pressure and wing area of the encountering aircraft. Taking strip theory as a basis, and supposing a rectangular wing planform, the rolling moment coefficient could be expressed as: 2 Cl = 2 bf

bf /2

CLα arctan −bf /2

w y dy , u

(1)

where bf was the wing span of the following aircraft and CLα = 2π. This method took into account the change in lift using a change in local induced angle of attack. The following wing was assumed to be rectangular. This always led to an overestimation of Cl compared to tapered wings due to greater forces towards the wing tips. The evaluation of the complete wake field gave a rolling moment distribution, of which the maximum was determined. All results are shown in figure 6b. The rolling moment coefficient was normalized by the lift coefficient, since vortex strength and lift are coupled via the circulation of the wing’s bound vortex. It can be seen that the clean configuration produced the highest rolling moment. This was not surprising, since the whole circulation was rolled up into a single vortex. At x/b the configurations with εif = εof = 20° exhibited a normalized rolling moment nearly double that of the remaining two configurations. This followed from the fact that the maximum induced rolling moment was correlated with the outer

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R. Schöll, R. Henke, and G. Neuwerth 0.12

rc / c

0.09

0.06

0.03

0

0

0.2

0.4

0.6

0.8

x/b

(a) Core radius of the wingtip vortex 0.5

cl,max / cL

0.4

0.3

0.2

0.1

0

0.2

0.4

x/b

0.6

0.8

(b) Maximum induced rolling moment. The span of the following wing was 20 % of the generating wing. Fig. 6 Core radius and maximum induced rolling moment

flap vortex, which was strongest for εof = 20°. Finally, the configurations with εof = 10° showed the most benign behaviour due to circulation being balanced between wingtip and outer flap vortex. Of those, the inboard loading configuration was more suitable for takeoff or landing due to higher lift reserves. As a conclusion, it followed that in order to achieve good vortex wake characteristics in the near field, it would be

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advisable to create a vortex system with three vortices, which should be of roughly equal strength. A more complete overview of these experiments was given by Özger et al. [14]. Based on these measurements, an extrapolation of results into the far field was conducted via a 2D-Navier-Stokes calculation. Those results are reported in the Far Field section of the current paper.

5.2 Extended Near Field In the investigation of the near field it became clear that the properties of the wake were a complex function of the downstream distance and could not be easily explained by linear relations. To characterize the wake further downstream, measurements in a circulating water tunnel were conducted which allowed measurements up to 4.0 full spans downstream. In order not to change the free wake by measuring probes, the non-intrusive 3C-PIV-technique was used. In investigations also reported in [14], a fin mounted on the suction side of the wing in front of the outer flap tip was found to be beneficial for the vortex wake. This finding was congruent with the findings of Rossow [4], who reported results of experiments in large wind tunnels. The same author showed that inboard loading configurations proved beneficial in flight tests. The inboard loading was achieved with differential flap settings. In addition to configurations with fins and conventional flap settings, the wakes of configurations with differential flaps and with/without a fin were investigated as well. Results from experiments with conventional flap settings were reported in [15]. A more detailed description of the experiments with differential flap settings was given in [16]. The lift coefficient was held constant at 1.33 for all measurements. Re was 1.9 × 105. Figure 7 illustrates the structure of the wake as measured in the circulating water tunnel. The coordinate system was the same for all measurements and is depicted in fig. 5, the wing root being at y/b = 0. It can be seen that the outer flap vortex is considerably weaker than both the wingtip vortex and the inner flap vortex. The counter-rotating fin vortex is also weaker, becoming quickly dispersed and from x/b = 0.8 onwards it is no longer visible. Figure 7b shows that after complete rollup of the shear layer the time-averaged vorticity of the discrete vortices is less concentrated. This is not mainly due to dispersion, but rather is the result of the vortices’ meandering motion. When comparing the influence of fin angles and settings of the outer flap, it was found that the flap setting was the most significant. It had to be noted, however, that when compared to the wind tunnel settings and results, a fin setting of ε = −35° or ε = 5°, led to considerably smaller local flow angles of incidence (±20°) due to the local flow direction. In those experiments, a fin angle ε = 20° led to an angle of incidence of 35°, which meant that there was a considerably higher amount of turbulence in the wake. This can also be seen in Fig. 8, which displays the development of the circulation of the inner flap vortex and the wingtip vortex. Circulation was calculated by taking

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R. Schöll, R. Henke, and G. Neuwerth

ω c / u∞ 16.5

0.1

14.5

fin vortex

12.5

z/b

wing tip vortex 0

10.5 8.5

outer flap vortex

inner flap vortex

6.5 4.5 2.5

-0.1

0.5 -1.5

-0.4

-0.3

-0.2

-0.1

0

0.1

y/b

(a) x/b = 0.0

ω c / u∞

wing tip vortex 0.1

16.5 14.5 0

12.5

z/b

10.5 8.5

-0.1

6.5

outer flap vortex

4.5

inner flap vortex

-0.2

2.5 0.5 -1.5

-0.3 -0.4

-0.3

-0.2

-0.1

0

y/b

(b) x/b = 4.0

Fig. 7 Time-averaged axial vorticity in the wake of a wing with fin and differential flaps, fin angle ε = −35°, ηif = 20°, ηof = 10°

the average of vorticity integrated over a circular area of both 1.5 and 2 core radii, following an approach similar to the Γ5−−15-method used in field measurements [5]. It was made dimensionless with the free stream velocity of 1.1 m/s and the mean aerodynamic chord. When looking at the wingtip vortex in fig. 8a, it can be seen that after the initial rollup, which led to a small increase in Γ between x/b = 0 and 0.4, there was a small decrease between x/b = 0.4 and 0.8. While there was a small decrease afterwards for the cases with a retracted outer flap, the other flap setting lead to a

Vortex Sheets of Aircraft in Takeoff and Landing

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0.3

Γ/(u∞ c)

0.25

0.2

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0.05 0

1

2

x/b

3

4

3

4

(a) wingtip vortex

0.3

Γ/(u∞ c)

0.25

0.2

0.15

0.1

0.05 0

1

2

x/b

(b) inner flap vortex Fig. 8 Circulation contained in the trailing vortices, values calculated from instantaneous measurements; solid lines denote ηof = 0°, dashed lines ηof = 10°, η = 5°, η = 0°, η = −35°

constant circulation between x/b = 0.8 and 4. Overall, standard deviation was well below 10 % of the average value, indicating steady conditions. Γ dropped gently and monotonically, which was to be expected. The inner flap vortex displayed a higher dependency on the fin setting, see fig. 8b. With the outer flap retracted there was a monotonous decrease up to x/b = 0.8 and a constant development downstream. For the differential flap setting, the fin’s influence was only apparent at x/b = 4, where a positive fin angle lead to an increase in Γ and a negative angle to a decrease as compared to the baseline without a fin. The bigger variation between fin settings was due to mutual interaction between

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R. Schöll, R. Henke, and G. Neuwerth

0.25 inner flap 20°, outer flap 0°, fin +5° inner flap 20°, outer flap 0°, no fin inner flap 20°, outer flap 0°, fin −35° inner flap 20°, outer flap 10°, fin +5° inner flap 20°, outer flap 10°, no fin inner flap 20°, outer flap 10°, fin −35°

ξ/c

0.2

0.15

0.1

0.05

0 0

1

2

x/b

3

4

3

4

(a) wingtip vortex 0.25 inner flap 20°, outer flap 0°, fin +5° inner flap 20°, outer flap 0°, no fin inner flap 20°, outer flap 0°, fin −35° inner flap 20°, outer flap 10°, fin +5° inner flap 20°, outer flap 10°, no fin inner flap 20°, outer flap 10°, fin −35°

ξ/c

0.2

0.15

0.1

0.05

0 0

1

2

x/b

(b) inner flap vortex Fig. 9 Meandering amplitude calculated from instaneous measurements

the closely spaced inner and outer flap vortices. It had to be noted that the standard deviation in these cases was considerably higher than in all others, being of the order of the average, which pointed to dynamic, instationary effects. Again, this behavior could be explained by the degree of interaction with the outer flap vortex, where the fin’s confined influence was large enough to change overall vortex behavior. The fin also had a certain influence on the meandering motion of the vortices. Figure 9 shows the average distance ξ between an instantaneous vortex position and the time-averaged centroid, non-dimensionalized with the mean aerodynamic chord. In general, the meandering amplitude did increase linearly with downstream distance, which was a behavior consistent with the meandering mechanism proposed by Rossow and Meyn [17]. However, the inner flap vortex displayed in fig. 9b displayed an influence of the fin setting on the meandering amplitude. This became apparent at x/b = 2 and 4, where the configurations with a fin and differential flap setting showed a distinct increase in ξ . In these cases, the outer flap vortex acted as a means to transmit the fluctuations produced by the interaction between the fin and

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the inner flap vortex. It had to be noted that the increase in meandering amplitude did not occur immediately behind the wing, so that a mere increase in turbulence in the wake is an unlikely source. From the data, it had to be concluded that the interaction between the fin and outer flap vortex caused the increase in ξ and that the orientation of the fin vortex had an influence on the onset of increased ξ . As to the influence of vortex meandering on the hazard posed to following airplanes, it was hard to give a definitive answer. While the rolling motion of an airplane moving into the vicinity of a vortex with a higher meandering amplitude would be decreased due to inertia, it would also mean that there is an increased risk of wake vortex encounters during takeoff and final approach. This is because the docile downward movement of a vortex system transporting the hazard away from the flight path gets superimposed with a random movement capable of moving individual vortices upwards again.

5.3 Far Field While experiments up to x/b = 4 gave more insight into the development of the wake than purely near field experiments did, there was still a need for examination of the far field. Also, due to promising results such as those published in [18], it was decided to exploit cooperative instabilities in the vortex system. In order to identify unstable vortex systems, linear stability theory was applied to wake data derived from measurements at x/b = 0. From these measurements the basic parameters of the vortex system such as position of the vortices, circulation and core radius were determined. These were used as input into the equations of motion, which were based on the linearized law of Biot-Savart with the addition of perturbation velocities [19, 7]. The stability equations were differential equations of the form 8 dηn = ∑ (v1mn ζn + v2mn ζm + v3mn ηn + v4mn ηm ) dt m=1

and

8 dζn = ∑ (w1mn ηn + w2mn ηm + w3mn ζn + w4mn ζm ) , dt m=1

(2) (3)

where the indices m and n denoted the individual vortices, t the time, v and w the horizontal and vertical velocities and η and ζ the dimensionless perturbation amplitudes. v and w were analytical solutions of the integrals of induced velocity. Equations (2) and (3) were formulated for an eight vortex system, because with differential flap and aileron settings there were four individual vortices in the near wake. This was an initial value problem where ηn (t = 0) = ηn0 and ζn (t = 0) = ζn0 [18]. It was solved numerically. The solution was interpreted with regard to the amplitude amplification of the individual vortices. A total of 27 individual configurations with conventional flap settings and outboard loading as well as varying aileron settings were measured in the water tunnel and subsequently analyzed for growth of instabilities as outlined above. Out of these, one configuration with conventional flap settings and one

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configuration with outboard loading were chosen for subsequent measurements in a towing tank. The stability analysis did not only reveal configurations with high transient growth rates of instabilities, but also the wavelength of those instabilities. In an effort to accelerate the growth of instabilities, the ailerons of the model were oscillated at a frequency equivalent to this wavelength. The ailerons were oscillated in antiphase to reduce lift and drag fluctuations. Finally, an inboard loading configuration was added. Table 2 summarizes the configurations examined in the towing tank. Table 2 Configurations investigated in the towing tank, η stands for slat and inner/outer flap deflection, δ for inner/outer aileron deflection, f is the oscillation frequency configuration

ηs ηif ηof

252020p10p10 252020p10p10_rpm87 252010p00p00 251020m10p00

25° 25° 25° 25°

20° 20° 20° 10°

20° 20° 10° 20°

δi 10° 10° 0° −10°

δo CL 10° 10° 0° 0°

f

1.25 — 1.25 1.45 Hz 1.25 — 1.25 —

Polars of all configurations had been measured in advance, so angle of attack α could be adjusted between 5° and 9° for a constant lift coefficient. Towing speed u∞ was 1.1 m/s to achieve the same Re-Number as in the water tunnel experiments. Since the measurement plane was 19 m from start and end point, end effects did not influence the measurement. Figure 10 shows the trajectories of the trailing vortices of the configuration with conventional loading between x/b = 0 and 63. Positions were averaged over at least 5 measurement runs. Towing tank experiments conducted before had shown that this number was sufficient to give a reliable result for the average. Directly downstream of the wing, the wingtip and the outer flap vortex started to rotate around each other, with the wingtip vortex describing a wider spiral. After vortex merging roughly 12–14 spans downstream, the remaining vortex moved downward. When comparing figs. 10a and 10b, it can be seen that there was an anomaly for the case with oscillation at z/b = −0.4 (corresponding to a downstream range of 20 up to 35 spans). This behavior was thought to be caused by the action of instabilities on the remaining vortex. The standard deviation of vortex positions in this range was also considerably higher than before and decreased afterwards, showing an interval of increased unsteadyness. After this interval, the vortex system showed a trajectory which was comparable with the case without oscillation. The circulation of the trailing vortices was calculated by fitting the vorticity distribution of a Lamb-Oseen vortex to the measured vorticity via a least squares method. Figure 11 shows the results for the conventional flap setting with and without oscillation. The error bars denote the standard deviation, the merger region was left out because the fitting algorithm was not robust enough to discriminate between the vortices during the merging.

17

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

z/b

z/b

Vortex Sheets of Aircraft in Takeoff and Landing

−0.3

−0.3

−0.4

−0.4

−0.5

−0.5

−0.6

−0.6

−0.7

−0.7

−0.8

−0.8 0

0.2

y/b

0.4

(a) without oscillation

0

0.2

y/b

0.4

(b) with oscillation

Fig. 10 Trajectory of the trailing vortices of the conventional loading configuration, the wing root was at x/b = y/b = 0. marks the wingtip vortex, marks the outer flap vortex, no symbol marks the merged vortex

It could be seen that the wingtip vortex had less circulation than the outer flap vortex for this flap setting, and that Γ of the wingtip vortex was similar for both configurations. The slight circulation decrease in the outer flap vortex before merging was more pronounced for the case with oscillation. There was an increased standard deviation in fig. 11a for x/b ≤ 5, which was caused by decreased signal to noise ratio of the PIV measurement due to increased particle displacement near the vortex core. The vortex merging set on earlier for aileron oscillation, which explained the higher Γof and the lower Γ of the merged vortex at x/b = 15. After vortex merging, both configurations displayed a decline in Γ , with the average gradient dΓ /dx decreasing about 15 spans downstream of the merging region. The steeper decline in the first segment after merging could be traced to the increased dissipation of the merged vortex with higher velocity gradients. Starting at x/b ≥ 20 the case with oscillation showed higher fluctuations, which were attributed to the action of instabilities on the vortex. There was no clear decrease in Γ , however. At x/b ≥ 50 the fluctuations went back to the level of the base configuration. Interestingly, starting from x/b = 45, a modulation of Γ could be seen with a wavelength of about 10b. Although this corresponded to the Crow instability, there was no accompanying movement of the vortex centroids. Measurements at higher x/b would have been necessary to investigate if this phenomenon lead to further changes downstream. Figure 12 shows the relative induced rolling moment of all configurations in the water towing tank as calculated according to eq. (1). As in fig. 6b, the global maximum at a given x-position was plotted. In addition, the same following wing span of 20 % of the generating wing was used. It had to be noted that the plot for the inboard

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R. Schöll, R. Henke, and G. Neuwerth

outer ﬂap vortex wing tip vortex merged vortex

0.25

Γ/(m2 /s)

0.2

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0.1

0.05

0 0

10

20

30

x/b

40

50

60

(a) without oscillation outer ﬂap vortex wing tip vortex merged vortex

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Γ/(m2 /s)

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20

30

x/b

40

50

60

(b) with oscillation Fig. 11 Circulation of the trailing vortices of the conventional loading configuration

loading configuration ended at x/b = 42. This was because the lift centroid in the spanwise direction was considerably closer to the mirror plane than for the other configurations, leading to a faster downward motion and the vortices left the observable area sooner than in the other cases. The modulation for all induced rolling moment curves in the region x/b ≤ 15 came from the orbital motion of the wingtip and outer flap vortices and ended with the merging of those vortices. A local minimum was associated with a horizontal position of the vortices with regard to each other, a maximum with a vertical position. Up to the maximum measured position at x/b = 42, the inboard loading configuration showed the weakest rolling moment Cl , especially for lower x/b. This was to be expected, since overall Γ could be distributed between three trailing vortices in the half-plane. However, at the end of the measurement, the inner flap vortex and the vortex resulting from the merging of outer flap and wingtip vortices were not

Vortex Sheets of Aircraft in Takeoff and Landing

19

0.4

251020m10p00 252010p00p00 252020p10p10 252020p10p10 RPM87

0.35

C l,max/C L

0.3 0.25 0.2 0.15 0.1 0.05 0 0

10

20

30

x/b

40

50

60

Fig. 12 Maximum induced rolling moment for all configurations examined in the towing tank; bf /b = 0.2

yet merged. A vortex with a combined circulation of both remaining vortices could well have an increased rolling moment, negating the decrease in Cl at higher x/b. When comparing the two conventional configurations without and with oscillation, it could be seen that neither differed much for x/b ≤ 17 and the latter had an only slightly elevated Cl for x/b ≥ 35. Between these regions, there was a region where the oscillation seemed to decrease Cl by at least 10 %. The fluctuations in this region were also higher than upstream or farther downstream, further indicating that there was an instability mechanism at work. This was also apparent for bf /b = 0.4. The results from stability theory suggested that instability growth would be maximal for the outboard loading case. This configuration lead to a counter-rotating vortex which did not merge and was thought to have a high potential for destabilizing the vortex system. On the other hand, the requirement for constant lift in combination with a partially retracted inner flap lead to a high α of 9° and strong outer flap and wingtip vortices. As a result, this configuration displayed the highest Cl at x/b = 20. After a steep drop until x/b = 30, Cl was lower than for the conventional configuration until the end of the measurement region. The same could be seen for bf /b = 0.4. The counter-rotating vortex was considerably weaker than the optimum for the counter-rotating vortex pair in a four vortex system according to Haverkamp et al. [20]. The ratio of circulation from the counter-rotating vortex to that of the other vortices (only the wingtip vortex for a four vortex system) was −0.17 as compared to −0.39 for optimal rolling moment reduction, but the effect was significant nonetheless.

20

R. Schöll, R. Henke, and G. Neuwerth

5.4 Force Fluctuations In order to judge whether the perturbations generated by the aileron oscillations did influence the wing, the fluctuations in lift and wing root bending moment were measured in the circulating water tunnel with the help of 3C-PIV. Strain gauge measurements were conducted in parallel, but since the inertia of the model and the model support and their respective eigenmodes changed the measured forces considerably, only time-averaged values were taken from those measurements. The configurations investigated were the same as in the towing tank experiments, but oscillation was used together with all configurations. In addition to static ailerons, the oscillation frequency f of 1.45 Hz and a frequency 1.5 times higher (2.18Hz) were used. Aileron amplitude was the same as in the towing tank experiments, δˆi = 5° and δˆo = 4°. The velocities were measured in the wingtip plane with a sampling frequency of 7 Hz, and the respective vorticity distribution was integrated to receive the sectional lift. Total lift was obtained by integration in the spanwise direction, wing root bending moment by calculating the first moment of the lift distribution. In the following, only results for the lift coefficient are presented. The results for the lift from PIV-measurements were within 4 % of the results from measurements with a 6 component strain gauge balance. The measured lift coefficient as a function of the time is plotted in fig. 13a, showing results for the conventional load distribution with different oscillation frequencies. No filtering was applied to the displayed time signal. Differences between the oscillation frequencies were small, with all measurements giving a lift 4 % higher than the lift measured with the strain gauge balance. This difference was probably caused by interference of the wing with a split blade installed to cut off the boundary layer, since this could not not be compensated for in the strain gauge measurements. The standard deviation was 5 % for 0 Hz and 1.45Hz and 4 % for 2.18 Hz. What was most interesting was the behavior of the lift in the frequency domain, since a high peak of fluctuations at the oscillation frequency would indicate considerable impact on the wing. To examine this, the lift signal was Fourier transformed. The time signal of 940 samples was split into overlapping groups of 256 elements each, the transformation applied and the results were averaged afterwards. Figure 13b displays the result for the conventional loading distribution, the power spectrum of the lift coefficient is plotted against the frequency, 3.5 Hz being the Nyquist frequency. Most significant was that there were no dominant peaks at the oscillation frequencies or at 2.9 Hz. The integral between 0 Hz and 3.5 Hz was considerably smaller than the integral in the time domain, indicating that there were many fluctuations at frequencies higher than the oscillation frequencies. This indicated that fluctuations caused by the water tunnel dominated the measurement. But it was also evident that the fluctuations caused by the oscillation were very small, |CL − C¯L |2 being on the order of 1 × 10−4 . This indicated that due to the small excitation, there would be only minor problems with wing bending oscillations or passenger comfort.

Vortex Sheets of Aircraft in Takeoff and Landing

21

(a) time domain

(b) frequency domain Fig. 13 Lift fluctuations derived from 3C-PIV measurements

6 Conclusion The main goal of this project was to characterize the vortex wake shed by a wing with a high-lift system and to investigate measures to reduce wake hazard. Experiments in a wind tunnel as well as in a circulating water tunnel showed the roll-up of the shear layer and the structure of the resulting vortices. The influence of various settings of the high-lift system were investigated as well. It could be shown that a

22

R. Schöll, R. Henke, and G. Neuwerth

distribution of overall lift over three individual vortices via differential flap settings with inboard loading did indeed decrease the hazard posed to following airplanes. In addition to near field and extended near field experiments, experiments in a towing tank facility were conducted as well with a maximum downstream distance of 63 spans. The beneficial influence of inboard loading could be shown there as well. In addition, an outboard loading configuration with a weak counter-rotating vortex emanating from between inner and outer flap was examined. This was shown to lead to a decreased wake hazard at x/b ≥ 30 as compared to a conventional flap setting, the difference being 30 % at 63 spans. In the near field and extended near field the influence of a fin installed on the suction side was investigated. It was shown that in the extended near field vortex meandering was increased, effectively spreading the area with high induced rolling moments and at the same time reducing the peak moment. As another measure, the active excitation of inherent instabilities was investigated. Therefore, split ailerons that could be oscillated in antiphase were installed. It was shown that while the force fluctuations on the wing itself were minor, the influence of the active excitation was only visible in a range between 17 and 35 spans downstream. Before and after this downstream range, active excitation did not make a difference in the hazard for following airplanes. It had to be noted that the conventional layout of the model, similar to a realistic transport aircraft wing, did not lend itself to the generation of a very unstable vortex system, where active excitation would have made a major impact on rolling moment reduction. The research reported in the present paper suggests that realistic configurations can reduce wake vortex separations mainly by differential flap settings. Highly unstable vortex systems displaying high growth rate elliptic instability and a significant reduction in airplane separation will most likely be restricted to experimental layouts with large outboard flaps unsuitable for real-world applications.

Acknowledgements The work in this field was funded by the Deutsche Forschungsgemeinschaft as project A2 of the Collaborative Research Centre 401. We are thankful for the generous support, which allowed 12 years of vortex research. The present authors would also like to acknowledge the contributions of D. Coors, D. Jacob, S. Kauertz and I. Schell (near field experiments), who worked on this project in the past.

References 1. N.N.: Aeronautical Information Manual—Official Guide to Basic Flight Information and ATC Procedures. Federal Aviation Administration (2008) 2. Kraft Jr., C.C.: Flight measurements of the velocity distribution and persistence of the trailing vortices of an airplane. Technical Note 3377, National Advisory Committee for Aeronautics (1955) 3. N.N.: Wake vortex minimization. Special Publication 409, National Aeronautics and Space Administration (1977) Symposium held at Washington, February 25–26 (1976)

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4. Rossow, V.J.: Lift-generated vortex wakes of subsonic transport aircraft. Progress in Aerospace Sciences 35, 507–660 (1999) 5. Gerz, T., Holzäpfel, F., Darracq, D.: Commercial aircraft wake vortices. Progress in Aerospace Sciences 38, 181–208 (2002) 6. Crow, S.C.: Stability theory for a pair of trailing vortices. AIAA Journal 8, 2172–2179 (1970) 7. Crouch, J.D.: Instability and transient growth for two trailing-vortex pairs. Journal of Fluid Mechanics 350, 311–330 (1997) 8. Rennich, S.C., Lele, S.K.: Method for accelerating the destruction of aircraft wake vortices. Journal of Aircraft 36, 398–404 (1999) 9. Ortega, J.M., Bristol, R.L., Sava¸s, Ö.: Experimental study of the instability of unequalstrength counter-rotating vortex pairs. Journal of Fluid Mechanics 474, 35–84 (2003) 10. Haverkamp, S.: Beeinflussung von Flugzeugnachläufen durch oszillierende Querruder. Dissertation, RWTH Aachen, Aachen (2004) 11. Veldhuis, L.L.M., Scarano, F., van Wijk, C.: Vortex wake investigation of an airbus a340 model using piv in a towing tank. In: 21st Applied Aerodynamics Conference, American Institute of Aeronautics and Astronautics (2003) 12. Albano, F., De Gregorio, F., Ragni, A.: Trailing vortex detection and quantitative evaluation of vortex characteristics by piv technique. In: 20th International Congress on Instrumentation in Aerospace Simulation Facilities, Institute of Electrical and Electronics Engineers, pp. 31–43 (2003) 13. N.N.: A selection of experimental test cases for the validation of cfd codes, vol. 2. Advisory Report AR-303-VOL-2, Advisory Group for Aerospace Research and Development (1994) 14. Özger, E., Schell, I., Jacob, D.: On the structure and attenuation of an aircraft wake. Journal of Aircraft 38, 878–887 (2001) 15. Kauertz, S., Neuwerth, G., Schöll, R.H.: Investigations on the Influence of Fins on the Extended Nearfield of a Wing in High-Lift Configuration, 1st edn. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 92, pp. 33–40. Springer, Heidelberg (2006) 16. Schöll, R.H., Buxel, C., Neuwerth, G.: Influence of spanwise loading and fins on extended near-field vortex wake. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, American Institute of Aeronautics and Astronautics (2006) 17. Rossow, V.J., Meyn, L.A.: Relationship between vortex meander and ambient turbulence. In: 6th AIAA Aviation Technology, Integration and Operations Conference, American Institute of Aeronautics and Astronautics (2006) 18. Kauertz, S., Neuwerth, G.: Excitation of instabilities in the wake of a wing with winglets. AIAA Journal 45 (2007) 19. Fabre, D., Jacquin, L., Loof, A.: Optimal perturbations in a four-vortex aircraft wake in counter-rotating configuration. Journal of Fluid Mechanics 451, 319–328 (2002) 20. Haverkamp, S., Neuwerth, G., Jacob, D.: Studies on the influence of outboard flaps on the vortex wake of a rectangular wing. Aerospace Science and Technology 7, 331–339 (2003)

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An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows about Elastic Configurations Gero Schieffer, Saurya Ray, Frank Dieter Bramkamp, Marek Behr, and Josef Ballmann

Abstract. In this paper the development of the new adaptive solver QUADFLOW is described. It is based on an integral concept and consists of a flow solver, a grid generation tool using parametric mapping based on B-splines, and local grid adaptation based on multiscale analysis. QUADFLOW has been designed to obtain a solution method for multidisciplinary problems in the field of aerospace engineering. The most important applications are the simulation of high-lift configurations and elastic wings in cruise configuration. A partitioned field approach is used for the solution to aeroelastic problems. In this article the finite volume solver and its inclusion into an aeroelastic solver are outlined. The results of the validation study and a new matrix-free Newton-Krylov method, offering potential to accelerate convergence, are presented.

1 Introduction The demand for high-capacity aircraft is increasing rapidly. In order to reduce time and cost during the development of next-generation fuel-efficient aircraft, engineers rely on numerical simulation as a design tool reducing the number of expensive wind tunnel experiments. To decrease the weight of the aircraft engineers rely on lightweight constructions. This involves a reduced stiffness of the wing. During the cruise of an airliner the wing can bend upwards, and due to positive sweep angle, a reduction of the effective angle Marek Behr · Gero Schieffer CATS, Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen e-mail: [email protected] Josef Ballmann · Frank Dieter Bramkamp · Saurya Ray Lehr- und Forschungsgebiet f¨ur Mechanik, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen e-mail: [email protected]

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 25–51. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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of attack is the consequence. By this mechanism, the generated lift decreases while the drag increases. Therefore, it is desirable for aircraft manufacturers to account for these deformations already during the design process. One particular challenging problem in the design of modern high-capacity aircraft is the high-lift configuration of the wing. In high-lift aerodynamics, a multitude of different flow phenomena is present which has to be resolved properly by the flow solver. The performance of high-lift configurations depends crucially on the riggings of the slat and the flap. As has been shown by Balaji et al. [3], changes in the slat and flap gaps as well as the changes in the deviations of both components strongly influence the generated lift and drag. Since in modern aircraft the wings are designed following the guidelines of lightweight construction, the riggings of flap and slat can change due to the deformation of the wing. In particular the flap might also exhibit deformations caused by its more slender construction. Experiments concerning elastic high-lift configurations are very expensive, and therefore, it is desirable to investigate these effects employing numerical methods. For the solution of the problems described above, an accurate computation of the forces generated by the flow around the wing is necessary. In the present study, compressible inviscid and viscous fluid flows modeled by the Euler and ReynoldsAveraged Navier-Stokes (RANS) equations are considered, respectively. Grid resolution remains still one of the most important factors in achieving accurate and reliable solutions of the RANS equations. The grid has to be fine enough to resolve all the relevant flow physics and to ensure a small remaining discretisation error. But fully grid-converged RANS solutions are not feasible in an industrial environment and a globally refined mesh would result in very high computational costs. To deal with these differing demands, a new adaptive solver QUADFLOW has been developed within the framework of the collaborative research center SFB 401 “Modulation of Flow and Fluid-Structure Interaction at Airplane Wings” [5]. The solver QUADFLOW is based on an integral concept and consists of a flow solver, a grid generation tool using parametric mapping based on B-splines and local grid adaptation based on multiscale analysis. For a detailed description of the general concept, please refer to [10, 5]. The flow solver in QUADFLOW is based on a surface oriented finite volume discretisation suited for arbitrary grid topologies. The grid generation and the grid adaptation are described in the articles by Brakhage et al. and Dahmen et al. in this volume, respectively. To enable QUADFLOW to solve the aeroelastic problems described above, the partitioned approach proposed by Braun [11] has been employed. Here, a flow solver, a structural solver and a grid deformer are called iteratively to solve the Fluid-Structure Interaction (FSI) threefield problem. For a detailed description of the used concept, the reader is referred to the article by Reimer et al. in this volume. The work presented in this article is concerned with the development and validation of QUADFLOW. The structure of the paper is as follows. In Chap. 2, the formulation of the Euler- and Navier-Stokes equations as well as turbulence models and transition models used throughout this work are described. In Chap. 3 the flow solver QUADFLOW and the FSI solution method are outlined.

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

27

In Chap. 4, QUADFLOW and the employed FSI method are validated against well-known test cases. Beside the purpose of validation a newly developed implicit time integration method offers a big potential to reduce the computational run time of QUADFLOW.

2 Physical Model 2.1 Governing Equations In the present study, compressible viscous fluid flow is described by the NavierStokes equations for a perfect gas. Neglecting viscous effects and the source term, the Euler equations are obtained. The conservation laws for any control volume V with boundary ∂ V and outward unit normal vector n on the surface element dS ⊂ ∂ V can be written in Arbitrary Lagrangian-Eulerian (ALE) formulation as

∂ ∂t

u dV + V (t)

∂ V (t)

Fc (u) − Fd (u) n dS = 0.

(1)

Here, u = (ρ , ρ vT , ρ etot )T denotes the vector of the unknown conserved quantities, and Fc and Fd represent the convective flux including pressure and the diffusive flux function, respectively. That is, ⎛ ⎛ ⎞ ⎞ 0 ρ vr (2) Fc = ⎝ ρ v ◦ vr + p I ⎠ , Fd = ⎝ Tv ⎠ , ρ etot vr + pv vTv − q where ρ denotes the density, p is the static pressure, v is the velocity vector of the fluid, and etot represents the total specific energy. The velocity relative to the mesh vr is defined as vr = v − x˙ where x˙ is the velocity of the faces of the control volume V (t). The symbol ◦ denotes the dyadic product, and I is the identity tensor. The viscous stress tensor Tv for an isotropic Newtonian fluid is defined as 2 Tv = μ grad v + (grad v)T − μ (div v) I . 3

(3)

Heat conduction is modeled by Fourier’s law q = −λ grad T , where the thermal conductivity is assumed as λ = c p μ /Pr, with Prandtl number Pr = 0.72. The molecular viscosity μ as a function of the temperature T is determined by the Sutherland formula. The static pressure is related to the specific internal energy according to the equation of state for a perfect gas p = ρ (γ − 1) etot − 1/2 |v|2 , where γ is the ratio of specific heats, which is 1.4 for air. High Reynolds number flows contain a large number of different scales. It is well-known, that the resolution of these scales for complex aerodynamic flows of engineering interest with Direct Numerical Simulation lies far beyond the capacity of current computer resources. To circumvent this problem it is a common practice

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to work with a time averaged formulation of the Navier-Stokes equations. For compressible fluid flows this is done using the Favre mass averaging procedure [44]. Due to this averaging, unknown correlations appear which have to be modeled.

2.2 Turbulence Models The turbulence modeling currently employed in aerodynamic flow simulations to close the Reynolds averaged Navier-Stokes equations is predominantly based on transport equations. Those turbulence models can be divided into two groups. Reynolds-stress transport models solve a transport equation for every component of the Reynolds stress tensor. They have a wide range of applicability but are considered a too complex approach for aerodynamic design work. Eddy viscosity models apply the Boussinesq hypothesis to compute the Reynolds stress tensor. This approximation assumes a linear dependency between the turbulent viscosity μt and the strain rate tensor Si j and is presented in a formulation suitable for compressible flows, following [44].

1 ∂ uk 2 δi j − ρ kδi j , (4) − ρ uiuj = 2 μt Si j − 3 ∂ xk 3 1 ∂ Ui ∂ U j Si j = ( + ). 2 ∂xj ∂ xi The turbulent viscosity and the turbulent kinetic energy k have to be provided by the eddy viscosity model. One- and two-equation models belong to the class of eddy viscosity models and are considered to be a good compromise between efficiency and accuracy for flows without or with only moderate separation. Turbulence models of this class are implemented in QUADFLOW and described in the following. The most popular one-equation model in external aerodynamics is the SpalartAllmaras (SA) model [38]. It was developed for aerodynamic applications and is especially suited for flows with attached boundary layers. Flows with boundary layer separation caused by an adverse pressure gradient are better described by two equation models. The k-ω model was one of the first two equation models. The implemented version is the one due to Wilcox [44]. The Local Linear Realizable (LLR) k-ω model includes realisability constraints and low-Reynolds damping terms to achieve a correct asymptotic near wall behaviour of the turbulent kinetic energy k [33]. Another turbulence model, developed by Rung, is the Linearised Explicit Algebraic Stress Model (LEA) k-ω model. It also includes realisability constraints. One of the most popular two-equation models is the SST (Shear Stress Transport) model of Menter [27]. It employs a blending function which converts the k-ω -model to a k-ε -model in the outer part of the boundary layer and in the free stream and thus combines the advantages of both models. The model equations are presented in Eq. (5).

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows D(ρ k) Dt D(ρω ) Dt

29

μ + σk μt ) ∂∂xk , ( j j

= ρ μγt P − β ρω 2 + ∂∂x ( μ + σω μt ) ∂xωj = P − β ∗ ρω k + ∂∂x

j

+ 2 (1 − F1) ρσω 2 ω1 ∂∂xk

k

∂ω ∂ xk

, (5)

P = −ρ ui uj ∂∂Ux ji ,

μt =

ρ ka1 max(a1 ω ;Ω F2 )

Ω =

2Ω i j Ω i j .

,

The set of model coefficients used by the k-ω - and by the k-ε -model are connected by the mixing function φ = F1 φ1 + (1 − F1) φ2 . Here F1 can take values between 0 and 1. The constants φ1 and φ2 represent one of the model coefficients in Table 1. Table 1 Model coefficients of the SST model

β

σk

σω

β∗

κ

1

0.075

0.85

0.5

0.09

0.41

β β∗

− σω √κ

2

2

0.0828

1.0

0.856

0.09

0.41

β β∗

− σω √κ

2

Set

γ β∗ β∗

Hellsten assumes that the linear dependency of the Reynolds stresses from the strain rate tensor as modeled by the Boussinesq hypothesis in Eq. (5) is a too restrictive assumption for high-lift aerodynamics. He proposed a new k-ω model especially developed for the flow around high-lift configurations [19]. The constitutive relation is based on an explicit algebraic Reynolds-stress model (EARSM). During the development of the new turbulence model Hellsten completely recalibrated Menter’s SST model to be used in conjunction with Wallinn’s EARSM [43]. The Reynolds stress tensor is expressed by a tensor polynomial of the strain rate tensor Si j and the vorticity tensor Ωi j . It seems to be generally accepted that Unsteady RANS (URANS) computations are not suited for the simulation of massively separated flows. Large Eddy Simulations (LES) on the other hand are considered a too expensive approach for industrial applications because of the grid resolution required for high-Reynolds-number flows. Hybrid RANS/LES methods have been developed to combine the advantages of both approaches. The presently most prominent of these approaches is the Detached Eddy Simulation (DES) proposed by Spalart et al. [39]. The DES model implemented in QUADFLOW is based on the SA model, see Eq. (6). The SA model can be written as follows:

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G. Schieffer et al.

˜∗ ∂ ρ ν˜ ∗ ∂ ρ ν˜ ∗ v j 1 ∂ ∗ ∂ν ˜ = QSA . + − ρ (ν + ν ) ∂t ∂xj σ ∂xj ∂xj

(6)

Here QSA denotes the source term, Production, QSA = ρ cb1(1 − ft2 )ω˜ ν˜ ∗ ∗ 2 ν˜ − ρ Cw1 fw − ccb1 Destruction, 2 ft2 d κ

∗ ∗ Diffusion, + σ1 ρ cb2 ∂∂νx˜ j ∂∂νx˜ j + ρ ft1 |v − vtr |2

(7)

Transition.

For the details of the SA-model we refer to the original publication [38]. By replacing the wall distance d in the destruction term in Eq. (7) by d˜ = min(d,CDES Δ ) with Δ = max(Δ x, Δ y, Δ z)

(8)

the DES model is obtained. In Eq. (8) Δ denotes the computational mesh size.

2.3 Transition Modeling The transition of laminar to turbulent flow remains of primary interest for an accurate prediction of the performance of aircraft in high-lift configuration. In a recent study performed by Moens et al. [28] the evaluation of the maximum lift of a high lift configuration could be significantly improved by the inclusion of transitional effects. Rumsey et al. [32] showed that the location of the transition point influences the boundary layer thickness, skin friction and wake profile shape. There have been several attempts to incorporate transitional effects into RANS solvers. One class of approaches applies turbulence models or turbulence/transition closure models. These models can be implemented into existing flow solvers without much effort, since they are based on transport equations. Wilcox used the k-ω turbulence model with low-Reynolds-number modifications to compute transition for an incompressible flow over a flat plate. Although his results were encouraging it is generally accepted that this approach lacks general applicability since the closure coefficients are taken constant [44]. Menter and Langtry [26] propose the use of transport equations in combination with experimental correlations. Their transition model has been calibrated for the use with the SST turbulence model and is based on two transport equations for the intermittency γ and the transition onset Reynolds number Reθ . It requires only local variables and is therefore especially suited for the incorporation into parallelized CFD solvers operating on unstructured grids. The physics of the transition process is not modeled by the proposed transport equations but entirely contained in experimental correlations. This γ -Reθ -model has been implemented into QUADFLOW by Krause in the framework of the GRK 1095 [16]. Two different sets of correlations are available in QUADFLOW. The first one is the

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

31

set generated by Menter et al. [26] which has been not yet completely published. The second set has been created by Krause on the basis of Menter’s ansatz to gain a transition model suited for the computation of hypersonic flows [23]. A second class of approaches is based on linear or nonlinear stability theory to model the physics of natural transition. Arnal and Casalis present a survey of transition prediction methods based on stability theory [2]. The most famous of these methods is the so called en -method, first developed by Smith and Gamberoni [37] and by van Ingen [40]. The en -method monitors the amplification of small disturbances, the so called Tollmien-Schlichting waves, as they propagate downstream. Once this amplification exceeds a critical value, the transition is assumed. To reduce the computational cost of en -methods, simplified stability methods have been investigated. One possibility is the use of analytical transition criteria as followed by Cliquet et al. [12] in a recent work. They presented a new approach using a combination of previously proposed transition criteria, the Arnal-Habiballah-Delcourt criterion (AHD) and the Gleyzes-Habiballah criterion (GH). In a first step the AHD part of this method has been implemented in QUADFLOW as an alternative to the γ -Reθ -model. This criterion is proposed as follows: Reθ T − Reθ cr = −106 exp(25.7λ¯ 2T )[ln(16.8Tu) − 2.77λ¯ 2T ] , 52 − 14.8 , Reθ cr f = exp Hi s 1 λ¯ 2 = λ2 dx , s − scr scr θ 2 dUe λ2 = , νe dx Ue θ Reθ = . νe

(9)

Tu denotes the free-stream turbulence intensity. Reθ cr and Reθ T are Reynoldsnumbers based on the momentum thickness θ at the critical point and the transition point. scr and Reθ cr are obtained when Reθ = Reθ cr f . The subscripts ” T ” and ” e ” correspond to values taken at the transition point and the boundary layer edge, respectively. The computation of Reθ cr f is very sensitive to the value of the shape parameter Hi . Arnal et al. expressed Hi as a function of the Pohlhausen parameter λ2 to solve this problem: Hi = 4.02923 − −8838.4λ24 + 1105.1λ23 − 67.962λ22 + 17.574λ2 + 2.0593.

(10)

The boundary layer thickness δ is searched along wall normal lines starting from the wall. Once the relation of shear stress and the maximum shear stress along the line decreases below a prescribed value, the edge of the boundary layer is found. One problem when using velocity profiles from RANS computation is the overestimation of the momentum thickness [12]. The other problem appears when the

32

G. Schieffer et al.

intermittency is changed at the transition point from zero to one in one step the computation might become unstable. To deal with both problems Cliquet decided to use the following parabolic intermittency function: Reθ − Reθ T 2 1 γ= . 0.152 Reθ T

(11)

It should be remembered that Eq. (11) does not try to model the physics of the transitional region. The AHD-GH criterion can be used with every eddy viscosity model.

2.4 Boundary Conditions To complete the problem formulation, initial values u (x,t0 ) = u0 (x), x ∈ V and boundary conditions u (x,t)|∂ V = B (x,t), x ∈ ∂ V are to be prescribed. For inviscid flows the kinematic condition vr · n = 0 , vr = v − vwall

(12)

has to be fulfilled at impermeable walls. In case of viscous flows the no-slip condition requires v = vwall . In QUADFLOW, isothermal walls are considered and the temperature at the wall is prescribed. At solid walls turbulent quantities as μt , ν˜ ∗ and k are zero. The same applies for the additional variables introduced by the γ -Reθ -model. The only difference is the value for the specific dissipation rate ω which is prescribed according to boundary conditions proposed by Menter [27] and Wilcox [44]. For the inviscid part of the governing equations, far-field boundary conditions are imposed by employing the theory of characteristics. For the solution of the Navier-Stokes equations, additional Neumann conditions for the flow gradients are required. For further details please refer to Bramkamp [8].

3 Numerical Methods 3.1 Finite Volume Scheme The flow computations in this study are performed using the finite volume solver QUADFLOW, which solves optionally the Euler- and Navier-Stokes equations for compressible fluid flow in two and three space dimensions. Quadrilateral and hexahedral meshes are preferred since they are widely accepted to facilitate best boundary fitted meshes for viscous fluid flow. A key idea is to represent such meshes with as few parameters as possible while successive refinements can be efficiently computed based on the knowledge of these parameters. This concept is embedded in a multiblock framework. The mesh in each block results from evaluating a parametric mapping from the computational domain into the physical domain. Such mappings are based on B-Spline representations. The mesh is locally adapted to

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

33

the solution according to the concept of h-adaptation. The adaptation strategy gives rise to locally-refined meshes of quadtree or octree type. A key role is played by reliable and efficient refinement strategies. In QUADFLOW, the adaptation criteria are based on recent multiresolution techniques [10]. Finally, a finite volume scheme that meets the requirements of the adaptation technique completes the concept. It is designed in a cell surface oriented manner to cope with fairly general cell partitions and allows, in particular, to handle hanging nodes in a unified fashion. The locally adaptive mesh is treated as fully unstructured and composed of simply connected elements with otherwise arbitrary topology. 3.1.1

Discretisation of Inviscid Fluxes

The discretisation of the inviscid fluxes is based on upwind methods. In the present study, the robust flux-vector splitting proposed by H¨anel and Schwane [18] is employed for solving the Euler equations. For solving the Navier-Stokes equations, flux-vector splitting methods are too diffusive. In this case we employ the HLLC flux-difference splitting according to Batten et al. [6] and the AUSMDV scheme proposed by Wada and Liou [42]. The higher-order extension of the scheme is crucial to obtain accurate solutions of the governing equations. To obtain second-order accuracy in space, a linear reconstruction of the primitive flow variables w ∈ {ρ , v, p} is defined as follows. w (x)|Vi := wi + φi (x − xi )T · ∇wi , x ∈ Vi .

(13)

Here, wi represents the solution at the centroid xi of Vi , and φi denotes a limiter function, with φi ∈ [0, 1]. To approximate the gradient ∇wi of the quantity in question, either a least-squares technique or the Green-Gauss method may be employed. At local extrema and discontinuities the reconstruction polynomial may generate new extrema and therefore cause oscillations in the numerical solution. In order to circumvent this phenomenon, the slope limiter by Venkatakrishnan [41] is employed. 3.1.2

Discretisation of Diffusive Fluxes

For the discretisation of the diffusive fluxes, the gradients of the velocity vector, ∇vi , and the temperature, ∇T , are required at the cell interfaces. The simplest procedure is to compute the gradients of the quantity in question, ∇w, within each cell and then average ∇w between the two cells that share a face on its left hand side (∇wL ) and on its right hand side (∇wR ), respectively. That is, the gradient ∇w at the cell interface is computed by ∇w| f ace =

1 (∇wL + ∇wR ) . 2

(14)

The gradients ∇wL and ∇wR are provided by the (unlimited) reconstruction procedure. This kind of discretisation supports undamped oscillatory modes that result from an odd-even point decoupling. A tighter coupling of the solution is obtained

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by approximating the gradient in the direction lLR = xR − xL , which connects the centroids of the left and right cell of the face by the divided difference wR − wL ∂ w . (15) = ∂ lLR f ace |lLR | Finally, the gradient is expressed by combining Eq. (14) and (15), i.e.,

wR − wL lLR lLR ∇w| f ace = ∇w − , − ∇w · f ace f ace |lLR | |lLR | |lLR | where ∇w is the averaged gradient according to Eq. (14).

(16)

f ace

3.1.3

Implicit Time Integration

After applying the spatial discretisation, we obtain a system of ordinary differential equations d dt

u dV + R(u) = 0 ,

(17)

V (t)

where R (u) denotes the residual vector defined by the sum of the discretised fluxes. The time integration in QUADFLOW relies beside an explicit Runge-Kutta scheme on an implicit two parameter family scheme, which can be used for steady and unsteady flow simulations. The resulting system of nonlinear equations is expressed as follows. := (1 + φ )un+1 − un + φ un−1 V + ϑ R(un+1 ) + (1 − ϑ )R(un ) = 0 . (18) R 1−φ Δt Table 2 summarises the different possible combinations of the parameters φ and ϑ . Table 2 Coefficients of implicit two-parameter family time integration scheme Type

Accuracy

φ

ϑ

Implicit Euler

O(Δ t)

0

1

Backward Difference (BDF)

O(Δ t 2 )

1/2

1

Trapezoidal

O(Δ t 2 )

0

1/2

The solution un+1 of the nonlinear system Eq. (18) is determined by a Newton iteration within each physical time step:

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

35

() ) , J(u() )Δ u() = −R(u

(19)

lim u() = un+1 .

(20)

with →∞

Here, Δ u() := u(+1) − u() denotes the change of the solution within each Newton step, indicated by the superscript (). The Jacobian of the system of equations, J(u() ), contains contributions of the temporal discretisation and the spatial discretisation, i.e., () ) ∂ R(u J(u() ) = = Jt (u() ) + J(u() ) , ∂ u()

(21)

with Jt (u() ) =

|V | I Δt

and J(u() ) =

∂ R(u() ) . ∂ u()

(22)

The initial guess is u(0) = un . For stationary flows, one Newton iteration to solve Eq. (19) is sufficient since convergence to steady state is enforced by the nonlinear (time) iteration. In this case, the Newton scheme for the implicit Euler time integration reduces to |V | ∂ R(u() ) (23) · Δ u() = −R(u() ) . + Δt ∂ u() Note that, for time step sizes approaching infinity, this results in a pure Newton scheme. To enhance convergence to steady state, a local time step within each cell is chosen with a constant CFL number in the domain. Implicit time integration schemes, based on Newton’s method, require the solution of the linear system of equations according to Eq. (19). The linear system is solved by a Krylov subspace method. The linearisation of the higher order accurate flux function R = Rhigh is usually prohibitive due to memory limitations. To reduce the number of non-zero entries in the Jacobian matrix J only an approximation to the exact Jacobian Jhigh is computed. Neglecting the reconstruction procedure and the enforcement of monotonicity, yields a first-order-accurate flux function Rlow . To linearise the flux, the automatic differentiation tool ADIFOR [30] is employed in QUADFLOW to compute the first order accurate approximation to the exact Jacobian Jlow . This procedure is described in detail in [9]. Employing Rhigh on the right hand side and Jlow on the left hand side of Eq. (19) yields an approximate Newton method

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high (u() ) . Jlow (u() )Δ u() = −R

(24)

The linear system is solved using Krylov subspace methods like GMRES or BiCGSTAB. To enhance the convergence of these methods, left preconditioning is employed. The preconditioning matrix is based on an incomplete LU factorisation ILU(p) of the Jacobian Jlow where p is the level of fill-in. The implementation of the Newton-Krylov method is based on the PETSc [4] library developed at Argonne National Library. The use of an approximate Newton method leads to a more robust and faster convergence at the early stages of the computation. But at the final stages of the computation, exact derivative information is required in order to achieve quadratic convergence of Newton’s method.

3.2 Matrix-Free Newton-Krylov Method Newton-Krylov methods require only the product of the Jacobian matrix with a given vector, rather than explicit access to the elements of the Jacobian. Krylov subspace methods do not require the system matrix J(u) in an explicit form but only the product of J(u) with a Krylov subspace vector v. Such Newton methods are called Jacobian-free Newton-Krylov methods. For a recent survey on Jacobianfree Newton-Krylov methods we refer to [21]. These Jacobian-vector products can be approximated by divided differences [17]. The divided differencing method has the conceptual disadvantage that it involves the choice of a suitable step size that is typically not known a priori. An alternative is provided by a technique called automatic differentiation, which allows the computation of derivatives without truncation error. In this work we employ automatic differentiation to obtain the matrix-free evaluation of Jacobianvector products in an efficient and accurate fashion. The second order accurate flux high is linearised which yields according to Eq. (19) the following system of linear R equations: high (u() ) . Jhigh (u() )Δ u() = −R

(25)

The matrix-free Newton-Krylov method represents an exact Newton method. Since the use of the exact Newton method in the early stages of the computation can slow down convergence, a hybrid scheme is proposed. The approximate Newton method is applied to generate a solution from which the exact Newton method can converge. The switch occurs when the relative residual of the density has decreased below a prescribed threshold. For simplicity, in the following the method employing the approximate Jacobians is called “first-order method”, whereas the exact Newton method is called “second-order method”. For further information on this method please refer to the articles by Pollul and Reusken and by Bischof et al. in this volume.

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

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3.3 Fluid Structure Interaction For the solution of aeroelastic problems a partitioned approach is followed which has been developed in the framework of the collaborative research center SFB 401, see Braun [11] and the article by Reimer et al. in this volume. Here, separate programs are called iteratively to solve the three-field problem. This problem consists of the structural deformation and its interaction with the potentially unsteady flow field which is to be computed on a deforming grid. The main difficulty of partitioned approaches lies in the coupling of the different solvers. The temporal and the spatial coupling problem have to be solved. The spatial coupling has to assure that the loads computed by the flow solver are transferred in an energy-conserving way to the grid of the structural solver. The same has to be valid for the transfer of the computed deformations to the grid of the flow solver. In this work two different grid deformers, depending on the format of the grid, are applied. Grids in POPINDA format are deformed by a tool developed by Boucke[7] and Hesse [20]. Here, a very coarse grid is extracted from the grid of the flow solver. This coarse grid is considered as a system of massless beams. It is assumed that these beams are clamped to the elastic structure and welded at the crossing points. The movement of the structure is transported into the frame of beams. By means of algebraic interpolation the positions of the remaining nodes are determined. Grids in B-Spline format are deformed by a tool developed in the framework of the collaborative research center 401, see the article by Brakhage et al. in this volume. At first, the displacements of the multiblock topology are interpolated from the surface deformation with radial basis functions. Afterwards the B-spline control points within each block are moved by means of transfinite interpolation. The temporal coupling controls the calls to the different solvers. Two different families of the temporal coupling are available, loose and strong coupling. For stationary solutions, the loose coupling method, a predictor-corrector scheme, is chosen. For nonstationary solutions, the strong coupling is preferred, where in a fixed-point iteration the flow solver and the structural solver are called iteratively several times before proceeding to the next time step. The tasks described above are solved using the Aeroelastic Coupling Module (ACM) developed by Braun [11]. In our work the flow solver QUADFLOW and the in house structural solver FEAFA are applied. For a more detailed description of the partitioned field approach please see the article by Reimer et al. in this volume.

4 Results In this chapter, results of numerical simulations are presented. The first two test cases have been defined to demonstrate the advantages of the chosen adaptive concept based on multiscale analysis of the full set of field quantities. One case represents shocked, transonic flow and the other boundary layer flow. Further results of the validation of QUADFLOW and the FSI solver are presented in the remaining part of the chapter.

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Simulations for turbulent flows have been computed for several well-known test cases like flow around the RAE2822 airfoil [13] and around the ONERA M6 wing [35] to validate the implemented turbulence models. The agreement between the predicted results and the experimental data was good. For the details of this validation study please refer to Schieffer et al. [34].

4.1 Fishtail Here, the transonic, inviscid flow over the NACA0012 airfoil is considered which has been specified as a AGARD reference test case 03 [1]. The free-stream conditions are Ma = 0.95 and angle of attack α = 0◦ . This configuration is well-known for the complicated shock pattern developing behind the trailing edge of the airfoil. Two oblique shocks are formed at the trailing edge. The remaining supersonic region behind the oblique shock is closed by a normal shock farther downstream. Because of this shock pattern the configuration is often referred to as fishtail. The computation has been performed on an adaptive grid, where the initial grid consists of 4 blocks where each block contains 20 × 20 cells. 8 cycles of adaptation have been performed with a maximum refinement level of Lmax = 8. Adaptation has been based on multiscale analysis of the set of primitive variables. On each grid, the solution has been integrated in time until the residual has been decreased by two orders of magnitude. On the final grid the residual is decreased by four orders of magnitude. Figure 1 shows computational grids and the corresponding Mach number distributions in the vicinity of the airfoil for the initial stage. Figure 2 presents a view of a large part of the computational grid after 8 adaptations and the corresponding Mach number distribution. The oblique shocks extend about 10 to 12 chord lengths into the flow domain. The position of the normal shock is located approximately 2.2 chord lengths behind the trailing edge of the airfoil. The adaptive grid contains 98320 cells and provides high resolution over the complete extent of the shocks. The discretisation of the shock region between x ∈ [1, 5],

Fig. 1 Transonic flow over the NACA0012 airfoil. Left Figure: Details of initial computational grid. Right Figure: Distribution of the Mach number

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

39

Fig. 2 Transonic flow over the NACA0012 airfoil. Left Figure: Computational grid after 8 adaptations. Right Figure: Distribution of the Mach number

y ∈ [−10, 10] using a uniform structured mesh according to a refinement level L = 8 would require about 29.5 × 106 grid cells. A uniform discretisation of the complete flow domain according to L = 8 would result in about 108 cells.

4.2 Laminar Flat Plate In the scope of this section, we apply anisotropic adaptation to resolve the boundary layer along a flat plate. The flow conditions are determined by the Mach number M∞ = 0.2 and the Reynolds number Re∞ = 104 , based on unit length. The wall is considered as isothermal, with Twall = T∞ = 273.0K. For purpose of validation, the similarity solution according to Blasius for an incompressible laminar fluid flow serves as a reference. The plate extends along the x-axis between x = 0.0 and x = 6.0, with 100 cells located on the plate itself, see left part of Fig. 3 for a partial view of the initial grid. Upstream of the leading edge, the lower boundary of the domain is modeled as an inviscid impermeable wall. The grid is clustered about the leading edge in stream wise direction, measuring a first grid spacing of 10−3 . In the y-direction, the initial resolution comprises only 10 grid points. In the following, 9 cycles of

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Fig. 3 Laminar subsonic flow over flat plate. Left Figure: Initial computational grid. Right Figure: Computational grid after 8 adaptations

cf 0.02 Blasius solution QUADFLOW

0.01

0 0

5000

10000

15000

20000

Re

Fig. 4 Distribution of friction coefficient for laminar subsonic flow over flat plate

adaptation in the y-coordinate direction are performed, with a highest refinement level permitted of Lmax = 8. The locally adapted grid after 9 cycles of adaptation is depicted in the right part of Fig. 3. The highest refinement level reached during the computation is L = 7, which is located in the vicinity of the leading edge of the plate. This fact indicates that even in the presence of steep gradients within the boundary layer, the multiscale analysis converges within the range of the prescribed threshold value. That is., the adaptation procedure was not truncated by reaching the maximum permissible refinement level. Figure 4 shows a good agreement between the computed skin friction coefficient along the plate after 9 adaptations and the theoretical solution, according to Blasius.

4.3 Transitional Flow over a Flat Plate To validate the different prediction methods for laminar-turbulent transition, a fundamental test case has been considered. The Schubauer-Klebanoff experiment, a viscous subsonic flow over a flat plate [36], has been simulated using QUADFLOW. The free-stream turbulence intensity Tu is low and corresponds to natural transition. The flow parameters are Ma = 0.14, Re = 3.3 × 106 and Tu = 0.18%. The nonadaptive block-structured computational grid consists of two blocks with 64 cells

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

41

Fig. 5 Predicted distribution of friction coefficient compared to measurements for the transitional, subsonic flow over flat plate. Applied transition models: Left Figure: AHD-GH criteron and γ -Reθ -model using correlations proposed by Krause. Right Figure: AHD-GH criterion and γ -Reθ -model using correlations proposed by Menter.

each in the direction normal to the wall. In flow direction, the domain is discretised with 200 and 100 cells, respectively, in the block containing the plate and the block upstream of the plate. The distance of the first cell off the wall is 10−5 of the chord length which assures a dimensionless wall distance of y+ < 1 over the plate. The prediction of transition has been performed applying the AHD-GH-criterion together with the SA-model. The computation has also been conducted using the γ -Reθ -model with both sets according to Menter [26] and Krause [23]. Figure 5 compares the computed friction coefficient to the one measured in the experiment as a function of the Reynolds number. It can be seen that all applied transition models predict the point where the flow changes from laminar to turbulent a bit downstream, compared to the point observed in the experiment. The length of the transitional region is underpredicted by all transition models. The computation employing the AHD-GH-criterion predicts higher values for the skin friction coefficient in the fully turbulent region.

4.4 High-Lift Configuration Here, the simulation of the subsonic, compressible viscous flow around the AGARD A-2 L1/T2 high-lift configuration is described. Experimental data for this test case has been published by Moir [29]. A fully-turbulent flow has been assumed with turbulence intensity in the free stream of Tu = 0.5%. Flow parameters are Ma = 0.197, Re = 3.52 · 106 and 0◦ < α < 23◦ . The experimental data consists of lift coefficients and the drag coefficients measured over the whole range of the angles of attack. At two different angles of attack, α = 4.01◦ and α = 20.18◦ , also the pressure distribution over the high-lift configuration is available. The computations for this test case have been conducted on an adaptive grid. 6 cycles of adaptation have been carried out with a maximum refinement level Lmax = 6. On each grid the solution has been integrated in time until the relative residual of

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Fig. 6 Subsonic flow around the AGARD A-2 L1/T2 high-lift configuration at α = 20.18◦ . Left Figure: Computational grid after 6 adaptations. Right Figure: Details

Fig. 7 Comparison of predicted and measured distribution of pressure coefficient for the flow around the AGARD A-2 L1/T2 high-lift configuration at two different angles of attack: Left Figure: α = 4.01◦ . Right Figure: α = 20.18◦

the density has been decreased about five orders of magnitude. The grid extends 25 chord lengths away from the configuration. The grid created by a simulation for the flow at α = 20.18◦ after six adaptations is displayed in the left part of Fig. 6. Details in the vicinity of the airfoil are shown in the right part of Fig. 6. It can be seen that most cells are inserted on the suction side of the configuration and in the wake. The convective fluxes are discretised by the AUSMDV(P) flux splitting scheme [15], and for the closure of the RANS equations the SA-model has been applied. Figure 7 shows a comparison of the computed and measured distribution of the pressure coefficient for the two angles of attack mentioned above. The predicted solutions match the experimental data very closely. It can be noticed from Fig. 7 that the suction peak on the main wing is underpredicted. This might be caused by neglecting the laminar-turbulent transition.

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

43

4.5 4

0.6

DES SA Original Experiment

3.5

DES SA Original Experiment

0.4

3 CL

Cd

2.5 0.2 2 1.5 0

5

10

15

α

20

25

30

0

0

5

10

15

α

20

25

30

Fig. 8 Comparison of computed and experimental lift coefficient of the AGARD A-2 L1/T2 high-lift configuration, (M∞ = 0.197, 0◦ < α < 24◦ ). Left Figure: Lift coefficient as function of angle of attack. Right Figure: Drag coefficient as function of angle of attack

Fig. 9 Mach number distribution of the flow around the AGARD A-2 L1/T2 high-lift configuration at α = 23.9◦

As has been noticed by Ying et al. [45], unsteady effects are present in the flow around high-lift configurations at the maximum-lift angle of attack. Therefore numerical simulations for this test case have been conducted applying DES in a timeaccurate way. Figure 8 shows a comparison between predicted and measured values of the lift and the drag coefficients for the whole range of angles of attack. An excellent agreement between computational and experimental data is achieved. The maximum lift coefficient is predicted at α = 22◦ which is close to the angle of attack α = 21.5◦ where maximum lift occurred in the experiment. At α > 22◦ the flow over the main element separates and causes lift break-down see Fig. 9.

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4.5 BAC 3-11 Airfoil - Shock Buffet In this section simulations to capture the shock buffet phenomenon are described. Measurement data were made available from pre-tests to the HIRENASD project in the G¨ottingen KRG to check the pressure measurement technique provided for the HIRENASD project under cryogenic conditions. In these tests, the behaviour of the BAC 3-11/RES/30/21 airfoil in cruise configuration was investigated in cryogenic transonic flow. At higher angles of attack periodic oscillation of the lift coefficient has been observed [22]. These oscillations were caused by an interaction between the shock on the upper surface and the boundary layer with flow separation. The flow parameters in the experiment considered here are Ma = 0.75, α = 4.0◦ and Re = 4.6 · 106 . The simulations are performed on a B-Spline grid with three different resolutions but without using adaptation. Here, only the results computed on the finest grid are presented. The flow field contains 24576 cells where the airfoil is dicretised with 640 cells. In wall normal direction 32 cells are located. The computation has been performed in a time accurate manner applying DES from the start of the computation. The predicted oscillation of the lift coefficient over time is displayed in Fig. 10. On the right hand side of Fig. 10 the distribution of the pressure coefficient over the chord length is shown. It can be concluded, that the oscillation of the lift coefficient is caused by the unsteady movement of the shock and flow separation behind the shock. The frequency of the shock movement is estimated to be 140Hz compared to the value 125Hz measured in the experiment.

4.6 FSI - HIRENASD In this section the simulation of the viscous, transonic flow around the elastic HIRENASD (High Reynolds Number Aero- Structural Dynamics) configuration is

0.8

2

1.5

0.6

1

CL - Cp

0.5

0.4

0

-0.5

0.2

-1

0

0.02

0.04

0.06

Physical time

0.08

0.1

-1.5

0

0.2

0.4

0.6

0.8

1

X

Fig. 10 Transonic, viscous flow over the BAC 3-11 airfoil. Left Figure: Variation of the lift coefficient as function of time. Right Figure: Movement of the shock

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

45

described. This configuration has been developed and built in the framework of the collaborative research center 401, see also the article by Ballmann et al. in this volume. The aim of the HIRENASD project is to gain new insight in the physics of transonic aero-structural dynamics at realistic Mach- and Reynolds numbers and to obtain experimental data to validate aeroelastic solvers. The experimental investigation of the HIRENASD configuration has been conducted in the ETW (European Transonic Wind Tunnel) under cryogenic conditions. An extensive computational investigation has been conducted using the structured flow solver FLOWer, which has been developed at DLR [24]. For further informations please see the article by Reimer et al. in this volume. To validate the aeroelastic coupling described in section 3.3, the computation has been performed on the same grid with QUADFLOW as flow solver in the nonadaptive mode. The parameters of the simulated experiment are Ma = 0.80, Re = 35 × 106 , T∞ = 136.5K and a loading factor Eq = 0.48 · 10−6. The angle of attack varies between −1◦ < α < 5◦ . The computation was performed on a non-adaptive, multiblock structured grid consisting of 46 blocks and 3.5 million cells. The grid on the surface of the configuration as well as the corresponding pressure distribution are depicted in Fig. 11. The convective fluxes have been discretised by the AUSMDV scheme, turbulence closure has been achieved by applying the SA model. The wing structure is modelled by Timoshenko beam elements. Figure 12 shows the computed deformation and the aeroelastic twist for three different angles of attack α = −1◦, 2◦ , 5◦ . On the left hand side, the comparison of the computed deformations of the wing as function of span is depicted. The computations applying QUADFLOW predict higher deformations compared to the ones applying FLOWer. On the right hand side of Fig. 12 the aeroelastic twist as function of span is displayed. Again, computations using QUADFLOW yield higher aeroelastic twist compared to computations using FLOWer. QUADFLOW seems to overpredict the deformation as well as the aeroelastic twist, independent of the angle of attack.

Fig. 11 Transonic, viscous flow over HIRENASD wing without deformation. Left Figure: Computational grid on the wing, every 4th grid line is shown. Right Figure: Corresponding distribution of pressure coefficient

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Fig. 12 Comparison of computed deformation and aeroelastic twist of the elastic HIRENASD-wing for three different angles of attack. QUADFLOW and FLOWer have been employed for this in the SOFIA code. The SA-model and the LEA-model have been applied. Left Figure: Deformation of the wing as function of span. Right Figure: Aeroelastic twist of the wing as function of span

Fig. 13 Comparison of computed deformations and aeroelastic twists of the elastic HIRENASD-wing for three different angles of attack. QUADFLOW and TAU have been employed. The SA-model and the LEA-model have been applied. Left Figure: Deformation of the wing as function of span. Right Figure: Aeroelastic twist of the wing as function of span

To compare the results of QUADFLOW with results obtained using another unstructured flow solver, a further computation has been performed by L. Reimer using the solver TAU [25] developed at the DLR. The parameters are Ma = 0.83, Re = 23.5 × 106, T∞ = 180K and Eq = 0.48 · 10−6. The angle of attack is α = −1.0 Figure 13 shows an excellent agreement for the computed deformation and the aeroelastic twist between these two solvers.

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

47

4.7 Performance of the Matrix-Free Newton-Krylov Method In this section, the simulation of inviscid flow around a swept wing is considered to investigate the run-time performance of the approximate Newton method (“first order”) and the hybrid Newton method (“second order”) described in Sect. 3.2. The swept wing has constant chord length, the BAC 3-11 profile and a sweep angle ϕ = 34.0◦ , see Fig. 14. This configuration has been designed and experimentally investigated within the collaborative research center SFB 401 [31]. The flow parameters are α = 4.64◦ and M∞ = 0.22. The implicit Euler scheme is used for the time integration. The non-adaptive, multiblock structured grid consists of 425984 cells. The computation is performed in parallel using 4 MPI tasks. We compare the number of time steps and the execution time of the approximate method and the hybrid method. In the latter case the exact Newton method is when the relative residual of the density R has been decreased by two orders of magnitude. Since the startup phase is identical for both methods, we focus on the final phase of the computation, i.e., R < 10−2. Results for different combinations of constant CFL-numbers and values of fill-in p are presented in Table 3. The computation is considered to be converged when the relative residual R is less than 10−4.

Table 3 Comparison of first- and second-order methods: Number of time steps #ts and CPU time for different values for CFL and different PBILU(p) preconditioners, neglecting the startup phase first-order CFL 102

103

104

second-order

p

# ts

CPU [s]

# ts

CPU [s]

1

142

1488.2

164

3672.7

2

142

1583.4

92

3747.5

3

142

2136.6

92

4159.3

1

77

1575.1

12

1858.5

2

77

1614.7

12

1493.2

3

77

1904.5

12

1468.4

1

71

1902.9

5

674.1

2

71

1859.5

4

558.5

3

71

2097.5

4

620.5

From Table 3 it can be observed that, for large CFL numbers, increasing the value of p can lead to a reduction of the overall execution time. However, the corresponding increased memory requirements have to be considered. The number of time steps is significantly decreased for both, the first- and the second-order method, if the CFL number is increased. The corresponding execution times increase for the first-order

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-Cp: -0.5

0

0.5

1

1.5

2

2.5

3

3.5

Fig. 14 Subsonic flow over swept wing. Upper Figure: contour plot of pressure distribution. Lower Figure: details of the wing tip

method and decrease for the second-order method. For CFL = 104 , the second-order method is significantly faster than the first-order method. On the other hand, with CFL = 102 , the first-order method is faster. This effect is visualized in Fig. 15, where the convergence histories for the parameters CFL = 102 and CFL = 104 , with p = 1, are presented. Note that for small values of CFL, the higher cost for a single Krylov iteration slows down the second-order method, whereas for large CFL numbers the time integration benefits from the higher accuracy in the Newton step. In the latter case, i.e., CFL = 104 , the execution time of the second-order method is significantly less than in all first-order computations.

An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows

49

Fig. 15 Comparison of results from first- and matrix-free second-order formulations: Residual history in terms of CPU time, neglecting the startup phase

5 Conclusion In this paper, the development of the adaptive flow solver QUADFLOW from its inception until today is described. The general concept of QUADFLOW is introduced and it is shown to work for several test cases concerning viscous and inviscid compressible flows. The coupling of QUADFLOW to the structural solver FEAFA via the coupling module ACM is explained. The resulting aeroelastic solver has been validated for steady and unsteady test cases. The closure of the RANS equations is improved by the implementation of several two-equation models. Two ways of modelling transition are now available in the flow solver to improve the simulation of flows where laminar-turbulent transition occurs as e.g. in high-lift aerodynamics. The implementation of an implicit matrix-free Newton-Krylov method for solving the Euler- and Navier-Stokes equations is described. Numerical experiments for the simulation of inviscid and laminar viscous show a big potential in reducing the computational runtime. Future plans include the further enhancement of the method for the simulation of turbulent flows.

References 1. AGARD-AR-211. Test cases for inviscid flow field methods (1985) 2. Arnal, D., Casalis, G.: Laminar-turbulent transition prediction in three-dimensional flows. Progress in Aerospace Sciences 36, 173–191 (2000) 3. Balaji, R., Bramkamp, F., Hesse, M., Ballmann, J.: Effect of Flap and Slat Riggings on 2-D High-Lift Aerodynamics. Journal of Aircraft 43(5), 1259–1271 (2006) 4. Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B., Zhang, H.: PETSc Web page (2001), http://www.mcs.anl.gov/petsc

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5. Ballmann, J.: Flow Modulation and Fluid-Structure-Interaction at Airplane Wings. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 84. Springer, Berlin (2003) 6. Batten, P., Leschziner, M.A., Goldberg, U.C.: Average-State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows. Journal for Computational Physics 137(1), 38–78 (1997) 7. Boucke, A.: Kopplungswerkzeuge f¨ur aeroelastische Simulationen. PhD thesis, RWTH Aachen University (2003) 8. Bramkamp, F.D.: Unstructured h-Adaptive Finite-Volume Schemes for Compressible Viscous Fluid Flow. PhD thesis, RWTH Aachen University (2003) 9. Bramkamp, F.D., B¨ucker, H.M., Rasch, A.: Using Exact Jacobians in an Implicit Newton-Krylov Method. Computers & Fluids 35(10) (2006) 10. Bramkamp, F.D., Lamby, P., M¨uller, S.: An adaptive multiscale finite volume solver for unsteady and steady state flow computations. Journal for Computational Physics 197(2), 460–490 (2004) 11. Braun, C.: Ein modulares Verfahren f¨ur die numerische aeroelastische Analyse von Luftfahrzeugen. PhD thesis, RWTH Aachen University (2007) 12. Cliquet, J., Houdeville, R., Arnal, D.: Application of Laminar-Turbulent Transition Criteria in Navier-Stokes Computations. AIAA Journal 46(5), 1182–1190 (2008) 13. Cook, P.H., McDonald, M.A., Firmin, M.C.P.: Aerofoil RAE 2822 - Pressure Distributions and Boundary Layer and Wake Measurements. AGARD-AR-138 (1979) 14. Deck, S.: Zonal-Detached-Eddy Simulation of the Flow Around a High-Lift Configuration. AIAA Journal 43(11), 2372–2384 (2005) 15. Edwards, J., Liou, M.S.: Low-Diffusion Flux-Splitting Methods for Flows at All Speeds. AIAA Journal 36(9), 1610–1617 (1998) 16. Gaisbauer, U., Weigand, B., Reinartz, B.: Research Training Group GRK 1095/1: AeroThermodynamic Design of a Scramjet Propulsion Sysytem. In: Proceedings of 18th ISABE Conference (2007) 17. Grotowsky, I.M.G., Ballmann, J.: Efficient time integration of Navier-Stokes equations. Computers & Fluids 28, 243–263 (1999) 18. H¨anel, D., Schwane, R.: An Implicit Flux–Vector Splitting Scheme for the Computation of Viscous Hypersonic Flow, AIAA Paper 1989–0274 (1989) 19. Hellsten, A.: New Advanced k-ω Turbulence Model for High-Lift Aerodynamics. AIAA Journal 43(9), 1857–1969 (2005) 20. Hesse, M.: Entwicklung eines automatischen Gitterdeformationsalgorithmus zur Str¨omungsberechnung um komplexe Konfigurationen auf Hexaeder-Netzen. PhD thesis, RWTH Aachen University (2006) 21. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications. Journal for Computational Physics 193(2), 357–397 (2004) 22. Ballmann, J., Dafnis, A., Braun, C., Korsch, H., Reimerdes, H.G., Olivier, H.: High Reynolds Number Aerostructural Dynamics Experiments in the European Transonic Windtunnel (ETW). In: 25th International Congress of the Aeronautical Sciences (2006) 23. Krause, M., Behr, M.: Modelling of Tranition Effects in Hypersonic Intake Flows Using a Correlation-Based Intermittency Model. In: 15th AIAA International Space Planes and Hypersonic Systems and Technologies (2008) 24. Kroll, N., Fassbender, J.K.: MEGAFLOW — Numerical flow simulation for aircraft design. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 89. Springer, Heidelberg (2005)

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25. Kroll, N., Schwamborn, D., Becker, K., Rieger, H., Thiele, F.: MEGADESIGN and MegaOpt — German Initiatives for Aerodynamic Simulation and Optimization in Aircraft Design. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 107. Springer, Heidelberg (2009) 26. Menter, F.R., Langtry, R.B., Likki, S.R., Suzen, Y.B., Huang, P.G., V¨olker, S.: A Correlation-Based Transition Model Using Local Variables Part I - Model Formulation. In: Proceedings of ASME Turbo Expo 2004, Power for Land, Sea and Air (2004) 27. Menter, F.R.: Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA Journal 32(8), 1598–1604 (1994) 28. Moens, F., Perraud, J., Krumbein, A., Toulorge, T., Iannelli, P., Eliasson, P., Hanifi, A.: Transition Prediction and Impact on a Three-Dimensional High-Lift-Wing Configuration. Journal of Aircraft 45(5), 1751–1766 (2008) 29. Moir, I.R.M.: Measurements on a Two-Dimensional Aerofoil with High Lift Devices. AGARD-AR-303, vol. 1 and 2 (1994) 30. Rall, L.B.: Automatic Differentiation: Techniques and Applications. LNCS, vol. 120. Springer, Heidelberg (1981) 31. Reimer, L., Braun, C., Ballmann, J.: Aanalysis of the Static and Dynamic Aero-Structural Response of an Elastic Swept Wing Model by Direct Aeroelastic Simulation. In: Proceedings of the 25th International Congress of the Aeronautical Sciences, ICAS 2006, Hamburg, Germany (2006) 32. Rumsey, C.L., Gatski, T.B., Ying, S.X., Bertelrud, A.: Prediction of High-Lift Flows using Turbulent Closure Models. Technical Report: NASA-AIAA 97-2260 (1997) 33. Rung, T., Thiele, F.: Computational modelling of complex boundary layer flows. In: Proc. 9th Intern. Symposium on Transport Phenomena in Thermal-Fluid Engineering, Singapore, pp. 321–326 (1996) 34. Schieffer, G., Ballmann, J., Behr, M.: Validation of advanced turbulence models in QUADFLOW. Turbulence, Heat and Mass Transfer 6, Begell House Inc. (2009) 35. Schmitt, V., Charpin, F.: Pressure Distributions on the ONERA M6 Wing at Transonic Mach Numbers. AGARD-AR-138 (1979) 36. Schubauer, G.B., Klebanoff, P.S.: Contributions on the Mechanics of Boundary Layer Transition. NACA TN 3489 (1955) 37. Smith, A.M.O., Gamberoni, N.: Transition, pressure gradient and stability theory. Report ES 26388, Douglas Aircraft Co., El Segundo, California (1956) 38. Spalart, P.R., Allmaras, S.R.: A One-equation turbulence Model for Aerodynamic Flows. AIAA-Paper 92-0439 (1992) 39. Spalart, P.R., Jou, W.H., Strelets, M., Allmaras, S.R.: Comments on the Feasability of LES for Wings and on a Hybrid RANS/LES Approach. In: 1st AFSOR Int. Conf. on DNS/LES. Greyden Press, Columbus (1997) 40. van Ingen, J.L.: A suggested semi-empirical method for the calculation of boundary layer transition region. Report UTH-74, Univ. of Techn., Dept. of Aero. Eng., Delft (1956) 41. Venkatakrishnan, V.: Convergence to Steady State Solutions of the Euler Equations on Unstructured Grids with Limiters. Journal for Computational Physics 118(1), 120–130 (1995) 42. Wada, Y., Liou, M.S.: A Flux Splitting Scheme with High-Resolution and Robustness for Discontinuities. AIAA Paper 94-0083 (1994) 43. Wallin, S., Johansson, A.: An Explicit Algebraic Reynolds Stress Model for Incompressible and Compressible Turbulent Flows. Journal of Fluid Mechanics 403, 89–132 (2000) 44. Wilcox, D.C.: Turbulence Modeling for CFD. DCW Industries, Inc., La Ca˜nada (1993) 45. Ying, S.X., Spaid, F.W., McGinley, C.B., Rumsey, C.L.: Investigation of Confluent Boundary Layers in High-Lift Flows. AIAA Journal 36(3), 550–562 (1999)

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Timestep Control for Weakly Instationary Flows Christina Steiner and Sebastian Noelle

Abstract. We report on recent work on adaptive timestep control for weakly instationary gas flows [16, 17, 18] carried out within SFB 401, TPA3. The method which we implement and extend is a space-time splitting of adjoint error representations for target functionals due to S¬ uli [19] and Hartmann [10]. In this paper, we first review the method for scalar, 1D, conservation laws. We design a test problem for weakly instationary solutions and show numerical experiments which clearly show the possible benefits of the method. Then we extend the approach to the 2D Euler equations of gas dynamics. New ingredients are (i) a conservative formulation of the adjoint problem which makes its solution robust and efficient, (ii) the derivation of boundary conditions for this new formulation of the adjoint problem and (iii) the coupling of the adaptive time-stepping with the multiscale spatial adaptation due to M¨uller [3, 12], also developed within SFB 401. The combined space-time adaptive method provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow, and becomes small when a perturbation enters the flow field. The efficiency of the Euler solver is investigated by means of an unsteady inviscid 2D flow over a bump.

1 Introduction For the aeroelastic problems studied in the SFB 401 project both the stationary and the instationary case are of interest. In either case, adaptive spatial grids are wellestablished and help to reduce computational time and storage. There has been a tremendous amount of research designing, analyzing and implementing codes which are adaptive in space, see e.g. [4, 11, 12, 13] and references therein. Christina Steiner · Sebastian Noelle Institut f¨ur Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany e-mail: {steiner,noelle}@igpm.rwth-aachen.de

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 53–75. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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Adaptive time-marching towards the stationary solution is an essential ingredient of most CFD codes. Here the timestep-sizes are adapted to the local CFL numbers. So far, it is not well-understood if adaptive timesteps might be useful for instationary flows. In the present paper we report on the work of SFB 401’s project TPA3b, where we explored adaptive explicit/implicit timesteps for weakly instationary flows. While time accuracy is still needed to study phenomena like aero-elastic interactions, large timesteps may be possible when the perturbations have passed. For explicit calculations of instationary solutions to hyperbolic conservation laws, the timestep is dictated by the CFL condition due to Courant, Friedrichs and Lewy [5], which requires that the numerical speed of propagation should be at least as large as the physical one. For implicit schemes, the CFL condition does not provide a restriction, since the numerical speed of propagation is infinite. Depending on the equations and the scheme, restrictions may come in via the stiffness of the resulting nonlinear problem. These restrictions are usually not as strict as in the explicit case, where the CFL number should be below unity. For implicit calculations, CFL numbers of much larger than 1 may well be possible. Therefore, it is a serious question how large the timestep, i.e. the CFL number, should be chosen. Some time adaptation stategies developed by other authors are those of Ferm and L¨otstedt [9] based on timestep control strategies for ODEs, and extended to fully adaptive multiresolution finite volume schemes, see [7, 6]. Alternatively Kr¨oner and Ohlberger [11, 13] based their space-time adaptivity upon Kuznetsov-type a-posteriori L1 error-estimates for scalar conservation laws. In this paper we will use a space-time-split adjoint error representation to control the timestep adaptation. For this purpose, let us briefly summarize the space-time splitting of the adjoint error representation, see [8, 1, 2, 19, 10] for details. The error representation expresses the error in a target functional as a scalar product of the finite element residual with the dual solution. This error representation is decomposed into separate spatial and temporal components. The spatial part will decrease under refinement of the spatial grid, and the temporal part under refinement of the timestep. Technically, this decomposition is achieved by inserting an additional projection. Usually, in the error representation, one subtracts from the dual solution its projection onto space-time polynomials. Now, we also insert the projection of the dual solution onto polynomials in time having values which are H 1 functions with respect to space. This splitting can be used to develop a strategy for a local choice of timestep. In contrast to the results reported in [19, 10] for scalar conservation laws we now investigate weakly instationary solution to the 2D Euler equations. The timestep will be very large in regions of stationary flow, and becomes small when a perturbation enters the flow field. Besides applying well-established adjoint techniques to a new test problem, we further develop a new technique (first proposed by the authors in [18]) which simplifies and accelerates the computation of the dual problem. Due to Galerkin orthogonality, the dual solution ϕ does not enter the error representation as such. Instead, the relevant term is the difference between the dual solution and its projection to the finite element space, ϕ − ϕh . In [18] we showed that it is therefore sufficient to

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compute the spatial gradient of the dual solution, w = ∇ϕ . This gradient satisfies a conservation law instead of a transport equation, and it can therefore be computed with the same conservative algorithm as the forward problem [18]. The great advantage is that the conservative backward algorithm can handle possible discontinuities in the coefficients robustly. A key step is to formulate boundary conditions for the gradient w = ∇ϕ instead of v. Generally the boundary conditions for the dual problem come from the weighting functions of the target functional, e.g. lift or drag. To formulate boundary conditions for w which are compatible with the target functional, one has to lift the well-established techniques of characteristic decompositions from the dual solution to its gradient. We will present details on that in Section 4. Starting with a very coarse, but adaptive spatial mesh and CFL below unity, we establish timesteps which are well adapted to the physical problem at hand. The scheme detects stationary time regions, where it switches to very high CFL numbers, but reduces the timesteps appropriately as soon as a perturbation enters the flow field. We combine our time-adaptation with the spatial adaptive multiresolution technique [12]. The paper is organized as follows. We start with a brief description of the fluid equations and their discretization by implicit finite volume schemes, see Section 2. The adjoint error control is presented in Section 3, including a complete error representation (Section 3.2) and a related space-time splitting (Section 3.3). Section 4 contains the conservative approach to the dual problem und its boundary conditions. In Section 5 we present the adaptive method in time. In Section 6 we present the instationary test case, a 2D Euler transonic flow around a circular arc bump in a channel. In Section 7 results of the fully implicit and a mixed explicit-implicit time adaptive strategy are presented to illustrate the efficiency of the scheme. In Section 8 we summarize our results. We refer the reader to [16, 18, 17] for further details and references.

2 Governing Equations and Finite Volume Scheme We will work in the framework of hyperbolic systems of conservation laws in space and time, Ut + ∇ · f (U) = 0

in ΩT

(1)

Here Ω ⊂ Rd is the spatial domain with boundary Γ := ∂ Ω ⊂ Rd and ΩT = Ω × [0 T ) ⊂ Ω × R0+ is the space-time domain with boundary ΓT := ∂ ΩT ⊂ Ω × R0+ . U is the vector of conservative variables and f the array of the corresponding convective d, in the ith coordinate direction. Our prime example are the fluxes fi , i = 1 multi-dimensional Euler equations of gas dynamics.

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We prescribe boundary conditions on the incoming characteristics as follows: P− (U + ) (fν (U + ) − g) = 0 on ΓT

(2)

Here f(U) := ( f (U) U) is the space-time flux, ν the space-time outward normal to ΩT , and fν (U) := f(U)· ν the space-time normal flux. U + is the interior trace of U at the boundary ΓT (or any other interface used later on). Given the boundary value U + and the corresponding Jacobian matrix fν (U + ), let P− (U + ) be the (d + 2)× (d + 2)matrix which realizes the projection onto the eigenvectors of fν (U + ) corresponding to negative eigenvalues. Then the matrix-vector product P− (U + ) fν (U + ) is the incoming component of the normal flux at the boundary, and it is prescribed in (2). See [17] for details. We approximate (1)–(2) by a first or second order finite volume scheme with implicit Euler time discretization. The computational spatial grid Ωh is a set of open cells Vi such that Vi = Ω i

The intersection of the closures of two different cells is either empty or a union of common faces and vertices. Furthermore let N (i) be the set of cells that have a common face with the cell i, ∂ Vi the boundary of the cell Vi and for j ∈ N (i) let Γi j := ∂ Vi ∩ ∂ V j be the interface between the cells i and j and ni j the outer spatial normal to Γi j corresponding to cell i. Since we will work on curvilinear grids, we require that the geometric consistency condition

∑

|Γi j |ni j = 0

(3)

j∈N(i)

holds for all cells. Let us define a partition of our time interval I := (0 T ) into subintervals Im = [tm−1 tm ], 1 ≤ m ≤ N, where 0 = t0

t1

tm

tN = T

The timestep size is denoted by Δ tm := tm − tm−1 . Later on this partition will be defined automatically by the adaptive algorithm. We also denote the space-time cells and faces by Vim := Vi × Im and Γi mj := Γi j × Im , respectively. Given this space-time grid the implicit finite volume discretization of (1) can be written as Uim +

Δ tm |Vi |

∑

|Γi j | Fimj = Uim−1 for m ≥ 1

(4)

j∈N (i)

It computes the approximate cell averages Uim of the conserved variables on the new time level. For interior faces Γi j , the canonical choice for the numerical flux is a Riemann solver, Fimj := Friem (Uimj U jim ni j )

(5)

Timestep Control for Weakly Instationary Flows

57

consistent with the normal flux fn (U) = f (U) · ni j . In the numerical experiments in Section 7 we choose Roe’s solver [15]. If Γi j ⊂ ∂ ΩT =: ΓT , then we follow the definition of a weak solution and define the numerical flux at the boundary by Fimj := P+ (Uimj ) fνi j (Uimj ) + P−(Uimj ) gm ij

(6)

m where gm i j is the average of g over Γi j . For simplicity of presentation we neglect in our notation that due to higher order reconstruction the numerical flux usually depends on an enlarged stencil of cell averages.

3 Adjoint Error Control - Adaptation in Time In order to adapt the timestep sizes we use a method which involves adjoint error techniques. We have applied this approach successfully to Burgers’ equation in [18], and to the Euler equations of gas dynamics in [17]. Since a finite volume discretization in space and a backward Euler step in time are a special case of a Discontinuous Galerkin discretization, techniques based on a variational formulation can be transferred to finite volume methods. The key tool for the time adaptive method is a space-time splitting of adjoint error representations for target functionals due to S¨uli [19] and Hartmann [10]. It provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in time regions of stationary flow, and become small when a perturbation enters the flow field.

3.1 Variational Formulation In this section we rewrite the finite volume method as a Galerkin method, which makes it easier to apply the adjoint error control techniques. Let us first introduce the space-time numerical fluxes. Let Vim = Vi × Im ∈ ΩT h be a space-time cell, and let γ ⊂ ∂ Vim be one of its faces, with outward unit normal ν . There are two cases: if ν points into the spatial direction, then γ = Γi j × Im and ν = (n 0). If it points into the positive time direction, then γ = Vi × {tm}, and ν = (0 1). Now we define the space-time flux by ⎧ Fimj from (5) if ν = ni j and γ ∈ ETinth ⎪ ⎪ ⎨ Fimj from (6) if ν = ni j and γ ∈ ETexth Fνm (Uh ) = (7) m−1 m ⎪ (1 − θ )Ui + θ Ui if ν = (0 1) and m ≥ 1 ⎪ ⎩ if ν = (0 1) and m = 0 Ui0 where ETinth are the interior faces and ETexth the boundary faces. In the third case, θ ∈ [0 1], so the numerical flux in time direction is a convex combination of the cell averages at the beginning and the end of the timestep. Different values of θ will yield different time discretizations, e.g. explicit Euler for θ = 0, implicit Euler for θ = 1.

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Let Vh := W 1 ∞ (ΩT h ) be the space of piecewise Lipschitz-continuous functions. Now we introduce the semi-linear form N by U × Vh → R

N:

N(U ϕ ) := ∑(Fνm (U) ϕ )∂ Vim im

− ∑ (U ϕh t )Vim + ( f (U) ∇ϕ )Vim

(8)

im

Here and below the sum is over the set {(i m) |Vim ∈ Ω T h }, i.e. all gridcells. Now we rewrite the finite volume method (4) as a first order Discontinuous Galerkin method (DG0): Find Uh ∈ Vh0 such that N(Uh ϕh ) = 0

∀ϕh ∈ Vh0

(9)

where Vh0 is the space of piecewise constant functions over ΩT h . Remark 1. For the DG0 method, Uh ϕ ∈ Vh0 are piecewise constant, so the last two terms in (9), containing derivatives of ϕ , disappear. Moreover, due to the geometric condition (3)

∑

m { j | Γi m j ⊂∂ Vi }

f(Uim ) · νimj = 0

holds for all cells Vim . Therefore, the DG0 solution may be characterized by: Find Uh ∈ Vh0 such that

∑(Fνm (U) − fν (U + ) ϕh )∂ Vim = 0

∀ϕh ∈ Vh0

(10)

im

This form is convenient to localize our error representation later on.

3.2 Adjoint Error Representation for Target Functionals In this section we define the class of target functionals J(U) treated in this paper, state the corresponding adjoint problem and prove the error representation which we will use later for adaptive timestep control. Before we derive the main theorems, we would like to give a preview of an important difference between error representations for linear and nonlinear hyperbolic conservation laws. For linear conservation laws (and many other linear PDE’s), it is possible to express the error in a user specified functional,

εJ := J(U) − J(Uh)

(11)

εJ = η

(12)

as a computable quantity η , so

Timestep Control for Weakly Instationary Flows

59

(see e.g. [1, 17, 20] and the references therein). In general, η will be an inner product of the numerical residual with the solution of an adjoint problem. Below we will see that such a representation does not hold for nonlinear hyperbolic conservation laws. The nonlinearity will give rise to an additional error εΓ− on the inflow boundary, an error εΓ+ on the outflow boundary, and a linearization error εΩ in the interior domain: Theorem 1. Suppose ϕ ∈ V solves the approximate adjoint problem

∂t ϕ + A˜ T ∇ϕ = ψ P˜+T (ϕ − ψΓ ) = 0

in ΩT

(13)

on ΓT

(14)

where A˜ := A(Uh ). Let

εΓ− := −((P− (U + ) − P−(Uh+ ))g ϕ )ΓT εΓ+ := −(P+ (U + )fν (U + ) − P+(Uh+ )fν (Uh+ ) ϕ − ψΓ )ΓT ˜ − Uh ) ∇ϕ )Ω εΩ := ( f (U) − f (Uh ) − A(U T

and

η := N(Uh ϕ ) = N(Uh ϕ − ϕh )

(15)

εJ + εΓ− + εΓ+ + εΩ = η

(16)

Then

Our adaptation is based on computing and equidistributing this η . Typical examples for the functional J are the lift or the drag of a body immersed into a fluid. To simplify matters we consider functionals of the following form: J(U) = (U ψ )ΩT − (P+(U + )fν (U + ) ψΓ )ΓT

(17)

where ψ and ψΓ are weighting functions in the interior of the space-time domain ΩT and at the boundary ΓT . For the proof of Theorem 1 as well as an illustrative example of the functional J we refer again to [17]. For the adjoint problem (13) and (14) the role of time is reversed and hence P˜+ plays the role of P− in (2). Here ψΓ comes from the weighting function in the functional (17).

3.3 Space-Time Splitting The error representation (16) is not yet suitable for time adaptivity, since it combines space and time components of the residual and of the difference ϕ − ϕh of the dual solution and the test function. The main result of this section is an error estimate whose components depend either on the spatial grid size h or the timestep k, but

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never on both. The key ingredient is a space-time splitting of (16) based on L2 projections. Similar space-time projections were introduced previously in [10, 19]. In [18] we adapted them to the finite element spaces and space-time Discontinuous Galerkin methods of arbitrary order. The splitting takes the form

η = ηk + ηh

(18)

For brevity, we only present the details for first-order finite volume schemes for which we obtain

ηk =

Δ tm 2

∑ im

(1 − θ )(Uim−1 − Uim−2) + θ (Uim − Uim−1) ψ − A˜ T w V

i

(19)

(we set Ui−1 = Ui0 in the first summand). Thus our temporal error indicator is simply a weighted sum of time-differences of the approximate solution Uh , and the weights can be computed from the data ψ and the solution w of the conservative dual problem (21). In our adaptive strategy, we will use the localized indicators

η¯ km :=

1 2

∑ ((1 − θ )(Uim−1 − Uim−2) + θ (Uim − Uim−1) ψ − A˜ T w)Vi

(20)

i

In the next section we present an example for the boundary conditions for the dual problem.

4 The Conservative Dual Problem In this section we present the conservative approach to the dual problem, which we introduced in [18] and derive boundary conditions for the gradient of the dual problem. The adjoint equation (13) is a system of linear transport equations with discontinuous coefficients. Therefore, numerical approximations may easily become unstable. Another inconvenience is that in order to obtain a meaningful error representation in (16), the approximate adjoint solution ϕ should not be contained in Vh0 . Therefore, ϕ is often computed in the more costly space Vh1 . In [18] we have proposed a simple alternative which helps to avoid both difficulties. Instead of computing the dual solution ϕ we will compute its gradient w := ∇ϕ which is the solution of the conservative dual problem wt + ∇(A˜ T w) = ∇ψ in Ω T

(21)

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61

This system is in conservation form, and therefore it can be solved by any finite volume or Discontinuous Galerkin scheme. Moreover, (21) may be solved in Vh0 , since a piecewise constant solution w already contains crucial information on the gradient of ϕ . The scalar problem treated in [18] was set up in such a way that the characteristic boundary conditions for the dual problem became trivial. In [17] we developed boundary conditions for the more general initial boundary value problem (1) – (2). Denoting the flux in (21) by H := A˜ T w, the boundary condition (14) becomes P˜+T (H − HΓ ) = 0 on ΓT

(22)

i.e. we prescribe the incoming component P˜+T H. Here HΓ is a given real-valued vector function, which depends on ψΓ . However, this characteristic boundary condition needs to be interpreted carefully. Using (21) and denoting the interior trace at the flux by Hint , we may introduce the boundary flux by H := P˜−T Hint + P˜+T HΓ

on ΓT

Note that all the projections P˜± used below depend on the point (x t) ∈ ΓT via the outside normal vector ν (x t). The value P˜−T Hint may be assigned from the trace wint at the interior of the computational domain, P˜−T Hint = P˜−T (A˜ T w)int The boundary values P˜+T HΓ are computed using the PDE

ϕt = −H + ψ

(23)

with boundary values (14), P˜+T HΓ = P˜+T (−ϕt + ψ )|Γ = −(P˜+T ψΓ )t + P˜+T ψ 1 ˜ T m m ˜ T m−1 m−1 ˜ T m−1 ≈− ψΓ ψ P ψΓ − P+ + P+ Δ tm +

(24)

This completes the definition of the numerical boundary conditions for the conservative dual problem.

5 Adaptive Concept Now we combine the multiscale approach in space [12] and the time adaptive method derived from the space-time splitting of the error representation to get a space-time adaptive algorithm:

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• Solve the primal problem (1) on a coarse adaptive spatial grid (e.g. level L = 2) using uniform CFL numbers (CFL = 0 8), • Compute the dual problem (13) and (14) and the space-time-error representation (18). In particular, compute the localized error indicators η¯ km using (20). • Compute the new adaptive timestep sizes depending on the temporal part of the error representation and the CFL number on the new grid, aiming at an equidistribution of the error, • Solve the primal problem using the new timestep sizes on a finer spatial grid (e.g. level L = 2). The advantage is, that the first computations of the primal problems and the dual problem are done on a coarse spatial grid, and therefore have low cost. These computations provide an initial guess of the timesteps for the computation on the finer spatial grid. We will restrict the timestep size from below to CFL = 0 8, since smaller timestep sizes only add numerical diffusion to the scheme and increase the computational cost. Note that all physical effects already have to be roughly resolved on the coarse grid in order to determine a reliable guess for the timesteps on the fine grid. We will deal with some aspects in detail in the numerical examples in Section 7.

5.1 Asymptotic Decay Rates Since the adaptive strategy outlined in Section 5 above depends on assumptions on the assymptotic behavior of the error, we first try to estimate these decay rates. There is no analytical result which shows how the error terms η¯ k and η¯ h depend on k and h. Therefore, we estimate this dependence numerically. We compute a perturbed shock of Burgers equation, for details see [18]. We compare the two approaches: • refinement only time • and refinement only space. Each of the plots in Figure 1 show the error estimators η¯ k (error in time) and η¯ h (error in space). In the Figure 1(a) we refined only in time. Here the spatial error remains constant, while the time error still decreases with first order. The second Figure 1(b) shows the refinement only in space. The time error η¯ k is almost constant, while the spatial error is decreasing with second order. Numerically the terms η¯ t and η¯ h behave as expected. They depend either on k or on h, but never on both. The behaviour of ηh and ηk is very similar, and not displayed here. Remark 2. The numerically validated results can be used for adaptive grid refinement. The error estimator η¯ h can be used as an indicator for spatial adaption and the estimator η¯ k for time step control.

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η¯k 1.96e-03 9.81e-04 4.81e-04 2.37e-04

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ηk 1.29e-04 1.83e-05 3.57e-06 1.30e-06

the error representation, CFL = 0 8

ηh 2.00e-01 4.75e-02 1.17e-02 2.89e-03

J(uh ) 1.72e+00 1.74e+00 1.75e+00 1.75e+00

ηh + ηk 2.01e-01 4.75e-02 1.17e-02 2.89e-03

θ 5.57e+00 5.49e+00 5.71e+00 7.06e+00

6 Setup of the Numerical Experiment An instationary variant of a classical stationary 2D Euler transonic flow [14], is investigated to illustrate the efficiency of the adaptive method. Steady State Configuration. First we consider the classical setup in the stationary case. The computational domain is a channel of 3m length and 2m height with an arc bump of l = 1m secant length and h = 0 024m height cut out, see Figure 2. At the inflow boundary, the Mach number is 0.85 and a homogeneous flow field characterized by the free-stream quantities is imposed. At the outflow boundary, characteristic boundary conditions are used. We apply slip boundary conditions across the solid walls, i.e., the normal velocity is set to zero. In the numerical examples in Section 7 the height of the channel is 2m and the length 6m. The threshold value in the grid adaptation step for the multiscale analysis is ε = 1×10−3 and computations are done on adaptive grids with finest level L = 2 and L = 5 respectively. In general, a smaller threshold value results in more grid refinement whereas a larger value gives locally coarser grids. In the stationary case at Mach 0.85 there is a compression shock separating a supersonic and a subsonic domain. The shock wave is sharply captured and the stagnation areas are highly resolved, see Figures 3 and 4.

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Slip Wall

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Fig. 2 Circular arc bump configuration of the computational domain Ω

Fig. 3 Adaptive grid L = 5, to steady state solution in Figure 4 of the circular arc bump configuration

We will use this steady state solution as initial data for the instationary test case. Instationary Test Case. Now we define our instationary test case prescribing a time-dependent perturbation coming in at the inflow boundary. First we keep the boundary conditions fixed, and prescribe the corresponding stationary solution as initial data. Then we introduce for short time periods perturbations of the pressure at the left boundary. The first perturbation is about 20 percent of the pressure at the inflow boundary and the second 2 percent, see Figure 5. The perturbations imposed move through the domain and leave it at the right boundary. Then the solution is stationary again, see Figure 6. The total time is t = 0 029s. The formulas of the pertubations are given in detail in [17]. The first computation is done on an adaptive grid with finest level L = 2. We also compute the dual solution and the error representation on this level. Using the time-space-split error representation (18) we derive a new timestep distribution aiming at an equidistribution of the error. Finally this is modified by imposing a CFL restriction from below.

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Fig. 4 Steady state solution of the circular arc bump configuration: Isolines of the density, L=5

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We aim to equidistribute the error and prescribe a tolerance Tol(5) = 2−3 η¯ kre f , where η¯ kre f is the temporal error from the computation on level L = 2. For this set-up we will show that the adaptive spatial refinement together with the time-adaptive method will lead to an efficient computation. The multiscale method provides a well-adapted spatial representation of the solution, and the dual solution will detect time-domains where the solution is stationary. In these domains, the equidistribution strategy will choose large timesteps. Target Functional. Now we set up the target functional. The functional J(U) is chosen as a weighted average of the normal force component exerted on the bump and at the boundaries before and behind the bump:

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J(U) = ∑

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x ∈ κi

Here Γ is only the bottom part of Γ . In all computations presented in this section the functional (25) is chosen, which is the pressure averaged at several points at the bump in front and behind the bump. The x-coordinates of these points at the bottom are xi = -3, -2, -1, 0, 1, 2, 3. At each of these points xi a smooth function ψi is given with support xi − 0 25 xi + 0 25. This functional measures the pressure locally. In Figure 6 we show a time-sequence of the instationary test case computed with uniform CFL = 1 on an adaptive grid with finest level L = 5. Note that the perturbation entering at the left boundary and moving through the boundary is resolved very well.

7 Computational Results 7.1 Numerical Strategies We gave an outline of the adaptive method in Section 5. Now we will present numerical computations, where we compare some variations of the adaptive concept. The first strategy is the one we proposed in [18]: We first compute a forward solution on a coarse grid (L = 2) and solve the adjoint problem on the adaptive grid of the forward solution. Then we use the information of the error representation based on the dual solution to determine a new sequence of timesteps. This sequence is used in the computation of the forward solution on a grid with finest level L = 5, where we additionally restrict the CFL number from below. We compare the results of the time adaptive strategy with uniform timestep distributions. In the second strategy we modify the fully implicit timestepping strategy and introduce a mixed implicit/explicit approach. The reason is that implicit timesteps with CFL 5 are not efficient, since we have to solve a nonlinear system of equations at each timestep. Thus, for CFL 5, the new implicit/explicit strategy switches to the cheaper and less dissipative explicit method with CFL = 0 5. We want to compare these strategies with respect to the following main aspects: • What is the quality and what are the costs determining the adaptive timestep sequence from computations on the coarse grid? • Is the predicted adaptive timestep sequence well-adapted to the solution on the fine grid? • How is the solution affected if we use uniform timesteps larger than the predicted adaptive timestep sequence?

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Fig. 6 Instationary solution of the circular arc bump configuration, uniform timestep CFL = 1, adaptive spatial grid on level L = 5, isolines of the density, perturbation entering on the left and leaving on the right side of the computational domain, from top to bottom: t = 0.0057s, 0.00912s, 0.01254s, 0.01596s, 0.01938s

In order to quantify the results we have to compare with a reference solution. Since the exact solution is not available we perform a computation with L = 5

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refinement levels using implicit timestepping with CFL = 1 amd explicit timestepping with CFL = 0 5.

7.2 Strategy I: Fully Implicit Computational Results 7.2.1

Adjoint Indicator and Adaptive Timesteps

Now we use the error representation on finest level L = 2 for a new time adaptive computation on finest level L = 5. The first computation is done on a mesh with finest level L = 2. We compute until time T = 0 0285s, which takes 1000 timesteps with CFL = 1. We use the results of the error representation of this computation to compute a new timestep distribution. The forward problem takes 329s and the dual problem including the evaluation of the error representation 619s on an Opteron 8220 processor at 2.86 GHz. The total computational costs are 948s, and in memory we have to save 1000 solutions (each timestep) of the forward problem which corresponds to 48 MB (total). This gives us a new sequence of adaptive timesteps for the computation of level L = 5. The error indicator and the new timesteps are presented in Figure 7. In time intervals where the solution is stationary, i.e. at the beginning, and after the perturbations have left the computational domain, the timesteps are large. In time intervals where the solution is instationary we get well-adapted small timesteps. Then we use the adaptive timestep sequence for a computation on level L = 5 and compare it with a uniform in time computation using CFL = 1. The uniform computation needs 8000 timesteps and the computational time is 21070s. The time adaptive solution is computed with 2379 timesteps and this computation takes 9142s. In Figure 8 we show a sequence of plots of the uniform, CFL = 1 computation on an adaptive spatial grid with finest level L = 5. In Figure 8 we compare the pressure distribution at the bottom boundary of the uniform solution and the time adaptive solution at several times. The two solutions on level L = 5 match very well. Remark 3. In [17] we also did some comparisions of the adjoint indicator with some ad hoc indicators, which estimate the variation of the solution from one timestep to the following. While these indicators also detect whether the solution is stationary or not, they lead to computations which are in general more expensive but not more accurate. In particular, most timesteps are smaller than in the case with adaptation via adjoint problems. One advantage may be that we do not need to compute a dual solution, which makes the computation of the variation indicator less expensive. But this is only a small advantage, since we compute the error indicators on a coarse mesh. On the other hand the ad-hoc indicator is more straight forward to implement. Another approach was to choose the maximum jump of the solution in one cell, both weighted and not weighted with the size of the cell. This was an approximation to the L∞ -norm. This indicator is not very useful, since it turned out to be highly oscillatory.

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Comparison with Uniform Timesteps

In many instationary computations where no a-priori information is known, one reasonable choice is to use uniform CFL numbers. Therefore we compared the computation with adaptive implicit timesteps with implicit computations using uniform CFL numbers. In Section 7.2 we have already done a computation with uniform CFL number, CFL = 1, on a grid with L = 2, to get timestep sizes for an adaptive computation on a grid with L = 5. As a reference solution we also computed with uniform CFL number, CFL = 1, a solution of the problem on a grid with L = 5. Now we compare these computations with computations using higher uniform CFL numbers.

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First we choose a uniform CFL number of approximately 3.2, which corresponds to 2500 timesteps. This equals roughly the number of timesteps in the adaptive method, and hence it should give a fair comparison. The uniform computation takes

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Table 2 Performance for computations on L = 5 using different fully implicit timestepping strategies

adaptive timesteps uniform CFL = 1 uniform CFL = 3 2 uniform CFL = 10

CPU [s] timesteps Newton steps linear steps 9142 2379 3303 16875 21070 8000 8282 32630 11509 2500 4904 26895 3290 800 1600 13500

about 11509s, more than the 9142s of the adaptive computation (see Table 2). In the uniform computation most of the timesteps are more expensive, since they need more Newton steps, and more steps for solving the linear problems. This shows that the understanding of the dynamics of the solution pays directly in the nonlinear and linear solvers. Moreover, it can be seen from Figure 8 that the quality of the solution is considerably worse than for the adaptive computation. Another computation with CFL number 10 takes only 3290s. However, as can be seen from Figure 8 the solution is badly approximated: In the beginning of the computation the solutions of the different methods match very well, which means that the inflow at the boundary is well-resolved. As time goes on, the solutions differ more and more. After the perturbation has passed the bump, the perturbations differ strongly. Only the time-adaptive method approximates the reference solution (CFL = 1) closely. Remark 4. Table 2 gives an overview of the CPU time and the number of Newton iterations and linear iterations for the computations. The costs for the computation of the indicator on level L = 2 are very low compared to the costs of computations on level L = 5. An adaptive computation including the computation of the indicator, i.e. 329s+619s+9142s, is cheaper than the computation using uniformCFL number, e.g. CFL = 1, that needs 21070s. Even the computation with CFL = 3 2 is more expensive than the time-adaptive computation, but leads to worse results, see Figure 8. Table 2 shows that the CPU time is roughly proportional to the number of Newton iterations and not to the number of timesteps or linear solver steps. The CPU time is about 2 5s per Newton iteration. This means that we have to minimize the number of Newton iterations in total to accelerate the computation. This is done very efficiently by the time-adaptive approach. For a large range of CFL numbers from 1 to more than 100, it needs only one or two Newton iterations per timestep, without sacrificing the accuracy, see [17]. In this example, the stationary time regions are not very large compared to the overall computation. If the stationary regions were larger, the advantage of the time adaptive scheme would be even more significant.

7.3 Strategy II: Explicit-Implicit Computational Results Now we modify the fully implicit timestepping strategy and introduce a mixed implicit/explicit approach. The reason is that implicit timesteps with CFL 5 are not

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Fig. 9 2D Euler equations, comparison of timestep sequence derived from error representation for implicit computation (dashed line) and for mixed explicit-implicit computation (bold line)

Table 3 Performance for computations on L = 5 using fully explicit timesteps and the timeadaptive explicit-implicit strategy CPU [s] timesteps timesteps (total) (implicit) explicit CFL = 0 5 18702 16000 adaptive expl.-impl. 7730 5802 259

efficient, since we have to solve a nonlinear system of equations at each timestep. Thus, for CFL 5, the new implicit/explicit strategy switches to the cheaper and less dissipative explicit method with CFL = 0 5. The timestep sequence is shown in Figure 9. Of course, we could choose variants of the thresholds CFL = 0 5 and 5. As we can see in Table 3, the new strategy requires 5802 timesteps, where 95% are explicit. The CPU time of 7730s easily beats the fully explicit solver (18702s), and is also superior to the fully implicit adaptive scheme (9142s, see Table 2). The computational results are presented in Figure 10. The results of the combined explicit-implicit strategy are very close to the results of the fully explicit method, and far superior to all fully implicit methods. Note that the explicit scheme serves as reference solution, since it is well-known that it gives the most accurate solution for an instationary problem.

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Fig. 10 2D Euler equations, comparison of solutions in time on adaptive grid with finest level L = 5, pressure p at the bottom boundary, zoom of the perturbation at time t=0.011975

8 Conclusion In the series of works [16, 18, 17] reviewed in this paper, explicit and implicit finite volume solvers on adaptively refined quadtree meshes have been coupled with adjoint techniques to control the timestep sizes for the solution of weakly instationary compressible inviscid flow problems. For the 2D Euler equations we study a test case for which the time-adaptive method does reach its goals: it separates stationary regions and perturbations cleanly and chooses just the right timestep for each of them. The adaptive method leads to considerable savings in CPU time and memory while reproducing the reference solution almost perfectly. Our approach is based upon several analytical ingredients: In Theorem 1 we state a complete error representation for nonlinear initial-boundary-value problems with characteristic boundary conditions for hyperbolic systems of conservation laws, which includes boundary and linearization errors. Besides building upon well-established adjoint techniques, we also add a new ingredient which simplifies the computation of the dual problem [18]. We show that it is sufficient to compute the spatial gradient of the dual solution, w = ∇ϕ , instead of the dual solution ϕ itself. This gradient satisfies a conservation law instead of a transport equation, and it can therefore be computed with the same algorithm as the forward problem. For discontinuous transport coefficients, the new conservative algorithm for w is more robust than transport schemes for ϕ , see [18]. Here we also derive characteristic

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boundary conditions for the conservative dual problem, which we use in the numerical examples in Section 7. In order to compute the adjoint error representation one needs to compute a forward and a dual problem and to assemble the space-time scalar product (20). Together, this costs about three times as much as the computation of a single forward problem. In our application, the error representation is computed on a coarse mesh (L = 2), and therefore it presents only a minor computational overhead compared with the fine grid solution (L = 5). In other applications, the amount of additional storage and CPU time may become significant. For instationary perturbations of a 2D stationary flow over a bump, we have implemented and tested both a fully implicit and a mixed explicit-implicit timestepping strategy. The explicit-implicit approach switches to an explicit timestep with CFL = 0 5 in case the adaptive strategy suggests an implicit timestep with CFL 5. Clearly, the mixed explicit-implicit strategy is the most accurate and efficient, beating the adaptive fully implicit in accuracy and efficiency, the implicit approach with fixed CFL numbers in accuracy, and the fully explicit approach in efficiency. Acknowledgement. We would like to thank Ralf Hartmann and Mario Ohlberger for stimulating discussions.

References 1. Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4(4), 237–264 (1996) 2. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001) 3. Bramkamp, F., Lamby, P., M¨uller, S.: An adaptive multiscale finite volume solver for unsteady and steady state flow computations. J. Comput. Phys. 197(2), 460–490 (2004) 4. CLAWPACK: Conservation law package (1998), http://www.amath.washington.edu/˜claw/ ¨ 5. Courant, R., Friedrichs, K., Lewy, H.: Uber die partiellen Differenzengleichungen der mathematischen physik. Math. Ann. 100(1), 32–74 (1928) 6. Domingues, M., Gomes, S., Roussel, O., Schneider, K.: An adaptive multiresolution scheme with local time stepping for evolutionary PDEs. J. Comput. Phys. 227(8), 3758– 3780 (2008) 7. Domingues, M., Roussel, O., Schneider, K.: On space-time adaptive schemes for the numerical solution of PDEs. In: CEMRACS 2005—computational aeroacoustics and computational fluid dynamics in turbulent flows, ESAIM Proc. EDP Sci., Les Ulis, vol. 16, pp. 181–194 (2007) 8. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal. 32(6), 1729–1749 (1995) 9. Ferm, L., L¨otstedt, P.: Space-time adaptive solution of first order PDEs. J. Sci. Comput. 26(1), 83–110 (2006) 10. Hartmann, R.: A posteriori Fehlersch¨atzung und adaptive Schrittweiten- und Ortsgittersteuerung bei Galerkin-Verfahren f¨ur die W¨armeleitungsgleichung. Master’s thesis, Universit¨at Heidelberg (1998) 11. Kr¨oner, D., Ohlberger, M.: A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions. Math. Comp. 69(229), 25–39 (2000)

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12. M¨uller, S.: Adaptive multiscale schemes for conservation laws. Lecture Notes in Computational Science and Engineering, vol. 27. Springer, Berlin (2003) 13. Ohlberger, M.: A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Numer. Math. 87(4), 737–761 (2001) 14. Rizzi, A., Viviand, H. (eds.): Numerical methods for the computation of inviscid transonic flows with shock waves. Notes on Numerical Fluid Mechanics, vol. 3. Friedr. Vieweg & Sohn, Braunschweig (1981) 15. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981) 16. Steiner, C.: Adaptive timestepping for conservation laws via adjoint error representation. Ph.D. thesis, RWTH Aachen University, Germany (2008), http://darwin.bth.rwth-aachen.de/opus3/volltexte/2009/2679/ 17. Steiner, C., M¨uller, S., Noelle, S.: Adaptive timestep control for instationary solutions of the euler equations. SIAM J. Sci. Comput. (2009) (submitted) 18. Steiner, C., Noelle, S.: On adaptive timestepping for weakly instationary solutions of hyperbolic conservation laws via adjoint error control. Comm. Numer. Meth. Eng. (2008) doi:10.1002/cnm.1183 19. S¨uli, E.: A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In: An introduction to recent developments in theory and numerics for conservation laws (Freiburg/Littenweiler 1997. Lect. Notes Comput. Sci. Eng, vol. 5, pp. 123–194. Springer, Berlin (1999) 20. S¨uli, E., Houston, P.: Adaptive finite element approximation of hyperbolic problems. In: Error estimation and adaptive discretization methods in computational fluid dynamics. Lect. Notes Comput. Sci. Eng., vol. 25, pp. 269–344. Springer, Berlin (2003)

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Adaptive Multiscale Methods for Flow Problems: Recent Developments Wolfgang Dahmen, Nune Hovhannisyan, and Siegfried M¨uller

Abstract. The concept of the new fully adaptive flow solver Quadflow has been developed within the SFB 401 over the past 12 years. Its primary novelty lies in the integration of new and advanced mathematical tools in a unified environment. This means that the core ingredients of the finite volume solver, the grid adaptation and grid generation are adapted to each others needs rather than putting them together as independent black boxes. In this paper we shall present recent developments and demonstrate their efficiency by numerical experiments for some representative basic configurations.

1 Introduction The work performed in the SFB 401 was motivated by two central questions arising from engineering applications, namely, (i) how to influence wake vortices generated by a lift-producing aircraft in order to reduce takeoff and landing frequencies at airports, and (ii) of better understanding the interaction of structural dynamics and aerodynamics to design new concepts for supporting wing structures. The accurate and reliable simulation of such processes pose challenging questions near or even beyond current simulation capabilities. The development of concepts that reduce computational complexity already on the level of mathematical algorithmic design appears to be indispensible. This has been the core objective of the new adaptive and parallel solver Quadflow [6, 7]. In order to exploit synergy effects, this solver has been designed as an integrated tool where the core ingredients, namely, (i) the flow solver concept based on a finite volume discretization, (ii) the grid adaptation concept based on wavelet techniques, and (iii) the grid generator based on B-spline mappings are adapted to each others needs, see Figure 1. In particular, the three tools Wolfgang Dahmen · Nune Hovhannisyan · Siegfried M¨uller Institut f¨ur Geometrie und Praktische Mathematik, RWTH Aachen University, D-52056 Aachen e-mail: {dahmen,hovhannisyan,mueller}@igpm.rwth-aachen.de

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 77–103. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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are not just treated as independent black boxes communicating via interfaces but are highly intertwined on a conceptual level mainly linking (i) the multiresolutionbased grid adaption that reliably detects and resolves all physical relevant effects, and (ii) the B-spline grid generator which reduces grid changes to just moving a “few” control points whose number is, in particular, independent of any local grid refinements. Block topology

Flow solver

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Over the past few years the above framework has been further enriched and extended by mathematical concepts, such as multilevel time stepping [21, 20, 24], multilevel solver strategies [25], time step control [30, 28, 29], adaptive local flux and source reconstruction [18], as well as techniques from computer science. In particular, parallelization of the multiresolution grid adaptation using space-filling curves [32], has been incorporated to further improve the efficiency of Quadflow. Originally, the multiresolution-based grid adaptation technique was kept separate from the treatment of discrete evolution equations, cf. [7]. However, the multiresolution analysis offers a much higher potential when applying it directly to the discrete evolution equations arising from the finite volume discretization rather than just using it as a data compression tool for the set of discrete cell data. In the present work we explain how to integrate the multiresolution anaysis in the flow solver by presenting recent developments on (i) a local strategy to compute numerical fluxes and sources on locally refined grids with hanging nodes, cf. [18], (ii) a multilevel time stepping strategy where refinement is also performed in time, cf. [24], and (iii) an FAS-like strategy to solve efficiently the nonlinear problems arising from an implicit time discretization of the underlying finite volume scheme, cf. [25]. Of course, in spite of the significant reduction of the accomplished computational complexity (Cpu time and memory) in comparison to computations on uniform meshes, efficiently performing 3D computations for complex geometries requires complementing efforts concerning parallelization techniques. The performance of a parallelized code crucially depends on the load-balancing. Since the underlying adaptive grids are unstructured, this task cannot be considered trivial. This issue will be addressed in more detail in [13, 10].

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The aim of the present work is to give an overview on recent developments regarding multiscale-based grid adaptation. For this purpose, we first summarize the basic ingredients of the grid adaptation concept starting with the underlying equations and their discretization using finite volume schemes, see Section 2. This is followed by the multiscale analysis of the discrete cell averages resulting from the finite volume discretization, see Section 3 and the construction of locally refined grids using data compression techniques, see Section 4. Applying the multiscale analysis to the original finite volume discretization on the uniform grid we obtain multiscale evolution equations, see Section 5. The crucial point is then to perform the time evolution on the adaptive grid where the accuracy of the uniform discretization is maintained but the computational complexity is proportional only to the number of cells of the adaptive grid. For this purpose, the computation of the local flux balances and sources has to be performed judisciously, see Section 6. The resulting scheme is further accelerated using multilevel time stepping strategies in case of an explicit time discretization for instationary flow problems, see Section 7, and FASlike multigrid techniques in case of an implicit time discretization for steady state flow problems, see Section 8. Finally, in Section 9, we present several computations that confirm the efficiency of the aforementioned concepts.

2 Governing Equations and Finite Volume Schemes The fluid equations are determined by balance equations

∂ ∂t

u dV + V

∂V

f(u) · n dS =

s(u) dV ,

(1)

V

where u is the array of the mean conserved quantities, e.g., density of mass, momentum, specific total energy, f is the array of the corresponding convective and diffusive fluxes, and s denotes a source term that may occur, for instance, in turbulence modelling. The balance equations (1) are approximated by a finite volume scheme. For this purpose the finite fluid domain Ω ⊂ Rd is split into a finite set of subdomains, the cells Vi , such that all Vi are disjoint at each instant of time and that their union covers Ω . To simplify notation, we will always assume that the grid does not move with time. Furthermore let N(i) be the set of cells that have a common edge with the cell i, and for j ∈ N(i) let Γi j := ∂ Vi ∩ ∂ V j be the interface between the cells i and j and ni j the outer normal of Γi j corresponding to cell i. For the time discretization we may use either explicit (forward Euler (θ = 0)) or implicit schemes (CrankNicholson (θ = 0.5), backward Euler (θ = 1)) where the time step may vary i.e., tin+1 = tin + τin+1 . These can be written in the form vn+1 +θ i

τ n+1 τin+1 n+1 (Bi − |Vi | Sin+1 ) = vni − (1 − θ ) i (Bni + |Vi | Sin ) |Vi | |Vi |

(2)

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to compute the approximated cell averages vi of the conserved variables on the new time level. Here the fluxes and the source terms are approximated by Bni :=

∑

|Γi j | F(vnij , vnji , ni j ),

Sin := S(vni ),

(3)

j∈N(i)

where the numerical flux function F(u, w, n) is an approximation for the flux f (u, n) := f · n in outer normal direction ni j on the edge Γi j . The numerical flux is assumed to be consistent, i.e., F(u, u, n) = f (u, n). For simplicity of presentation we neglect the fact that, due to higher order reconstruction, it usually depends on an enlarged stencil of cell averages. Moreover, to preserve a constant flow field we assume that the geometric consistency condition ∑ j∈N(i) |Γi j |ni j = 0 holds.

3 Multiscale Analysis A finite volume discretization typically works on an array of cell averages. In order to realize a certain target accuracy at the expense of a possibly low number of degrees of freedom, viz. a possibly low computational effort, one should keep the size of the cells large whereever the data exhibit little variation, reflecting a high regularity of the searched solution components. Our analysis of the local regularity behavior of the data is based on the concept of biorthogonal wavelets [11]. This approach may be seen as a natural generalization of Harten’s discrete framework [16]. The core ingredients are (i) a hierarchy of nested grids, (ii) biorthogonal wavelets and (iii) the multiscale decomposition. In what follows we will only summarize the basic ideas. For the realization and implementation see [23]. Grid Hierarchy. Let be Ωl := {Vλ }λ ∈Il a sequence of different meshes corresponding to different resolution levels l ∈ N0 where the mesh size decreases with increasing refinement level. The grid hierarchy is assumed to be nested, i.e., each cell λ ∈ Il on level l is the union of cells μ ∈ Mλ0 ⊂ Il+1 on the next higher refinement level l + 1, i.e., Vλ = Vμ , λ ∈ Il+1 , (4) μ ∈Mλ0 ⊂Il+1

where Mλ0 ⊂ Il+1 is the refinement set and, hence, Ωl ⊂ Ωl+1 . A simple example is shown in Figure 2 for a dyadic grid refinement of Cartesian meshes. Note that the framework presented here is not restricted to this simple configuration but can also be applied to unstructured grids and irregular grid refinements, cf. [23].

Fig. 2 Sequence of nested grids

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Box Function and Cell Averages. With each cell Vλ in the partitions Ωl we associate the so–called box function 1 1/|Vλ | , x ∈ Vλ φ˜λ (x) := χ (x) = , λ ∈ Il (5) 0 , x ∈ Vλ |Vλ | Vλ defined as the L1 –normalized characteristic function of Vλ . By |V | we denote the volume of a cell V . Then the averages of a scalar, integrable function u ∈ L1 (Ω ) can be interpreted as an inner product, i.e., uˆλ := u, φ˜λ Ω

with u, vΩ :=

Ω

u v dx.

(6)

Obviously the nestedness of the grids as well as the linearity of integration imply the two–scale relations

φ˜λ =

∑

μ ∈Mλ0 ⊂Il

˜ ml,0 μ ,λ φ μ

and

uˆλ =

∑

μ ∈Mλ0 ⊂Il

ml,0 μ ,λ uˆ μ ,

λ ∈ Il−1 ,

(7)

:= |Vμ |/|Vλ | for each cell μ ∈ Mλ0 where the mask coefficients turn out to be ml,0 μ ,λ in the refinement set. Wavelets and Details. In order to detect singularities of the solution we consider the difference of the cell averages corresponding to different resolution levels. For this purpose we introduce the wavelet functions ψ˜ λ as linear combinations of the box functions, i.e., ˜ ψ˜ λ := (8) ∑ ml,1 μ ,λ φμ , λ ∈ Jl , μ ∈Mλ1 ⊂Il+1

with mask coefficients ml,1 μ ,λ that only depend on the grids. The construction of the wavelets is subject to certain constraints, namely, (i) the wavelet functions Ψ˜l := (ψ˜ λ )λ ∈Jl build an appropriate completion of the basis system Φ˜ l := (φ˜λ )λ ∈Il . By this we mean (ii) they are locally supported, (iii) provide vanishing moments of a certain order and (iv) there exists a biorthogonal system Φl and Ψl of primal functions satisfying analogous two-scale relations. The last requirement is typically the hardest to satisfy. It is closely related to the Riesz basis property of the infinite ˜ collection Φ˜ 0 ∪ ∞ l=0 Ψl of L2 (Ω ), say. For details we refer to the concept of stable completions, see [11]. Aside from these stability aspects the biorthogonal framework facilitates an efficient change of bases. While the relations (7), (8) provide expressions of the coarse scale box functions and detail functions as linear combinations of fine scale box functions, the mask coefficients in the analogous two-scale relations for the dual system Φl , Ψl give rise to the reverse change of bases between Φ˜ l ∪ Ψ˜l and Φ˜ l+1 , i.e., ˜ ˜ φ˜λ = ∑ gl,0 (9) ∑ gl,1 μ ,λ φμ + μ ,λ ψ μ , λ ∈ Il+1 , μ ∈G0λ ⊂Il

μ ∈G1λ ⊂Jl

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where we rewrite the basis function φ˜λ on level l + 1 by the scaling functions φ˜μ and the wavelet functions ψ˜ μ on the next coarser scale l. Here again the mask l,1 coefficients gl,0 μ ,λ and g μ ,λ depend only on the grid geometry. Biorthogonality also yields a data representation in terms of the primal system. The corresponding detail coefficients are given by dλ := u, ψ˜ λ Ω =

∑

μ ∈Mλ1

ml,1 μ ,λ uˆ μ ,

λ ∈ Jl ,

(10)

whose two-scale format follows from its functional counterpart (8). Note that, by biorthogonality, the dλ are the expansion coefficients with respect to the basis Ψ , obtained by testing u by the elements from Ψ˜ . Therefore, Ψ is called primal system, while Ψ˜ is used to expand the cell averages which are functionals of the solution u whose propagation in time gives rise to the finite volume scheme. The primal basis itself will actually never be used to represent the solution u. Instead the enhanced accuray of the approximate cell averages can be used for higher order reconstructions commonly used in finite volume schemes. Cancellation Property. It can be shown that the details become small with increasing refinement level when the underlying function is smooth |dλ | ≤ C 2−l M u(M) L∞ (Vλ ) .

(11)

in the support of the wavelet ψ˜ λ . More precisely, the details decay at a rate of at least 2−l M provided the function u is differentiable and the wavelets have vanishing moments of order M, i.e., p, ψ˜ λ Ω = 0 for all polynomials p of degree less than M. Here we assume that the grid hierarchy is quasi-uniform in the sense that the diameters of the cells on each level l is proportional to 2−l . If coefficient and function norms behave essentially the same, as asserted by the Riesz basis property, (11) suggests to neglect all sufficiently small details in order to compress the original data. In fact, the higher M the more details may be discarded in smooth regions. Multiscale Transformation. In order to exploit the above compression potential, the idea is to transform the array of cell averages uL := (uˆλ )λ ∈IL corresponding to a finest uniform discretization level into a sequence of coarse grid data u0 := (uˆλ )λ ∈I0 and details d l := (dλ )λ ∈Jl , l = 0, . . . , L − 1, representing the successive update from a coarser resolution to a higher resolution. In summary, according to (7) and (10), the change of bases provides two–scale relations for the coefficients inherited from the two–scale relations of the box functions and the wavelet functions uˆλ =

∑

μ ∈Mλ0 ⊂Il+1

and, conversely,

ml,0 μ ,λ uˆ μ , λ ∈ Il ,

dλ =

∑

μ ∈Mλ1 ⊂Il+1

ml,1 μ ,λ uˆ μ , λ ∈ Jl ,

(12)

Adaptive Multiscale Methods for Flow Problems: Recent Developments

Fig. 3 Two-scale Transformation

uˆλ =

∑

μ ∈G0λ ⊂Il

gl,0 μ ,λ uˆ μ +

83

Fig. 4 Multiscale transformation

∑

μ ∈G1λ ⊂Jl

gl,1 μ ,λ d μ ,

λ ∈ Il+1 ,

(13)

which reflects the typical cascadic format of a wavelet transform. The two-scale relations are illustrated for the 1D case in Figure 3. A successive application of the relations (12), see Figure 4, decomposes the array uˆ L into coarse scale averages and higher level fluctuations. We refer to this transformation as multiscale transformation. It is inverted by the inverse multiscale transformation (13).

4 Multiscale-Based Spatial Grid Adaptation To determine a locally refined grid we employ the above multiscale decomposition. The basic idea is to perform data compression on the vector of detail coefficients using hard thresholding as suggested by the cancellation property. This will significantly reduce the complexity of the data. Based on the thresholded array we then perform local grid adaptation where we refine a cell whenever there exists a significant detail, i.e. a detail coefficient with absolute value above the given threshold.. The main steps in this procedure are summarized as follows: Step 1: Multiscale Analysis. Let be vnL the cell averages representing the discretized flow field at some fixed time level t n on a given locally refined grid with highest level of resolution l = L. This sequence is encoded in arrays of detail coefficients d nl , l = 0, . . . , L − 1 of ascending resolution, see Figure 4, and cell averages on some coarsest level l = 0. For this purpose the multiscale transformation (12) need to be performed locally which is possible due to the locality of the mask coefficients. Step 2: Thresholding. In order to compress the original data we discard all detail coefficients dλ whose absolute values fall below a level-dependent threshold value εl = 2l−L ε . Let DnL,ε := λ ; |dλn | > εl , λ ∈ Il , l ∈ {0, . . . , L − 1} be the set of significant details. The ideal strategy would be to determine the threshold value ε such that the discretization error of the reference scheme, i.e., difference between exact solution and reference scheme, and the perturbation error, i.e., the

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Fig. 5 Grid adaptation: refinement tree (left) and corresponding adaptive grid (right)

difference between the reference scheme and the adaptive scheme, are balanced. For a detailed treatment of this issue we refer to [12]. Step 3: Prediction and Grading. Since the flow field evolves in time, grid adaptation is performed after each evolution step to provide the adaptive grid at the new time level. In order to guarantee the adaptive scheme to be reliable in the sense that no significant future feature of the solution is missed, we have to predict all significant details at the new time level n + 1 by means of the details at the old time level n. Let D˜ n+1 L,ε be the prediction set satisfying the reliability condition ˜ n+1 DnL,ε ∪ Dn+1 L,ε ⊂ DL,ε .

(14)

Basically there are two prediction strategies (i.e. ways of choosing D˜ n+1 L,ε ) discussed in the literature, see [15, 12]. Moreover, in order to manage grid adaptation this set is additionally inflated somewhat so that the grid refinement history, i.e. the parent-child relations of subdivided cells correspond to a graded tree, i.e. transitions between cells of different levels are sufficiently gradual. The connection with trees whose leaves form the grid partition and whose interior nodes are the refined cells will be addressed later again in some more detail. Step 4: Grid Adaptation. By means of the set D˜ n+1 L,ε a locally refined grid is determined along the following lines. We check the transformed flow data represented on D˜ n+1 L,ε proceeding levelwise from coarse to fine whether the detail associated with any cell marked by the prediction set is significant. If it is we refine the respective cell. We finally obtain the locally refined grid with hanging nodes represented by the index set G˜ n+1 L,ε . The flow data on the new grid can be computed from the detail coefficients in the same loop where we apply locally the inverse multiscale transformation (13).

5 Adaptive Multiresolution Finite Volume Schemes The rationale behind our design of adaptive multiresolution finite volume schemes (MR-FVS) is to accelerate a given finite volume scheme (reference scheme) on a

Adaptive Multiscale Methods for Flow Problems: Recent Developments

85

uniformly refined mesh (reference mesh) through computing actually only on a locally refined adapted subgrid, while preserving (up to a fixed constant multiple) the accuracy of the discretization on the full uniform grid. We shall briefly indicate now how to realize this strategy with the aid of the ingredients discussed in the previous section. The conceptual starting point is to rewrite the evolution equations (2) for the cell averages, i.e., vλ , λ ∈ IL , of the reference scheme in terms of evolution equations for the multiscale coefficients. For this purpose we apply the multiscale transformation (12) to the set of evolution equations (2). Then we discard all equations that do not correspond to the prediction set D˜ n+1 L,ε of significant details. Finally we apply locally the inverse multiscale transformation (13) and obtain the evolution equations for the ˜ n+1 cell averages on the adaptive grid G˜ n+1 L,ε which is obtained from DL,ε as explained before: n+1

vn+1 λ + Θ λλ (Bλ

n+1

n

n

− |Vλ | Sλ ) = vnλ − (1 − Θ ) λλ (Bλ + |Vλ | Sλ ),

(15)

n

n+1 /|V |. Here the flux balances B , the numerical for all λ ∈ G˜ n+1 λ λ L,ε where λλ := Δ t n n fluxes F λ and the source terms Sλ are recursively defined from fine to coarse scale via

∑

n

Bλ =

Γλl,μ ⊂∂ Vλ

∑

l,n

F λ ,μ = n

Sλ =

l,n

|Γλl,μ | F λ ,μ ,

l Γμl+1 ,ν ⊂Γλ ,μ

∑

l+1,n

|Γμl+1 ,ν | F μ ,ν = . . . =

ΓμL,ν ⊂Γλl,μ

|ΓμL,ν | F(vnL,μν , vnL,ν μ , nL,μν ), (17)

|Vμ | n |Vμ | n S(v μ ). Sμ = . . . = ∑ ,μ ∈Il+1 |Vλ | Vμ ⊂V , μ ∈IL |Vλ |

∑

Vμ ⊂Vλ

(16)

(18)

λ

We refer to (17) and (18) as exact flux and source reconstruction, respectively. Since in (18) we have to compute all sources on the finest scale, there is no complexity reduction, i.e., we still have the complexity of the reference grid. In order to gain in efficiency we therefore have to replace the exact flux and source reconstruction by some approximation such that the overall accuracy is maintained. This will be discussed in detail in Section 6. The complete adaptive scheme consists now of the following three steps: Step 1. (Refinement) Determine the prediction set D˜ n+1 L,ε from the data of the old time time step tn and project the data of the old time step to the pre-refined grid G˜ n+1 L,ε of the new time level, i.e., {vnλ }λ ∈Gn → {vnλ }λ ∈G˜ n+1 . Step 2. (Evolution) Evolve the cell averages associated to the pre-refined grid G˜ n+1 L,ε according to (15) where the numerical fluxes and sources are not necessarily determined by (17) and (18), respectively, i.e.,

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W. Dahmen, N. Hovhannisyan, and S. M¨uller

{vnλ }λ ∈G˜ n+1 → {vn+1 λ }λ ∈G˜ n+1 . L,ε

L,ε

Step 3. (Coarsening) Compress the data of the new time level by thresholding the corresponding detail coefficients and project the data to the (somewhat coarsened new) adaptive grid Gn+1 L,ε , i.e., n+1 {vn+1 λ }λ ∈G˜ n+1 → {vλ }λ ∈Gn+1 . L,ε

L,ε

The performance of the adaptive MR-FVS crucially depends on the threshold parameter ε . With decreasing value the adaptive grid becomes richer and, finally, if ε tends to zero, we obtain the uniform reference mesh, i.e., the adaptive scheme coincides with the reference scheme. On the other hand, the adaptive grid becomes coarser with increasing threshold value, i.e., the computation becomes faster but provides a less accuracte solution. An ideal choice would maintain the accuracy of the reference scheme at reduced computational cost. For a detailed analysis we refer to [12, 18] and explain here only the main ideas. In order to estimate the error, we introduce the averages uˆ nL of the exact solution, the averages vnL determined by the reference FVS and the averages vnL of the adaptive scheme prolongated to the reference mesh by means of the inverse multiscale transformation where non-significant details are simply set to zero. Ideally one would like to choose the threshold ε so as to guarantee that ˆunL − vnL ≤ tol where tol is a given target accuracy and · denotes the standard weighted l 1 -norm. Since vnL can be regarded as a perturbation of vnL this is only possible if L is chosen so as to ensure that the reference scheme is sufficiently accurate, i.e. one also has ˆunL − vnL ≤ tol. Again ideally, a possibly low number of refinement levels L should be determined during the computation such that the error meets the desired tolerance ˆunL − vnL ≤ tol. Since no explicit error estimator is available for the adaptive scheme, we try to assess the error by splitting the error into two parts corresponding to the discretization error τ nL := uˆ nL − vnL of the reference FVS and the perturbation error enL := vnL − vnL . We now assume that there is an a priori error estimate of the discretization error, i.e., τ nL ∼ hαL where hL denotes the spatial step size and α the convergence order. Then, ideally we would determine the number of refinement levels L such that hαL ∼ tol. In order to preserve the accuracy of the reference FVS we may now admit a perturbation error which is proportional to the discretization error, i.e., enL ∼ τ nL . From this, one can derive a suitable level L = L(tol, α ) and ε = ε (L). Therefore it remains to verify that the perturbation error can be controlled. To this end, note that in each time step we introduce an error due to the thresholding procedure. Obviously, this error accumulates in each step, i.e., the best we can hope for is an estimate of the form enL ≤ C n ε . However, the threshold error may be amplified in addition by the evolution step. In order to control the cumulative perturbation error, we have to prove that the constant C is independent of L, n, τ and ε . For this purpose, we have to choose a prediction strategy satisfying the reliability condition (14). In [15], a heuristic approach was suggested, taking into account the finite speed of propagation and the steepening of gradients that are characteristic

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87

for hyperbolic problems. So far the reliability condition (14) could not be rigorously verified for this approach. However, in [12] a slight modification of Harten’s prediction strategy was shown to lead to a reliable prediction strategy in the sense of (14) for homogeneous conservation laws using exact flux reconstruction. Recently, in [18] this could be extended to inhomogeneous conservation laws where an approximate flux and source reconstruction strategy was used.

6 Approximate Flux and Source Approximation Strategies As already mentioned above, the adaptive MR-FVS with exact flux and source reconstruction (17) and (18) will have the same complexity as the reference scheme performed on the reference mesh. If there is no inhomogeneity, i.e., s ≡ 0, then the complexity of the resulting algorithm might be significantly reduced from the cardinality of the reference mesh to the cardinality of the refined mesh. To see this we note that, due to the nestedness of the grid hierarchy and the conservation property of the numerical fluxes, the coarse-scale flux balances are only computed by the finescale fluxes corresponding to the edges of the coarse cell, see (17). Those in turn, have to be determined by the fine scale data. However, the internal fluxes cancel and, hence, the overall complexity is reduced. For instance, for a d-dimensional Cartesian grid hierarchy we would have to compute 2d 2(L−l)(d−1) fluxes corresponding to all fine-scale interfaces μ ∈ IL with ∂ Vμ ⊂ ∂ Vλ where λ ∈ Il , due to the subdivision of the cell faces. Note that in both cases missing data on the finest scale have to be determined where we locally apply the inverse two-scale transformation. This is illustrated in Figure 6. On the other hand, the coarse scale sources can be computed similarly with the aid of the recursive formulae (18). Here, however, we have to compute all sources on the finest scale which at the first glance prevents the desired complexity reduction. Hence the adaptive scheme with both exact flux and source reconstruction is useless for practical purposes. However, in the reliability analysis one may perform the adaptive scheme with some approximate flux and source recosntruction to be considered as a further perturbation of the “exact” adaptive scheme.

Fig. 6 Exact (left) versus local (right) flux and source computation

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W. Dahmen, N. Hovhannisyan, and S. M¨uller

In order to retain efficiency we therefore have to replace the exact flux and source reconstruction by some approximation such that the overall accuracy is maintained. A naive approach would be to use the local data provided by the adaptive grid, i.e., l,n

F λ ,μ = F(vnl,λ μ , vnl,μλ , nl,λ μ ),

n

Sλ = S(vnλ )

(19)

for λ , μ ∈ Il . So far, this approach is applied in Quadflow. Obviously, the complexity of the resulting adaptive MR-FVS is reduced to the cardinality of the adaptive grid. Unfortunately, this approach may suffer from serious loss in accuracy in comparison with the reference scheme. This is demonstrated by a performance study in Section 9.3 for a model problem. Recently, in [18] a new approach was suggested using an approximate flux and source reconstruction strategy along the following lines: Step 1. Determine for each cell Vλ , λ ∈ G˜ n+1 L,ε , a higher order reconstruction polynomial RNλ of degree N using only local data corresponding to the adaptive grid. Step 2. Approximate the boundary and volume integrals in (17) and (18) by some appropriate quadrature rules. Step 3. Compute fluxes and source terms in quadrature knodes by determining point-values or cell averages on level L of the local reconstruction polynomial RNλ , respectively. In [18], this concept has been analyzed in detail for the 1D case. In particular, it was proven that the accuracy of the reference scheme can be maintained when using the prediction strategy in [12] and judisciously tuning the parameters such as the reconstruction order and the quadrature rules. In Section 9.3, we will demonstrate that this new approach is superior to the naive approach with respect to accuracy and efficiency.

7 Multilevel Time Stepping For an explicit time discretization the time step size is bounded due to the CFL condition by the smallest cell in the grid. Hence Δ t is determined by the highest refinement level L, i.e., Δ t = τL . However, for cells on the coarser scales l = 0, . . . , L − 1 we may use Δ t = τl = 2L−l τL to satisfy locally the CFL condition. In [24] a multilevel time stepping strategy has been incorporated into the adaptive multiscale finite volume scheme. This strategy has been extended to multidimensional problems in [21, 20]. Here ideas similar to the predictor-corrector scheme [26] and the Adaptive Mesh Refinement (AMR) technique [5, 4] are used. The differences between the classical approaches and the multilevel strategy are discussed in detail in [24]. The basic idea is to save flux evaluations where the local CFL condition allows a large time step. The precise time evolution algorithm is schematically described by Fig. 7: In a global time stepping, i.e., using Δ t = τL for all cells, each vertical line section appearing in Fig. 7 (left) represents a flux evaluation and each horizontal

Adaptive Multiscale Methods for Flow Problems: Recent Developments

89 t n + τl−2 t n + 3 τl t n + τl−1 t n + τl tn

Fig. 7 Synchronized time evolution on space-time grid

line (dashed or solid) represents a cell update of u due to the fluxes. In the multilevel time stepping a flux evaluation is only performed at vertical line sections that emanate from a point where at least one solid horizontal line section is attached. If a vertical line section emanates from a point, where two dashed horizontal sections are attached, then we do not recompute the flux, but keep the flux value from the preceeding vertical line section. Hence fluxes are only computed for the vertical edges in Fig. 7 (right). In case of the multilevel time stepping we perform 2L intermediate time steps with step size τL , i.e., one macro time step corresponds to the time interval τ0 = 2L τL . On each intermediate time level (horizontal lines) u is updated for all cells. Since not all fluxes have to be recomputed, we can save significantly in CPU time. Furthermore, for each even intermediate time level, i.e., at t n + k τL for k ∈ {2, 4, . . . , 2L } we perfom the multiscale-based grid adaption but only for the levels l = lk , . . . , L where lk = min{l : 0 ≤ l ≤ L, k mod 2L−l = 0} is the smallest synchronization level. This partial grid adaptation procedure ensures that a discontinuity can be tracked on the intermediate time levels instead of a–priori refining the whole range of influence, see Fig. 7 (right).

8 FAS-Like Multilevel Scheme In the present work, we are interested in combining the multiscale-based grid adaptation with multigrid techniques to solve efficiently the nonlinear system (16) arising from the implicit time discretization of the underlying finite volume scheme. First work on adaptive multigrid techniques has been reported by Brandt [8, 9] who introduced the so-called multilevel adaptive technique (MLAT) that is an adaptive generalization of the full approximation scheme (FAS). The fast adaptive composite grid method (FAC) [14, 22] can be regarded as an alternative to the MLAT approach. An overview on multigrid methods can be found in the review book [31]. In contrast to classical adaptive multigrid schemes we employ the multiscale transformation (12) and (13) using biorthogonal wavelets to define the restriction and prolongation operators, respectively. Since the underlying problem is nonlinear we choose the FAS [8] for the coarse grid correction. Note that similar investigations have been published in [19] where classical AMR techniques are used for grid adaptation and the standard FAS method is extended to locally refined grids. The definition of composite residuals turned out to be crucial

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in this concept whereas they are easily determined from the multiscale analysis in our strategy. In order to solve the nonlinear system (16) arising from the implicit time discretization on locally refined grids in one evolution step we combine the FAS strategy [8] with the multiresolution analysis. The main ingredients are (i) the smoother to damp high frequencies, (ii) the restriction and prolongation operator to transfer data from coarse to fine and vice versa and (iii) the coarse grid problem to perform the coarse grid correction. All of them are operating on adaptively refined grids that are composed of cells in the underlying grid hierarchy. To describe them properly we have to distinguish between (i) the cells of the adaptive grid that are levelwise characterized by the index sets Gl ⊂ Il , l = 0, . . . , L, and (ii) the cells in the grid hierarchy that are being refined during the adaptation procedure; these are characterized by the significant details that are levelwise determined by the index sets Dl , l = 0, . . . , L − 1, and DL := 0. / Then the adaptive grid G is the union G = l=0,...,L Gl of the index sets Gl , l = 0, . . . , L. Furthermore the composite set T is composed of all cells in the adaptive grid and the cells characterized by significant details, i.e., it is the union T := l=0,...,L Tl of the composite index sets Tl := Gl ∪ Dl on level l = 0, . . . , L with Gl ∩ Dl = 0. / The above collection of cells and index sets, respectively, can be interpreted in terms of a graded tree where the adaptive grid G corresponds to the leaves of this tree and the non-leaves (interior nodes) correspond to the significant details D. The composite collection T is the union of both, i.e., the tree itself. For an illustration see Figure 5 (left). Note that we suppress the time index n for simplification of representation. Smoothing. To smooth the data on level l we perform μ Newton steps, i.e., N l (v(i) ) Δ v(i) = −N l (v(i) ) vnl ,

v(i+1) = v(i) + Δ v(i) ,

i = 0, . . . , μ − 1,

(20)

with initial data v(0) = vm l given by the mth FAS cycle . Here the nonlinear operator N l is determined by the discrete evolution equations (16) of the implicit finite volume scheme for the data corresponding to the composite grid Tl on level l, i.e., (N l v)λ = vλ +

τl (B − |Vλ | Sλ ), |Vλ | λ

λ ∈ Tl ,

where we usually use the naive flux and source reconstruction strategy (19). The linear systems is solved iteratively using GMRES with ILU(2) pre-conditioner. For this purpose we employ the PETSc software library of Argonne National Laboratory [3, 1, 2]. The iteration terminates when the residual is dropping below the tolerance tol = 1.e − 8 or the maximum number of 100 relaxation steps is exceeded. Restriction. Due to the nestedness of the underlying grid hierarchy the restriction l : Tl+1 → Dl is naturally defined by operator Il+1 vλ =

∑

μ ∈Mλ0

|Vμ | vμ |Vλ |

(21)

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according to (7). This relation holds for all cells. However, the restriction is performed on level l only for those cells that have been refined since we are working on locally adapted grids. These are characterized by the set Dl of significant details. Furthermore we note that by the restriction the adaptive grid Il on level l is inflated by the new data corresponding to Dl . This is the composite grid Tl on level l. Prolongation. For the prolongation of data μ ∈ Il+1 from level l to level l + 1 we employ the inverse two-scale transformation (13) where we put the details to zero, i.e., v μ = ∑ gl,0 (22) λ ,μ vμ . μ ∈G0μ

This prolongation can be considered a higher order polynomial reconstruction of fine grid data by coarse grid data provided the underlying wavelets have sufficiently high vanishing moments. Note that the prolongation operator Ill+1 : Dl → Tl+1 is only applied to cells of the composite grid Tl on level l that are refined according to the significant details Dl . Coarse Grid Problem. Let us assume that we have some approximation vl = (vλ )λ ∈Tl and vl−1 = (vλ )λ ∈Il−1 on level k = l − 1, l and some right hand side f l = ( fλ )λ ∈Tl . To set up the nonlinear problem on the coarser level l − 1 we first have to determine the residual of the nonlinear problem on level l, i.e., the defect. For this purpose we compute the nonlinear operator N l by means of the given data vl , i.e., τl (N l vl )λ = vλ + (B − |Vλ | Sλ ), λ ∈ Tl . |Vλ | λ Note that for the computation of the flux balances Bλ we resort also to data of the adaptive grid on coarser scales. Then the defect on level l is determined by d λ = fλ − (N l vl )λ ,

λ ∈ Tl .

The defect data should not be confused with the detail coefficients of the multiscale decomposition. Next we apply the restriction operator Ill−1 to the defect (array) d l and to the data vl , i.e., d λ = (Ill−1 d l )λ and vλ = (Ill−1 vl )λ , λ ∈ Dl−1 . Note that the restriction of the latter will not interfere with the given data vl−1 because Dl−1 ∩ Gl−1 = 0. / Therefore, we may concatenate the data on level l − 1, i.e., vl−1 = (vλ )λ ∈Gl−1 ∪Dl−1 . Furthermore we employ the same restriction operator for both the defect and the data. In other approaches, it is suggested to use different operators. We then determine the right hand side f l−1 on the coarse scale l − 1 by means of the coarse grid data vl−1 . For this purpose we first compute the nonlinear operator N l−1 τl−1 (N l−1 vl−1 )λ = vλ + (B − |Vλ | Sλ ), λ ∈ Tl−1 , |Vλ | λ

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where again we may access to data of the adaptive grid on coarser scales to compute the flux balances. Then the right hand side f l−1 is determined by fλ = d λ + (N l−1 vl−1 )λ ,

λ ∈ Dl−1 ,

(23)

for the cells on level l − 1 that are being refined and fλ = rλn ,

λ ∈ Il−1 ,

(24)

for the non-refined cells in the adaptive grid. Here rλn is the residual corresponding to the old time level t n , i.e., rλn = vnλ ,

λ ∈ Gl−1 .

(25)

Then the coarse grid problem is given by (N l−1 wl−1 )λ ≡ wλ +

τl−1 (B − |Vλ | Sλ ) = fλ , |Vλ | λ

λ ∈ Tl−1 .

(26)

Adaptive FAS Cycle. Finally, we describe one iteration step m → m + 1 of the adaptive multilevel cycle m vlm+1 = ADAPCYCLE(l, γ , vm l−1 , vl , N l , f l , μ1 , μ2 )

in terms of the above ingredients. Here we restrict ourselves to the adaptive twoscale case with given data vm k on level k = l (fine grid) and on level k = l − 1 (coarse grid) corresponding to Gl and Gl−1 , respectively. The iteration cycle is initialized by the data on the adaptive grid at time level t n . From these we compute the residuals rλn , λ ∈ Gk , according to (25) that are stored in the right hand side terms f k : We start with performing μ1 smoothing steps (20) on the data vm l of level l. Next we m perform the coarse-grid correction. For this purpose, we first compute the defect d l from the relaxed data vm l . The defect as well as the relaxed data are restricted from Tl to Dl−1 according to (21). Note that there exist data vm l−1 of the adaptive grid on level l − 1 that are complemented by the restricted data on Tl−1 . From this we compute the right hand side f m l−1 where we have to distinguish between cells of the adaptive grid Gl−1 and the refined cells Dl−1 on level l − 1, see (23) and (24). The coarse grid problem (26) is then iteratively solved by the Newton scheme (20) or, if there are additional scales, we recursively apply the algorithm again to the coarser scale l − 1. The current solution on the adaptive grid on level l − 1 is then replaced by the coarse grid solution wˆ m l−1 whereas for the refined cells Dl−1 the correction m m is computed. The latter is interpolated to T using (22) and the vˆ m = w ˆ − v l l−1 l−1 l−1 relaxed data are updated by the interpolation vˆ m . On the corrected approximation l m m+v vm,cgc = v ˆ we again perform μ smoothing steps. The algorithm is sketched 2 l l l in Figure 8.

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Fig. 8 FAS (l, l − 1) two-grid method

9 Numerical Results The performance resulting from the above concepts, namely, the multilevel time stepping, the multilevel solver and the flux and source reconstruction strategies, will be investigated now by means of different test configurations in 1D and 2D: an instationary flow over an oscillating plate, a steady state computation over a bump and the solution of the inhomogeneous, inviscid Burgers’ equation, respectively. For all computations the multiscale analysis is based on biorthogonal wavelets with M = 3 vanishing moments. For the prediction we apply Harten’s original strategy, cf. [15].

9.1 Multilevel Time Stepping: Oscillating Plate The multilevel time stepping strategy in combination with the naive flux reconstruction strategy (19) is investigated for an inviscid flow over an oscillating plate with prescribed deformation in time. The deformation is determined by w(t, x) =

α sin(2 π t β ) ∗ sin(π x/l) l

with amplitude α = 0.2, panel length l = 1 and frequency β = 1/2π . The flow domain extends from -5 to 5 in x-direction and from 0 to 5 in y-direction. At time t = 0 a periodic oscillation in the interval [0,1] is initiated at the the lower boundary. The simplicity of the geometry allows us to employ transfinite interpolation techniques for deforming the grid. Although the multiscale-based grid adaptation and the multilevel time stepping strategies have been outlined here only for stationary flow domains, these can be extended to moving grids by using an ALE formulation of the Euler equations, cf. [7, 20]. The flow enters the domain from the left hand side with free-stream conditions ρ∞ = 1.2929 [kg/m3 ], p∞ = 101325 [Pa], v∞ = (165.619, 0) [m/s]. The reference time is determined by tre f = 1./ p∞ /ρ∞ = 279.947 [m/s]. At the boundaries we impose slip conditions at the lower boundary and characteristic boundary conditions elsewhere because of the subsonic free-stream conditions (M∞ = 0.5). The grid is adapted after every timestep. The maximum refinement level is Lmax = 5, the threshold ε = 0.002, the coarsest grid consists of 1375 cells. After two cycles of

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1.4 multilevel global

1.2

Mach number

1 0.8 0.6 0.4 0.2 0 0

2

4

6

8

10

12

14

t/t_ref

Fig. 9 Mach number at bump midpoint

the boundary oscillation the number of grid cells varies around 40.000 grid points depending on the phase of the boundary movement. The bump is moving periodically up and down. When the bump is moving upwards a shock occurs at the leeward side because of the acceleration of the flow. The shock weakens and moves in upstream direction when the bump moves downward. This can be deduced from Figure 9 where the Mach number at the midpoint of the bump is plotted versus the dimensionless time t/tre f . When the shock is passing a steep gradient can be seen. The computation was carried out with both the global and the multilevel time stepping strategy. Although we performed no grid deformation step for the intermediate time levels in the latter case the accuracy of the solution is not affected as can be concluded from Figure 9. On the other hand, in comparison to a global time stepping strategy, we gain a factor of 3.7 in efficiency.

9.2 FAS-Like Multilevel Scheme: Bump Results of a 2D Euler transonic flow, considered in [27], are presented next in order to illustrate the convergence and efficiency of the multilevel strategy. The computational domain is defined by a circular arc bump in a channel with a secant of length l = 1[m] and a thickness of h = 0.024[m], see Figure 10. At the inlet boundary, the Mach number is 0.85 and a homogeneous flow field characterized by the free-stream quantities is imposed. At the outlet boundary, characteristic boundary conditions are used. We apply slip boundary conditions across the solid wall. Again, the multiscale analysis employs biorthogonal wavelets with the order M = 3 of vanishing moments. The threshold value in the grid adaptation step is ε = 2.5 × 10−3 and L = 5. Since we are dealing with a steady state problem, the time stepsize is determined locally for each cell by a time-dependent CFL number. For

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Slip Wall

Outflow

Inflow Slip Wall

Fig. 10 Circular arc bump configuration

Fig. 11 Temporal variation of the residual (left) and CFL number

the local flux computation we use the naive strategy (19). In each time step, we perform one FAS cycle to approximate the solution of the nonlinear problem. Our computation started on a structured grid corresponding to refinement level 1 that is determined by uniformly refining once the cells of the coarsest resolution of 24 × 8 cells, which span the entire computational domain, and we run to steady state. Additional refinement levels are added in response to time residual dropping or after a fixed number of time steps. In the present computation we enforced grid adaptation after 20 iterations for the first five adaptation steps. Then additional adaptations are performed as soon as the averaged residual of the density has dropped by a factor of 10−5 . Figure 13 (right) shows the computed pressure distributions after each adaptation step. At Mach 0.85 there is a compression shock separating a supersonic and a subsonic domain. The shock wave is sharply captured and the stagnation areas are highly resolved as can be concluded from the adaptive meshes shown in Figure 13 (left).

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Figure 11 shows the corresponding convergence history of the computation. The measure of convergence to steady state is the averaged residual for the density, i.e., |Vλ | n+1 (ρλ − ρλn ). | Ω | n+1

∑

λ ∈GL,ε

At the beginning the residual oscillates and decreases almost monotonically between two adaptation steps. After each grid adaptation it increases by several orders of magnitudes. This is caused by the thresholding that is performed within the multiscale analysis. After the 5th adaptation, the flow pattern is already established and the residual decreases more strongly. In total, the residual was reduced from 10−4 to 10−16 . For steady flows, the CFL number is controlled and varied between a minimum and maximum value during the computation. In the presence of shock waves, it is not possible to start the computation with a large CFL number directly due to the instationary behavior of the shock development. In the present work, computations were initiated with CFLmin = 10 and increased after each time step by a constant factor β = 1.05 until a maximum CFLmax is reached, i.e., CFL(t n ) = β CFL(t n−1 ). Figure 11(right) shows the history of the CFL number.

9.3 Local Versus Exact Flux and Source Reconstruction: Burger’s Equation In order to investigate the performance of the different flux and source reconstruction strategies discussed in Section 6, we conduct some parameter studies for a simplified 1D configuration. For this purpose, we consider the inhomogeneous, inviscid Burgers’ equation with flux f (u) = 0.5u2 , source s(u) = u (u − 0.5) (u − 1) and initial data u0 (x) = sin(2 π x). The computational domain Ω = [0, 1] is discretized by N0 = 10 cells on the coarsest level, i.e., h0 = 0.1. Hence the resolution for higher refinement levels is Nl = 2l N0 and hl = 2−l h0 . At the boundaries we use periodic boundary conditions. For the time discretization, we have to respect the CFL condition. Here we choose τ0 = 0.016. The final integration time is T = 0.24. Since we use global time stepping, the CFL condition has to hold for the smallest cells corresponding to the highest refinement level L, i.e., τ = 2−L τ0 . The explicit reference FVS (2) is determined by the Godunov flux. In order to improve spatial and temporal accuracy, we employ a piecewise linear ENO reconstruction, cf. [17]. For the source term, we apply the first order approximation (3). Computations have been carried out for several threshold values ε and different flux and source reconstruction strategies: (i) exact reconstruction strategy according to (17) and (18), (ii) flux and source computation on unstructured meshes using only local data corresponding to the adaptive grid according to (19) as is frequently

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Ma 1.50 1.40 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

t = 1.2680 · 2π Ma 1.50 1.40 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

t = 1.4090 · 2π Ma 1.50 1.40 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

t = 1.5499 · 2π Ma 1.50 1.40 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

t = 1.6907 · 2π Fig. 12 Time evolution of Mach number distribution and adaptive grid for flow over moving bump (multilevel time stepping, L = 5)

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L = 1: number of cells = 768

L = 2: number of cells = 2112

L = 3: number of cells = 4137

L = 5: number of cells = 7551 Fig. 13 Adaptive grid (left) and pressure contours over the bump (right) after each adaptation

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used in applications, cf. [7], and (iii) approximate reconstruction strategy using the midpoint rule and reconstruction polynomials of degree N = 2 with central stencil that are detailled in [18]. In the following, these are referred to as the original, naive and modified adaptive MR-FVS. The solution develops a shock at time t = 1/π at position x = 0.5 which is moving at negative speed due to the inhomogeneity. In Figure 14 we present the solution for the modified adaptive scheme for L = 10, ε = 10−3 by points at the cell center of the adaptive grid and the exact solution computed by the reference scheme on a uniform grid corresponding to L = 14. To investigate the influence of the different flux and source reconstruction strategies on the efficiency of the adaptive schemes, we have to consider the computational effort (memory and CPU time) and the accuracy (discretization and perturbation error) for various different threshold values. All adaptive computations are performed with L = 10 refinement levels. According to the ideal strategy in Section 5, the threshold value ε has to be chosen such that the discretization error τ L = uˆ L − vL of the reference scheme and the perturbation error eL = vL − vL are balanced. For L = 10 we obtain τ L = 5.8 × 10−4 where the “exact ” solution is obtained by the FVS on a uniform mesh corresponding to L = 14 refinement levels. First we consider the perturbation error due to thresholding plotted in Figure 17 for various threshold parameters. Obviously, the perturbation error is decreasing with smaller threshold values. In particular, eL → 0 for ε → 0+ , i.e., the modified adaptive scheme converges to the reference solution obtained on the reference grid with L refinement levels. Of course, we do not gain in accuracy when choosing a very small threshold value because the discretization error is fixed by the number of refinement levels. To determine the optimal threshold value, we plot the error ˆuL − vL of the adaptive scheme for different threshold values, see Figure 16. From this, we conclude that an optimal choice would be εopt ∈ [10−5 , 10−4 ] because the error of the adaptive schemes is decreasing with decreasing threshold value ε as long as ε > εopt whereas it stalls for ε < εopt . Hence, for ε > εopt the perturbation error due to thresholding dominates whereas for ε < εopt the discretization error dominates. The above observations concerning the discretization and perturbation error hold true independently of the adaptive scheme. However, for a threshold value εopt in the optimal range we see in Figures 17 and 16 that the highest accuracy is obtained with the original adaptive scheme. The modified adaptive scheme looses a bit in accuracy, but for the naive adaptive scheme the loss is much more severe. To draw any conclusions concerning the efficiency of the different adaptive schemes, we have to take the computational cost into account. First we discuss the size of the adaptive grids that determine the memory requirements, see Figure 18. We observe that the minimal grid size is usually obtained for the original adaptive scheme whereas for the naive and the modified adaptive scheme we need more cells. This might be caused by small oscillations induced by the reconstruction error. This becomes more severe in the case of the naive adaptive scheme when the threshold value is chosen too small, i.e., ε < εopt .

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1

-1

0

-1.5

u

log(error)

-1 -2

naive modified original

-2

-2.5

-3 -3

exact adaptive

-4 0

0.2

0.4

0.6

0.8

-3.5

1

0

100

200

x

300 400 CPU time [s]

500

600

Fig. 14 Comparison of adaptive solution Fig. 15 Comparison of CPU time and er(L = 10, ε = 10−3 ) and exact solution (L = ror of adaptive scheme for different threshold 14, ε = 0) values -1

-1 naive modified original

-1.5

naive modified original

-2

log(error)

log(error)

-3 -2

-2.5

-4 -5

-3

-6 -7

-3.5 1

2

3

4

5 -log(eps)

6

7

8

1

9

2

3

4

5 -log(eps)

6

7

8

9

Fig. 16 Error of adaptive solution with L = Fig. 17 Perturbation error of adaptive solution (L = 10, varying threshold value ε ) and 10 and varying threshold value ε reference solution (L = 10, ε = 0) on reference grid (L = 10)

10000

1000 naive modified original

CPU [s]

# cells

100 1000

10 naive modified original

100 1 1

2

3

4

5 6 -log(eps)

7

8

9

1

2

3

4

5 -log(eps)

6

7

8

9

Fig. 18 Number of cells: Adaptive compu- Fig. 19 Computational time: Adaptive comtations with L = 10 and varying threshold putations with L = 10 and varying threshold value ε value ε

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Finally, we consider in Figure 19 the computational time. The CPU time needed for the original adaptive scheme is much higher as long as the threshold value is not too small. This is caused by the source term computation on the uniform reference grid dominating the overall cost for grid adaptation and time evolution. In case of the naive and the modified adaptive scheme, the adaptive grid becomes more dense with decreasing threshold values, i.e., more cells are refined, and the cost approaches the cost of the reference computation on the reference grid. Of course, this behavior is expected for any adaptive scheme. To summarize the above observations, we conclude that for an optimal threshold value εopt the exact strategy is most accurate but at the cost of the reference computation, i.e., there is no gain at all. For the naive adaptive scheme, we observe a severe loss in accuracy at lower computational cost in comparison to the the modified adaptive scheme. This loss can only be compensated by a smaller threshold value at higher computational cost. From this point of view, the approximate strategy is more efficient when fixing the target accuracy by the discretization error, i.e., log(τ L ) = −3.24, see Figure 15. Finally, we wish to point out that in practice the optimal threshold value εopt can only be roughly estimated and, hence, the use of the local strategy cannot be recommended: we either (i) loose significant accuracy if ε εopt , see Figures 16, or (ii) the computational cost (memory) is significantly higher due to instabilities triggered by the increasing influence of the reconstruction error if ε εopt , see Figure 18.

References 1. Balay, S., Buschelmann, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: Petsc web page, Technical report (2001), http://www.mcs.anl.gov/petsc 2. Balay, S., Buschelmann, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: Petsc users manual. Technical report anl-95/11 - revision 2.1.5, Argonne National Laboratory (2004) 3. Balay, S., Gropp, W.D., McInnes, L.C., Smith, B.F.: Efficient management of parallelism in object oriented numerical software libraries. In: Arge, E., Bruaset, A.M., Langtangen, H.P. (eds.) Modern Software Tools in Scientific Computing, pp. 163–202. Birkh¨auser Press, Basel (1997) 4. Berger, M.J., LeVeque, R.J.: Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35(6), 2298–2316 (1998) 5. Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comp. Physics 53, 484–512 (1984) 6. Bramkamp, F., Gottschlich-M¨uller, B., Hesse, M., Lamby, P., M¨uller, S., Ballmann, J., Brakhage, K.-H., Dahmen, W.: H-adaptive Multiscale Schemes for the Compressible Navier–Stokes Equations — Polyhedral Discretization, Data Compression and Mesh Generation. In: Ballmann, J. (ed.) Flow Modulation and Fluid-Structure-Interaction at Airplane Wings. Numerical Notes on Fluid Mechanics, vol. 84, pp. 125–204. Springer, Heidelberg (2003) 7. Bramkamp, F., Lamby, P., M¨uller, S.: An adaptive multiscale finite volume solver for unsteady an steady state flow computations. J. Comp. Phys. 197(2), 460–490 (2004)

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Interaction of Wing-Tip Vortices and Jets in the Extended Wake Frank T. Zurheide, Guido Huppertz, Ehab Fares, Matthias Meinke, and Wolfgang Schr¨oder

Abstract. The interaction of wing-tip vortices and jets in the extended wake is experimentally and numerically investigated. The measurements focus on the unsteady wake of a swept-wing half-model equipped with an engine jet and on the analysis of meandering vortex. The aircraft engine is modeled by a cold jet driven by pressurized air. To investigate the influence of the location of the engine jet on the vortex wake, it is mounted in two different positions under the wing model. The spatial development of a vortex wake behind a wing is simulated up to the extended near field. The measurements are used as inflow distribution for a large-eddy simulation (LES) of the wake region. To better capture the motion of the wake vortices a method for hexahedral block structured adaptive mesh refinement with vertex-centered fluxes is introduced. The numerical simulations of the wake are able to predict trajectories and instabilities of the vortex core. The closer the engine is located near the root of the wing, the smaller is the deflection of the vortex and the fewer wave modes of the vortex are excited. The meandering motion of the vortex core is triggered by the engine jet.

1 Introduction The limited airport capacity due to wake vortex fields is of growing importance due to the expected passenger increase and aircraft traffic. Heavy transport aircraft generate vortex wakes which endanger the following airplane, since the high momentum flow structure cannot be precisely detected. The decay of vortex wakes by diffusion and dissipation mechanisms is weak such that the frequency of taking-off and landing aircraft is determined by the decay of the vortex wake. Instabilities of wake vortex systems, however, can lead to a more rapid decay of the vortex strength. Frank Thomas Zurheide · Guido Huppertz · Ehab Fares Matthias Meinke · Wolfgang Schr¨oder Institute of Aerodynamics, RWTH Aachen University, W¨ullnerstr. 5a, 52062 Aachen, Germany e-mail: {f.zurheide,g.huppertz, m.meinke,office}@aia.rwth-aachen.de W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 105–135. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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For many years, scientific research focuses on the evolution of instabilities inherent to or artificially introduced into the wake structure [53]. The aim is the excitation of mechanisms which lead to an amplification of different types of instabilities and therefore reduce the time required for the onset of rapid vortex decay. In vortex systems several three-dimensional instabilities are known. Mutual straining of the vortices leads to an amplification of asymmetric Kelvin waves if three-dimensional perturbations exist. Due to this amplification vortex systems are generally unstable. Using a few assumptions concerning the vortex core radius and vortex spacing, the vortex system behind wings can be described by a set of parallel vortex filaments. The perturbations are Kelvin waves at small amplitude. From this model a system of stability equations can be derived. The results of which were presented by Crow [10] for a single pair of counter-rotating vortex filaments. The wave length of the Crow instability is λ /b ≈ 7.85. Crouch [9] and Fabre et al. [13, 14] performed a similar stability analysis for multiple vortex pairs. Stability analyses of a vortex pair using the vortex filament method were presented by Waleffe [60]. The basic findings is a higher growth rate for short-wave elliptical instabilities than for long-wave instabilities. The existence of this instability has been shown in experiments for counter-rotating vortex pairs by Leweke and Williamson [34] and co-rotating vortex pairs by Meunier et al. [41, 42, 43]. Numerical simulations of this instability were performed by Laporte and Corjon [30], Laporte and Leweke [31] as well as Le Diz`es and Laporte [33]. Sipp [52] studied the nonlinear dynamics of the instability. It goes without saying that viscous effects change the behavior of the vortex system. The sinusoidal instability saturates at a low level and does not lead to a breakdown of the system. Consequently, the slower growing long-wave instability, i.e., Crow instability, is of primary interest. One possible technology to influence the vortex wake, which is more or less readily available, is the engine jet. There are, however, only few studies in which the interaction of an engine jet and a trailing vortex in the near field of the wake is analyzed. The growth of instabilities in a jet is well understood. Loiseleux et al. [35] considered the effects of swirl on jets and wakes and Lu and Lele [36] investigated the inviscid instabilities for swirling mixing layers. Miake-Lye [44] identified two distinct phases of the jet-vortex interaction. In the first phase, the jet regime, the engine jets mix with the ambient air and the vortices roll up to a vortex pair. In the second phase, the interaction regime, the entrainment of jets into the vortex flow dominates. Unlike this model, experiments from Brunet et al. [8] showed that at a distance of only one semispan downstream of the wing the jet was already strongly entrained. The influence of the jet on the tip vortex was analyzed with temporal simulations by Labb´e et al. [29], Paoli et al. [47], Ferreira Gago et al. [19], and Holz¨apfel et al. [24]. Unlike these temporal investigations, Stumpf [56, 57] and Fares and Schr¨oder [16] simulated a spatially developing wake-jet interaction in the near field based on the inviscid or viscous conservation equations. Vortex meandering, also known as vortex wandering, was introduced by Baker et al. [3] in 1974. Devenport [12] developed strategies to filter out the unsteadiness and to correct the time-averaged vortex position. Jacquin et al. [26] explain

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vortex meandering by the amplification of bending waves through external freestream fluctuations and linear co-operative instabilities. Beninati and Marshall [4] found no evidence for vortex meandering, but a bending wave resulting from the interaction with vortex turbulence. The study of the literature shows that there is a vast amount of papers available dealing with fundamental and applied phenomena in aircraft wakes. However, compared with the total number of references there is little research results available as far as the impact of the engine jet on the wake dynamics and the role of perturbations generated in the very near wake on the overall vortex decay are concerned. These issues are experimentally and numerically addressed in this contribution. This article is organized as follows. The governing equations, the LES method, and the preconditioning are briefly described in Sect. 2. In Sect. 3 we will present a one-equation turbulence model that is derived from the k-ω turbulence model. A data communication method for adaptive mesh refinement (AMR) with vertexcentered fluxes in hexahedral block-structured meshes is introduced in Sect. 4. This method exchanges the fluxes among different blocks of the mesh while the control volumes are kept hexahedral and the stencils are not changed. A Reynolds-averaged Navier-Stokes (RANS) simulation of the wake of a rectangular wing with engine jets mounted at different positions and operating at different velocities is performed. The influence of different positions and velocities is investigated in Sect. 5.1. In Sect. 5.2 experimental methods and the measurement set-up used to capture the unsteady vortex movement are described. Averaging methods customized for unsteady vortex measurements are presented. Results based on large-eddy simulations (LES) are discussed in Sect. 5.3. The development of short wave and long wave instabilities for a counter-rotating vortex pair are described in Sect. 5.3.1. The LES of spatially developing wake-jet interaction in the near field and the extended near field are presented in Sect. 5.3.2. Measurements from the wake are used as inflow data for the simulations. The flow fields for two different positions of an engine jet are analyzed and compared. With the spatially simulation of the wake a strong meandering of the vortex core can be found. Finally, the major findings are summarized in Sect. 6.

2 Governing Equations and Numerical Method The Navier-Stokes equations formulated in tensor notation and in dimensionless conservative variables read in Cartesian coordinates

∂Q + (FCβ − FD β ),β = 0 , ∂t

Q = [ρ , ρ uα , ρ E]T .

The quantity Q denotes the vector of the conservative variables, FCβ represents the vector of the convective, and FD β of the diffusive fluxes ⎛

⎞ ⎞ ⎛ ρ uβ 0 1 ⎝ ρ uα uβ + pδαβ ⎠ − ⎠. ⎝ σ αβ FCβ − FD β = Re uα σ αβ + qβ uβ (ρ E + p)

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The stress tensor σ αβ is written as a function of the strain rate tensor Sαβ , i.e., 1 σ αβ = −2 μ (Sαβ − Sγγ δ αβ ) 3 with

1 Sαβ = (uα ,β + uβ ,α ) . 2 The dynamic viscosity μ is assumed to be a function of the temperature only. Fourier’s law of heat conduction is used to compute the heat flux qβ qβ = −

k T , Pr(γ − 1) ,β

where Pr = μ c p /k is the Prandtl number. The system is closed using the equation of state for ideal gases 1 p = ρT , γ

ρ p = (γ − 1) ρ e − uβ uβ , 2

where γ is the ratio of specific heats and T is the temperature. To achieve a high accuracy of the unsteady flow motion the LES approach is applied. Most of the anisotropy and flow-specific motion are contained in the large scales and are directly calculated whereas the small scales are represented by a subgrid-scale (SGS) model, which ensures the energy drain from the resolved scales. The scale separation is achieved by applying a low-pass filter to the Navier-Stokes equations for an ideal gas. The filtering operation is implicitly performed by the discretization method, where the filter width is twice the grid spacing [50]. Similar to the monotone integrated large-eddy simulation (MILES) approach [18], where the scheme is constructed such that its truncation error mimics a SGS model, the inherent dissipation of the present scheme which is a mixed central-upwind advective upstream splitting method (AUSM) scheme [40] is used to remove energy from the smallest resolved scales. The viscous fluxes are discretized by a central scheme and the solution is advanced in time using an explicit five-stage Runge-Kutta scheme. Overall, the approximation is second-order accurate in space and time. The temporal development of vortical wakes are characterized by low Mach numbers. This low Mach number results in an inefficient solution of the compressible Navier-Stokes equations with the above described explicit temporal schemes. To achieve a more efficient and likewise accurate temporal development of the vortex systems implicit dual time-stepping, multigrid methods, and preconditioning methods [2] were developed and implemented in the LES algorithm. With this method convincing results could be reached for a variety of internal [50] and external turbulent flows [20].

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3 New One-Equation Turbulence Model The intention of the one-equation turbulence model of Fares and Schr¨oder [15, 17] is the introduction of a one-equation turbulence model which reliable descirbes vortical jet flows. The model correctly predicts a wide class of fundamental flows like boundary layer, wake, jet, and vortical flows. The basis of the model is the mature and validated two-equation k-ω turbulence model of Wilcox [62, 63]. A transport equation for νt can be derived using the definition of the eddy viscosity k = νt ωt . The turbulent dissipation rate ωt 1 is reconstructed according to the Bradshaw hypothesis [7]. This hypothesis states for a 2-D flow that the Reynolds shear-stress tensor τ txy is proportional to the turbulent kinetic energy k

τtxy = βb k

with

βb = 0.3 .

This hypothesis was validated by Townsend [59] through various measurements in boundary layers, mixing layers, and wakes. The hypothesis states that the turbulent dissipation rate ωt is given by k 1 du ωt = = νt βc∗ dy and generalized using the norm of the strain tensor Si j

1 ∂ ui ∂ u j . Si j = + 2 ∂ x j ∂ xi According to Nagano et al. [46] ωt can be determined by 1 ωt = 2Si j Si j . βc∗ The closure coefficients and functions are similar to the k-ω model. The near-wall behavior is like that of the Spalart-Allmaras [54] model, but no wall distance d is needed. Additionally, the decay of isotropic turbulence is modeled [17]. The turbulence model calculates the kinematic eddy viscosity

νt = νˇ fv1 where νˇ and fv1 are defined by fv1 =

χ3 χ 3 + c3v1

and

χ=

νˇ . ν

The transport equation for the dependent variable νˇ is given by 1

The turbulent dissipation rate ω is denoted by ωt . This is done to prevent confusion with the vorticity ω = 1/2∇ × v. Nevertheless, the original model is still called the k-ω turbulence model.

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Dνˇ νˇ ∂ ui 2 ∂u ˇ ij i − (1 − α )νδ = 2(1 − α ) Si j Dt ωt ∂ x j 3 ∂xj ! "# $ production

νˇ 5 ˇ t − kinit φ − (β ∗ − β )νω ! "# $ νinit ! "# $ destruction decay

∂ ∂ νˇ ν + σ νˇ ∂ νˇ ∂ ωt + +2 (ν + σ νˇ ) ∂xj ∂xj ωt ∂ x j ∂ x j "# $ ! "# $ ! diffusion 1

diffusion 2/destruction

with the closure coefficients

σ = 0.5

α = 0.52

βc = 0.072

cv1 = 7.1

βc∗ = 0.09 φ = (βc − βc∗ )1/5

and the closure functions

β ∗ = βc∗ fβ∗

1 + 1610ψ fβ∗ = 1 + 1195ψk k

1 + 64ψω fβ = 1 +80ψω & 2 ∂ ω ∂ ω ˇ 1 ∂ ν t t ψk = max 0, 3 νˇ + ωt ∂xj ∂xj ∂xj ωt ω ω ω ψω = i j ∗ jk 3ki ωt = 1 ∗ 2Si j Si j . (β0 ωt ) βc

β = βc f β

%

The initial and boundary conditions for νˇ are also comparable to those used by the Spalart-Allmaras model ∂ νˇ νˇ init ≈ 0.1ν νˇ wall = 0 =0 ∂ n in f low/out f low with the initial turbulent kinetic energy kinit = 3/2u2∞Tu2 |init . In Table 1 the right-hand side terms of the Spalart-Allmaras [54] and the new turbulence model are compared. In the Spalart-Allmaras model turbulence is mainly produced through vorticity, whereas in the new model it is produced through shear stress. The destruction terms are completely different, since there is no dependence on the wall distance d in the new model. The new model possesses a more difficult formulation of the destruction term which dependends on closure functions. The first diffusion term is almost identical in both models except for the coefficients. In the Spalart-Allmaras model a second diffusion term depends only on the gradient of νˇ but in the new model it is a function of the gradient of ω . Unlike in the

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Table 1 Comparison between right-hand side terms of the Spalart-Allmaras [54] and the new turbulence model

Term Production Destruction Diffusion 1 Diff. 2 / Destr. Decay

Spalart-Allmaras Model new Model

Si j ∂ u i 1 ∂ u i νˇ cb1 νˇ 2ωi j ωi j + 2 2 2(1 − α )νˇ − δij κ d ωt ∂ x j 3 ∂ x j

2 ˇν ˇ t cw1 fw (β ∗ − β )νω d 1 ∂ ∂ νˇ ∂ ∂ νˇ (ν + σ νˇ ) (ν + σ νˇ ) σ ∂xj ∂xj ∂xj ∂xj cb2 ∂ νˇ ∂ νˇ ν + σ νˇ ∂ νˇ ∂ ωt 2 σ ∂xj ∂xj ωt ∂ x j ∂ x j

νˇ 5 — kinit φ νinit

Spalart-Allmaras model the quantity ω is reconstructed through shear stress and plays a major role in the production, destruction, and diffusion terms. Finally, the new model is extended to account for the decay of turbulence, a feature that is missing in the Spalart-Allmaras model.

4 Adaptive Mesh Refinement The concept of adaptive mesh refinement (AMR) is useful for an efficient numerical solution of partial differential equations (PDE). The highest required mesh resolution is given by the highest gradient in the computational domain. Without a partially refined mesh the entire computational domain has the highest mesh resolution. At mesh at a higher mesh resolution than required results in a waste of computer resources, i.e., CPU time, while the use of a mesh at lower mesh resolution than required causes a bad or even wrong solution. Therefore, the mesh has to be adapted to the solution [55]. AMR can be divided into four parts. The error estimation function decides which nodes of the mesh have to be refined or coarsened. The error estimate can be computed directly, which is, however, expensive or a cheaper heuristic function can be used instead [11]. The Domain decomposition determines the number of levels, the number of subgrids, the size of the subgrids, the used clustering technique, shape and form of the grids, and the skewness of the subgrid. Some highly efficient domain decomposition algorithms have been presented by Berger and Oliger [5] and Berger and Rigoutsos [6]. Parallelization and partitioning covers data distribution (partitioning) over the processors. The aim is to achieve a good load balance for the dynamically changing grid structure [28, 49, 61]. The last part of AMR, data communication, arranges the communication pattern, which will most definitely be quite

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complex and varying. In the following, we will focus on the data communication. For the other three parts of AMR, the reader is referred to the above given references.

4.1 Data Communication 4.1.1

Discontinuous Internal Boundary Conditions

If a mesh is split in several blocks the solution at the block boundaries must be exchanged between the neighboring blocks. Usually ghost cells are added on the boundaries of the mesh to solve the discretized equations on the mesh with an uniform stencil. Adaptive refinement in structured hexahedral meshes is achieved through element sub-division and refinement or coarsening of the sub-blocks. If the new parted mesh should consist only of hexahedral meshes, this refinement of a mesh causes hanging nodes at the interfaces between refined and non-refined mesh regions, as shown for 2-D in Fig. 1 and indicated for 3-D in Fig. 2. In the following, the nomenclature introduced by T´oth and Roe [58] is used. Values on the coarse mesh are labeled with uppercase letters and on the fine mesh with lowercase letters. The coarse vertex I, J, K and the fine vertex i, j, k are at the same position on the hanging node interface. To simplify the description of the method it is assumed that the interface in 3-D has constant minimum values for k = kmin and K = Kmin although the position and alignment of the interface can be arbitrary on the surface of the mesh. Coinciding vertices of the coarse and fine mesh, e.g. N1 , N7 , N8 in Fig. 2, should have the same value for the conservative variables Q and q on the coarse and fine mesh QI,J,K = qi, j,k . The values of the hanging nodes on the fine mesh are linearly interpolated from the values of the surrounding nodes. For the node N2 the linear interpolation of the four surrounding nodes N3 , N4 , N5 , N6 results in

J

Q

Q

I,J

I+1,J

I q

i,j

q

i+1,j

q

i+2,j

J j I i

N7

N6

N3

N2

N4

N1

N5

N8

j i

Fig. 1 2-D hanging node interface. Lower block with finer resolution than upper coarser block. Hanging node at e.g. qi+1, j

Fig. 2 3-D hanging node interface at constant k-plane. Finer block in gray and coarser block in black. Hanging nodes at N2 , N3 , · · · , N6

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qi, j,k =

113

1 qi−1, j,k + qi+1, j,k + qi, j−1,k + qi, j+1,k . 4

It is required that the method conserve fluxes, so the fluxes on the fine side of the interface must add up to the fluxes on the coarse side FβI,J,K = ∑ fβi, j,k , A

where A is the area of the control volume of node I, J, K of the coarse mesh. While the above described formulas hold for both, the cell-centered and the vertex-centered fluxes, the summation of fluxes must distinguish between these two. 4.1.2

Cell-Centered Fluxes

If the discretization of the fluxes is cell centered the handling of these discontinuous interfaces is straightforward. The fine fluxes are stored at i + 12 , j − 12 and I + 12 , J + 12 at the coarse mesh. In 3-D the coarse cell has four refined neighboring cells, so the fluxes sum up to Fβ

I+ 12 ,J+ 21 ,K+ 12

= fβ fβ

i+ 21 , j+ 21 ,k+ 21

+ fβ

i+ 21 , j+ 21 +1,k+ 12

i+ 21 +1, j+ 12 ,k+ 21

+ fβ

i+ 21 +1, j+ 12 +1,k+ 12

.

To calculate the flux of a fine cell from the flux of a coarse cell the fluxes are just split, e.g. 1 = Fβ 1 1 1 . fβ 1 i+ 2 +1, j+ 21 ,k+ 21 4 I+ 2 ,J+ 2 ,K+ 2 Other sub-division results for isotropic refinement of hexahedral element (eight smaller hexahedral elements) and non-isotropic refinement (hexahedra, pyramids, prisms, and tetrahedra) are presented in detail by Mavriplis [38]. 4.1.3

Vertex-Centered Fluxes

For vertex-centered fluxes control volumes for the fluxes Fβ , fβ at the hanging node interface must be constructed in a more tedious way than for the cell-centered fluxes. Few approaches can be found for hanging node interfaces on structured meshes with vertex-centered fluxes [1, 32]. Unfortunately, these lead to control volumes which are not hexahedral or have non-uniform stencils. We will now introduce a new algorithm for the hanging node interface on structured multi-block meshes that use hexahedral control volumes. The location and extent of hanging node interfaces are known a-priori. Thus, the computation of combined control volumes for the boundary cells of the hanging node interface is possible, although there are some difficulties to overcome. The fluxes in one cell on the interface are exchanged at the same time with cells from up to seven neighboring sides and meshes, respectively. The exchange depends on the adjacent sides of the mesh and not only – like in the case of the cell-centered

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(a) Control volumes for a point inside the interface

(b) Control volume for a point at a boundary point of the interface

Fig. 3 3-D vertex-centered fluxes in a hanging-node interface at constant k-plane. Displayed are control volumes for the fine mesh (///) and the computed control volume for the coarse mesh (×××)

fluxes – from the cell itself. Hence, the exchange algorithm has to distinguish between three different types of connections: physical boundary condition (e.g. wall, inflow, outflow), (internal) mesh connection, and (internal) hanging node interface. To keep the modification in the flux exchange routines in the existing flow solver at a minimum a new algorithm has been developed. First, the fluxes are coarsened on the interface of the fine mesh to obtain a flux which is equivalent to the flux on a coarsened grid. The coarsening of fluxes is performed independently for each side of a block. This step is necessary since neither the exact order of the exchange of the fluxes from fine to coarse or vice versa nor the execution order of the interfaces is exactly known when this step is executed. At a second step the fluxes and the conservative variables are exchanged. Due to the coarsening of the fluxes this step requires nearly no adaptation in the exchange routines. At a third step, the fluxes on the fine grid are refined to define the fluxes at the hanging nodes. First, the computation of a coarse control volume inside a hanging node interface without consideration of boundaries and corners is presented. This case is displayed in a view from the finer to the coarser mesh in Fig. 3a. The flux of the coarsened control volume consists of the following parts: the flux at position i, j, k, half of the flux of the neighboring cell fluxes in the orthogonal direction, and a quarter of the flux of the neighboring diagonal cells 1 FβI,J,K = fβi, j,k + fβi−1, j,k + fβi+1, j,k + fβi, j−1,k + fβi, j+1,k 2 1 + fβi−1, j−1,k + fβi+1, j−1,k + fβi−1, j+1,k + fβi+1, j+1,k . 4 Fluxes at a left interface point i = imin are displayed in Fig. 3b, where the edge imin is a physical boundary. The new flux FβI,J,K consists of the flux fβi, j,k and half of the

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lower and upper-cell fluxes at j ± 1 plus parts of the fluxes of the three cell neighbors in the next row i + 1 1 1 FβI,J,K = fβi, j,k + fβi, j−1,k + fβi, j+1,k + fβi+1, j,k 2 2 1 + +f . f 4 βi+1, j−1,k βi+1, j+1,k The fluxes for a corner point with hanging node interfaces at i = imin and j = jmin and an internal block connection at k = kmin sum up to 1 1 1 1 FβI,J,K = fβi, j,k + fβi+1, j,k + fβi, j+1,k + fβi+1, j+1,k 2 2 4 4 1 1 + fβi+1, j,k+1 + fβi+1, j+1,k+1 . 8 16 Note that only some simple configurations are described. The other cases are computed likewise. The given parameters for the flux computation are in a few cases comparable to the basis functions given by McDill et al [39]. However, the construction of fluxes at the hanging node interface is much more complicated than the computation of the basis functions for a finite-element method (FEM) since they depend on the topology of the neighboring meshes and the boundary conditions. After the equivalent coarse fluxes are computed from the fine fluxes the exchange can be performed. Then, the fluxes have to be split to get the correct fluxes on the fine mesh. At a regular point inside the interface the value of the coarse flux is four times larger than the fluxes on the fine mesh (Fig. 3a), so the fine flux is computed by 1 fβi, j,k = FβI,J,K 4 and the values on the hanging nodes are interpolated. The points on the boundaries and corners of the interface are handled similarly. With this hanging node interface it is possible to massively reduce the number of mesh points. Therefore the influence of the boundaries on the solution could be minimized through increasing the computational domain while keeping the resolution of the vortex core constant.

5 Results Based on today’s computer perfomance a full LES of the flow over a wing-engine configuration is definetely beyond any acceptable computing time. Therefore, a RANS simulation of the flow over a wing with engine jets is presented in Chap. 5.1. Measurements and analysis of unsteady jet-vortex interaction are presented in Chap. 5.2. Chapter 5.3 covers temporal and spatial development of wake vortices.

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5.1 RANS Simulation of the Flow over a Rectangular Wing with Engine Jets 5.1.1

Flow Configuration

A RANS simulation of the flow over a rectangular wing and its wake is performed. The rectangular wing has a BAC 3-11 profile [45] and an aspect ratio of Λ = 9.0. The half wingspan is b/2 = 4.5c where c defines the chord. The engine geometry is similar to the Trent 900 engine. In the experimental investigations which are discussed in Chap. 5.2 three positions of the model engine can be chosen, each 0.4c apart, with a minimum distance of 1.1c from the wingtip (Fig. 4). The positions are numbered “A”, “B”, and “C”, where “A” is nearest the wing tip. A structured multiblock grid is used for the simulation of the wing and the engine flow. The computational grid consists of 74 blocks and 1.1 million nodes. The freestream Mach number is set to Ma∞ = 0.18, the Reynolds number based on the chord length is Rec = 268250, and the angle of attack is α = 8.0◦ . The ratio of jet velocity to freestream velocity is u jet /u∞ = 1.77 and 2.87. To compute the wake of the wing the results of the simulation of the flow over the rectangular wing with engine jet is interpolated onto a rectangular grid starting at x/b = 1.1 consisting of 50×334×601 grid points.

Fig. 4 Geometry, coordinate system and positions of the engine

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5.1.2

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Wake Flow

The three essential flow structures in the wake are the shear layer, the wingtip vortex, and the engine jet. Qualitative representations of these structures are presented using contours of streamwise velocity in Figs. 5 and 6 for the two engine positions “A” and “C”. These figures evidence the roll-up of the shear layer and thus, the formation of the wing-tip vortex. The shear layer undergoes a strong deformation especially near the engine jet. The engine jet, which does not have a circular shape in the inflow plane, also deforms continuously and interacts with the shear layer. The illustrations, e.g. Fig. 6a, indicate through the wavy shape of the shear layer an unstable behavior. From the visualization it is evident that a more perturbed shear layer is observed for the engine in position “C” than in position “A” (Figs. 5b and 6b). However, the comparison of the varying jet velocities in Figs. 5 and 6 show that the essential difference is not caused by the engine positions but by the jet velocity. The direction of the jet is hardly influenced by the position of the engine whereas the vortex core is slightly impacted. It can be summarized that as far as the interaction between the shear layer and the engine jet is concerned the distance to the wing-tip vortex plays a minor role. The engine jet velocity plays the major role, i.e., a stronger interaction between the engine jet and the shear layer occurs at higher engine jet velocities. The RANS simulation provides results of the averaged wake. To get a more detailed insight in the structure and unsteady behavior of the wake a large-eddy simulations of the wake is required. A large-eddy simulation over a complete is not feasible on today’s computers. Therefore, experiments will be performed to provide reliable input data for an LES which will focus on the analysis of the wake.

a u jet /u∞ = 1.77

b u jet /u∞ = 2.87

Fig. 5 Contours of the streamwise velocity u in the wake at x/c = const. for engine position “A”

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a u jet /u∞ = 1.77

b u jet /u∞ = 2.87

Fig. 6 Contours of the streamwise velocity u in the wake at x/c = const. for engine position “C”

5.2 Measurements of the Flow over a Wing with Engine Jets 5.2.1

Experimental Methods and Set-Up

A reduced swept-wing half-model is used resembling a geometry of a wing of a large transport aircraft with a sweep angle of 34◦ . At the trailing edge the wing has three sections of different angles. The wing possesses a BAC 3-11/RES/30/21 [45] profile and a three-dimensional wing-tip geometry. In order to produce reasonably large vortices the wing model consists of the outer two of the three wing segments with a semispan s = 607 mm. The mean chord length is cm = 150 mm. The flow field is tripped by a zig-zag tape. The model is equipped with one flap and two slats, which during measurements are either retracted or deployed at an angle ϕ = −20◦ and ϕ = 25◦ , respectively. Throughout the measurements the angle of attack of the wing is set to α = 8◦ . Fig. 7 shows a sketch of the wing with the engine model that can be mounted at three different spanwise positions to the wing and the orientation of some measurement sheets downstream of the wing. The engine model works as a jet apparatus when operated with pressurized air otherwise as a through-flow nacelle. This model is identical to that used in [25], except for the pylon geometry which had to be adapted to the 34◦ swept leading edge. The measurements are performed at a jet speed of u jet /u∞ = 1.74. The experiments are carried out in the low-speed wind tunnel of the Institute of Aerodynamics, RWTH Aachen University at a flow speed of u∞ = 27 m/s yielding a Reynolds number based on the mean chord length of Recm = 2.8 × 105. The 3D velocity distributions in the wake of the wing/engine configuration are measured by the stereoscopic particle-image velocimetry (PIV) method. The measurement technique is chosen to capture unsteady flow phenomena in the wake with high spatial and temporal accuracy. The unsteady phenomena are mainly instabilities in the near field which grow in the far field and cause a breakdown of the

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Fig. 7 Sketch of the wing model in flap-down configuration. Vorticity distributions in several vertical planes in the wing-tip and flap edge wake are shown. The wind tunnel coordinate system x, y, z is centered at the wing-tip trailing edge.

vortex system. Like mentioned before, these measurement data are used as input for numerical simulations of the wake-jet interaction. 5.2.2

Unsteady Vortex Center Position

Figure 8 shows the normalized wandering amplitudes of the vortex core as a function of time. Compared are the two engine positions “A” and “C”. The meandering of the vortex core has a random behavior and is for all investigated cases different in the wandering amplitude and the frequency of the movement. From the results of the study it can be stated that for the clean wing configuration the engine position inboard of the trailing edge kink leads to higher wandering amplitudes of the wing-tip vortex while the measurements for the engine position outboard of the flap edge show an increased distance between wing-tip and flap-edge vortex, which indicates a longer coexistence of the two-vortex system. In high-lift configuration the engine position outboard the flap edge and closer to the wing-tip creates the highest amplitudes of vortex movement. 5.2.3

Centered Average

At steady flows the PIV-vector maps are ensemble averaged [48]. To analyze the unsteady behavior of the vortex the individual PIV-vector maps are centered with respect to their concentric vorticity distribution to separate the unsteady vortex

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Fig. 8 Normalized vortex wandering amplitudes over time. The amplitude is defined as the absolute value of the distance between the time averaged vortex center position and the center position of the individual PIV vector distribution of the data series.

a Ensemble average

b Centered average

Fig. 9 Comparison of ensemble and centered average of the circumferential velocity vθ of the vortex core

behavior and the instantaneous flow structure. The two-stage processing of the PIV data, i.e., in a first step the data are centered relative to the vortex core and then, in the second step averaged, is called centered averaging. The combination of the ensemble and the centered averaging process allows the determination of the vortex center position, wandering amplitudes, and accurate vortex structure. In Figs. 9 and 10 the results of the ensemble and the centered average are displayed for the circumferential velocity vθ and the vorticity ω . The centered average gives a more exact representation of the vortex structure with a higher circumferential velocity vθ at the vortex core radius and therefore a higher vorticity ω at the center of the vortex compared to the pure ensemble average. The ensemble average of a meandering vortex gives only a smeared representation. Therefore, the centered average is suitable for a structural analysis of the vortex.

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a Ensemble average

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b Centered average

Fig. 10 Comparison of ensemble and centered average of the vorticity ω of the vortex core

5.2.4

Ratio of Fluctuations

The averaging methods of the measurement data are used to derive a measure for the vortex meandering regarding amplitudes and fluctuations of spatial motion. The turbulence intensity Tu of the measured flow field is used. The centered average of the turbulence intensity is divided by the ensemble average of the turbulence intensity resulting in the ratio of fluctuations RTu . The turbulence intensity in the vortex core for the centered average is smaller than for the ensemble average since the flow field is dominated by the meandering of the vortex core, i.e., the ratio is less than 1. The region away from the vortex core is dominated by the wake of the wing and the meandering of the vortex has a minor impact. The movement of the measurement data by the centered average leads to a higher turbulence level than for the ensemble average. Hence, the ratio RTu has values greater than 1. The vortex buffer region is the region which is influenced by the vortex movement and has values RTu ≤ 1. In Fig. 11 the ratio of fluctuations is displayed for the two engine positions “A” and “C”. Furthermore, for position “A” the jet is considered under on and off conditions. Flaps and slats are retracted. The vortex buffer region is marked by a round dashed line. The comparison of the operation mode for the engine position “A” (Figs. 11a and 11b) shows that the “on” mode stabilizes the vortex meandering. The “off” mode has higher fluctuations indicated by the darker color. The influence of the engine position on the vortex meandering can be seen by comparing Figs. 11a and 11c. The engine position towards the wing root creates higher vortex fluctuations and larger amplitudes of spatial motion. Generally, the vortex meandering behaves nonlinear as a function of parameters like wing-tip vortex strength, jet stream velocity, engine jet position, high-lift devices. In other words, the results from the variation of the engine position on the vortex meandering cannot be directly transferred to the vortex properties being also determined by the engine operation mode.

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a “A”, engine ON

b “A”, engine OFF

c “C”, engine ON

Fig. 11 Ratio of fluctuations for engine position “A” and “C”. Engine operation mode is ON and OFF. Vortex buffer region is marked by a dashed line.

5.3 Large-Eddy Simulation of Wake Vortices Results of an LES of the temporal development of short and long-wave instabilities are presented in Sect. 5.3.1. Sect. 5.3.2 covers the spatial development of the wake vortices interacting with an jet in the extended near field. As inflow data the PIV measurements described in Sect. 5.2 are used. 5.3.1

Temporal Development of Wake Vortices

Flow Configuration Leweke and Williamson [34] 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. Two cases of two-vortex systems are simulated [65]. A co-operative elliptical short-wave instability and a long-wave Crow instability [10]. The two-vortex system consists of a pair of counter-rotating vortices which is sketched in Fig. 12. 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 leads to a mutual induction of velocity directed downward, i.e., in the negative z-direction. The vortices of the short-wave instability have a vortex-core radius rc /b0 = 0.2 relative to the vortex separation b0 and the Reynolds number ReΓ = Γ /ν based on the circulation Γ and the viscosity ν is ReΓ = 2400. The values for the Crow instability are rc /b0 = 0.085 and ReΓ = 7400. To initialize the growth of instabilities in the vortex systems white noise perturbations are added. The Mach numbers are very low which is why preconditioning, implicit dual time-stepping, and multigrid methods are used for the LES of the wake. Periodic boundary conditions are used in each direction. The influence of the boundaries can be neglected when the distance of the vortices is small compared to the distance between a vortex core and the boundary [21, 22]. The integration domain of the short-wave instability has a size of 4.5 × 5.5 × 4.5b30 and that of the Crow instability of 8.68 × 4 × 4b30.

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b0 z

Γ rc

y

−Γ rc

Fig. 12 Description of two-vortex system

Fig. 13 Temporal development of short-wave instability of a two-vortex system. From left to right λ2 contours for non-dimensional times t ∗ = 0.1, 6.7, 7.2, 7.8

Results of a Two-Vortex System The temporal development of the short-wave instability is shown in Fig. 13 where λ2 contours [27] are displayed. The non-dimensional time t ∗ is related to the sink rate of the vortex system. Fig. 14 depicts the behavior of the Crow instability. At the short-wave instability the vortices are deformed with a phase shift of 180 degrees resulting from mutual and self-induction of the vortices. The interaction of the vortices leads to secondary vortices (t ∗ > 7.8). The Crow instability in Fig. 14 causes a deformation of the two vortices and an interaction, which in turn generates vortex rings. Note that short-wave instabilities exist although they saturate at a very low amplitude [52] (e.g. t ∗ = 7.6, 8.6 in Fig. 14). The type of instabilities which leads to an interaction of the vortices depends on the ratio of core radius to vortex distance rc /b0 . The short-wave instabilities saturate at a low level for the case rc /b0 = 0.085 < 0.1 such that the weakly growing Crow instability causes an interaction of the vortices. In the case of rc /b0 = 0.2 the strongly growing co-operative elliptical instability leads to an interaction of the vortices. Aircraft at larger wing span have a ratio of rc /b0 0.1 such that fast growing small instabilities saturate at a lower level. 5.3.2

Jet-Vortex Interaction

A schematic of the wake of an airplane is sketched in Fig. 15. The wing span b can be regarded as a typical length scale of the simulation of the wake in the near and extended near-field since the instabilities in the far field depend on the distance b0 between the vortex centers. The tip vortices roll up and the separation distance is reduced in the extended near field of the wake [51].

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Fig. 14 Temporal development of the Crow instability of a two-vortex system. From left to right λ2 contours for non-dimensional times t ∗ = 0.1, 6.4, 7.6, 8.6

Fig. 15 Schematic of the wake of an airplane in the extended near field. The wake rolls up into a two-vortex system

Inflow Boundary Conditions The three components of the velocity distribution of the wake are measured by PIV one mean chord length cm downstream of a half model. The data is used to develop an inflow boundary condition for the spatial computation of the wake in the near field and in the extended near field. The experimental set-up is described in Sect. 5.2.1. The dimensions of the measurement plane are 453 mm × 215 mm, which is covered by a uniform Cartesian mesh with 480 × 156 points. The x-axis of the coordinate system defines the downstream direction, the y-axis is in the symmetry axis of the wing and defines the spanwise direction while the z-axis coincidences with the normal direction. This frame of reference is shown in Fig. 16. In Fig. 17 the nondimensional velocity distribution of the measurement at engine position “A” is shown. The high velocity at (y/b, z/b) ≈ (0.2, −0.01) describes the cross section of the jet. The shear layer of the wing can be identified by the low velocity region at z/b ≈ 0.02. Near the wing tip the shear layer forms the tip vortex the center of which is located at (y/b, z/b) = (0.35, 0.022). The velocity distribution for engine position “C” is not displayed since no pronounced differences occur. That is, in the very near field neither the location of the center of the wing-tip vortex nor its overall structure is affected by the position of the engine. The measured data comprise the three velocity components u, v, w and the root mean square (RMS) value of the fluctuations. From the velocities the initial pressure distribution p is computed by the Poisson equation for the pressure. The fluctuating components u , v , w are determined using the RMS value of the fluctuations and assuming a Gaussian distribution of the oscillations. Based on the mean velocity

Interaction of Wing-Tip Vortices and Jets in the Extended Wake

Fig. 16 Schematic of the measurement setup. The BAC 3-11 wing with the engine mounted on the lower side is shown. The dark plane downstream of the wing indicates the location of the PIV measurement cross section

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Fig. 17 PIV measurements of the nondimensional velocity distribution for engine position “A” at x/b = 1

components u, ¯ v, ¯ w¯ and the fluctuations the instantaneous velocity components u, v, w are obtained. Numerical set-up The Reynolds number based on the wing span b = 11.56cm is Reb = 3.2368 × 106. The domain of integration has a size of x/b × y/b × z/b = 12.48 × 0.35 × 0.34. The streamwise extension of the domain is 144cm = 12.48b and its spanwise extension covers only one half of the wake. The number of mesh points is 1249 × 225 × 225 equally decomposed into 24 blocks leading to approximately 64 million points. The radius of the wing tip vortex is rc = 1.93 × 10−3b, which is 44 times smaller than the radius of the vortex used by Holz¨apfel et al. [22, 24, 23] or Zurheide and Schr¨oder [65]. While Labb´e et al. [29], Paoli et al. [47], and Gago et al. [19] use a temporal approach to simulate a jet-vortex interaction, we use a spatial approach described by Fares and Schr¨oder [16] and Stumpf [56, 57]. In the temporal approach, the velocity deficit in the vortex core in the axial direction is neglected although it is known to have a strong effect on the absolute and convective instability of a vortex [64]. Additionally, the deformation of the vortex is not convected downstream and parts of the engine jet wrapped around the tip vortex influence the upstream flow field, since periodic boundary conditions in the axial direction are used for the temporal development. On the other hand, the spatial approach allows to compute only relatively small spatial wake extensions due to the vast number of mesh points for a sufficient resolution of the computational domain. Spatial Structure of the Wake In Fig. 18, the wake is displayed by a sketch of the wing. The thick dark solid line visualizes the vortex by the λ2 -criterion [27] and the remaining light lines illustrate

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Fig. 18 Computed wake at engine position “A”. The vortex is visualized by the λ2 -criterion and streamlines illustrate the position of the jet and the surrounding flow field. The wing with engine is displayed for illustration reasons

a Engine position “A”

b Engine position “C”

Fig. 19 Computed wake at engine position “A” and “C” for non-dimensional time t = 29.5. The vortex is visualized by the λ2 -criterion and streamlines illustrate the position of the jet and the surrounding flow field

the streamlines of the developing jet. The vortex moves towards the root of the wing, while the jet is wrapped around the vortex. In Fig. 19 the wake at engine positions “A” and “C” are displayed for the non-dimensional time t = 29.5. The vortex reaches the maximum deflection downstream of the wing at x/b ≈ 4. Including symmetry the separation of the vortices is y/b = 0.8 for engine position “A” and y/b = 0.84 for position “C” (Fig. 23). This is close to the theoretical separation downstream of a wing with an elliptical circulation distribution of y/b = π /4 = 0.785 [51]. More details of the vortex core position are given in Sec. 5.3.2 Vortex Core Position. When the jet is wrapped around the vortex halfway small perturbations can be identified on the vortex tube at x/b = 3, Figs. 18, 19, and 23. These perturbations change their shape and peak in the streamwise direction They are convected downstream while they decay. The perturbations have a 3D shape and turn clock wise like the rotation of the vortex. The change of the shape of the disturbances can be explained by investigating the incompressible form of the azimuthal component of the vorticity-transport equation

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a x/b = 0.1

b x/b = 0.5

c x/b = 1.0

d x/b = 2.0

e x/b = 4.0

f x/b = 6.0

Fig. 20 Lines: vorticity ωx for engine position “A” at different x positions. 11 levels for each plot. x/b = 0.1 : 0.5 < ωx < 45, x/b = 0.5, 1.0 : 0.5 < ωx < 30, x/b > 2.0 : 0.2 < ωx < 16. Contours: nondimensional velocity u/u∞ , 13 levels: u/u∞ = 0.9 (white) < u/u∞ < u/u∞ = 1.1 (black)

Dωθ ∂ vθ ωθ ∂ vθ ∂ vθ = ωr + , + ωx Dt ∂r r ∂θ ∂x

(1)

where D(·)/Dt is the total derivative. In the downstream direction close to the wing the radial and azimuthal vorticity components are zero (ωr = ωθ = 0). Axial variations of the azimuthal velocity component are induced when the jet is wrapped around the vortex. According to Eq. (1), this leads to an axial vorticity distribution which is transferred into an azimuthal vorticity ∂ ωθ /∂ t ≈ ωx ∂ vθ /∂ x. In Fig. 19b, the wake of the wing with the engine at position “C” is shown. Since the initial distance between the vortex and the jet is larger than for case “A” the interaction between them occurs further downstream. The comparison of the two wakes for engine positions “A” and “C” shows the perturbations at position “A” to have a slightly bigger amplitude and to occur a little further upstream. Quantities along the Vortex Core In Figs. 20 and 21, the vorticity ωx and the velocity u for several locations in the streamwise direction are displayed. Figure 20a contains the wing-tip vortex and the wake. The jet is wrapped around the vortex as illustrated in the cross section further downstream (Figs. 20a–f). One part of the jet can be seen at (y/b, z/b) =

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a x/b = 0.1

b x/b = 0.5

c x/b = 1.0

d x/b = 2.0

e x/b = 4.0

f x/b = 6.0

Fig. 21 Vorticity ωx for engine position “C” at different x positions. 11 levels for each plot. x/b = 0.1 : 0.5 < ωx < 45, x/b = 0.5, 1.0 : 0.5 < ωx < 30, x/b > 2.0 : 0.2 < ωx < 16. Contours: nondimensional velocity u/u∞ , 13 levels: u/u∞ = 0.9 (white) < u/u∞ < u/u∞ = 1.1 (black)

(0.3, −0.07) for engine position “A” in Fig. 20d. In Fig. 20f parts of the jet are moved to (y/b, z/b) = (0.3, 0.016) at x/b = 6.0. For engine position “C” in Fig. 21, the behavior is similar to that in position “A”. At x/b = 2.0 (Fig. 21d) the jet occurs outside the illustrated cross section at (y/b, z/b) = (0.25, −0.1). Further downstream, at x/b = 6.0, the jet forms a line from (y/b, z/b) = (0.37, −0.01) to (0.32, 0.0). The structure of the wake is simular to that found by Margaris et al. [37] in experiments on jet-vortex interaction. The jet is wrapped around the vortex core without intrusion of the vortex core. The roll up of the wake in the extended near field does not change the velocity distribution of the wing-tip vortex inside the vortex core. Nevertheless, the structure of the vortex outside the vortex core is changing dramatically. Fluid at low velocity and high vorticity from the wake of the wing and a higher velocity from the jet is wrapped around the vortex. Distribution of the Circulation The circulation Γ (r) is calculated at radius r using the vorticity ωx

Γ (r) =

2π r 0

0

ωx (ζ )ζ d ζ d θ .

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Γ / Γ0 [-]

Γ / Γ0 [-]

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x/b=1.0 x/b=4.0 x/b=8.0 x/b=12.0 0

1

2 3 r/rc [-]

a Engine position “A”

4

5

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x/b=1.0 x/b=4.0 x/b=8.0 x/b=12.0 0

1

2 3 r/rc [-]

4

5

b Engine position “C”

Fig. 22 Circulation Γ /Γ0 at selected positions x/b

In Figs. 22a and 22b, the nondimensional circulation Γ /Γ0 , where the root circulation Γ0 is defined by the circulation at x/b = 12.0 is displayed vs. the nondimensional radius r/rc . For a single vortex, the circulation Γ grows until a constant and maximum value is reached. At the initial position this is Γ0 . In Figs. 22a and 22b the circulation grows beyond the maximum Γ0 , since the vorticity in the wake is non zero at r > rc as can be seen in Figs. 20 and 21 for the vorticity. The circulation reaches a local maximum at 1 < r/rc < 2. Then, at higher values of r/rc , the vorticity of the wake is added to the circulation and as such the value of Γ increases. At x values downstream of x/b ≥ 8.0, Γ rises more strongly than for smaller x values, since a larger part of the wake is already wrapped around the vortex core. This is evidenced in Figs. 20f and 21f. Vortex Core Position In Fig. 23, the streamwise distributions of the positions of the center of the wing tip vortex are shown. To get a more exact position an interpolation of the vortex core position between the mesh points is used. Figure 23a depicts the deflection along the y-axis and Fig. 23b along the z-axis. In both figures the vortex core positions of simulations under jet-off conditions are also shown. For engine position “A”, i.e., the engine is located near the wing tip, the deflection is larger than for position “C”. At x/b = 4.3 the difference between the two trajectories is Δ y = 0.023b, which is in the order of the distance of the two engine positions (Δ y = 0.0346b). The differences in the z-direction are relative to the y-deflection even larger. For case “A” the vortex sinks from x/b = 8 on, while at position “C” the vortex still moves upward. One possible explanation for the different wing-tip vortex trajectories is the influence of the engine and its pylon on the flow field of the wing. The pylon behaves as a barrier on the lower side of the wing. Up to a distance of x/b = 3 the differences between the engine-on and engine-off conditions are marginal. This is valid for both engine positions. Further downstream the differences of the vortex core position for the cases when the jet is turned on and off can be clearly seen for the two engine

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0.5

0.02

On A Off A On C Off C

0.48

0

0.46

-0.01

z/b [-]

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0.01

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0.4

-0.05 0.38

-0.06

0.36

-0.07 2

4

6

8

10

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2

x/b [-]

4

6

8

10

12

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a y axis

b z axis

Fig. 23 Positions of wing-tip vortices determined by least squares method for cases “A” and “C” at jet-on and jet-off conditions

1.4

A C

1.2

H(k)

H(k)

1 0.8 0.6 0.4 0.2 0 5 10 15 20 25 30 35 40 45 50 k

45 40 35 30 25 20 15 10 5 0

A C

5 10 15 20 25 30 35 40 45 50 k

Fig. 24 Averaged Fourier modes H(k) of the scaled Fourier modes k of vortex position for engine position “A” and “C”. Left: FFT of radius r, right: FFT of angle θ

positions. Moreover strong meandering on the vortex core can be found when the engine is turned on. For the cases without jet, meandering at such a large amplitude does not occur. For engine position “A” the vortex is bent towards the wing tip for x/b > 4.0 and the meandering starts at x/b = 3.5. For “C” the vortex is influenced by the jet from x/b = 5.0 on and the meandering also starts at x/b = 5.0. At position “A”, the jet is rolled up over a shorter distance than at engine position “C”. At engine position “A” the angle of the jet-vortex interaction is more acute. That is, at engine position “C”, where the angle is more obtuse, the jet possesses a larger y-momentum causing a more pronounced deflection in the y-direction. Vortex Meandering To perform a spatial FFT of the vortex core position, the position was transformed from Cartesian (x, y, z) to cylindrical (r, θ , z) coordinates. The FFT was performed for 40 time steps of the unsteady flow field at a constant increment Δ t = 0.1 in a range of 5.0 ≤ x/b ≤ 8 and afterwards the data was averaged. The scaled Fourier modes for the radius r and the angle θ are illustrated in Fig. 24 for the engine

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positions “A” and “C”. The Fourier mode of r and θ peaks at k = 6 for position “A” and at k = 7 for position “C”. These scaled Fourier modes k correspond to a wavelength λ = 0.5b and λ = 0.429b, respectively. The spatial analysis of the vortex core trajectory shows larger values of the Fourier coefficients for engine position “A” than for position “C”. These modes describe the deformation of the vortex core through vortex meandering. The higher influence of the jet on the vortex meandering for engine position “A” is consistent with the findings in Figs. 18 and 19 where the overall flow structure in the wake is illustrated.

6 Conclusions The interaction of wing-tip vortices and engine jets in the extended wake was experimentally and numerically investigated. To perform this analysis several novel numerical and experimental tools hat to be developed. A new one-equation turbulence model was derived to reliably describe vortical jet flows. An adaptive mesh refinement (AMR) method for vertex-centered fluxes on structured meshes was introduced to ensure a more efficient method of solution. The measurements of the unsteady jet-vortex interaction required the development of the centered average of the vortex core. The temporal development of a short-wave co-operative elliptical instability and the Crow instability was discussed based on the flow field of the generic two-vortex system which was created by superposing perturbed Lamb-Oseen vortices. The short-wave instabilities saturated at a low level for the case rc /b0 = 0.085 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. Furthermore, spatial LES of a vortex-jet interaction in the wake of a wing was presented. The inflow distribution for the spatial simulation of the wake was based on PIV measurements in the wake of a wing at an engine mounted in two positions. The simulation showed vorticity distribution in the wake to be influenced by the engine position. The main difference was illustrated by the trajectory of the vortex core. The engine at the most inboard position generated a larger deflection of the vortex core than the engine position most outboard. The vortex meandering analyzed by Fourier transformation of the vortex core position revealed for the most inboard engine position higher Fourier modes, i.e., stronger shifts than for the more outboard position. It is fair to conclude that the impact of the interaction of engine jets and wingtip vortices on the wake is not yet fully understood. To complete the understandings more experimental and numerical investigations, for instance, on the influence of deployed high-lift devices especially on the generation and development of short-wave perturbations in the very near-field downstream of the wing are to be conducted.

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Acknowledgements. The support of this research by the Deutsche Forschungsgemeinschaft (DFG) in the frame of SFB 401 is gratefully acknowledged.

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Experimental and Numerical Investigation of Unsteady Transonic Airfoil Flow Atef Alshabu, Viktor Hermes, Igor Klioutchnikov, and Herbert Olivier

Abstract. A summary of experimental and numerical results concerning the phenomenon of upstream moving pressure waves in the transonic flow regime is presented. As experimental and numerical time-resolved shadowgraphs show, such waves initiate near the sharp trailing edge of a supercritical airfoil and in the wake, propagate upstream in the subsonic region of the flow, and strengthen before becoming apparently weaker and almost disappear near the leading edge. Using high order numerical simulation several mechanisms of wave generation based on vortex dynamics as well as vortex/trailing edge interactions and wake fluctuations are distinguished. These waves on the upper side of the airfoil are also captured with pressure transducers mounted in the airfoil model as pressure oscillations of dominant frequencies ranging between 1 to 2 kHz. Furthermore, the statistical analysis of the pressure histories allowed for the determination of wave propagation direction, strength and speed.

1 Introduction Upstream moving pressure waves are observed in transonic airfoil flows already for several decades [4], [12]. Upon investigating the phenomenon of periodic shock motions on airfoils upstream moving waves originating at and closely behind the trailing edge in the wake have been experimentally observed by several authors [5], [13]. These waves are associated with wake fluctuations due to unsteady shock motions and are called ”Kutta waves” by Tijdeman [14]. Nevertheless, the phenomenon is not fully understood yet and is still subject of numerous experimental and numerical investigations [1], [2], [9], [11]. Especially, its interaction with the undisturbed flow field and its influence on the transition, Atef Alshabu · Viktor Hermes · Igor Klioutchnikov · Herbert Olivier Shock Wave Laboratory of RWTH University, Templergraben 55, 52062 Aachen, Germany e-mail: {alshabu,hermes,klioutchnikov, olivier}@swl.rwth-aachen.de

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 137–151. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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turbulence and on the shock/boundary layer interaction are of great interest. For instance, the recent theories about the buffet phenomenon involves the strong pressure wave generation at the trailing edge, their movement upstream and their interaction with the recompression shock. Hereby the shock strength, its position and the intensity of the shock/boundary layer interaction [9], [11] changes. In this paper the most important results of an experimental and numerical investigation of the aforementioned phenomenon in the transonic flow regime are summarized.

2 Experimental Setup The test facility used is a modified shock tube with a rectangular test section (280 x 200 mm) to perform airfoil testing at transonic Mach numbers and relatively high Reynolds numbers extending up to 38 · 106 based on a chord length of 100 mm. The flow behind the incident shock wave provides the testing flow for a measurement period of about 5 ms.. A full description of the facility and its working principle can be found in [10]. The tested model is a BAC3-11 airfoil with a 200 mm span, 80 mm chord length and a sharp trailing edge. For pressure measurements, 11 pressure transducers have been mounted directly beneath pressure taps of 0.6 mm diameter to minimize the influence of the pressure-transmitting volume on the pressure signals. For flow visualization, high-speed photography is used to obtain highly time-resolved shadowgraphs and schlieren pictures of the flow. The optical setup corresponds to the classic Toepler-Z-configuration. Shadow as well as schlieren techniques were applied for visualizing. The applied camera is a highspeed Shimadzu HPV-1 digital camera. This camera can capture up to 1 million frames per second with a maximum of 100 frames. A constant spatial resolution of 312 x 260 pixels is maintained at all recording speeds. Frames presented in this paper were obtained at a frequency of 125 kHz with a chip exposure time of 1 μ s.

3 Numerical Method Two different solvers for numerical simulation of the transonic airfoil flow are available at the Shock Wave Laboratory. The solver were compared with each other and it is found that the WENO (Weighted Essentially Non-Oscillatory) based solver is superior to the FCT (Flux-Corrected Transport) based solver because of higher stability properties in transonic regime. Detailed results of the comparison are reported in [8]. The solver is based on a finite-difference WENO method of high order accuracy in space and time and shock capturing capability [8]. The transonic flow contains different complex interactions, large gradients like shocks, but also weak pressure waves. For a precise simulation of such a type of flow numerical methods with a small amount of numerical dissipation are essential. The used numerical method meets these demands and will be introduced in the following. The Navier-Stokes equations in weak conservative form for the compressible flow in general coordinates ξ = ξ (x, y), η = η (x, y), ζ = ζ (z) are employed:

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1 1 1 1 1 1 Ut + Fξ + Gη + Hζ = Fξv + Gvη + Hζv . J J J J J J Here U is the solution vector of the conservative variables, F, G, H and F v , Gv , H v are the inviscid and viscous fluxes respectively. For two-dimensional simulations the fluxes in the third spatial dimension H, H v are removed. sub−stencil i +1/2

i −2 i −1

i

sub−stencil

i+2 i+1

sub−stencil stencil

Fig. 1 Schematic representation of stencil and sub-stencil configuration for N = 5 (p = 3)

The inviscid fluxes F, G and H are approximated using the WENO scheme formulation according to Jiang and Shu [6]. This scheme is of formally odd order (N > 5) and uses an N-point stencil. The stencil is subdivided into p sub-stencils (N = 2p − 1), that uses p points (Fig. 1). The approximation of the fluxes is calculated first for each sub-stencil separately and recombined using non-linear weighting coefficients. The weighting coefficients are determined depending on the smoothness of the approximation. If the approximation of all sub-stencils is smooth, the weighting coefficients are chosen to achieve the formal order of accuracy. In the vicinity of large gradients and discontinuities, such as shocks, smooth flux approximations are weighted heavily and numerical dissipation is added to avoid spurious oscillations of the solution. Beside an adaptive stencil the WENO scheme has some upwind properties and is combined here with a flux-vector splitting technique. It can be rewritten as a central and a dissipation term according to Balsara and Shu [3] by applying the fact that the sum of all weighting coefficients is equal to unity: Fi+1/2 = Θ (Fi−M/2+1 , · · · , Fi+M/2 ) 5

+

m− −1 m− ∑ [ϕ (R−1 m,i+1/2 Δ Fi−N/2 , · · · , Rm,i+1/2 Δ Fi+N/2−2 )

m=1

m+ −1 m+ −ϕ (R−1 m,i+1/2 Δ Fi−N/2+1 , · · · , Rm,i+1/2 Δ Fi+N/2−1 )]Rm,i+1/2

Here, the approximation is obtained by two difference operators Θ and ϕ . Θ is the central difference operator of even order M = N − 1, whereas in the function ϕ

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upwinding is taken into account and the non-linear weights are calculated and deployed. R−1 m and Rm are the mth left and right eigenvectors of the Jacobian matrix, respectively and m is an index running from one to the number of independent variables of the system of equations (m = 1, · · · , 5 for three-dimensional Navier-Stokes m± equations). For the calculation of the neighbour grid point fluxes Δ Fi+1/2 local Lax-Friedrichs flux-vector splitting is used: m± m± Δ Fi+1/2 = Fi+1 − Fim± ,

Fim± =

1 m m Ui Fi ± Λi,max 2

m Λi,max is the mth maximum eigenvalue inside the stencil. The viscous fluxes F v , Gv and H v are discretised using central difference operators of high even order (N + 1). The derivatives of the viscous fluxes contain second derivatives of the velocity vector (u, v, w)T and temperature T . The approximation of the second derivative of the x-component of the velocity vector u will be explained in detail exemplary for all variables. For the approximation of the second derivative with respect to one spatial direction an approximation operator Mi+1/2 () is applied. For mixed second derivatives an interpolation operator Li+1/2 () is applied to the first and then to the second spatial direction. The operators Li+1/2 () and Mi+1/2 () of even order N + 1 are defined as [7]: (N+1)/2

∑

Li+1/2 (u) =

(−1)m+1 am (ui+m + ui−m+1)

m=1 (N+1)/2

Mi+1/2 (u) =

∑

(−1)m+1 bm (ui+m − ui−m+1)Δ ξ −1

m=1

The coefficients am can be determined analytically for every even order (N + 1) by solving the following linear system [7]: ' (N+1)/2 1/2 for s = 1 m+1 p p ∑ (−1) (m − (m − 1) )am = 0 for s > 1 m=1 Here s is an index running from one to the maximal number of equation of the system s = 1, 2, ..., (N + 1)/2 and p = 2s − 1. Analogous, the coefficients bm are determined by solving [7]: ' (N+1)/2 1 for s = 1 m+1 p p p ∑ (−1) (m − (m − 1) )bm = 0 for s > 1 m=1 with p = 2s. It can be shown by the Taylor expansion that using the coefficients am and bm the difference quotient approximates the derivative with order N + 1: Li+1/2 (u) − Li−1/2(u) ∂u = |i + O(Δ ξ (N+1) ) Δξ ∂ξ

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Mi+1/2 (u) − Mi−1/2(u) ∂ 2u = |i + O(Δ ξ (N+1)). Δξ ∂ξ2 The time integration is explicit and performed with a third order, low-storage Runge-Kutta-TVD-Scheme [6]: Ui = Uin − Qni 3 1 1 Ui = Uin + Ui − Qi 4 4 4 1 2 2 Uin+1 = Uin + Ui − Q 3 3 3 i 3

with Qi =

v v + Fr,i−1/2 ) ∑ λr (Fr,i+1/2 − Fr,i−1/2 − Fr,i+1/2

r=1

and (λ1 , λ2 , λ3 ) = (

Δt Δt Δt , , ) and (F1 , F2 , F3 ) = (F, G, H). JΔ ξ JΔ η JΔ ζ

(N + 1)/2 fictitious points are added in each direction to maintain the stencil in the vicinity of boundaries. Non-reflecting boundary conditions based on Riemann invariants are used for the subsonic inflow and outflow boundary. No-slip boundary conditions are applied on the airfoil surface. If wind tunnel walls are simulated, then Euler boundary conditions are applied on them to reduce the otherwise necessary spatial resolution near wind tunnel walls. The effect of the boundary layer displacement thickness on the pressure distribution around the airfoil is investigated in numerical simulations with no-slip boundary conditions applied to the wind tunnel walls but found negligible. The used numerical method is computational time-consuming, but well parallelisable. Therefore the solver is massively parallelised with means of MPI (Massage Passing Interface). A linear scalability up to 8192 cores for a test problem with 5120x512 grid points is achieved on the JUGENE supercomputer of Forschungszentrum J¨ulich.

4 Results The experiments have been conducted at chord Reynolds numbers Rec ranging from 1.0 to 5 · 106 . The Mach numbers Ma∞ are between 0.60 and 0.8 at zero incidence. Figure 2 shows exemplary time-resolved shadowgraphs for an experiment with a Mach number of 0.71 and a Reynolds number of 3 · 106. The wave structure on the suction side of the airfoil as well as the upstream propagation of the waves can be easily seen. An increase in the wave intensity is also seen in the region of maximum thickness of the airfoil. Near the leading and trailing edges the intensity of the waves is apparently lower. Here, they become almost invisible depending on the sensitivity of the shadowgraph system. Figure 3 shows some selected pressure histories on the suction side of the BAC3-11 airfoil for the above experiment. Pressure fluctuations

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Fig. 2 Time-resolved shadowgraphs showing the upstream wave propagation on the upper side of the airfoil BAC3-11; Ma = 0.71, Re = 3.0 × 106 , α = 0o

Fig. 3 Observed pressure fluctuations on the upper side of the BAC3-11, Ma = 0.71, Re = 3.0 × 106 , α = 0o

in the pressure histories resulting from the aforementioned wave processes can be seen. Starting from the trailing edge, the amplitude of these fluctuations increases in the region of maximum thickness of the airfoil before decreasing strongly near the leading edge. This confirms the conclusion made from the shadowgraphs. Figure 4 a) and b) show plots of the relative wave intensity measured as the normalized standard deviation of the pressure fluctuations at different Mach and Reynolds numbers respectively. In all plots the relative wave intensity reveals a maximum in the region of the maximum flow velocity and decreases rapidly towards leading and trailing edges. Further, as seen in Figure 4 a), higher Mach numbers result in larger wave intensities. Figure 4 b) reveals that the influence of the Reynolds number on the relative wave intensity is negligible. Figure 5 shows a Fourier analysis (power spectral density) exemplary for two pressure histories at the positions x/c = 0.37 and x/c = 0.73. Figure 5 reveals two predominant frequencies ranging from 1 to 2 kHz. The same holds for other

Experimental and Numerical Investigation of Unsteady Transonic Airfoil Flow

4.5

4.5

4

4

3.5

3.5 3 σ / p [%]

∞

∞

σ / p [%]

3 2.5 2 1.5 6

Ma=0.68, Re=1.9x10 , 6 Ma=0.71, Re=2.0x10

1

a)

2.5 2 1.5

0.5 0

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Ma=0.73, Re=2.1x10 0.3

0.4

0.5 x/c [−]

0.6

0.7

6

Ma=0.71, Re=1.9x10 6 Ma=0.71, Re=2.9x10

1 0.5

6

0

b)

Ma=0.71, Re=5.1x10 0.3

0.4

x/c [−]

0.5

6

0.6

Fig. 4 Wave intensity at different Mach a) and different Reynolds numbers b)

Fig. 5 Power spectral densities of two pressure histories at x/c = 0.37 (left) and x/c = 0.73 (right), Ma= 0.71, Re = 3.0 · 106 , α = 0o

pressure histories. No significant Mach and Reynolds numbers dependence of the wave frequencies was found in the investigated range of flow conditions. In Figure 6 a) and b) the plots of the obtained wave speed - obtained using crosscorrelation technique- with respect to the airfoil at different Mach and Reynolds numbers respectively are shown. Regardless of the rough spatial resolution, the figure shows an expected behaviour of the wave speed. It can be easily seen that in all plots the wave speed is lowest in the region of high local flow velocities and increases further downstream in the recompression zone of the airfoil. As depicted in Figure 6 a), higher Mach numbers result in an overall decreased wave speed with respect to the airfoil. Furthermore, for the considered Reynolds number range, the Reynolds number does not significantly influence the wave speed as it is depicted in the right part of Figure 6 b). The small deviation in the wave speeds shown in the right part of Figure 6 might be explained by the small deviation in the Mach number. In analogy to the experiment the influence of the inflow parameters Mach Ma∞ , Reynolds number based on chord length Rec and the angle of attack α are investigated by numerical simulations in the following range: Ma∞ = 0.69 to 0.80, Rec = 0.1 · 106 to 3 · 106 and α = −3◦ to + 4◦ .

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160

Ma=0.68, Re=1.9x106, u∞=283 m/s

Ma=0.71, Re=1.9x106, u∞=289 m/s Ma=0.71, Re=2.9x106, u∞=291 m/s

Ma=0.73, Re=2.1x106, u∞=300 m/s

100

velocity [m/s]

120

velocity [m/s]

120

Ma=0.71, Re=2.0x106, u∞=291 m/s

140

100 80 60

Ma=0.71, Re=5.1x106, u∞=290 m/s

80

60

40 40 20

20 0 0.25

a)

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0 0.25

0.7

x/c [−]

b)

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

x/c [−]

Fig. 6 Wave speed at different Mach a) and different Reynolds numbers b)

For reasons of time and computational costs predominant two-dimensional flow simulations are performed. Here grids with ca. 1280 x 130 points are used. The numerical simulation is carried out with a spatial approximation of ninth order and temporal approximation of third order accuracy. Laminar inflow conditions are assumed. No-slip boundary conditions are applied at the airfoil surface with boundary layer related quantities Δ x+ ≈ 10 − 100 and Δ y+ ≈ 1 − 10. For comparison with the experimental data, the effect of wind tunnel walls was taken into account by applying slip boundary conditions at the wall boundaries, because the contraction of the wind tunnel area by the displacement thickness of the wind tunnel boundary layer has been studied and found negligible. The computational domain for the simulation of the airfoil flow inside the wind tunnel is −3 < x/c < 5 and −1.63 < y/c < 1.87. For the variation of the Mach number, Reynolds number, the incidence and the spatial approximation order of the scheme the computational domain was stretched to −20 < x/c < 20 and −20 < y/c < 20, but in the vicinity of the airfoil the resolution for both grids is kept equal. Outside the computational domain fictitious points are added, to keep the formal order of spatial approximation and to use the same stencil formation on the domain boundaries. The analysis of the flow field is performed using instantaneous as well as timeaveraged flow variables. Besides the analysis and comparison of the results between the different available orders of spatial approximation, the numerical results are compared with experimental ones.The obtained numerical results revealed a quite good agreement with the experimental results presented above. Due to space limitations these results are not shown below, rather, the wave/shock interactions, the influence of the angle of attack, the mechanisms of the wave generations and three dimensional effects will be discussed in some detail.

4.1 Wave/Shock Interactions At high inflow Mach numbers, a well-established supersonic region on the suction side of the airfoil terminated by a shock is observed (Figures 7, and 8). Depending on the shock strength different wave/shock interactions are observed. Figure 7 presents two shadowgraphs of an experiment for a Mach number of 0.76

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b)

Fig. 7 Time-resolved shadowgraph showing wave/shock interaction, Ma = 0.76, Re = 1.0 × 106 , α = 0o , time delay between the pictures is 0.26 ms

a)

b)

Fig. 8 Schlieren picture showing a wave/shock interaction on the upper side of the BAC3-11, a) Ma = 0.80, Re = 3.4 × 106 , α = 0o and b) schematic drawing of wave propagation in the sub- and supersonic flow region

and a Reynolds number of 1.0 ·106. In Figure 7 a) a lambda-shock, typical for a laminar shock/boundary-layer interaction, can be seen. Time-resolved shadowgraphs of this experiment show a strong wave/shock interaction in which the shock for short periods of time is altered and degenerated into compression waves as depicted in Figure 7 b). This observation is of great importance and it might be interesting to further investigate this kind of interactions aiming at drag reduction. First numerical simulations performed for a simple wave/shock interaction model indicate that the time-averaged shock intensity can be reduced by suitable wave interaction. Further increase in the inflow Mach number results in an increase in the shock strength as well as downstream movement of the shock (Figure 8 a)). Figure 8 a) shows a schlieren picture for an experiment with a Mach number of 0.80 and a Reynolds number of 3.4x106 . Here, a large supersonic region visible by a field of Mach lines and terminated by a cambered shock can be clearly seen. At this Reynolds number, the boundary layer upstream of the shock is turbulent so that no lambda-shock forms. The strong shock causes a significant flow separation as

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seen in Figure 8 a). A dark region is also observed immediately downstream at the mid-height of the shock. A corresponding coloured schlieren picture −not shown here− indicates that this region represents an expansion region. The expansion region might be formed by the observed strong wave/shock interaction or more likely by the shock shape itself. Due to the cambering of the shock, the shock part ahead of the dark region can be considered as a normal shock with the highest pressure jump. To adjust this high pressure region to the lower pressure regions adjacent to the normal shock part, the flow has to expand in this region. Moreover, one notices that the upstream propagation of the waves is hindered by the shock. The waves can only propagate around the shock in the subsonic flow field. At the sonic line separating the subsonic flow field from the supersonic flow field, the waves generate weak pseudo-upstream moving waves in the supersonic region as depicted in Figures 8 b). In the supersonic part of the flow field, the wave-induced disturbances are certainly travelling downstream with respect to flow.

4.2 Variation of the Angle of Attack The study of the influence of the angle of attack on observed upstrem moving waves is performed numerically for a constant Mach and Reynolds number Ma∞ = 0.70 and Rec = 1 · 106 . For positive incidences (Fig. 9 a)) a supersonic region is moving towards the leading edge on the upper airfoil side and a strong interaction between pressure waves and the recompression shock is observed. The position of the recompression shock for α = 3◦ is at x/c ≈ 0.3 to 0.4. The vortices on the upper airfoil side start to develop further upstream in the region of the shock/boundary layer interaction. For negative incidences (Figure. 9 b)) the flow over the entire upper airfoil side is subsonic. The flow over the lower airfoil side contains a lot of different pressure waves that are generated by the shock/vortex interaction near the leading edge, the vortex/vortex interaction in the middle part of the airfoil and by the

a)

b)

Fig. 9 Numerical shadowgraph visialisation of the flow field for Ma∞ = 0.70, Rec = 1 · 106 , a) α = +3◦ , b) α = −3◦

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b)

Fig. 10 a) Wave velocity relative to the airfoil and b) wave Mach number relative to the flow for incidence variation with Ma∞ = 0.70 and Rec = 1 · 106

vortex/trailing edge interaction. On the upper airfoil side small vortices are observed far downstream at x/c ≈ 0.75. The wave velocity relative to the airfoil and the wave Mach number with respect to the flow on the upper airfoil side for different angles of attack is presented in Figure. 10. Because of strongly different flow fields along the upper airfoil side, the plot for the wave velocity is different for different incidences. In analogy to the variation of the Mach number, strong differences in the wave speed are not reflected in the wave Mach number. For the calculation of the wave Mach number the mean velocity in the flow field is added to the local wave speed. The wave Mach number on the upper airfoil for all incidences is determined to Mawave = 1.03 ∼ 1.05.

4.3 Mechanisms of Pressure Wave Generation The mechanisms of pressure wave generation is investigated by two-dimensional numerical simulations. In the numerical simulation vortices are observed in the boundary layer on the upper and lower side of the airfoil (Figure. 11). As the boundary layer at the leading edge is laminar, formation of vortices inside the boundary layer is associated with transition to turbulence. The position of laminar/turbulent transition is investigated experimentally by means of infrared thermography. Good agreement between numerical results and experimental data is found. The position of vortex formation depends on the adverse pressure gradient, provoked by the airfoil shape and by the recompression shock. The vortices are convected downstream in the boundary layer. Three different mechanisms of pressure wave generation are recognised in the simulations of the two-dimensional flow: • vortex/trailing edge interaction and wake fluctuation • vortex/shock interaction • vortex dynamics

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Fig. 11 a) Numerical shadowgraph visualisation and b) Mach number contours of the trailing edge flow for Ma∞ = 0.73, Rec = 2 · 106 and α = 0◦

Which of these mechanisms dominates, depends on the incoming flow Mach number Ma∞ , Reynolds number Rec and incidence α . For flow without or with small, not well-established supersonic region the vortex/trailing edge interaction is the dominant wave generation mechanism. Here the boundary layer vortices from the upper and lower airfoil side are passing the trailing edge and interacts with each other as well as with the trailing edge and the wake (Figure. 11). The phenomenon of pressure wave excitation by vortices passing a trailing edge is well known in literature and traced back to acoustic excitation. Compared to the subsonic flow a different development of upstream moving part of acoustic waves is observed in the transonic regime. In transonic airfoil flow the waves are moving upstream against increasingly faster flow under almost sonic condition. Therefore merging of contiguous pressure waves to weak shocks while propagating upstream is observed. If supersonic regions are present in the flow, the waves merge with the recompression shocks, as they cannot move upstream inside a supersonic region. Nevertheless disturbances are observed inside the supersonic domain. In this case the waves can only propagate in the subsonic part of the flow further upstream. Hereby they induce disturbances similar to Mach lines that are detectable inside the supersonic region by schlieren and shadowgraph visualisation. A strong fluctuation in shock position is observed for weak shocks, whereas stronger shocks are influenced less by pressure waves generated at the trailing edge. This is in a good agreement with experimental observations (see above). For transonic flow with a well-established supersonic region, corresponding to higher incoming flow Mach number, pressure waves generated by a vortex/shock interaction are observed. As the incomig boundary layer is laminar a strong λ -shaped shock is formed. The interaction of the oblique part of the shock with the boundary layer leads to a boundary layer separation. Vortex formation is observed in the separated shear layer due to Kelvin-Helmholtz instabilities. The vortices interacts with the normal part of the shock (Fig. 12). The phenomenon of vortex/shock interaction is also well-investigated and described in literature. When a vortex is passing

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Fig. 12 Numerical shadowgraph visualisation of the shock region in the flow with Ma∞ = 0.76, Rec = 2 · 106 and α = 0◦ for a) t = t0 b) t = t0 + 0, 01 ms

a)

b)

Fig. 13 λ2 visualisation of the vortices in the boundary layer on the a) upper and b) lower airfoil side for Ma∞ = 0.73, Rec = 1 · 106 and α = +3◦ (2560 x 450 x 280 grid points)

through a shock wave the shock is deformed whereas the vortex form and strenght is nearly uneffected. Furthermore upstream moving acoustic waves are generated by this interaction. In the transonic airfoil flow shock movement due to the vortex/shock interaction is observed. The vortices inside the boundary layer are convected downstream. As the velocity in the boundary layer is slower than the main stream velocity, the flow is accelerated on the top of each vortex (Fig. 11 b)). Depending on the inflow Mach number small supersonic regions are observed on the top of each vortex. This effect is comparable with the flow over small bumps on a airfoil surface. During an earlier phase it was assumed that the shock waves terminating the small local supersonic region on top of each vortex is the dominant pressure wave generation mechanism. The results of numerical simulations showed that these shocks are present only for high inflow Mach numbers. Furthermore they are transported downstream together with the corresponding vortex. The main mechanisms of the wave generation are described by the vortex/trailing edge and the vortex/shock interaction. Furthermore, upstream moving pressure waves are observed for a case when two vortices are directly interacting with each other (merging) while moving downstream. The vortex/vortex interaction is found non-periodic, so it is concluded that vortex/vortex interaction is a minor wave generation mechanism under investigated flow conditions. Finally, the simulation of the three-dimensional flow is performed for selected inflow parameter to investigate the influence of the third dimension on the observed wave phenomenon. Especially, the wave generation at the trailing edge and their

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Fig. 14 a) Instantaneous Mach number distribution, b) instantaneous pressure coefficient compared with 2D case and experimental data for an airfoilflow with Ma∞ = 0.73, Rec = 1 · 106 and α = +3◦ (2560 x 450 x 280 grid points)

development changed compared to the simulation of the two-dimensional flow. The boundary layer at the trailing edge is turbulent and simulations of three-dimensional flows are therfore essential, as two-dimensional flow simulations are not capable to resolve turbulence properly. Differences between two- and three-dimensional flow simulations are discussed in the following. In Figure 13 the vortices in the boundary layer are visualised by the λ2 criterion. The vortices in the rear part of the upper and lower airfoil side are complete three-dimensional as expected. Compared to the two dimensional flow simulation they are smaller and cause therefore less pronounced pressure gradients on the airfoil surface (Fig. 14 b)). The amplitude of the upstream moving pressure waves decreases compared to two-dimensional simulation, but the pressure waves are still detectable in the simulation results (Fig. 14 a)). The pressure wave fronts are oriented normal to the main velocity and consists of small parts three-dimensionally distributed in spanwise direction. Nevertheless the footprint of the recompression shock is nearly two-dimensional in spanwise direction. Acknowledgements. The authors would like to thank the German Research Foundation (Deutsche Forschungsgemeinschaft) for funding this work as a part of the collaborative research center SFB 401 ”Flow Modulation and Fluid-Structure Interaction at Airplane Wings” at the RWTH Aachen University, Germany.

References 1. Alshabu, A., Olivier, H., Klioutchnikov, I.: Investigation of Upstream Moving Pressure Waves on a Supercritical Airfoil. Aerospace Science and Technology 10, 465–473 (2006) 2. Alshabu, A., Olivier, H.: Unsteady Wave Phenomena on a Supercritical Airfoil. AIAA Journal 46, 2066–2073 (2008) 3. Balsara, D., Shu, C.W.: Monotonicity Preserving Weighted Essentially Non-Oscillatory Schemes with Increasingly High Order of Accuracy. Journal of Computational Physics 160(2), 405–452 (2000)

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4. Finke, K.: Stoßschwingungen in schallnahen Str¨omungen. VDI-Forschungsheft Nr. 580, D¨usseldorf (1977) 5. Gibb, J.: The Cause and Cure of Periodic Flows at Transonic Speeds. In: Proceedings of the 16th Congress of the Int. Council of Aeronautical Science, Jerusalem, Israel, AugustSeptember 1988, pp. 1522–1530. AIAA, Washington (1988) 6. Jiang, G.S., Shu, C.W.: Efficient Implementation of Weighted ENO Schemes. Journal of Computational Physics 126(1), 202–228 (1996) 7. Klioutchnikov, I.: Direct Numerical Simulation of Turbulent Compressible Fluid Flow. Habilitation Thesis, Rus. Academy of Sciences, Moscow (1998) (in rus.) 8. Klioutchnikov, I., Ballmann, J.: DNS of Transitional Transsonic Flow about a Supercritical BAC3-11 Airfoil using High-Order Shock Capturing Schemes. In: Lamballais, E., Friedrich, R., Geurts, B.J., Mtais, O. (eds.) DLES VI. ERCOFTAC Series, vol. 10, pp. 737–744. Springer, Heidelberg (2006) 9. Lee, B.H.K.: Self-Sustained Shock Oscillations on Airfoils at Transonic Speeds. Progress in Aerospace Sciences 37, 147–196 (2001) 10. Olivier, H., Reichel, T., Zechner, M.: Airfoil Flow Visualization and Pressure Measurements in High-Reynolds-Number Transonic Flow. AIAA Journal 41(8), 1405–1412 (2003) 11. Soda, A.: Numerical Investigation of Unsteady Transonic Shock/Boundary-Layer Interaction for Aeronautical Application. In: DLR-Forschungsberichricht 2007-03 (2007), ISSN 1434-8454 12. Srulijes, J., Seiler, F.: A Study on Upstream Moving Pressure Waves Induced by Vortex Separation. In: Gr¨onig, H. (ed.) Shock Tubes and Waves - Proceedings of the 16th International Symposium on Shock Tubes and Waves, pp. 621–628. VCH Publishers (1988) ISBN:3-527-26874-X 13. Stanewsky, E., Basler, D.: Experimental Investigation of Buffet Onset and Penetration on a Supercritical Airfoil at Transonic Speeds. In: Aircraft Dynamic Loads Due to Flow Separation, AGARD-CP-483, pp. 4.1–4.11 (September 1990) 14. Tijdeman, H.: Investigation of the Transonic Flow around Oscillating Airfoils. NLR TR 77090, Amsterdam (1977)

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Enabling Technologies for Robust High-Performance Simulations in Computational Fluid Dynamics Christian H. Bischof, H. Martin B¨ucker, and Arno Rasch

Abstract. In computational science and engineering, the role of computer science includes the mechanical generation of programs for the fast computation of accurate derivatives and the efficient utilization of parallel computer architectures. Automatic differentiation and parallel computing are two technologies enabling robust highperformance simulations in various scientific and engineering disciplines. This article gives a survey of selected results of an interdisciplinary research project where automatic differentiation, parallel computing, and their interplay are investigated in the context of two computational fluid dynamics packages developed at RWTH Aachen University.

1 Introduction The complexity of numerical simulations in computational fluid dynamics is growing rapidly. For real-world problems involving three-dimensional phenomena on different length and/or time scales, there is need to bring together knowledge from different scientific and technical disciplines. Engineering disciplines are developing sophisticated mathematical models for the physical phenomena of interest. Mathematics makes available advanced numerical algorithms for the accurate solution of the underlying equations. Computer science offers a rich set of enabling technologies to harness the ever increasing computational power of modern computer architectures. In practice, successful solutions to real-world problems in computational fluid dynamics rely on a synergetic interaction of tools and techniques from various disciplines. The aim of this article is to report on computer science technologies that are developed and put into action within the collaborative research center SFB 401 Christian H. Bischof · H. Martin B¨ucker · Arno Rasch Institute for Scientific Computing, RWTH Aachen University, Seffenter Weg 23, D–52074 Aachen e-mail: {bischof,buecker,rasch}@sc.rwth-aachen.de

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 153–180. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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“Modulation of flow and fluid-structure interaction at airplane wings” [2] at RWTH Aachen University, Germany. More precisely, we review selected results of the project A8 focusing on automatic differentiation, parallel computing, and their interplay in the context of two fluid dynamics solvers. We demonstrate that carefully designed techniques involving these two enabling technologies improve the efficiency and robustness of simulations in computational aerodynamics. The structure of this article is as follows. In Sect. 2, we introduce notations and some background on automatic differentiation and parallel computing. The combination of these two technologies leads to a notable progress in the performance of the computation of accurate sensitivity information which, in turn, helps to increase the robustness of numerical simulations. For different computational fluid dynamics problems, this is demonstrated taking the two CFD solvers TFS [36, 37, 48, 58] and QUADFLOW [14, 15, 16, 17] as illustrating examples. We start with a discussion of the work related to TFS in Sect. 3. Here, we summarize selected results of [10, 29] and also give some more details on our strategy of delaying derivative computations than available in [29]. In addition, this section contains selected topics of automatic differentiation for parallel programs previously investigated in [63, 67, 68]. Furthermore, we present new results on hybrid MPI/OpenMP parallelization of derivative computations. Concerning TFS, we are primarily interested in derivatives of the flow field with respect to some parameters without altering TFS at all. The situation is different for QUADFLOW whose robustness and algorithms are improved by bringing automatic differentiation into the solver itself. This is described in Sect. 4 which summarizes selected results of [18, 19]. There are additional results of the project A8 that are not covered in this survey. The reader is referred to [6, 9, 11, 12, 21, 22, 23, 24, 25, 26, 27, 28, 30, 72] for the results that are omitted because of lack of space.

2 Automatic Differentiation and Parallel Computing From an abstract point of view, a computer program with n scalar input values that computes m scalar output values represents a function f : Rn → Rm . Automatic differentiation (AD) is a set of techniques to transform a given program capable of evaluating y = f (x) into a new program capable of evaluating the m × n Jacobian matrix J := ∂ y/∂ x. The input variables representing x are called independent variables; the output variables y are called dependent variables. The basic AD techniques are the forward mode and the reverse mode computing the products J · S and S · J, respectively, without explicitly assembling the Jacobian J. Here, the symbol S denotes some user-specified seed matrix of suitable dimension. For instance, initializing S to the n × n identity matrix, the forward mode computes J. So does the reverse mode when initializing S to the m × m identity matrix. Compared to the time of the original program, the time of the AD-generated program is slower by a constant factor. In the forward mode, this factor is roughly given by the number of columns of the seed matrix. The corresponding factor for

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the reverse mode is approximately the number of rows of the seed matrix. A suitable initialization of the seed matrix can be exploited in several different ways. For instance, Jacobian-vector products are efficiently computed using the forward mode with the seed matrix initialized to the column vector. Another example is the computation of the Jacobian of a composition of two functions f (g(x)). This is of interest if the output of a program evaluating a function g is taken as the input to a program evaluating a function f . By the chain rule, the Jacobian is given by ∂ f /∂ x = (∂ f /∂ g) · (∂ g/∂ x). So, the output of the AD-generated program for g can be taken as the input for the AD-generated program for f . More information on AD is given in [66, 45, 47, 4, 35, 20, 8]; see also the community portal www.autodiff.org.

2.1 Automatic Differentiation of OpenMP Programs Automatic differentiation of MPI-parallelized programs is investigated in [49, 50, 38, 39, 33, 32, 59]. Here, we consider programs that are already parallelized with OpenMP [64]. In [31] it was demonstrated that, in principle, AD could easily be applied to OpenMP-parallelized codes. Current AD tools mostly ignore OpenMP directives in the program code, i.e., they are treated as comments. If the comments are generally preserved by the AD transformation, the resulting differentiated program still contains the original OpenMP directives, but additional scoping of the derivative variables into private and shared is required. The corresponding data scoping clauses for the newly introduced variables related to the derivative computation could be inserted automatically in a postprocessing step since the decision whether a derivative variable g x is private or shared depends on the scoping of the corresponding original variable x. For example, consider the OpenMP-parallelized code fragment given in Fig. 1. Now suppose that this code fragment is differentiated propagating n scalar directional derivatives. The resulting differentiated code is depicted in Fig. 2, where the scoping clauses in line 2 are added manually. Note that the differentiated program is parallelized in the same way as the original program. Other OpenMP worksharing constructs like, e.g., sections, could be analogously handled in a straightforward manner. Parallel reduction operations, however, require special attention, as pointed out in [32, 50]. C$OMP parallel do private(i,tmp) shared(L,x,y,z) do i = 1,L tmp = sin(x(i)) z(i) = y(i) * tmp enddo C$OMP end parallel do Fig. 1 Example code with OpenMP directives for loop parallelism

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C$OMP parallel do private(i,tmp) shared(L,x,y,z) C$OMP+private(j,g_tmp) shared(n,g_x,g_y,g_z) do i = 1,L C-------do j = 1,n g_tmp(j) = cos(x(i)) * g_x(j,i) enddo tmp = sin(x(i)) C-------do j = 1,n g_z(j,i) = y(i) * g_tmp(j) + tmp * g_y(j,i) enddo z(i) = y(i) * tmp C-------enddo C$OMP end parallel do Fig. 2 Example code from Fig. 1 after differentiation

2.2 Automatic Scoping As an alternative to explicitly classifying variables into shared and private one could use the so-called autoscoping mechanism [52], an extension to the OpenMP specification currently available only in the Fortran, C, and C++ compilers from Sun Microsystems, Inc. By attributing variables with the new data-sharing clause auto the user can specify those variables to be scoped automatically by the compiler according to certain scoping rules. The default( auto) clause specifies each variable within the active parallel region to be scoped automatically, if no other scoping clause is used. In [52] scoping rules for scalar and array variables are proposed, and most of these rules are implemented in the Sun compilers in order to be applied to variables that are subject to autoscoping. The autoscoping feature is especially useful when differentiating OpenMP-parallelized code. If the default( auto) clause is employed in the original code, it is in general also working for the differentiated code, except for reduction operations. Consider, for example, the OpenMP-parallelized loop in Fig. 1 with the first line replaced by C$OMP parallel do default( auto) enabling automatic scoping for all variables in the parallel region. That is, the variables x, y, z, and L are shared, whereas tmp is private. The loop index variable, i, is scoped implicitly by the OpenMP specification. Applying AD to this code is discussed below; see Fig. 2. The additional scoping clauses for derivative variables introduced by the AD process are no longer necessary if the original code employs autoscoping. In the current Sun compiler suite (Sun Studio 11), only a subset of the autoscoping rules proposed in [52] is implemented. All rules for scalar variables are implemented, and array variables are automatically scoped shared if the use of

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these variables in the parallel region is free of data race conditions. However, if there are data races on array variables, the current implementation does not automatically make these variables private. Instead, these array variables must be privatized explicitly. In the vector forward mode of AD, a scalar variable is associated with a derivative object which is typically represented by an array variable. This implies that manual intervention is required in the differentiated code whenever a scalar variable is automatically scoped private in the original code. Considering the differentiated code in Fig. 2 this means that the variable g tmp must be scoped explicitly, while the remaining variables are correctly handled by the autoscoping mechanism.

3 Automatically-Generated Sensitivities in TFS In this section, we review some results of automatically differentiating the flow solver TFS [48, 58, 36, 37] in several different situations. Although the resulting AD-generated programs may compute different derivatives depending on the present situation, we use the same term TFS.AD hereafter to denote any of these different AD-generated programs. Which derivatives are actually computed by TFS.AD will become clear from the context. Derivatives of the flow field with respect to geometric shape parameters are considered in Sect. 3.1; those with respect to the angle of attack and yaw angle are presented in Sect. 3.2. The software package TFS is developed at the Institute of Aerodynamics, RWTH Aachen University, and is capable of solving the Navier–Stokes equations of a compressible ideal gas in two or three space dimensions with a finite volume formulation. The spatial discretization of the convective terms follows a variant of the advective upstream splitting method (AUSM+ ) proposed by Liou [53, 54]. The viscous stresses are centrally discretized to second-order accuracy. The implicit method uses a relaxation-type linear solver whereas the explicit method relies on a multigrid scheme.

3.1 Sensitivities with Respect to Geometric Parameters In this subsection, we are concerned with the computation of derivatives arising from a shape optimization problem that is solved using derivative-free algorithms in [57, 56]. After introducing a sequence of different programs we demonstrate that automatic differentiation of this computational chain is feasible enabling gradientbased optimization algorithms. The crucial observations is that an appropriate initialization is necessary. Significant further performance improvements are obtained from delaying the derivative computations. 3.1.1

Transforming a Computational Chain by Automatic Differentiation

The shape optimization problem consists of maximizing an objective function f (ξ1 , . . . , ξn ),

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where the free variables ξ j ∈ R denote geometric parameters defining the airfoil, and f is some integral quantity obtained from the flow around the airfoil, e.g., the ratio of the lift and drag coefficients, cL /cD . The objective function f is computed in three stages as illustrated in Fig. 3. By superposing a given “base airfoil” A0 with so-called “mode functions” A j , the first stage generates the actual airfoil n

A(ξ ) = A0 + ∑ ξ j A j . j=1

The number of mode functions is small, a typical value being n = 8. The airfoil A is described via the coordinates of roughly 200 points lying on its boundary, and these coordinates are then fed into a second program, an elleptic grid generator, which base airfoil A0 (exaggerated) geometric parameters

ξ1 , . . . , ξn

airfoil generator (MATLAB)

?

H Y H

actual airfoil A

H “modes” A j

grid generator (Fortran)

control parameters

?

grid G

CFD solver TFS (Fortran)

fluid parameters etc.

? simulation results p, cL /cD , . . .

Fig. 3 Computational chain from the geometric parameters to the simulation results

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produces a multi-block structured grid G. A section of the grid is shown in Fig. 3. It consists of four blocks whose boundaries are also shown in the bottommost picture. Finally, TFS is used to determine the pressure and velocity fields of the flow around the airfoil, from which the objective value f = f (G(A(ξ ))) with ξ = (ξ1 , . . . , ξn )T is easily obtained. A gradient-based numerical optimization algorithm requires the evaluation of the derivatives ∂ f /∂ ξ of the whole computational chain consisting of three separate programs. Thus, its derivative can be obtained by differentiating each of the three stages and combining these derivatives according to the chain rule:

∂f ∂ f ∂G ∂A = . · · ∂ξ ∂G ∂A ∂ξ A naive approach would first explicitly generate the three Jacobian matrices ∂ f /∂ G, ∂ G/∂ A, and ∂ A/∂ ξ by applying automatic differentiation to each of the three programs separately. The second step would then perform two matrix-matrix multiplications. The airfoil generator is written in MATLAB for which we developed a new AD tool called ADiMat [11, 6, 72]. In contrast to previous AD tools that are based on operator overloading [73, 41], the tool ADiMat is based on a combination of source transformation and overloaded operators. AD tools following a source transformation approach are capable of applying the chain rule not only on elementary operations but also on higher-level operations, potentially increasing the performance of the AD-generated programs [7]. The grid generator and the TFS flow solver are written in Fortran, comprising roughly 5,000 and 24,000 lines of code, respectively. After some preparations in order to force strict adherence to the Fortran 77 standard, ADIFOR [5] was used to augment these two codes with derivative computations. That is, the transformed grid generator returns not only the coordinates of the grid points but also their derivatives with respect to the coordinates of the roughly 200 points on the airfoil, and the augmented TFS program computes the derivatives of p, cL /cD , etc., with respect to the grid coordinates in addition to the original results. Since ADIFOR analyzes the whole data flow of each code, it automatically detects that only 112 out of the 227 subroutines of TFS can contribute to the requested derivatives and must be transformed. The augmented routines contain approximately 19,000 lines of code, which — together with the remaining 115 original routines — give a total of roughly 30,000 lines of code for the augmented flow solver. The problem with this naive approach when using a forward mode-based AD tool is the large number of independent variables of the flow solver. Recall that, for computing ∂ f /∂ G, the number of independent variables equals the number of grid points. That is, the execution time (and storage requirement) of the AD-generated code for the computation of ∂ f /∂ G would be proportional to the execution time (and storage requirement) of the flow solver times the large number of grid points.

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Since we are only interested in the gradient of the scalar function cL /cD, the reverse mode of AD would alleviate this problem for the flow solver. However, the problem would remain for the grid generator whose number of dependent variables is large. In summary, this naive approach would be prohibitively expensive in terms of time and storage. 3.1.2

Performance Improvement by Seeding Derivative Objects

To make feasible the computation of ∂ f /∂ ξ , we employ the forward mode with appropriate seeding. That is, we take the AD-generated programs sketched above but use different initializations of the derivative objects. If the seed matrix in the augmented grid generator is initialized with the matrix ∂ A/∂ ξ instead of the identity matrix then this program computes the derivative (∂ G/∂ A) · (∂ A/∂ ξ ) = ∂ G/∂ ξ instead of ∂ G/∂ A. Here, length-n derivative objects are used throughout, where the number of geometric parameters, n, is small. Analogously, initializing the seed matrix in the augmented flow solver TFS.AD with ∂ G/∂ ξ leads to the required derivatives ∂ f /∂ ξ being computed. Again all derivative objects are of length n. Compared to the underlying function f , this AD approach theoretically increases the execution time and storage requirement by a factor of roughly n and is summarized in Fig. 4. To quantify these factors for time and storage in practice, we set n = 8 and compute the objective function cL /cD on a computational grid consisting of 17,428 grid points. The transonic flow conditions are described by a Mach number of 0.74 and a Reynolds number of 2.4 × 106. The angle of attack is given by 1.5593◦ corresponding to a lift coefficient of 0.5. On the airfoil a no slip condition and an adiabatic wall is assumed, the flow variables on the far field boundary are computed with one-dimensional linearized characteristics. Numerical experiments are performed in double precision arithmetic on a 1.7 GHz Pentium 4 Linux machine. Compared to the original program TFS, the storage requirement of the differentiated program TFS.AD increases by a factor of 8.3. The execution time of TFS.AD increases by a factor of 16.5, compared to TFS. So, the approach with appropriate seeding which is detailed in [10] is computationally feasible, though expensive in terms of execution time. 3.1.3

Performance Improvement by Delaying Derivative Computations

For small problems in two dimensions, the AD approach described in the previous section is sufficient. However, when more complex and realistic problems in three dimensions are considered, the efficiency of the computation of accurate derivatives should be increased. This is extremely important for shape optimization problems where the CFD problem and its derivatives are needed multiple times in an iterative optimization algorithm. To further improve the efficiency of TFS.AD—and therefore the efficiency the solution of the overall shape optimization problem—we delay the derivative computations. That is, the original code TFS is started to compute the lift-over-drag ratio and, after a suitable number of iteration steps, TFS is stopped to continue the computation with TFS.AD, where the parts of the code associated

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∂ ξ /∂ ξ = In augmented airfoil generator: A and (∂ A/∂ ξ ) · (∂ ξ /∂ ξ )

?

∂ A/∂ ξ augmented grid generator: G and (∂ G/∂ A) · (∂ A/∂ ξ )

?

∂ G/∂ ξ augmented CFD solver TFS.AD: p, cL /cD , . . . and (∂ p/∂ G) · (∂ G/∂ ξ ), (∂ (cL /cD )/∂ G) · (∂ G/∂ ξ ), . . .

?

∂ p/∂ ξ , ∂ (cL /cD )/∂ ξ , . . . Fig. 4 Derivative computation in the chain using the forward mode with appropriate seeding

to the original simulation are initialized using the data from the last iteration of the TFS run. The basic idea behind this approach is the following. As long as the original function is not converged and hence may vary from iteration to iteration, the corresponding derivatives have little relevance anyway. Hence it is sufficient to start the derivative computation when the original function of which derivatives are sought is already converged to a certain precision. The theory behind this approach is outlined in [43, 45], and it is also applied in [34, 44, 60, 42]. In [3] it was shown that, under mild conditions, automatic differentiation of an iterative algorithm such as implemented in TFS yields a correct iterative scheme for the derivative computation. However, the convergence rate of the iterative scheme for the derivative is not necessarily the same as for the original function, as pointed out in [40, 43, 46]. Therefore, we designed the following two-stage strategy: 1. Run TFS until the solution f is converged to a certain precision. The convergence check is performed in intervals of T iterations. It compares the values of f at iteration (m + 1)T and the “sliding window” of the preceding S iterations with the solution obtained after the previous check, i.e., the value of f at iteration mT + 1. More precisely, we switch to the second stage if | f(m+1)T − j − fmT +1 | < ε1 ,

∀ j = 0, . . . , S − 1.

Here, ft denotes the solution of the original function f at iteration number t, and the parameter ε1 controls the desired precision for f . 2. Run TFS.AD until all derivatives are converged to a certain precision. The derivatives are converged if ∂ f(m+1)T − j ∂ fmT +1 < ε2 , ∀ j = 0, . . . , S − 1 , ∀i = 1, . . . , n , − ∂ ξi ∂ ξi

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where ξ1 , . . . , ξn denote the independent variables, and the parameter ε2 controls the desired precision for the derivatives of f . In practice we typically set ε1 < ε2 because the range of numerical oscillations in the derivatives is often wider than in the original solution. The convergence history of a subset of the derivative computations is depicted in Fig. 5 for T = 100 and S = 10. Let k denote the number of iterations the original TFS is run before switching to TFS.AD to start the derivative computations. Then, Fig. 5 shows, on a logarithmic scale, the convergence history of the derivative of the drag coefficient, cD , w.r.t. to the first geometry parameter, ξ1 , for three different values of k. The first curve, k = 0, describes the behavior of the derivative that is computed from the beginning together with the original function, i.e., a blackbox-like AD approach. The second curve represents the delayed computation of the derivatives after the original function has already iterated for k = 500 iterations and has a precision of ε1 = 10−2 . Finally the third curve, depicts the delay of k = 1700 iterations, where the original function has converged within a precision of ε1 = 10−4 . The figure demonstrates the strong influence of the parameter k on the convergence behavior of ∂ cD /∂ ξ1 . All curves converge to the same derivative value within the given limit of ε2 = 10−2 but the convergence speed tends to be increased when increasing the delay of starting the derivative computations. In order to better quantify the improvement in efficiency that the delayed derivative computation provides, a series of experiments is conducted in which the computation of derivatives is started after k iterations of the original function. The corresponding results are depicted in Fig. 6. At the bottom of each column, the

6

10

k=0 k = 500 k = 1700

4

10

2

∂cD / ∂ξ1

10

0

10

−2

10

−4

10

−6

10

1

1000 2000 3000 4000 5000 6000 7000 8000 9000 iterations

Fig. 5 Convergence history for the derivative of the drag coefficient cD w.r.t. the geometric shape parameter ξ1 where the derivative computation starts after k = 0, k = 500, and k = 1700 iterations

Enabling Technologies for Robust High-Performance Simulations in CFD

16x10^4

TFS.AD TFS

8750 7500

9000

14 12

8800

iterations

8400

6250

10

5000

8

6000

3750

6

4500

2500

300

0

300

500

500

700

700

900

900

4

2500

3000

1250 0

2400

3300

1100

1100

1300

1300

1500

1700

1500 1700

estimated execution time

10000

163

2 0

k

Fig. 6 Number of iterations required to achieve convergence with precision ε2 = 10−2 , and (left scale), where the derivative computation with TFS.AD starts after k iterations of TFS. The line with circle markers, corresponding to the right scale, indicates an estimation of the overall execution time (TFS and TFS.AD), given in number of TFS iterations

number of iterations representing the computation of the original function without any derivatives is presented as a TFS bar, labeled with the respective number of iterations. Above each TFS bar, there is a TFS.AD bar, representing the number of iterations that are performed by the differentiated code, in order to satisfy the convergence criterion with ε2 = 10−2 . For example, the first experiment illustrated by the leftmost column represents the computation of the derivatives from the beginning, together with the original function, i.e., k = 0. The rightmost experiment is composed of k = 1700 iterations of TFS and then another 2400 iterations of TFS.AD. As the figure shows, the number of iterations required for the computation of the desired derivatives decreases as the number of initial original TFS iterations increases, with the exception of the experiment where k = 1300. Recall that the execution time of a single iteration of TFS.AD is approximately 16.5 times that of a single time step of TFS. Therefore, the optimal efficiency is obtained with a minimal number of TFS.AD iterations. An estimate of the execution time of the approach starting the derivative computation after k TFS iterations is given by k + 16.5 · , where is the number of TFS.AD iterations. This estimated execution time, given in the corresponding number of TFS iterations, is indicated by the line with circle markers which corresponds to the right scale in Fig. 6. It turns out that the best combination is obtained in the last experiment corresponding to k = 1700, with the smallest overall execution time of 41, 300 TFS iterations. Compared to the 9000 TFS.AD iterations for the black-box-like approach, which correspond to an execution time equivalent to 148, 500 TFS iterations, this is a speedup of 3.6. Hence, the use of

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the delayed computation of derivatives starting from a converged original function, results in a significant increase in efficiency with respect to the execution time and in this way accelerates the overall numerical optimization process that makes use of the derivatives. In summary, as reported in [29], the factor by which the execution time of TFS.AD compared to TFS is increased is 16.5 for the black-box-like approach and 16.5/3.6 ≈ 4.6 for the approach delaying the derivative computations. Thus, the gradient used in the solution of the shape optimization algorithm with n = 8 geometric parameters is obtained in roughly the time it takes to compute 5 evaluations of the objective function f .

3.2 Sensitivities with Respect to Angle of Attack and Yaw Angle Since the computation of derivatives of a flow field is typically more time-consuming than the computation of the flow field itself, we are interested in parallelizing the AD-generated programs. After sketching a three-dimensional configuration for which derivatives are required, we show fine- and coarse-grained parallelizations and their combination. 3.2.1

Transforming a Parallel Version of TFS by Automatic Differentiation

We consider an inviscid three-dimensional flow around the BAC 3-11/RES/30/21 transonic airfoil which is reported in the AGARD Advisory Report No. 303 [61] and used as reference airfoil for the collaborative research center SFB 401 [70]. The Mach and Reynolds numbers are M∞ = 0.689 and Re = 1.969 × 106, respectively. The angle of attack is −0.335◦ and the yaw angle is 0.0◦ . The computational grid for TFS consists of approximately 77,000 grid points and is divided into three different blocks. We present results from numerical experiments on a Sun Fire E2900 equipped with 12 dual-core UltraSPARC IV processors running at 1.2 GHz clock speed. The SUN Fortran 95 compiler, version 7.1, is used. Details on differentiating the serial version of TFS with respect to various flow parameters, like the angle of attack or the Mach number, are given in [9, 22, 23]. Here, we are interested in derivatives of the MPI-parallel version of TFS with respect to the angle of attack and the yaw angle. 3.2.2

Coarse-Grained Parallelization of Derivative Computations Using MPI

In [68], we outline a strategy to transform MPI-parallelized programs with automatic differentiation and take TFS as an illustrating example where MPI send and receive calls are transformed in order to communicate the derivatives in addition to the original function values. The performance results are summarized in Table 1 where the execution times for the parallel code running on one and three processes is given. The table also shows the resulting speedup, taking the execution time of the parallel code with one MPI process as reference. The second column refers to the original TFS code while the third column contains the data of its differentiated

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Table 1 Execution times required for performing 100 iterations with TFS and its differentiated version TFS.AD. The serial and parallel execution time is measured on a Sun Fire E2900 using one and three MPI processes, respectively Metric

TFS

TFS.AD

Ratio

Serial execution time [s] Parallel execution time [s] Speedup

338 157 2.15

2197 969 2.27

6.50 6.17 1.05

version denoted by TFS.AD. The ratio of the third to second column is displayed in the fourth column of this table. While the original TFS code achieves a speedup of 2.15 employing three MPI processes, TFS.AD yields a slightly better speedup of 2.27 for the same configuration. However, optimal speedup cannot be achieved because the domaindecomposition strategy applied to the multi-block structured grid distributes blocks of different sizes to the three processes. While the largest of the three blocks consists of 35,640 grid points, the remaining two blocks comprise 24,687 and 17,091 grid points, respectively. This leads to a work load imbalance where one process is assigned about 46% of the total amount of computational work, while the other two processes perform only 32% and 22% of the computational work, respectively. As a consequence two of the three MPI processes spend a certain amount of time waiting for the process with the higher work load, assuming identical processing elements. A run-time analysis of the execution trace using the VAMPIR [62] performance analysis and visualization tool reveals that about 23.7% of the overall execution time is spent in MPI communication. 3.2.3

Fine-Grained Parallelization of Derivative Computations Using OpenMP

We employ an incremental approach using the OpenMP shared-memory parallel programming paradigm. To this end, we identify the six most time-consuming subroutines given in Table 2. That is, we parallelize only these six routines and execute all remaining routines serially. In total, 70.72% of the execution time of TFS is spent in these routines. The corresponding percentage 84.8% is even higher for TFS.AD. The drawback of this incremental approach is that, according to Amdahl’s law [1], the potential speedup of a parallel code is limited by the fraction of the code that can not be parallelized. Hence, for the original TFS code, the potential speedup obtained cannot exceed 3.36, even when employing an infinite number of processing elements. For TFS.AD, the theoretical limit of the speedup is larger and is given by 6.58 for an infinite number of processors. So, compared to the original code, the differentiated code has potentially a greater benefit from the OpenMP-parallelization. We first parallelize the six most time-consuming routines of TFS using OpenMP and autoscoping where possible. Then, a combination of ADIFOR [5] and manual scoping of private arrays as described in Sect. 2.2 is carried out leading to an

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Table 2 Six most compute-intensive subroutines of TFS (left column) and their percentage of the total serial execution time (middle column). In the rightmost column, the percentage of the corresponding differentiated subroutines, related to the serial execution time of TFS.AD, is given Routine

%CPU time (TFS)

%CPU time (TFS.AD)

ausm visc viscsrc funcq vort strain ∑

33.3 12.5 9.9 5.7 5.2 3.6 70.2

55.3 12.1 8.3 1.7 3.7 3.7 84.8

8 TFS TFS.AD 7

6

Speedup

5

4

3

2

1

0

1

2

4

8 Threads

12

16

20

Fig. 7 Speedup for OpenMP-parallelized TFS and TFS.AD with varying number of threads on a Sun Fire E2900 with 1.2 GHz UltraSPARC IV, 12 × dual core CPU (24 PE), 48 GB main memory. The speedup is related to the parallel version with a single thread

OpenMP-parallelized version of TFS.AD where the parallelization strategies for TFS and TFS.AD are identical. The speedup for 1 to 20 threads is presented in Fig. 7. Here, the speedup is related to the parallel version with a single thread. As predicted by the theoretical analysis, the differentiated code TFS.AD yields better speedup than the original code TFS. Also the scaling properties for TFS.AD are better.

Enabling Technologies for Robust High-Performance Simulations in CFD

3.2.4

167

Hybrid Parallelization of Derivative Computations Using MPI/OpenMP

The granularity of parallelism in the coarse-grained MPI version is limited to the number of blocks in the multi-block structured grid. Combining the fine-grained OpenMP parallelization with the coarse-grained block-oriented MPI parallelization yields a hybrid MPI/OpenMP parallelized version of TFS with two levels of parallelism. The advantage of this two-level parallelization strategy is its increased flexibility compared to a single level of parallelization. If the computational grid consists of several blocks with different sizes, the pure MPI parallelization typically yields a load imbalance on the different MPI processes assigned to each of these blocks. This implies that, in practice, the achieved speedup depends not only on the number of blocks and MPI processes but also on the ratios of the different block sizes. Figure 8 gives the speedup of the hybrid versions of TFS and TFS.AD where each of the three MPI processes spawns an additional level of parallelism using a varying number of threads. The speedups are moderate due to the load imbalance of the MPI processes and the incremental OpenMP parallelization executing parts of the program serially. Once more, TFS.AD scales better than TFS. With the additional level of OpenMP parallelization, the load imbalances caused by the MPI processes can be compensated by assigning a different number of OpenMP threads to each of the MPI processes, depending on the assigned work

8 TFS TFS.AD 7

6

Speedup

5

4

3

2

1

0

3 = 3x1

6 = 3x2 9 = 3x3 12 = 3x4 15 = 3x5 18 = 3x6 Threads = Processes x Threads per Process

21 = 3x7

Fig. 8 Speedup for MPI/OpenMP-parallelized TFS and TFS.AD with 3 MPI processes and varying number of threads on a Sun Fire E2900 with 1.2 GHz UltraSPARC IV, 12 × dual core CPU (24 PE), 48 GB main memory. The speedup is related to the OpenMP-parallelized version with a single thread

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Table 3 Distribution of threads to 3 MPI processes using the DTB library Number of Threads

3

4

5

6

7

8

9

10

Process 1 Process 2 Process 3

1 1 1

2 1 1

2 1 2

3 1 2

3 2 2

3 2 3

4 2 3

5 2 3

8 TFS TFS.AD 7

6

Speedup

5

4

3

2

1

0

3

4

5

6

7

8

9

10

11 12 13 Threads

14

15

16

17

18

19

20

21

Fig. 9 Speedup for MPI/OpenMP-parallelized TFS and TFS.AD with 3 MPI processes and varying number of threads dynamically balanced using the DTB library on a Sun Fire E2900 with 1.2 GHz UltraSPARC IV, 12 × dual core CPU (24 PE), 48 GB main memory. The speedup is related to the OpenMP-parallelized version with a single thread

load. By measuring the user time for each MPI process at run-time, the dynamic thread balancing (DTB) library [71] is capable of dynamically adjusting the number of threads assigned to each of the MPI processes. Table 3 demonstrates how the threads are distributed over the different MPI processes. For instance, a total number of 9 threads is not evenly distributed. More precisely, the threads are distributed over the three MPI processes by the relation 4 : 3 : 2. This is similar to the relation of the three block sizes 35, 640 : 24, 687 : 17, 091 representing the computational load of the three MPI processes. Figure 9 shows the speedup of the hybrid versions using the DTB library. The speedups for thread numbers given by multiples of three are comparable to those of the hybrid approach given in Fig. 8. However, the main advantage is the efficient utilization of an arbitrary number of processing elements.

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169

4 Automatically-Generated Sensitivities in QUADFLOW Within the collaborative research center SFB 401 at RWTH Aachen University, the flow solver QUADFLOW [14, 15, 16, 17] is developed. After a sketch of the underlying finite volume scheme in Sect. 4.1 we show in Sect. 4.2 that replacing numerical differentiation by automatic differentiation improves the robustness and efficiency of the solver. In Sect. 4.3, we demonstrate that automatic differentiation enables the transition from an approximate Newton method to an exact Newton method.

4.1 Finite Volume Scheme Implemented in QUADFLOW The package QUADFLOW solves the Euler and Navier–Stokes equations for compressible fluid flow in two and three space dimensions. QUADFLOW implements an adaptive finite volume method based on a multiresolution adaptation strategy to solve flow problems for complex geometries. The quadrilateral and hexahedral mesh concept treats meshes as fully unstructured and is embedded in a multi-block framework. The implicit time integration leads to a nonlinear system of equations which is solved by a Newton method. In QUADFLOW, the Newton scheme for the implicit Euler time integration is given by [Jt (un ) + J(un )] Δ un = −R(un ),

(1)

where Δ un = un+1 − un is used to update the vector of conservative variables un and R(un ) denotes the residual vector defined by the sum of the discretized fluxes. The Jacobian in (1) consists of a term Jt (un ) denoting the contribution of the temporal discretization and ∂ R(un ) J(un ) := ∂ un comprising the contribution of the spatial discretization. The system of linear equations (1) is solved iteratively by a Krylov subspace method. So, the Jacobian matrix Jt (un ) + J(un ) is solely needed in the form of matrix-vector products in which the term involving Jt is immediately available whereas the term involving J is difficult.

4.2 Increasing the Robustness of an Approximate Newton Method We replace approximations to the Jacobian J obtained from numerical differentiation by exact derivatives via automatic differentiation. The numerical experiments presented in more detail in [18] show that the nonlinear system of equations is then solved more reliably. 4.2.1

First-Order Accurate Jacobian-Vector Products

QUADFLOW makes use of a lower-order approximation of the Jacobian, Jlow , of the higher-order operator Rhigh . More precisely, the linearization of the fluxes is based on a first-order accurate method in space, meaning that the linearization takes into

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account only immediately-adjacent neighboring cells. That is, the linear system (1) is given by [Jt + Jlow ] Δ u = −Rhigh . (2) Because of the approximation J ≈ Jlow the approach represents an approximate rather than an exact Newton method. Therefore, quadratic convergence is not expected. Numerical differentiation is used to compute approximations to Jlow . That is, divided differencing is applied to the elementary flux functions resulting in the corresponding local Jacobians. The overall Jacobian Jt + Jlow is assembled in a block-wise fashion, where the local Jacobians are appropriately inserted into a global sparse data structure. In [18], numerical differentiation is replaced by automatic differentiation, transforming given Fortran source code for the elementary flux functions of various upwind schemes into Fortran code for evaluating the corresponding local Jacobians. More precisely, ADIFOR is used to generate derivative code for the AUSMDV(P) and HLLC upwind schemes. About 500 lines of Fortran 77 source code have been transformed yielding approximately 1400 lines of differentiated code for the inviscid 2D cases considered below. Various flux functions for two dimensional as well as three dimensional flows, consisting of roughly 1600 lines of Fortran 77, have been transformed automatically by ADIFOR yielding approximately 7700 lines of differentiated code. The advantage of using automatic differentiation is that there is no need to experiment with different step sizes and the Jacobian Jlow is evaluated without truncation error. 4.2.2

Numerical Experiments

The following selected numerical results from [18] are concerned with inviscid fluid flow about an airfoil according to SFB 401 cruise configuration which is assembled from the BAC 3-11/RES/30/21 airfoil reported in the AGARD Advisory Report No. 303. Two distinct flight conditions are selected to represent different classes of flow problems of relevance to aircraft aerodynamics: Quasi-incompressible fluid flow with M∞ = 0.001 and transonic flow with M∞ = 0.75. All computations are carried out on a 1.7GHz Pentium IV processor, using the GCC-3.3.2 C/C++ and Portland Group Fortran77 compiler version 5.1 with -O3, resp. -fast optimization. Table 4 summarizes performance results for the quasi-incompressible flow using an angle of attack of α = 2.0◦ . The table shows the ratios of the required CPU time to reach steady state using Jacobians derived by divided differences (DD), relative to the required CPU time using Jacobians obtained with automatic differentiation (AD). The selected step sizes for DD range between 10−3 ≤ h ≤ 10−9 and the CFL number is varied between 2 ≤ CFL ≤ 1000. An empty entry ’–’ within the table indicates that no stable solution could be obtained for this case, i.e., the nonlinear iteration broke down due to the occurrence of negative densities/pressures in the flow field when using Jacobians derived with DD. It is important to note that, employing AD, a converged solution could be obtained for all CFL numbers. The results indicate that for the considered test case the proper selection of the step size for DD has a large impact on the robustness of the method. Only for h = 10−8 it was possible to increase the CFL number up to CFL = 1000. In all other cases, only very low CFL

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Table 4 Ratios of CPU time using DD with step size h and AD, for varying h and CFL numbers for SFB 401 cruise configuration, M∞ = 0.001, α = 2.0◦ h

CFL = 2

CFL = 4

CFL = 8

CFL = 10

CFL = 100 CFL = 1000

10−3 10−4 10−5 10−6 10−7 10−8 10−9

1.48592 1.51657 1.51686 1.51500 1.51365 1.52245 1.49661

– 1.45524 1.45571 1.45521 1.45126 1.45433 1.44768

– – – 1.52203 1.52230 1.52437 –

– – – – – 1.62403 –

– – – – – 1.53962 –

10

Residual (density)

10

10

10

10

10

0

– – – – – 1.22028 –

DD(h=1.0E−2) DD(h=1.0E−3) DD(h=1.0E−4) DD(h=1.0E−5) AD

−1

−2

−3

−4

−5

0

200

400

600

800

1000

1200

CPU time [sec]

Fig. 10 Convergence histories for SFB 401 cruise configuration, M∞ = 0.75, α = 2.0◦ , CFL=500

numbers could be used to obtain a stable solution procedure. Besides its superior robustness, using AD is about 22% to 62% faster compared with DD. Figure 10 displays the convergence histories for the transonic case. The convergence of the residual versus the CPU time is compared for AD as well as for DD using step sizes between 10−2 ≤ h ≤ 10−5 . In all cases, CFL = 500 is selected. For DD, the convergence history strongly depends on the selected step size. AD outperforms DD with h = 10−2 by a factor of 2.6, and DD with h = 10−5 by a factor of 1.4. For step sizes smaller than h = 10−5, no further improvement was achieved.

4.3 Enabling an Exact Newton Method Due to the convincing results reported in the previous subsection, automatic differentiation is now routinely used in the solution of the nonlinear system of equations whenever numerical experiments are carried out in QUADFLOW. We now

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illustrate that automatic differentiation also enables the transition from an approximate Newton method to an exact Newton method. The corresponding numerical results reported in [19] indicate that there are situations where this is advantageous. 4.3.1

Second-Order Accurate Jacobian-Vector Products

The approximate Newton method based on (2) does not achieve quadratic convergence. To arrive at such a high convergence behavior, we would need an exact Newton method based on [Jt + Jhigh ] Δ u = −Rhigh ,

(3)

where Jhigh represents a linearization of the fluxes based on a second-order accurate method in space. However, the storage requirement for Jhigh is often prohibitively large so that one can not afford to assemble Jhigh in some sparse storage format. Fortunately, automatic differentiation is capable of efficiently performing Jacobianvector products in a matrix-free fashion, i.e., avoiding the need for an explicit data structure for the Jacobian. In [19], we show that an exact Newton method based on (3) is indeed computationally feasible. Although the exact Newton method has the advantage of local quadratic convergence, it is not reasonable if the initial guess is too far away from the solution. To circumvent this problem, we start with an approximate Newton method based on the first-order accurate Jacobian Jlow . When the relative residual of the density first drops below a prescribed threshold ν , we switch to the exact Newton method using Jhigh . This hybrid approach switching between (2) and (3) is similar to the strategy employed in [13, 55]. 4.3.2

Numerical Experiments

We present selected results from [19] comparing the performance of the NewtonKrylov method employing approximate Jacobian-vector products based on a firstorder accurate discretization in space and exact Jacobian-vector products based on a second-order accurate discretization in space. For simplicity, the method employing the approximate Jacobian-vector products is called “first-order method,” whereas the exact Newton method is called “second-order method.” For the firstorder method, we usually compute the Jacobian-vector product by relying on an explicit sparse representation of the Jacobian Jlow because this matrix is assembled anyway for the purpose of preconditioning. A matrix-free implementation of the first-order method is available for performance comparison. For the second-order method we have to use the matrix-free implementation as the storage requirement of Jhigh would exceed the available memory resources. Using the numbering of test cases from [19], we consider the four test problems 1B, 1C, 2 and 3 whose characteristics are summarized in Table 5. In two space dimensions, we consider inviscid transonic stationary flow around the NACA0012 airfoil and the SFB 401 cruise configuration. As a three-dimensional problem, we consider the subsonic inviscid flow around a swept wing of constant chord length

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Table 5 Summary of characteristics of the test cases 1B, 1C, 2 and 3 Test case

Dimension Airfoil

M∞

α

1B 1C 2 3

2D 2D 2D 3D

0.95 1.20 0.77 0.22

0◦ 0◦ 0◦ 4.64◦

NACA0012 NACA0012 BAC 3-11/RES/30/21 Swept wing, BAC 3-11/RES/30/21

with the BAC 3-11/RES/30/21 profile in cruise configuration and a sweep angle ϕ = 34.0◦. This configuration has been experimentally investigated within SFB 401 [69, 51]. The corresponding flow parameters, M∞ and α , are given in the table for all four test cases. Table 6 gives an overview of the simulations concerned with test cases 1B, 1C and 2, where the local time step is determined by an exponential rule for the CFL number. More precisely, the CFL number in time step k is given by CFLk = CFLFAC ·CFLk−1 , where CFLFAC is a constant factor, and CFL0 = 1. The CFL number is limited by a fixed upper bound, CFLMAX . In test cases 1B and 1C, an aggressive CFL strategy is used, whereas a more cautious strategy is used in test case 2 in order to avoid divergence to a non-physical state during the startup phase which is based on Jlow . The reader is referred to [19] for more details on the actual numerical parameters used in these experiments. The table presents the number of time steps and the corresponding CPU times for the matrix-based firstorder method, the matrix-free first-order method, and the hybrid method with switch parameter ν = 10−5 . The number of time steps is almost identical for both first-order implementations, while the hybrid method needs significantly fewer iterations. Comparing the execution times for the first-order methods it turns out that the matrix-based implementation is always faster than the matrix-free variant. This is caused by the fact that the matrix is explicitly built for preconditioning in both cases. For the hybrid method, the same preconditioner is used as in the first-order method. In particular, the preconditioner designed for the first-order approach is reused for the part of the hybrid approach involving second-order information. The table indicates that, when using the hybrid method, the overall execution times are reduced by 43% in test case 1B and 16% in test case 2, compared to the first-order matrix-based method. In test case 1C, no performance improvement can be observed. Next, we focus on test case 1C for the same situation as in Table 6. The residual histories in terms of time steps are given in Fig. 11 for this test case. More precisely, the residual histories for the first-order method and the hybrid method with switch parameters ν = 10−2 , ν = 10−5 , and ν = 10−6 are presented. The figure illustrates the change in the rate of convergence of the hybrid method. When the switch from first-order to second-order takes place, the rate of convergence is immediately increased. This leads to fewer time steps compared to the first-order method. It can also lead to a significant reduction of the overall execution time of the simulation, if

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Table 6 Test cases 1B, 1C, and 2: Number of time steps k and CPU time for the firstorder matrix-based, first-order matrix-free, and hybrid methods. The parameters for the CFL strategy are CFLFAC = 1.5,CFLMAX = 106 in test cases 1B and 1C, and CFLFAC = 1.2,CFLMAX = 105 in test case 2

Test case

First-order Matrix-based First-order Matrix-free k CPU [s] k CPU [s]

Hybrid with ν = 10−5 k CPU [s]

1B 1C 2

539 74 101

61 51 70

503.1 158.3 595.8

534 74 101

576.8 163.9 672.6

287.9 167.5 500.4

Test case 1C −2

10

−4

10 R

−6

10

−8

10

−10

10

first−order hybrid (ν=10−2) −5

hybrid (ν=10 ) hybrid (ν=10−6) 10

20

30 40 50 Time step k

60

70

Fig. 11 Test case 1C: Residual histories of the first-order and hybrid methods varying the switch parameter ν

the switch parameter ν is chosen appropriately. However, if the switch to the secondorder method is performed too early, i.e., the parameter ν is chosen too large, the execution time increases as shown in [19]. In test case 3, we consider a three-dimensional problem where we compare the number of time steps and the execution time of the first-order method and the hybrid method with switch parameter ν = 10−2 . Since the startup phase of the hybrid approach is based on Jlow , both methods coincide until the hybrid method switches to the final phase which is based on Jhigh . Therefore, we focus on the final phase of the computation in Table 7 presenting performance results for different values of constant CFL numbers, CFL ∈ {102 , 103 , 104 }. For each CFL number we vary the parameter p ∈ {1, 2, 3} denoting the level of fill-in of the point-block-ILU(p) preconditioner [65]. From Table 7, we observe that, for large CFL numbers, increasing the level of fill-in p can lead to a reduction of the overall execution time. However, the corresponding increased memory requirements have to be considered. The number of

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Table 7 Test case 3: Number of time steps k and CPU time for different values of CFL and different PBILU(p) preconditioners, neglecting the identical startup phase of both methods

CFL

p

First-order k CPU [s]

Second-order k CPU [s]

102 102 102

1 2 3

142 142 142

1488.2 1583.4 2136.6

164 92 92

3672.7 3747.5 4159.3

103 103 103

1 2 3

77 77 77

1575.1 1614.7 1904.5

12 12 12

1858.5 1493.2 1468.4

104 104 104

1 2 3

71 71 71

1902.9 1859.5 2097.5

5 4 4

674.1 558.5 620.5

time steps is significantly decreased for both, the first- and the second-order method, if the CFL number is increased. The corresponding execution times increase for the first-order method and decrease for the second-order method. For CFL = 104 , the second-order method is significantly faster than the first-order method. On the other hand, with CFL = 102 , the first-order method is faster. Note that for small values of CFL, the higher cost for a single Krylov iteration slows down the secondorder method, whereas for large CFL numbers the time integration benefits from the higher accuracy in the Newton step. For CFL = 104 , the execution time of the second-order method is significantly less than for all first-order computations.

5 Conclusions Automatic differentiation is a technology to transform a given computer program into a new computer program for the computation of derivatives. In contrast to numerical differentiation via divided differencing, the derivative values obtained from automatic differentiation are truncation-error free. We give evidence of the successful transformation of the two flow solvers TFS and QUADFLOW developed at RWTH Aachen University. For TFS, we are interested in derivatives of the flow field with respect to some parameters. These derivatives are computed efficiently. For instance, the derivatives of the lift-over-drag ratio with respect to 8 scalar geometric parameters takes roughly 5 times the time to compute the lift-over-drag ratio. Furthermore, a hybrid MPI/OpenMP parallelization approach is shown to economically utilize any number of processing elements, compensating for the inherent load imbalances caused by an existing coarse-grained MPI parallelization. In QUADFLOW, automatic differentiation not only yields a more robust and reliable overall computational scheme, compared to an approach based on divided

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differences, but is also faster in terms of CPU time. Therefore, it has replaced divided differences and is now routinely used in the computation of the Jacobian during the Newton scheme for the solution of the underlying nonlinear system of equations. Furthermore, automatic differentiation makes possible the transition from an approximate to an exact Newton method by relying on a matrix-free approach avoiding the assembly of the second-order accurate Jacobian. So, automatic differentiation enables a Newton method achieving local quadratic convergence. Acknowledgements. This research is supported by the Deutsche Forschungsgemeinschaft (DFG) within SFB 401 “Modulation of flow and fluid-structure interaction at airplane wings,” RWTH Aachen University, Germany. We would like to thank our local collaborators at the Institute of Aerodynamics, the Chair for Computational Analysis of Technical Systems, the Chair for Numerical Mathematics, and Lehr- und Forschungsgebiet Mechanik. In particular, we appreciate the discussions with J. Ballmann, F. Bramkamp, M. Meinke, B. Pollul, A. Reusken, G. Schiefer, W. Schr¨oder, and F. Zurheide. The Aachen Institute for Advanced Study in Computational Engineering Science (AICES) provides a stimulating research environment for our work.

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Influencing Aircraft Wing Vortices R. Hörnschemeyer, G. Neuwerth, and R. Henke 1

Abstract. Results presented here were obtained within a project which was part of the Collaborative Research Centre SFB 401 “Flow Modulation and FluidStructure Interaction at Airplane Wings” funding from the Deutsche Forschungsgemeinschaft (DFG). The goal of this project was to gain better understanding of aircraft wake vortices in order to investigate possibilities to mitigate the hazard posed by these to following aircraft. To this end wind tunnel testing was undertaken in which the vortex wakes of various wings were measured using hot wire anemometry. It was shown that in the near field, the rolling moment induced on a following aircraft can be significantly reduced by introducing additional turbulence into the wake. Another focus point was the investigation into excitation of short wave instability mechanisms in the vortex wake and their effects in the far field. For these purposes experiments with various models and oscillating control surfaces were conducted in water towing tanks in which the vortex wakes were measured using particle image velocimetry. The results show that for an appropriate multi-vortex system, inherent instabilities can be excited leading to a more rapid vortex decay within the first 30 span lengths behind the model. The effects of these mechanisms further out in the far field are, however, minimal.

1 Introduction In their wakes aircraft generate vortices that result directly from lift and which can pose a hazard to aircraft following at close distances. Such circumstances are present around airports where aircraft takeoff and land at short intervals. Therefore, in order to prevent from accidents safety clearances have been specified, but these also limit airports’ capacities. In order to be able to handle the annually increasing air traffic, intensive research has been conducted for some years in the field of vortex reduction and manipulation. Already in 1978, Rossow showed that the rolling moment induced on a following aircraft by wake vortices can be reduced by one third using fins mounted vertically on the wings [35]. He categorized this alleviating effect as being due to the R. Hörnschemeyer . G. Neuwerth . R. Henke Institute of Aerospace Engineering, RWTH Aachen University, Wüllnerstraße 7, 52062 Aachen, Germany 1

W. Schröder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 181–204. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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convective dispersion of vorticity, which is caused by the wing-mounted fins. A detailed analysis of the vortex wake and vortex parameters was not conducted. In [1] and [8] detailed investigations of the vortex structures for various unconventional flap positions have been carried out, but a quantitative characterization of the hazard posed to following aircraft was not undertaken. Yet another method to influence wake vortices takes advantage of inherent instabilities within the vortex system. In the 1970’s, Crow [6][7] discovered the existence of instability mechanisms for counter-rotating vortex pairs. Building on this work, Crouch [5] investigated the stability of vortex systems with two corotating vortex pairs. In addition to the instability forms found by Crow, Crouch observed additional symmetrical and antisymmetric instabilities at short wavelengths that led to markedly faster vortex decay. In their investigations with two counter-rotating vortex pairs, Rennich and Lele [29] showed a strong increase in the Crow-instability with the presence of another vortex pair. The first flight tests attempting to excite the inherent instabilities with oscillating control surfaces were undertaken shortly after Crow’s findings [7]. Chevalier [2] analysed the behaviour of contrails of a DeHavilland Beaver DHC-2 and determined the disturbance frequency from their sinusoidal oscillations. The excitation of instabilities was accomplished by oscillating the elevator with the calculated disturbance frequency and led to an accelerated dissipation of the vortex wake. However, the oscillation also led to an unallowable lift fluctuations. Crouch [4] therefore suggested using oscillating ailerons to excite the instability mechanisms. He recreated the typical wake of a large airliner in its start configuration in theoretical and experimental investigations, and through oscillating deflections of the inner and outer ailerons, he was able to minimize the lift fluctuations. The disturbance amplification with the method proposed by Crouch led to notably faster dissipation of the vortex system. Earlier and current research is summarized in various overview publications, for example by Spalart [39], Rossow [31] and Gerz [13]. The results presented in this report build directly upon these earlier works. They offer an overview of the work carried out within this project. For more detailed information, references to the relevant publications are given.

2 Coordinate Systems The coordinate systems used are right-handed, orthogonal systems. For wind tunnel half-model investigations during the first two funding periods, the coordinate system origin was located at the wingtip with the x-axis pointing in the direction of flow, the y-axis pointing along the wingspan direction and the z-axis in the lift direction (cp. Fig. 3). For investigations in the water towing tank the same orientation was retained. The x-axis points opposite to the towing direction, the y-axis runs along the wingspan, and the z-axis points in the lift direction. The origin, however, is now located in the wing’s plane of symmetry at the centre of the wing trailing edge (cp. Fig. 14).

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3 Test Facilities 3.1 Wind Tunnel The wind tunnel at the RWTH Institute of Aerospace Engineering (ILR) is a closed circuit type with an open test section having a length of L = 3 m and a nozzle diameter of D = 1.5 m. The flow velocity can be continuously adjusted from U∞ = 0-70 m/s. A half-model experimental rig was constructed for investigations (see Fig. 1). The floor plate served as the symmetry plane and has boundary layer suction built-in at the leading edge. A wing model can be attached to an electronically controlled rotating assembly via the half-model strain gauge that can measure the forces and moments at any angle of attack α. The wind tunnel traverse equipment (WITRA) is located above the test table and can, for example, automatically traverse sensors in all three spatial dimensions. In this manner the velocity fields in the wing model wake can be determined.

3.2 Water Towing Tank at the ILR The water towing tank at the RWTH Institute of Aerospace Engineering has a length of 9 m, a height of 1 m, a width of 1.5 m, and the sides and floor are paned allowing for good optical access (see Fig. 13). The free surface is covered with plastic mats that damp surface waves which are generated during towing runs. The towing rig itself is mounted on tracks that are decoupled from the tank so that vibrations cannot be transferred from the tracks to the tank structure during towing runs. For all discussed investigations, the towing speed is u∞ = 1 m/s, resulting in a Reynolds number ranging from Re = 50000 to Re = 140000 (based on the wing chord length) depending on the specific model. At the beginning of a towing run the model is continuously accelerated at about 0.4 m/s2, and the final deceleration at the end of the tank occurs over a length of about 0.5 m. This deceleration stage produces a disturbance wave, which propagates in the direction opposite to the towing direction with a speed of ca. v = 0.1 m/s. Depending on the model used in the investigations, the time remaining before this disturbance reaches the measurement plane allows for investigation of the wake vortex for up to 80 span lengths behind the model.

3.3 Water Towing Tank at the DST External investigations into wake vortices were undertaken in the deep water tank belonging to the Development Centre for Ship Technology and Transport Systems (DST) in Duisburg. The tank has a width of 3 m over a length of 73 m, whereas 45 m were used. The water depth during the investigations was 2.63 m. The model mount was realized with an extension arm, manufactured at the RWTH Institute of Aerospace Engineering, attached to the side of the towing rig. The towing rig accelerates and decelerates with a = 0.5 m/s2 and reached a velocity of u∞ = 1 m/s in the investigations discussed here. Since none of the infrastructure required for the experiments is available at the DST, other than the towing rig, a sinkable tower

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was custom-made at the ILR that can accommodate the cameras needed to observe the wake vortices. Additionally, the ILR designed and built equipment to accommodate the lasers and make possible the required well-defined laser light sheet in the measurement plane.

4 Measurement Techniques and Instrumentation 4.1 Particle Image Velocimetry In recent times, particle image velocimetry has proven itself capable for investigations of wake vortices. The positions of particles suspended in the fluid medium are recorded at defined discrete time intervals, and by using a correlation function in conjunction with this information both magnitude and direction of the most probable velocity vector of a particle group can be calculated. In order to infer the flow velocity from the particle velocity, the densities of the particles and fluid medium should be as close to one another as possible. In these experiments polyamide particles with a mean diameter of dP = 55 μm and a density of ρP = 1,016 g/cm3 were used, resulting in good fluid-following characteristics while simultaneously guaranteeing a high light reflectance factor. In order to illuminate the particles, up to two pulsed Nd:YAG lasers each with a pulse energy of 125 mJ were used. Using optical lenses, the point-shaped laser beam was expanded to a 3 mm thick light sheet. A synchronizer ensures proper sequencing of the recordings and the in sync triggering of the laser and CCD cameras. In the case of two-component PIV (2C-PIV), the particle reflections in the measurement area are recorded by a single camera. This allows flow velocities to be resolved in two dimensions as the projection on the plane formed by the laser light sheet. In order to capture all three velocity components in the measurement plane, the previously described PIV-technique can be expanded upon by using an additional camera. Both cameras look at the same measurement area but from two different angles θ1 and θ2. The recorded particle images are correlated separately and the subsequent integration of these results delivers the three-dimensional velocity vectors within the measurement plane (3C-PIV).

4.2 Flow Visualization In order to make the vortex core visible in the towing tank, a method to generate tiny air bubbles was used. The method, introduced by G. Neuwerth [21], consists of a tank partly filled with water, and whose internal air pressure can be increased. Under higher pressures, part of the air is dissolved in the water until saturation is reached. With a pressure of pA = 6 bar, the achievable concentration is on the order of c ≈ 20 mg air / kg water. Through tubes inside the model this water with high concentrations of dissolved air can be transported to and discharged from locations where vortices are generated. Since these locations have lower pressures the dissolved air is freed in the form of tiny, yet highly visible bubbles. Due to the bubbles’ low density in comparison to the surrounding fluid they collect at the vortex cores. The bubbles

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have a high reflectivity and can therefore be photographically recorded under appropriate lighting.

5 Results 5.1 Investigations of Wing Wakes in the Near Field with Additional Fin Vortices Building on experiments of Rossow [32][33][34][35] experimental and theoretical work was undertaken to investigate the influence of fins on the wakes of straight rectangular and swept wing planforms, both with and without flaps, in the wind tunnel (Fig. 1). In order to quantify the effects, the rolling moment induced on a trailing wing is used. Additionally the rolling moment coefficient induced on an imaginary trailing wing can be calculated behind the model according to the following formula:

2 cl = 2 bf

bf 2

∫

−b f

⎛w⎞ c Aα arctan ⎜ ⎟ y dy ⎝u⎠ 2

(1)

Fig. 2 shows the rectangular wings which have the SFB reference airfoil section (BAC 3-11/RES/30/21). The rectangular wing without flaps has a span length of b = 1 m and a chord length of c = 100 mm and is segmented along the spanwise direction. In the investigations with fins, flat plates (thickness d = 0.5 mm, chord cFinne = 25 mm, height h = 25 mm and 50 mm) were clamped between the segments. These fins each have a defined angle of incidence εF (0°, 8.53°, 34° depending on Position) relative to the flow. Fig. 3 shows the position of the coordinate system together with the wing and fin. The rectangular wing with flaps, shown in Fig. 2 and Fig. 3, has a span length of b = 1.11 m and a chord length of

Fig. 1 Wind tunnel test rig

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Fig. 2 Rectangular planform wing without flaps (left) and with flaps (right). Definitions for fin position and fin angle setting εF

Fig. 3 Coordinate system position for the rectanguFig. 4 Swept reference wing with fin lar wing with and without flaps

c = 120 mm and is also segmented in the spanwise direction. The wing’s flaps are held in place by support plates located on the sides of the individual segments. The fins are attached to the wing (εF = ±15°) and have a rectangular planform (height h = 29 mm, chord cFin = 29 mm, thickness = 7 mm) with a symmetrical NACA airfoil section. Another model, is the swept reference wing shown in Fig. 4. Scaled at 1:53.3, this wing has a span length of b = 1.35 m, a chord length of c = 276 mm at the wing root, and a chord length of c = 75 mm at the wingtip. Its airfoil section is the defined SFB reference section. The wing is equipped with both slats and trailing edge flaps. The slats can be attached along the entire span length, and two separate settings (extended and retracted) are possible by using two different slat sets. The trailing edge flaps can only be mounted on the inner area of the wing while the outer trailing edge wing section represents ailerons that are not deflected. For the flaps three positions are possible (retracted and extended by η = 10° und η = 20°) by using one of three available flap attachments, analogous to the slats. The first investigation on the rectangular wing without flaps shows that a large fin located near the tip vortex and with a large angle of incidence leads to the largest reduction in induced rolling moment (Fig. 5). This reduction is caused by the strong enlargement of the vortex due to the separated and turbulent fin wake. Similar investigations for the rectangular wing with flaps show that placement of the fin near the outer flap vortex leads to the greatest reduction in the induced rolling moment (Fig. 6).

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Fig. 5 Plots of the maximum induced rolling Fig. 6 Plots of the maximum induced moment coefficient near the tip vortex for the rolling moment coefficient for the rectanrectangular wing (α = 6°, cFinne/b = 0.025, gular wing with flaps (α = 4°, η = 20°, cFinne/b = 0.026, h/b = 0.026, bf/b = 0.2) bf/b = 0.2)

Based on these results, more detailed investigations were undertaken on the swept reference wing with various flap settings and fin positions [24][25][26][27][28][36][37][38]. Using oil streak flow visualizations on the reference wing in the wind tunnel, the effective fin angle of incidence was determined. In Fig. 7 (a) the results of the oil streak experiments are shown for the reference wing without and with the fin εF = ±20°. The area of the outer flap end, where the fins have been attached, is shown. For the configuration without fin, a local flow angle of βlok = 15° relative to the wing root can be identified from the oil streak pattern. This results in an effective angle of incidence of βeff = 35° for a positive fin angle setting (εF = 20°) and βeff = -5° for a negative fin angle setting (εF = -20°). The fin will have a positive angle when the fin vortex has the same rotational direction as the tip vortex. The large flow angle of incidence for the positive fin setting causes a large separation on the fin’s suction side, made clear by the hatching in Fig. 7 (b). In contrast, the angle of incidence for the negative fin setting is very small resulting in only small separation due to the fin (see Fig. 7 (c)). Furthermore, a horse shoe vortex forms at the base of the fin, as shown in Fig. 7 (b) and (c). The flow visualization with smoke in Fig. 8 und Fig. 9 on the reference wing with and without fin clearly shows that the fin with a positive angle setting (εF = 20°) causes a break-up of the concentrated flap vortex structure. In contrast the tip vortex is seemingly undisturbed by the fin, and remains stable and concentrated within the investigated area of the wake. The results from both the oil flow investigation and the smoke visualization indicate that the fin, with a positive angle setting, produces a turbulent wake due to flow separation, strongly influencing the flap vortex. In Fig. 10 and Fig. 11 the vorticity distribution ωc/U∞ in the reference wing’s wake without and with a fin with positive fin angle (εF = 20°) is shown. The vorticity distribution is calculated either from the measured or from the calculated velocity components in the wake of the wing configuration:

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Fig. 7 Oil streak pattern for the outer flap area of the reference wing (a): without fin, local flow angle of incidence βlok = 15° (b): with fin at εF = 20°, effective flow angle of incidence βeff = 35° (c): with fin at εF = -20°, effective flow angle of incidence βeff = -5°

Fig. 8 Smoke visualization of the tip and flap Fig. 9 Smoke visualization of the tip and vortices in the reference wing’s wake without flap vortices in the reference wing’s wake fin with fin at εF = 20°

ωc

⎛ ∂w ∂v ⎞ c = ⎜⎜ − ⎟ . U ∞ ⎝ ∂y ∂z ⎟⎠ U ∞

(2)

The tip and flap vortices in the wake of the configuration without a fin are clearly visible in Fig. 10. Through the fin’s influence, the flap vortex expands and its vorticity values decrease when compared to the configuration without a fin, confirming the results from the flow visualization. Due to the flow separation, the fin produces a turbulent wake that enlarges the flap vortex structure. The strength and size of the turbulent fin wake depends on the fin angle setting and the local flow conditions on the wing. Furthermore, depending on the angle setting, the fin generates a fin vortex which can be detected in the first measurement plane at x/b = 0.0 in Fig. 11. The strength and rotational direction of the fin vortex also depends on the fin angle setting and the local flow angle of incidence at the fin. If the effective flow angle of incidence βeff is positive, the fin vortex’s rotational direction is the same as the flap or tip vortex. The results from flow visualization and velocity measurements make clear that the strength of the turbulent fin wake and the fin vortex varies depending on the effective angle of incidence βeff. At a small effective angle of incidence βeff the separation at the fin is minimal and the fin wake’s turbulence is relatively small while the fin vortex is well formed. For large effective angles of incidence βeff the flow separation is greater at the fin so that the turbulent fin wake is much more pronounced and only a weak fin vortex is generated.

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Fig. 10 Vorticity distribution ωc/U∞ in reference wing wake without fin

189

Fig. 11 Vorticity distribution ωc/U∞ in reference wing wake with fin at εF = 20°

Fig. 12 Comparison of the vorticity distribution ωc/U∞ at x/b = 0.4 in reference wing wake with flaps and fin at εF = -20° from measurements and from simulation with the 2D-NavierStokes und 3D-Navier-Stokes techniques (α = 8°, δi = δa = 20°)

The near field of the reference wing with flaps has been simulated in numerical investigations using 2D-computations by vorticity transport equations developed by Fell et al. [11][12] and Türk et al. [40][41][42][43] as well as 3D-Navier-Stokes computations. In both methods the measured velocity and vorticity distributions at the wing trailing edge were used for the initial conditions. In Fig. 12 the measured and the numerically calculated vorticity distribution ωc/U∞ in the reference wing’s wake with a fin angle of εF = -20° at x/b = 0.4 are shown for comparison. The vortex structure has been qualitatively well reproduced with the numerical techniques. However, it can clearly be seen that small structures such as the tip vortex could not be resolved very well due to the large grid size. This is especially problematic for the simulation using the 3D technique because of the significantly larger grid (3D technique ca. 2.5×106 grid points, 2D technique ca. 2.8 ×105 grid points). A detailed comparison of the numerical and experimental results is given in [24]. One conclusion is that the circulation distribution of larger vortex structures such as the flap vortex are rendered better by

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the numerical simulation than smaller vortex structures. Qualitatively, the maximum induced rolling moment is reproduced well, whereby the induced rolling moment for a trailing aircraft calculated by the 3D simulation is closer to the experimental results than the 2D simulation due to the turbulence model and the consideration of three dimensions. The 2D technique indicates considerably larger values for the maximum induced rolling moment than has been determined from experiments.

5.2 Investigations of Multi-vortex Systems in the Far Field In order to address the effects in the near far field and to investigate the sustainability of wake manipulation, measurements were conducted at the ILR’s water towing tank (see Fig. 13). The PIV technique was used, collecting velocity information in planes behind the model. This way, the wake can be observed up to the near far field. The measurements are limited by the boundary influence from walls which mostly affects the trajectory while the vortex structure remains unaffected. Disturbances stemming from surface waves can effectively be minimized with a stiff surface cover on top of the tank. 5.2.1 Rectangular Wing with Vortex Generators The first measurements were carried out using a rectangular planform wing model with a NACA0012 airfoil section. The model’s span length is b = 0.37 m with an aspect ratio of Λ = 7. This model can be equipped with triangular vortex generators (TVG), as shown in Fig. 14, based on the experiments carried out by Ortega et al. [22][23]. The vortex generators are extensions of the wing trailing edge tangential to the suction side. In order to investigate various circulation and span length ratios, the vortex generators’ geometries were varied over a wide range (see Table in Fig. 14). The wing’s angle of attack was set to α = 10°. The goal of these investigations was not to generate a turbulent wake as was done in wind tunnel experiments using fins, but rather to create an inherently instable vortex system. In this manner the investigations, dealing with the stability of counter-rotating vortex pairs, build upon those conducted by Rennich and Lele [30], as well as by Fabre and Jaquin [10]. The particular variation of the vortex generators with bTVG/b = 0.5 and cTVG/c = 0.7 resulted in the plot of the induced rolling moment coefficient, normalized with the lift coefficient, shown in Fig. 15. The hazard posed to following aircraft is markedly reduced compared to the clean wing configuration without TVGs. The rolling moment begins with very high values and initially decreases due to the rotation of the vortices around each other. At x/b = 0 the two counterrotating vortices of the half plane are situated horizontally next to each other. The maximum rolling moment that results from flying into the wingtip vortex is therefore increased by the additionally generated vortex. As time progresses the TVG generated vortex rotates about the wingtip vortex and reduces the induced rolling moment that results from flying into the wingtip vortex.

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Fig. 13 Water towing tank

bTVG/b: Span length ratio cTVG/c: Chord length ratio

bTVG/b 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.8 0.8 0.8 0.8

cTVG/c 0.4 0.5 0.6 0.7 0.4 0.5 0.6 0.7 0.4 0.5 0.6 0.7 0.4 0.5 0.6 0.7

Fig. 14 Left: Rectangular wing model with triangular vortex generators, Right: Table 1 Investigated TVG geometries

Data was evaluated up to a distance of 80 spans behind the model. An effectiveness criterion is defined as the distance xec/b behind the model for which the following condition is met:

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C l , max ( xec b) CL

=

Cl , max, ref ( x b = 80) C L , ref

.

(3)

This means that at a distance of xec behind the wing with vortex generators, the same hazard to following aircraft is present as that which exists 80 span lengths behind the reference wing. Since the motivation of this work was to investigate possibilities to decrease the required air traffic safety clearances, the above equation is a crucial criterion. For the case that the hazard to following aircraft 80 span lengths behind the reference wing has reached an acceptable level, then this same acceptable degree of hazard has also been reached at the distance xec. In the case shown for a wing with vortex generators the distance is xec/b = 22, so that the safety clearance could be reduced by about one-fourth under the assumed boundary conditions. From the measurements it can be ascertained that large circulation ratios and small span length ratios lead to the required reductions in the rolling moment coefficient. Good results can therefore be expected for configurations that have circulation ratios of Γi/Γo < -0.29 at span length ratios of bi/ba < 0.62. The indices i and a denote the inner and the outer vortex pairs, respectively. All investigated configurations that fulfilled these criteria led to an effective weakening of the wake. Considering the area with bi/ba ≈ 0.6 in Fig. 16, it seems that the essential condition for an effective impact on the wake is the existence of a strong second vortex. However, these conditions concerning span and circulation ratio would be hardly achievable at realistic aircraft configurations. A detailed description of the results is given by Haverkamp et al. in [16][17].

Fig. 15 Induced rolling moment on a follow- Fig. 16 Position of the investigated cases ing wing with span length bf/b = 0.2 in the Donaldson-Bilanin-Diagram (bTVG/b = 0.5, cTVG/c = 0.7)

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5.2.2 Stability Analysis A stability analysis of the investigated vortex system provides information as to at which frequency range the vortex system in the wake of various configurations becomes instable. The stability of a vortex system comprising two vortex pairs has been investigated with the linearized Biot-Savart law, whereby the model and notation stem from Crouch [5]. A detailed description of the underlying theoretical basis is given by Haverkamp [14] and Kauertz [19]. With the circulation of the TVG-generated vortex Γi and the circulation of the wingtip vortex Γa, the resulting total circulation in one half-plane is given by Γ0 = Γi + Γa. The four vortices are numbered according to Fig. 14. For the stability investigation, the vortices are excited by periodic disturbances that exhibit small amplitudes η and ζ in the y- und z- directions. With the wavelength λ, the wave number for a disturbance is given by k = 2π/λ, indicating the number of waves per unit of length. The wave number is made dimensionless by multiplication with the distance between vortex centroids b0. The helical progression of the vortices can be neglected under the assumption that the wave length of the vortices’ rotation around each other is much larger than the disturbance’s wavelength. This implies that the computation can be carried out in the time domain rather than in the space-time domain. The validity of this assumption has been shown by Fabre et al. [9]. Input values for the stability analysis are the circulation ratios (Γi/Γa), the span length ratios (bi/ba), and the core radii (rc,a/b; rc,i/b), which can be ascertained from the measured velocity distribution directly behind the wing. Results of the stability analysis for the same configuration as in Fig. 15 are shown in Fig. 17. For the investigated vortex system, numerous movement patterns become instable in various frequency ranges, whereby a maximum growth rate is reached at a wave number of kb0 = 8.0. The instability of an isolated vortex pair (Crow-instability) has also been identified for the case of a four-vortex system.

Fig. 17 Growth rate as a function of the wave number (bTVG/b = 0.5, cTVG/c = 0.7)

Fig. 18 Flow visualization in the wake of a wing with vortex generators, cTVG/c = 0.7, bTVG/b = 0.5

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With increasing Eigenmodes, the probability increases that only the weaker vortex will be perturbed since the amplitude ratio decreases with an increasing wave number. Additionally, another problem arising at high frequencies is that the curvature of the vortex lines become very pronounced even at small amplitudes so that the flap vortex, which is more strongly deflected, becomes locally dispersed before the wingtip vortex can be significantly disturbed. Without external excitation of a vortex system, the most instable Eigenmode will prevail. To illustrate this, results from the flow visualization in the wake of the investigated wing with TVGs is shown in Fig. 18. At a distance of x/b ≈ 20 the oscillation of the right wingtip vortex is especially noticeable and the left wingtip vortex has already decayed to a large degree. The wavelength of the oscillation is about λ = 0.4 m. The wave number at which the vortex system is most instable, as determined from the stability analysis, corresponds to a wavelength of λ = 0.43 m. The minimal deviation between these two values means that the stability analysis can identify the frequencies of the instable natural oscillations quite well despite the approximations used. 5.2.3 Rectangular Wing with Oscillating Control Surfaces More realistic models having rectangular planform wings with integrated control surfaces were designed. By setting the control surface deflection angles, vortex systems similar to those generated by the model with vortex generators can be generated. The model shown in Fig. 19 also has a NACA0012 airfoil section. The wing’s span length is b = 0.5 m, and it is equipped with two ailerons on each side, which can both be preset and/or oscillated periodically. The wing chord length is

Fig. 19 Model of a rectangular planform Fig. 20 Growth rate as a function of wave wing with oscillating ailerons number and investigated wave numbers for α = 4°, δV = 20°

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Table 2 Experimentally investigated wave numbers and the corresponding wave lengths for a wing with preset aileron deflection angle, α = 4°, δV = 20° kb0 λ in m

0.79 4.80

2.38 1.60

4.76 0.80

7.93 0.48

11.11 0.34

12.69 0.30

14.28 0.27

15.87 0.24

c = 75 mm, and the four ailerons each have a span of bR = 50 mm. The Reynolds number is Re ≈ 75000 with respect to the chord length, and for the experiments the wing’s angle of attack was chosen to be α = 4°. The two ailerons on each side can be oscillated in opposite phase so that lift fluctuations resulting from their movement is small. The preset deflection angle for the ailerons is δV = 20°, generating an additional vortex pair in the wake. The initial span length ratio is bi/ba ≈ 0,6 and the initial circulation ratio is Γi/Γa ≈ -0.35 so that the vortex system is positioned in the lower periodic area of the Donaldson-Bilanin Diagram (see Fig. 16). Therefore, strong instabilities can be expected to be artificially excited using aileron oscillation. Application of the previously discussed stability analysis to the vortex system described here results in the plot of growth rate over wave number shown in Fig. 20. The wave numbers and growth rates of the symmetrical oscillation modes have been included in the picture. The corresponding wave lengths are given in Table 2. In Fig. 21 the coefficients of the maximum induced rolling moment are given for the cases of statically deflected ailerons and preset, oscillating ailerons. The excitation is carried out at a wave number of kb0 = 12.7 and leads to a markedly accelerated reduction in the induced rolling moment coefficient. The vortices are already strongly disturbed at the outset, and the instabilities cause a rapid decay of the vortex structures, leading to a reduction in the hazard pose to following aircraft. The aileron oscillation however does not result in a reduction of the rolling moment coefficient in the far field when compared to the static case. The acceleration in the reduction of the rolling moment coefficient depends on the excitation frequency. Fig. 22 shows the parameter ξCl/2, which is defined in Equation (4) and is the distance at which a reduction of the rolling moment coefficient to half of the original static value is reached, with respect to the corresponding distance in the static case. Equation (4) clarifies this parameter, as the ratio of the two distances:

ξC

l

/2

=

( x / b) kb0 ( x / b) static

(4) C l / C A = 12 ( C l / C A ) 0 , static

It is apparent that the hazard decreases more rapidly with increasing wave numbers. The largest acceleration in the reduction of the induced rolling moment coefficient is reached at kb0 ≈ 14. Significant hazard mitigation can therefore be attained with the use of oscillating ailerons, so that a safe hazard level could possibly be reached at shorter distances behind aircraft. This can lead to a reduction in the safety clearances. Detailed information on these investigations can be found in Haverkamp [15].

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Fig. 21 Maximum induced rolling moment Fig. 22 The measure ξCl/2 characterising the coefficient behind the wing for statically de- acceleration of the alleviation of the induced flected ailerons and for preset oscillating ai- rolling moment, α = 4°, δV = 20° lerons, α = 4°, δV = 20°

Fig. 23 View and dimensions of the rectangular wing with winglets

5.2.4 Winglet-Equipped Wing with Oscillating Control Surfaces In order to investigate the possibilities of using additional control surfaces to influence wake vortices, another rectangular planform wing was designed which is equipped with winglets (see Fig. 23). Both the wing section and the winglets themselves have NACA0014 airfoil sections. The winglets are installed with a dihedral angle of ν = 45° relative to the main wing and have a leading edge sweep of φ = 25°. The trailing edges of both the wing and winglets have integrated control surfaces which can be independently deflected. Additionally, the winglet rudders and the ailerons can each be oscillated separately or collectively opposite in phase over a wide frequency range about a chosen control deflection angle. The amplitude of the aileron oscillations was set to ±5° while that for the winglet rudders was set to ±10°. As opposed to the vortex systems discussed so far, three vortices per half plane are usually generated in the wake of this model. Fig. 24 shows the vorticity distribution of the left half plane. One can easily identify the outer and inner winglet vortices and the aileron vortex. An additional vortex is generated by the sting mount, but this vortex drifts downwards within the first few span lengths behind the wing and therefore doesn’t influence the vortex behaviour thereafter.

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Fig. 24 Vorticity distribution due to winglet rudder and aileron deflection (χ = 10°, δ = 20°, α = 4°)

Fig. 25 Plot of the induced rolling moment coefficients (χ = -10°, δ = 20°, α = 4°)

Results presented here build upon the investigations by Kauertz, which were carried out with the same wing model. Kauertz [19][20] considered just the excitation of instabilities with the winglet-integrated rudders, and this work followed by additional investigations on the effect of ailerons. Fig. 25 to Fig. 28 summarize the results for the cases of (1) statically deflected control surfaces, (2) oscillating winglet rudders, (3) oscillating ailerons and (4) both oscillating winglet rudders and oscillating ailerons. Fig. 25 and Fig. 26 respectively show the induced rolling moments for an imaginary following wing with a span length bF and the vorticity distributions at x/b = 0.1 for a winglet rudder deflection of χ = -10°, an aileron deflection of δ = 20° and an angle of attack of α = 4°. Similarly, Fig. 27 und Fig. 28 show the analogous results for a configuration with a winglet rudder deflection of χ = 0°, an aileron deflection of δ = 20° and an angle of attack of α = 4°. Comparing the two configurations, a significant change in the vorticity distribution directly behind the wing is apparent. The vortices at the winglet tip and at the winglet-main wing junction are both pronounced with the winglet rudder not deflected. This leads to lower vorticity values of the individual vortices and therefore results in rolling moments with relatively low initial values. Overall the plots for the case with χ = 0° are flatter than those in the

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Fig. 26 Vorticity distribution in the plane x/b = 0.1 (χ = -10°, δ = 20°, α = 4°)

Fig. 27 Plot of the induced rolling moment coefficients ( = 0°, = 20°, = 4°)

previously considered case and after about 35 span lengths behind the model are at the same levels as those for the configuration with χ = -10°. For numerous investigated configurations, it can be said that the excitation of the vortex wake with aileron oscillations is considerably more effective than exciting the strong wingtip vortex using the winglet rudder [18]. However, the winglet rudders do cause a shift of the vorticity distribution depending on their deflection. It is therefore possible to manipulate the evolving vortices with winglet rudders to create a vortex system that alleviates the induced rolling moment for following aircraft. Furthermore, by concurrently oscillating winglet rudders and ailerons in opposite phase, the lift fluctuations may be minimized. In general, the curves of the induced rolling moments on the following wing for the different excitation forms exhibit declining trends with differing local gradients. Within the first few span lengths behind the model the excitations prove to be highly effective and the induced rolling moment declines fairly rapidly. However, after about 20 span lengths behind the model this decline due to control

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Fig. 28 Vorticity distribution in the plane x/b = 0.1 (χ = 0°, δ = 20°, α = 4°)

surface oscillation subsides. In contrast, the reference curve for the case with statically deflected control surfaces shows a relatively constant gradient throughout the entire measurement area. After about 40 span lengths, the relative hazard posed to a following wing is therefore at about the same level for all investigated cases. 5.2.5 Investigations with a Realistic Wing-Body Configuration A yet more realistic model was designed with the SFB reference wing and a fuselage, whereby the wing has an angle of incidence of ε = 4° relative to the fuselage. Modular construction allows the model to be used as either a half-model with a half-span of b/2 = 0.6 m, or as a full model with a span length of b = 1.2 m (see Fig. 29). The wingtips are equipped with winglets having integrated rudder control surfaces, and the wing itself has ailerons in the outer wing section. Both the winglet rudders and the ailerons can be statically deflected as well as oscillated in opposite phase to each other. Wake investigations were conducted at the ILR’s water towing tank with the half-model, whereby the model was positioned such that the tank floor constituted the symmetry plane of the full model. The transition between model and tank floor was realised with bristles. Further measurements were also carried out for the chosen configurations at the larger towing tank at the DST in Duisburg using both the full model and the half-model. Results for the configuration with a winglet rudder deflection of χ = 0°, an aileron deflection of δ = 15° and a fuselage angle of attack of α = 4° are shown in Fig. 30 to Fig. 32. Fig. 30 shows the induced rolling moment on a trailing wing for the various models with statically deflected control surfaces in the different towing tanks. Good agreement among the results is observed up to 12 span lengths behind the model. This shows that the results from different towing tanks are comparable and that simple vortex systems for the estimation of the hazard posed to trailing wings can be investigated with half-models. After about 12 span lengths

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Fig. 29 Wing-Body combination, as half-model or full model

Fig. 30 Development of the induced rolling moment coefficient in the wake of the half- and full model in various water towing tanks

behind the model the individual curves begin to diverge. Most notably the curve for the induced rolling moment coefficient of the half-model in the smaller towing tank decreases sharply. A comparison of the development of the core radii in Fig. 31 also shows a notable increase in core size after this distance for the ILR towing tank. This effect is related to the geometry of the smaller tank, specifically the shorter measurement section. The acceleration and deceleration phases cause disturbances that migrate into the measurement plane, contaminating the results. On the basis of the stability analysis, investigations with this model were also conducted with oscillating control surfaces. Fig. 32 shows a comparison of coefficient plots of the rolling moment induced on a following wing for statically deflected and oscillating control surfaces. The two curves are nearly overlapping— the additional excitation of the vortex wake shows no reduction in the hazard posed to a following wing. This is mainly due to a very weak aileron vortex,

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Fig. 31 Comparison of the core radii development in the wake of the half- and full model in various water towing tanks

Fig. 32 Comparison of the rolling moment induced on a following wing in the wake of the full model with statically deflected control surfaces and with oscillating control surfaces

which, although it is shifted and perturbed, decays before it can influence the much stronger and dominating wingtip vortex. More promising would be an aileron situated further inboard, which decreases the span length ratio while increasing the circulation ratio. For a conventional wing, however, an aileron fulfilling this requirement would have to be positioned in the area of the high-lift system.

6 Conclusion Within the framework of this project numerical and experimental investigations were carried out to analyse vortex wakes of wing models and to find measures which minimize the hazard posed by wake vortices to following aircraft. Initially, the near field of various wings was investigated in a wind tunnel. Results show that the introduction of additional turbulence into the vortices, for example with fins, reduces the hazard to a following aircraft within the measurement area. This assertion is supported by Navier-Stokes computations that were undertaken in parallel to the experiments. To investigate the effects of vortex manipulation measures in the far field, experiments were carried out in water towing tanks. In this case the primary goal was to create an inherent instable wake vortex system and to

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obtain a faster decay of this system through excitation by selectively impinging small disturbance velocities. A stability analysis was used to identify frequencies at which inherent short wave instabilities within the vortex system become excited. For certain ratios of the vortex parameters, such an artificially impinged excitation does actually lead to a markedly more rapid decrease of the rolling moment induced on a following wing within the first 30 span lengths behind the model. Further downstream, however, the excitation shows no alleviating effect on the hazard posed to a following aircraft. The adherence of the span length and circulation ratios would be problematic for a realistic wing configuration since the ailerons would be in the span position of the high-lift system. One possibility could be to actively influence vortices using the elevator control surfaces. Thus far though, the results indicate that the excitation of short wave instabilities is not promising in the far field. Better prospects may be achieved by exciting the long wave Crow-instability.

Acknowledgments This work has been performed with funding from the Deutsche Forschungsgemeinschaft. The presented results have been accomplished within Subproject A1 (later A9) which was part of the Collaborative Research Centre SFB 401 “Flow Modulation and Fluid-Structure Interaction at Airplane Wings”. During the 12 year funding period numerous students from RWTH Aachen were involved in the work and made valuable contributions, usually in the form of project theses and final theses. Although their names cannot be included here, they deserve special thanks. The results presented here are documented in more detail in publications and dissertations and were in large part due to the following individuals: Dr.-Ing. E. Özger, Dr.-Ing. I. Schell, Dr.-Ing. S. Haverkamp und Dr.-Ing. S. Kauertz. This Subproject was led by Prof. Dr.-Ing. D. Jacob (emerit.), Prof. Dipl.-Ing. R. Henke, Dr.-Ing. G. Neuwerth and Dr.-Ing. D. Coors.

Literature [1] Chen, A.L., Jacob, J.D., Savas, Ö.: Dynamics of corotating vortex pairs in the wake of flapped airfoils. Journal of Fluid Mechanics 382, 155–193 (1999) [2] Chevalier, H.: Flight Test studies of the Formation and Dissipation of Trailing Vortices. Journal of aircraft 10(1), 14–18 (1973) [3] Corsiglia, V.R., Dunham, R.E.: Aircraft Wake Vortex Minimization by Use of Flaps. In: NASA SP-409, Symposium held at Washington D.C., February 1976, pp. 305– 338 (1977) [4] Crouch, J.D., Miller, G.D., Spalart, P.R.: An Active-Control System for Break-up of Airplane Trailing Vortices. AIAA Journal 39(12), 2374–2381 (2001) [5] Crouch, J.D.: Instability and transient growth for two trailing-vortex pairs. J. Fluid Mech. 350, 311–330 (1997) [6] Crow, S.C., Bate, E.R.: Lifespan of Trailing Vortices in a Turbulent Atmosphere. Journal of Aircraft 13(7), 476–482 (1976) [7] Crow, S.C.: Stability Theory for a Pair of Trailing Vortices. AIAA Journal 8(12), 2172–2179 (1970)

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[8] de Bruin, A.C., Hegen, S.H., Rohne, P.B., Spalart, P.R.: Flowfield survey in trailing vortexsystem behind a civil aircraft model at high lift. In: Paper presented at the AGARD FDP Symposium on The Characterization & Modification of Wakes from Lifting Vehicles in Fluids, held in Trondheim, Norway, May 20-23 (1996), published in CP-584 [9] Fabre, D., Cossu, C., Jacquin, L.: Spatio-ltemporal development of the long and shortwave vortex-pair instabilities. Physics of Fluids 12(5), 1247–1250 (2000) [10] Fabre, D., Jacquin, L.: Stability of a four vortex aircraft wake. Physics of Fluids 12(10), 2438–2443 (2000) [11] Fell, S.: Formierung und Struktur von Randwirbeln verschiedener Flügelkonfigurationen, VDI-Fortschrittsberichte, Reihe 7, Nr. 279, Dissertation RWTH Aachen (1995) [12] Fell, S., Staufenbiel, R.: Formation and Structure of Vortex Systems Generated by Flapped and Unflapped Wing Configurations. Z. Flugwiss. Weltraumforschung 19(6) (1995) [13] Gerz, T., Holzäpfel, F., Darracq, D.: Commercial Aircraft Wake Vortices. Progress in Aerospace Sciences 38(3), 181–208 (2002) [14] Haverkamp, S.: Beeinflussung von Flugzeugnachläufen durch oszillierende Querruder. Shaker Verlag Aachen, Dissertation ILR RWTH Aachen (2004) [15] Haverkamp, S., Neuwerth, G., Jacob, D.: Active and passive wake vortex mitigation using control surfaces. Aerospace Science and Technology 9(1), 5–18 (2005) [16] Haverkamp, S., Neuwerth, G., Jacob, D.: Untersuchungen zum Einfluss von Außenbordklappen auf den Nachlauf eines Rechteckflügels, Deutscher Luft- und Raumfahrtkongress, Stuttgart (2002) [17] Haverkamp, S., Jacob, D., Neuwerth, G.: Studies on the Influence of Outboard Flaps on the Vortex Wake of a Rectangular Wing. Aerospace Science and Technology 7(5), 331–339 (2003) [18] Hörnschemeyer, R., Rixen, C., Kauertz, S., Neuwerth, G., Henke, R.: Active Manipulation of a Rectangular Wing Vortex Wake with Oscillating Ailerons and WingletIntegrated Rudders. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 96, pp. 44–51. Springer, Heidelberg (2007) [19] Kauertz, S.: Beeinflussung des Wirbelnachlaufs eines Tragflügels mir aktiven Winglets. Shaker Verlag Aachen, Dissertation ILR RWTH Aachen (2006) [20] Kauertz, S., Neuwerth, G.: Excitation of Instabilities in the Wake of an Airfoil with Winglets. AIAA Journal 45(3), 577–592 (2007) [21] Neuwerth, G.: Strömungssichtbarmachung in Wasserkanälen mittels eines Verfahrens zur Erzeugung kleinster Luftbläschen. Z. Flugwiss. Weltraumforsch. 9(3), S187– S189 (1985) [22] Ortega, J.M., et al.: Wake Alleviation Properties of Triangular-Flapped Wings. AIAA Journal 40(4) (2002) [23] Ortega, J.M., Savaş, Ö.: Rapidly Growing Instability Mode in Trailing MultipleVortex Wakes. AIAA Journal 39(4) (2001) [24] Özger, E.: Abschwächung des Wirbelnachlaufs von Flugzeugen mit Hilfe von Finnen, VDI-Fortschrittsberichte, Reihe 7, Nr. 422, Dissertation RWTH Aachen (2001) [25] Özger, E., Schell, I., Jacob, D.: Experimental Analysis and Modulation of Vortices. In: Flow Control and Fluid-Structure Interaction at Airplane Wings in der Reihe. Notes on Numerical Fluid Mechanics, vol. 84, pp. 7–38. Springer, Heidelberg (2003) [26] Özger, E., Schell, I., Jacob, D.: On the Structure and Attenuation of an Aircraft Wake. AIAA-Paper 2000-4127 (2000)

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[27] Özger, E., Schell, I., Jacob, D.: On the Structure and Attenuation of an Aircraft Wake. In: 18th AIAA Applied Aerodynamics Conference, Denver, CO, August 14-17 (2000) [28] Özger, E., Schell, I., Jacob, D.: Wirbelbeeinflussung an Tragflügeln mit Hilfe von Finnen, DGLR Jahrestagung 2000-053 (2000) [29] Rennich, S.C., Lele, S.K.: Method for Accelerating the Destruction of Aircraft Wake Vortices. Journal of Aircraft 36(2), 398–404 (1999) [30] Rennich, S.C., Lele, S.K.: Method for Accelerating the Destruction of aircraft Wake Vortices. Journal of Aircraft 36(3), 398–404 (1999) [31] Rossow, V.J.: Lift-generated Vortex Wakes of Subsonic Transport Aircraft. Progress in Aerospace Sciences 35(6), 507–660 (1999) [32] Rossow, V.J., et al.: Vortex Wakes of two Transports Measured in 80 by 120 Foot Wind-Tunnel. Journal of Aircraft 33(2) (1996) [33] Rossow, V.J., et al.: Wind Tunnel Measurements of Hazards Posed by Lift-Generated Wakes. Journal of Aircraft 32(2) (1995) [34] Rossow, V.J.: Effect of Wing Fins on Lift-Generated Wakes. Journal of Aircraft 15(3) (1978) [35] Rossow, V.J.: Experimental Investigation of Wing Fin Configurations for Alleviation of Vortex Wakes of Aircraft, NASA TM 78520 (1978) [36] Schell, I., Özger, E., Jacob, D.: Influence of Different Flap Settings on the WakeVortex Structure of a Rectangular Wing with Flaps and Means of Alleviation with Wing Fins. Aerosp. Sci. Technol. 4(2) (2000) [37] Schell, I., Özger, E., Jacob, D.: Influence of Different Flap Settings on the Hazard Posed to Following Aircraft. In: 12. DGLR Fach-Symposium der AG STAB, November 15-17, 2000. erschienen in: Notes on Numerical Fluid Mechanics, vol. 77, pp. 66–73. Springer, Heidelberg (2000) [38] Schell, I., Özger, E., Jacob, D.: Simulation of Vortex Sheets at Take Off and Landing. In: Flow Control and Fluid-Structure Interaction at Airplane Wings in der Reihe. Notes on Numerical Fluid Mechanics, vol. 84, pp. 39–56. Springer, Heidelberg (2003) [39] Spalart, P.R.: Airplane Trailing Vortices. Annual review of Fluid Mechanics 30, 107– 138 (1998) [40] Türk, L., Coors, D., Jacob, D.: Behaviour of Wake Vortices near the Ground over Large Range of Reynolds Numbers. Aeospace Science and Technology 3(2) (1999) [41] Türk, L., Coors, D.: Numerical Simulation of the Vortex Sheet Roll-Up behind Wings with Different Lift Distributions. In: 10th AG-STAB/DGLR-Symposium, Braunschweig (November 1996). Notes on Numerical Fluid Mechanics, vol. 60, pp. 359– 366. Vieweg Verlag (1997) [42] Türk, L.: Verschmelzen von Wirbeln, VDI-Fortschrittsberichte. Reihe 7, Nr. 393, Dissertation RWTH Aachen (2000) [43] Türk, L.: Wake Vortex AAAF-DGLR Meeting, The Large Aircraft Operational Challenge Wake Vortices - Aerodynamics and Noise Effects, St. Louis (ISL), January 2122 (1999)

Development of a Modular Method for Computational Aero-structural Analysis of Aircraft Lars Reimer, Carsten Braun, Georg Wellmer, Marek Behr, and Josef Ballmann

Abstract. This paper outlines the development of the aero-structural dynamics method SOFIA over the duration of the Collaborative Research Center SFB 401. The algorithms SOFIA applies for the spatial and the temporal aero-structural dynamics coupling are presented. It is described in particular how SOFIA’s load and deformation transfer algorithms suitable for non-matching grids at the coupling interface were enhanced towards the application to complete aircraft configurations. The application of SOFIA to various subsonic and transonic aeroelastic test cases is discussed.

1 Introduction The design of high-performance wings for large commercial aircraft requires the inclusion of their aeroelastic properties into the aerodynamic and structural design process. During preliminary design, the geometry of the wing is defined as a compromise between good flight performance during take-off, landing and cruise flight on the one hand and load capacity and weight of the structure on the other hand. In an iterative fashion, the aerodynamic shape, the loads, the construction of the wing assembly and the deformation are studied sequentially and more or less independently. Aerodynamic wind tunnel testing with rigid or nearly-rigid reduced-scale models plays a key role. But in those tests, similarity with the full scale body can only be achieved in a very limited manner, primarily with respect to the aerodynamic Josef Ballmann · Carsten Braun LFM, Lehr- und Forschungsgebiet f¨ur Mechanik, RWTH Aachen University, Schinkelstrasse 2, 52062 Aachen e-mail: [email protected] Lars Reimer · Georg Wellmer · Marek Behr CATS, Chair for Computational Analysis of Technical Systems, CCES, RWTH Aachen University, Schinkelstrasse 2, 52062 Aachen e-mail: {reimer,wellmer,behr}@cats.rwth-aachen.de

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 205–238. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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parameter Mach number and to a certain extent also with respect to the Reynolds number. Aeroelastic similarity is usually not achieved. Based on the aerodynamic analysis, wing loads, deformations and particularly the aerodynamic twist are determined. Then the wing geometry and construction of the wing assembly are modified a posteriori so that after taking into account the static aeroelastic deformation in cruise flight sufficient lift and minimum drag are ensured. The described design and construction procedure requires several iterations because in every step the aeroelastic coupling and the nonlinearity of the problem cannot be captured completely. Besides that, nonlinear flutter possibly occuring in the transonic flow regime cannot be predicted with such a procedure. Therefore it is necessary to develop numerical methods, which reliably predict the interaction between aerodynamic, structural and inertial forces. Such a numerical method has been progressively developed in the past four funding periods of the Collaborative Research Center SFB 401 Flow Modulation and Fluid-Structure Interaction at Airplane Wings at RWTH Aachen University. This paper gives an overview about this numerical method named SOFIA and its past and present development stages. The organization of the subsequent sections of this paper is as follows. The general concept of SOFIA, some numerical components and their development over the past years are described in Chapter 2. Furthermore, the section explains in detail the Aeroelastic Coupling Module as core of our method and its progress during the final funding period. In Ch. 3 selected results for two different aeroelastic test cases are presented which demonstrate SOFIA’s development status after the first three funding periods. Steady and unsteady aeroelastic results obtained for the HIRENASD test case with the current version of SOFIA are discussed in Ch. 4. The paper concludes with statements about the validation status of SOFIA and an outlook about ongoing work.

2 General Concept and Applied Numerical Methods The computational aero-structural dynamics (CASD) problem considered here consists of the flow field, which surrounds the investigated wing or aircraft configuration, the deformation field of its structure and the deformation field of the volume flow grid. In general, the described coupled problem is solved preferably by methods with multi-field formulation, because these offer several advantages over methods with monolithic formulation (see e.g. [1]). One major advantage is that separate single-field solvers can be employed which are reliable, well-tested and specialized regarding the particular needs of the respective field. For instance, if one strives to compute more complex geometries or use a more detailed physical model in one field, the overall multi-field solver can be customized by simply replacing one of the single-field solvers with an improved one. Based on the objective of numerically solving the kind of CASD problem that is described above, the development of the SOFIA package was initiated with the beginning of the SFB 401 more than a decade ago. During its development, which is still ongoing, the following objectives have been the most prominent:

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• Since the accurate prediction of various aero-structural dynamics phenomena (e.g. transonic dip, buffeting, LCO) depends on the consideration of flow nonlinearities, SOFIA focusses on the application of flow solvers which are based on the Reynolds-averaged Navier-Stokes (RANS) equations. This orientation requires simulations in the time domain instead of in the frequency domain. • As a consequence of a high computational cost on the flow side, reduced-order structural methods are favoured, e.g. beam or shell models. Ideally, these should be a good compromise between accurate computations of the unsteady deformation field of the structure and low computational cost. • A multi-field formulation requires an adequate coupling of the single-field solvers. The coupling algorithm must ensure the correct mutual energy transfer between the flow and structural field, such that neither energy is generated nor destroyed at the common interface. To realize a CASD solver in multi-field formulation which complies with the aforementioned objectives, the Aeroelastic Coupling Module (ACM) is being developed as SOFIA’s core. The actual CASD solver originates from coupling a CFD (Computational Fluid Dynamics) solver with a CSD (Computational Structural Dynamics) solver via the ACM (see Fig. 1). Since the flow grid deformation is closely linked to the flow solver, the grid deformation method is not coupled together with the ACM. In the following three sections the single-field solvers which are currently employed in SOFIA, i.e. the CFD solver, the CSD solver and the flow grid deformation method are outlined. The fourth section describes the current features of the ACM in depth.

Fig. 1 General concept of the CASD solver SOFIA. CFD and CSD solvers exchange the coupling data via the ACM

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2.1 Flow Solver Although all results presented in this paper were obtained using the flow solver FLOWer as part of SOFIA, two other flow solvers were recently coupled with SOFIA, which are the QUADFLOW and the TAU code. The FLOWer and the TAU code were both developed under the leadership of the German Aerospace Centre (DLR) during the project MEGAFLOW I/II [2]. All mentioned codes have in common that they are finite volume methods of second order accuracy in time and space and they all solve the 3D time-dependent RANS equations for perfect gas. All solvers are based on arbitrary Euler-Lagrangian (ALE) formulation and extended for the operation on deformable grids such that the discrete geometric conservation law is satisfied. While FLOWer operates on multi-block structured grids, TAU is designed for hybrid unstructured grids. In comparison, the QUADFLOW code developed within the framework of the SFB 401 follows a completely different grid concept. To reduce the computational cost regarding grid deformations, QUADFLOW’s grid is based on the evaluation of B-spline transformations at a few grid control points only. Just as TAU, QUADFLOW can also be seen as an unstructured solver due to its face-based formulation for polyhedral cells. Details about QUADFLOW’s B-spline grid concept and its benefits compared to the conventional approach can be found in several articles of this volume. Each CFD solver uses a different method to describe the discrete wetted surface, as can be seen from Fig. 3. In order to be compatible with each of them, unique requirements had to be met by the ACM’s coupling interface. Furthermore, both TAU and QUADFLOW offer grid adaptation schemes, which promise a very accurate tracing of shock motions occurring during steady and unsteady aeroelastic solution procedures. While QUADFLOW determines a very sensitive adaptation indicator based on multiscale analysis, (described in other papers of this volume in detail), TAU uses predominantly a gradient-based indicator. Since the boundary layer is of importance for the accurate prediction of the aeroelastic equilibrium configuration on the one hand and aeroelastic instabilities on the other hand, adequate turbulence closure models must be available in the flow solver. Therefore, all of the applied flow solvers provide various one- and two-equation turbulence models based on eddy-viscosity approaches, as well as Reynolds stress models. In this paper only aeroelastic results are presented which have been achieved with FLOWer. Corresponding results with QUADFLOW and TAU are presented by Schieffer et al. in this volume.

2.2 Structural Solver So far, the CSD problem is solved in SOFIA by the in-house code FEAFA (Finite Element Analysis for Aeroelasticity). It is based on the finite element (FE) method which is applied to the structural theory of small strains and linear-elastic isotropic material behaviour. Meanwhile FEAFA offers a full range of structural FE types comparable to commercial CSD codes, which include various shell and solid

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elements, translatory and rotatory springs, point masses and multi-degree-of-freedom constraints. But because of the demand for reduced-order structural models with low computational cost, the multi-axial Timoshenko beam element [3, 4] remains the idealization of choice for slender structures such as aircraft wings. Multi-axial means that the mass, bending and shear axes do not have to coincide. Thereby, structural coupling between bending and torsional vibrations of aircraft wings can be captured. In contrast to the Euler-Bernoulli beam theory, the Timoshenko theory considers shear deformation and thereby provides physically reasonable wave propagation through the structure, which is significant for unsteady aeroelastic analysis. Even though a beam element does not provide complete insight into the stress distribution of the structure, it allows an accurate computation of the unsteady deformation state of slender aircraft wings. To assess the applicability of the beam discretization in load application regions, such as intersections of structural components, shell or solid elements can be employed in the aeroelastic simulation for a more detailed structural analysis. FEAFA provides different solution schemes for the structural dynamics equations which arise after spatial discretization with the FE method. Unconditionally stable implicit time integration schemes afford an increased flexibility regarding the choice of the time step size. Thus, a Newmark-like time integrator like the Bossak scheme is one option in FEAFA. Depending on one parameter of the Bossak scheme which controls the amount of active high-frequency damping, the remaining parameters are chosen such that second order time accuracy is achieved. The second, preferred option in FEAFA is the modal scheme. By applying a transformation with the eigenmodes of the structure, the governing equations of motion can be decoupled and each of the resulting modal equations can be integrated in time separately. This is done by applying the undamped Bossak scheme with subcycling such that the oscillation period of each mode is resolved with 800 time steps. The evolution of the right-hand side is interpolated between the current and the previous time step. The number of considered modal degrees of freedom can be adjusted such that a good compromise is achieved between the following competing aspects: (i) the number of considered modes should be sufficiently large for an accurate computation of the structural deformation state; (ii) high-frequency modes should be excluded to prevent from unphysical oscillations during the time integration; and (iii) fewer modes resp. lower eigenfrequencies enable larger time steps because the time step size must fulfill the Shannon theorem for all considered modes.

2.3 Flow Grid Deformation Method The flow grid has to be updated in every time step of the aeroelastic computation according to the change of the aerodynamic surface shape. In general, two competing concepts can be found in literature: (i) purely algebraic interpolation methods which mostly lack conservation of the original orthogonality properties of the grid and (ii) methods of structural analogy which model the flow grid as a linear-elastically deformable body. In this project a hybrid of both methods has been developed for multi-block structured grids [3, 5], e.g. FLOWer grids. It combines the numerical

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Fig. 2 Principle scheme of the grid deformation method MUGRIDO [4, 3, 5]. For presentation purposes, the grid deformation is only shown for those blocks close to the wing surface, but in fact the deformation computation includes all grid blocks

efficiency of algebraic methods with the robustness of structural analogy methods. The resulting method MUGRIDO (Multiblock Grid Deformation Tool) determines the flow grid deformation in three successive steps which are visualized in Fig. 2 and described in the following: (i) MUGRIDO generates a fictitious framework of beams by modelling the CFD block boundaries and a given percentage of additional grid lines as massless linear elastic Timoshenko beams; (ii) the deformation of the beam framework is computed based on the deformation of the surface relative to the undeformed grid; and (iii) the new positions of the grid points which do not belong to the beam framework are determined efficiently via algebraic interpolation from the deformed beam framework. The resulting FE problem of the beam framework model of the flow grid has a lot fewer degrees of freedom than the corresponding solid body problem. MUGRIDO is thus much more efficient in terms of memory consumption and computation time. Another advantage results from the fact that the beam elements are considered rigidly attached to each other in points of intersection and are clamped to the aerodynamic surface. The angles between grid lines of the undeformed grid can be preserved at these intersection points in the deformed grid. Thereby, common drawbacks of purely algebraic interpolation methods are resolved since the beam framework retains the orthogonality properties of the originally undeformed grid during interpolation.

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2.4 The Aeroelastic Coupling Module As already indicated in the beginning of this chapter, the ACM represents the core of SOFIA. All tasks which are relevant to the aeroelastic coupling are carried out in the ACM. This way, code changes in the single-field solvers participating in the overall aero-structural dynamics solver can be largely avoided. There are basically two tasks which must be accomplished by the ACM in each coupling step: 1. The so-called spatial coupling which comprises the conservative projection of aerodynamic loads and, in opposite direction, the projection of the structural deformation; and 2. the temporal coupling, i.e. the synchronization of the single-field solver calls. The coupling algorithms of the ACM are described in the following, according to their development status at the end of the final funding period of the SFB 401 in 2008. The spatial coupling is the subject of Sec. 2.4.1. The algorithms implemented in the ACM for the temporal coupling are outlined in Sec. 2.4.2. 2.4.1

Load and Deformation Transfer between Non-matching Interface Grids

When using reduced-order structural models, almost inevitably the problem of nonmatching grid interface has to be faced. Fig. 3 exemplarily depicts situations of different non-matching grids on both sides of the coupling interface between fluid and structure. Shown to the left is a beam discretization on the structural side which is faced by a structured surface grid (e.g. a FLOWer grid). The same aerodynamic surface predominantly meshed with triangles is combined with a shell structure in the figure in the middle. Even in the right figure, where the tetrahedral solid element discretization of the structure comes into contact with an aerodynamic surface mesh consisting mainly of quadrilaterals, a non-matching grid interface may exist. To manage these different mesh situations at the coupling interface, the transfer of coupling data should be independent from the surface cell geometry, at least on the side of the flow domain. Therefore, the input data to the ACM in each time step is the pointwise distribution of the surface mesh and the aerodynamic forces acting on that surface mesh. The aerodynamic force distribution should be derived consistently from the aerodynamic stress distribution by the flow solver, e.g. according to the approach proposed in [6]. Since most of today’s flow solvers determine the velocity state of the aerodynamic surface on their own, just as the ones employed in SOFIA so far, only the distribution of surface mesh points of the deformed configuration needs to be returned by the ACM. In the following, the coupling algorithm realized in the ACM is presented. It is based on a relationship between one point of the aerodynamic surface mesh and the closest structural element for one-component configurations. For configurations with multiple structural or aerodynamic components (e.g. wing-fuselage or whole aircraft configurations) the algorithm had to be extended to resolve several issues regarding the contiguity of the aerodyanmic surface mesh. These issues and the ensuing algorithmic extensions are covered subsequently.

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Fig. 3 Examples of non-matching grids at the interface between CFD and CSD domains

Transfer Algorithm for the Nearest Structural Element The ACM’s algorithm for the transfer of loads and deformations is based on finite interpolation elements (FIE). This method is also known as inverse isoparametric mapping (IIM) and has been formulated and successfully applied by many authors, e.g. [6, 7, 8]. In this method the FE shape functions are used to interpolate loads and deformations between a structural element and a CFD surface point. It leads to independent algebraic expressions that can be evaluated efficiently for each surface or load incidence point. Therefore, the FIE approach can be easily parallelized. This is also an advantage in the case of a grid adaptation of the CFD solver, as in this case a recomputation of the coupling data is essential, at least for the new points of the CFD surface. Furthermore, a good verification of the physical consistency of the transfer can be ensured due to the locality inherent to the FIE approach. It has also proven to be very robust at highly non-matching grids at the coupling interface. A disadvantage of the FIE method is that the access to the FE shape functions of the involved structural elements is not available for commercial CSD codes. In the ACM, the actual FIE method is realized based on the following three steps, which can also be traced in Fig. 4: 1. Find the projection point of each aerodynamic surface point onto the nearest structural element. The aerodynamic surface point is either the mesh point itself or the point subjected to an aerodynamic force. 2. Determine the weighting coefficients between the projection point and the nodes of the nearest element.

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Fig. 4 Load/deformation transfer between a node of the aerodynamic surface mesh and the nearest structural element which can be a beam, shell or solid element. (The projection on shell/solid elements is depicted here just for the sake of presentation for elements with quadrilateral face. But the same principle applies to shell/solid elements with triangular face.)

3. Interpolate the aerodynamic load or the structural deformation by applying the determined weighting coefficients. Steps 1 and 2 are done only once for the initial configuration, i.e. both the structure and the aerodynamic surface are undeformed. Step 1 Although the projection algorithm onto the nearest structural element face is derived in the following for 2D element faces, it applies also to 1D element faces. 2D element faces represent either shell elements or originate from solid elements, whereas 1D element faces are either edges of plane stress elements or beams. The projection algorithm uses the parameterization of the respective element face by means of the FE method. Therefore, the projection algorithm depends either on two natural coordinates (2D element face) or one natural coordinate (1D element face). Hence, the governing equations for 1D element faces result from those for 2D element faces, when the equations corresponding to the second natural coordinate are removed.

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The actual algorithm begins with the assumption that the location of the projection point can be expressed in terms of the FE shape functions which are assembled in matrix H for the respective (2D) element face: x(r, s) = H(r, s) XCSD .

(1)

The column matrix XCSD contains the coordinates of all corner nodes of the element face. As already indicated before, the 2D element face is parameterized by the two natural coordinates r and s. In Fig. 4, the projection point of the aerodynamic surface point xCFD onto the depicted structural element is marked with PF for the load transfer and Pu for the deformation transfer. The nearest projection point is found if the distance vector Δ x = xCFD − x is orthogonal to all tangential vectors to the face of the structural element. The tangential vectors are local derivatives of x with respect to the natural coordinates. For a 2D element face, the following function ⎛ ⎞

∂ x(r, s) [xCFD − x(r, s)] · fr (r, s) ∂r ⎠ =⎝ f= (2) ∂ x(r, s) fs (r, s) [xCFD − x(r, s)] · ∂s ⎞ ⎛ ∂ H(r, s) XCSD [xCFD − H(r, s) XCSD ] · ∂r ⎠ (3) =⎝ ∂ H(r, s) XCSD [xCFD − H(r, s) XCSD ] · ∂s vanishes at the natural coordinates r = r pro j and s = s pro j of the projection point x(r pro j , s pro j ) = x pro j . The resulting equation system f = 0 may be linear or nonlinear in terms of r and s and thus is solved iteratively for r and s using Newton’s method. If the final projection point onto an element face lies outside of the element, an ensuing projection onto the respective edge of the element face must be carried out using the same technique as described before, but now in a 1D subspace. Step 2 The resulting weighting coefficients of the aforementioned process can be assembled in the FE shape function matrix evaluated at the projection point H(r pro j , s pro j ). These weighting coefficients are used in the following until one of the involved surface meshes changes. This way, step 2 of the coupling algorithm is concluded. Step 3 After having received the aerodynamic surface FCFD from the CFD solver, the ACM initiates step 3 and actually applies the weighting coefficients. In the load transfer branch, it computes at first an equivalent force-moment group (F pro j , M pro j ) for FCFD at the projection point (cf. Fig. 4): F pro j = FCFD , M pro j = Δ x ∧ FCFD .

(4)

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(The symbol ∧ denotes a vector product.) To distribute this force-moment group on the nodes of the nearest CSD element, the mapping coefficients computed in step 2 are applied: PCSD = HTu (r pro j , s pro j ) F pro j + HTϕ (r pro j , s pro j ) M pro j (5) with PCSD = F(1) , M(1) , . . . , F(n) , M(n) , n=number of element nodes Hu and Hϕ are the matrices of FE shape functions of the translatory and resp. rotatory nodal degrees of freedom which are evaluated at the projection point x(r pro j , s pro j ) (cf. step 2). The vector PCSD comprises all nodal CSD loads of the current element and is added to the global structural load vector. Equations (4) and (5) are applied for each aerodynamic surface point separately. In case of structural element types which do not have rotatory degrees of freedom, e.g. solid elements, M pro j is replaced by an equivalent group of nodal forces to preserve local and global conservation of moments. Having computed the structural solution, the structural deformation is projected onto the aerodynamic surface as shown in the right part of Fig. 4. For this purpose the FE solution is interpolated from the nodes of the nearest element to the projection point using the FE shape functions. This can be expressed as u pro j = Hu (r pro j , s pro j ) UCSD , ϕ pro j = Hϕ (r pro j , s pro j ) UCSD (6) with UCSD = u(1) , ϕ (1) , . . . , u(n) , ϕ (n) , n=number of element nodes for the displacement u pro j and the rotation ϕ pro j . The vector UCSD consists of all nodal displacements and rotations of the respective structural element. A rigid link between the projection point and the aerodynamic surface point is established via the following kinematic relationship: uCFD = u pro j + ϕ pro j ∧ Δ x.

(7)

The equivalence of work can be verified by inserting Eqs. (4) to (7) in the principle of virtual displacements applied on both sides of the coupling interface: the section after the next oneumber.

δ WCFD = FCFD δ uCFD = PCSD δ UCSD = δ WCSD .

(8)

The equivalence of forces and moments holds in the aforementioned procedure by the fact, that the FE shape functions sum up to one. In case the structural and the aerodynamic meshes remain unchanged during the whole computation, the computed weighting coefficients can be applied in each coupling step. In case of an adaptation of the CFD mesh, steps 1 and 2 must be repeated to determine new weighting coefficients based on the deformed configuration. The projected structural deformation is always applied to the undeformed aerodynamic surface stored in the first coupling step. The location of those aerodynamic surface points which were added to the aerodynamic surface by the adaptation algorithm is

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unknown in the undeformed configuration. Accordingly, this surface must be reconstructed from the undeformed structure and the adapted, deformed surface mesh. For the reconstruction of the undeformed aerodynamic surface, we employ the deformation projection algorithm described below in opposite direction. Once the undeformed aerodynamic surface and the corresponding new weighting coefficients are computed, these can be applied until one of the meshes changes again. Transfer Algorithm for Multi-Component Configurations Discretized by Beam Frameworks For structural models which have large offsets to the aerodynamic surface, for instance beam idealizations like the one depicted in Fig. 5, non-uniqueness problems arise if a load/deformation projection algorithm is used which is based solely on relations between a single CFD surface point and its closest structural element as is derived in the section before. Fig. 5 exemplarily points out three arising issues which are discussed in the following. The load and deformation projection has to be prevented between surface segments and structural components that are not physically connected. For instance, with a wing-flap configuration, a CFD surface point of the wing located close to the trailing edge can be erroneously projected onto those parts of the structural idealisation which represent the flap. But uniqueness can be preserved by explicit assignment of CFD surface components to components of the structural idealization as is illustrated in Fig. 6. Structural parts which do not have any mechanical contact with the aerodynamic surface, e.g. flap-tracks, can be explicitly excluded from the load/deformation projection by omitting an assignment.

Fig. 5 Several issues in the load/deformation transfer at aircraft configurations of which the structure is modelled by a beam framework

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Fig. 6 Definition of aircraft components. The structural model is partitioned into element groups which can either be assigned to an aerodynamic surface part for load/deformation transfer or explicitly excluded from load/deformation transfer by omitting the assignment

These explicit assignments between CFD and CSD components may cause other issues in areas close to intersections of CFD surface components. Here, gaps can emerge which would later result in a termination of the CFD solution process. The left image of Fig. 7 demonstrates this issue at the intersection between the wing and the adjacent fuselage. Gaps can be prevented if the displacement of a given surface point close to an intersection of CFD surfaces is not only determined by the deformation of the directly assigned structural component, but also by all structural components which are assigned to adjacent surfaces. Latter are called the indirectlyassigned components. This approach, also followed by Badcock et al. [9] in a similar fashion, is presented in more detail here. In Fig. 8 (left), the structural components assigned directly to the considered CFD surface point are element groups 2 and 3, because both are assigned to the wing. An indirectly-assigned structural component is element group 1 belonging to the fuselage. The final displacement of the CFD surface point is blended between dir ) and indirectly (uindir ) deformations resulting from projections on directly (uCFD CFD assigned components: uCFD = with

1 wa udir + uindir 1 + wa CFD 1 + wa CFD

a . wa = w alimit

(9)

In contrast to Badcock et al., the blending function w applied here is a polynomial of degree five with one inflection point within the interval [0, 1]. It has zero slope at both interval ends. As depicted in Fig. 8 on the left, the coefficient a is the distance

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Fig. 7 Prevention of gaps at intersecting aircraft surfaces by considering the deformation contributions of adjacent structural components

of the CFD surface point to the intersection curve of both adjacent CFD surface components. a is normalized by alimit which defines the user-defined width of the blending region. The right part of Fig. 7 demonstrates the benefit of this projection with consideration of adjacent structural components. There is no gap at the intersection between wing and fuselage. Another issue becomes apparent in Fig. 5. In areas close to kinks of the beam axis more than one orthogonal projection is possible and another problem of nonuniqueness arises. To resolve it, the final displacement of the CFD surface point is determined from a weighted average of the contributions of all possible orthogonal projections: uCFD =

∑ wG,i uCFD,i ,

i = 1, . . . , number of proj. points

(10)

i

with

wG,i =

wd,i wβ ,i ⇒ ∑ wG,i = 1. ∑ j wd, j wβ , j i

As shown in Fig. 8 on the right, the weighting coefficients wG,i are determined based on two influence factors: (i) the deviation angle β from an orthogonal projection; and (ii) the distance between the CFD point and the projection points, which is finally normalized by the minimal distance dmin among all projections:

|βi | di /dmin − 1 wβ ,i = w , wd,i = w . (11) βlimit dlimit − 1

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Fig. 8 Left: Assignment of a CFD surface node close to the fuselage to both of the structural element groups representing the wing and the fuselage. The influence of the deformation from the fuselage structure on the surface node is blended with respect to the relative distance a/alimit . Right: Interpolation in non-unique mapping situations

The extent of the interpolation region can be controlled with βlimit and dlimit . Fig. 9 demonstrates the effect of the described interpolation for a deformed wing. For a structural model consisting of two element groups with a kink, the CFD point located in the dark region in the upper right part of the figure is in a region of nonunique projection. If the displacement of the CFD surface point is not interpolated from all orthogonal projections, the lower left part of Fig. 9 reveals the unacceptable result, i.e. kinks in the CFD surface. In contrast, the application of the interpolation concept preserves the smoothness of the CFD surface which can be seen on the right. The load/deformation algorithm remains conservative if the same weighting coefficients are applied to the aerodynamic loads to be transferred as used for the interpolation and blending of deformations. All described transfer rules are given explicitly without solving any system of equations and thus can be evaluated in a very efficient manner. In applications with complex configurations, the concept of the aforementioned weighting coefficients retains the required smoothness of the CFD surface and preserves the locality of the load/deformation transfer. 2.4.2

Temporal Coupling

Several temporal coupling schemes are available in the ACM. These control the synchronization of the single-field solvers in case of steady or unsteady aeroelastic problems.

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Fig. 9 Example of the deformation transfer between a wing surface and a beam model which is composed of two element groups. The surface which results from considering exclusively the deformation of the nearest neighbouring beam element is shown to the left. The interpolation of the deformation contributions from both element groups yields the improved surface shown to the right

Steady Temporal Coupling Steady aeroelastic computations are performed in the sense of an under-relaxed Block-Gauss-Seidel scheme. One field is iterated after exchanging the coupling data while the others are held constant. Therefore, one complete coupling step reads as follows: 1. The flow solver computes the steady aerodynamic load field F(k+1) of iteration k + 1. This load field depends on the CFD grid X(k) which in turn depends on the structural displacement field U(k) : F(k+1) = f (X(k) ).

(12)

It is not necessary to push the flow solver to a fully converged solution. Instead, the computation of a rough estimate of the aerodynamic load field is sufficient in each coupling step.

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2. The aerodynamic load field F(k+1) is projected onto the CSD nodes by the ACM P(k+1) = p(F(k+1) ).

(13)

The resulting steady structural solution U(k+1) is determined by the structural solver: U(k+1) = g(P(k+1) ). (14) 3. To improve the convergence behaviour of the steady aeroelastic solver the structural solution is under-relaxed in the ACM: ( (k+1) = ω U(k+1) + (1 − ω )U(k) U

(15)

The under-relaxation coefficient ω is set to a user-defined constant value between zero and one. The value 0.7 has proven to be good for this coefficient. ( (k+1) onto the aerodynamic 4. After the ACM has projected the structural solution U surface ( (k+1) ), S(k+1) = p(U (16) the CFD grid X(k+1) of the next coupling step is computed by the grid deformation method: (17) X(k+1) = h(S(k+1) ). 5. The aeroelastic equilibrium configuration (AEC) is finally reached if the difference between two successive structural deformations falls below a user-defined threshold. Otherwise the counter k of the coupling step is increased by one and another coupling step is performed as just described. In this sequence f (), g(), and h() denote the operations of the respective single-field solvers, i.e. the flow solver, the structural solver and the grid deformation method. p() represents the projection algorithm of the ACM. Unsteady Temporal Coupling Several unsteady temporal coupling schemes are available in the ACM. The implemented schemes were motivated by the results presented in [10, 11]. Based on the assumption that the structural solution Un , the aerodynamic load field Fn and the deformed CFD grid Xn are known at time step t n , all unsteady coupling schemes for the computation of the coupled solution at time t n+1 can be summarized by the following sequence of steps: (please note that the index k only serves as an iteration counter within one time step now.) 1. Compute an approximation of the structural solution at time t n+1 by extrapolation. Either the structural deformation state can be extrapolated directly with first order accuracy or the structural loading is extrapolated with second order accuracy. In the latter case the resulting deformation state is computed by calling the structural solver with the extrapolated loads. These two actions can be expressed as ( n+1 ← (Un , U ˙ n) U (k)

or resp.

n ˙n ( n+1 ( n+1 U (k) ← F(k) ← (F , F ).

(18)

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( n+1 2. After the ACM has projected the structural deformation U (k) onto the CFD surface grid ( n+1 Sn+1 (19) (k) = p(U(k) ), the flow grid Xn+1 is deformed according to this surface: (k+1) n+1 Xn+1 (k+1) = h(S(k) )

(20)

˙ n+1 3. The flow solver computes the grid velocities X (k+1) under consideration of the discrete geometric conservation law. This is followed by the computation of the aerodynamic load field Fn+1 depending on the current flow grid Xn+1 and (k+1) (k+1) n+1 ˙ (k+1) : the corresponding grid velocities X n+1 ( n+1 ˙ n+1 F (k+1) = f (X(k+1) , X(k+1) ).

(21)

4. After the ACM has projected these aerodynamic loads onto the structural nodes ( n+1 Pn+1 (k+1) = p(F(k+1) ),

(22)

the structural solution Un+1 is computed by the CSD solver: (k+1) n+1 n Un+1 (k+1) = g(P(k+1) , P(k+1) , . . .).

(23)

5. In the sense of a fixed-point iteration, the iteration over the aforementioned steps 2 to 4 within the current time step realizes a numerically tight coupling scheme. In this case, the structural solution can also be under-relaxed to ease the approach to convergence within one time step: n+1 n+1 ( n+1 U (k+1) = ω U(k+1) + (1 − ω )U(k) .

(24)

6. The tight coupling loop is left if the difference between the structural deformations of two successive inner iterations k and k + 1 falls below a certain threshold. Otherwise the counter k is increased by one and the tight coupling loop proceeds with step 2 for the same time step. In order to comply with the kinematic and the dynamic coupling conditions simultaneously, it is essential to perform instationary aeroelastic computations on the basis of a strong coupling. Apart from that very good experiences were made with a loose predictor-corrector scheme in regard to numerical stability, computational efficiency and accuracy. All results presented in this paper were computed with this coupling scheme. It extrapolates the aerodynamic loads in the predictor step 1 and the structural solution is computed based on these. The corrector step results from performing step 4.

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3 Selected Results of Preceding Funding Periods In this chapter the development of SOFIA during the second and third funding period of the SFB 401 from 2000 to 2005 is outlined. A major milestone was reached in 2003. SOFIA’s simulation capabilities for unsteady transonic aerodynamics were verified for the NLR 7301 airfoil moving in prescribed harmonic heave and pitch motion [12, 13]. In addition, first 3D results were presented for a straight flexible wing depicted in Fig. 10 on the left. The predicted decaying vibrational behaviour agreed very well with the measurements from the corresponding subsonic wind tunnel tests [12, 13, 14]. Despite of these achievements several limitations of SOFIA were present at that time. The simulation of the aforementioned tests, though satisfactory in this case, were still performed on the basis of the Euler equations. The structure of the wind tunnel model was idealized with a singular, at that time necessarily straight Timoshenko beam. No other structural element types were available at that time. The flow grid was deformed using an algebraic interpolation algorithm, which was restricted to one block topologies. Only first concepts of the MUGRIDO method, of which the final version is described in Sec. 2.3, had been developed yet.

Fig. 10 Development stages of the aeroelastic solver by means of its applications in the previous three funding periods

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The simulation capabilities of SOFIA were successively expanded between 2003 and 2008, and more challenging validation cases were simulated accordingly. The central image of Fig. 10 shows the flexible swept wing. On the right is the HiReTT test case, which provided the first experimental data of a 3D elastic two-component configuration exposed to transonic flow. It was also the first validation case for the blending technique available in the ACM’s spatial coupling method, which is described in the paragraph Transfer Algorithm for Multi-Component Configurations Discretized by Beam Frameworks of Sec. 2.4.1. Exemplarily, some of SOFIA’s numerical predictions for the swept wing and the HiReTT cases are compared with the respective wind tunnel measurements in the subsequent two sections.

3.1 Static and Dynamic Validation for an Elastic Swept Wing in Subsonic Flow The flexible wing illustrated in Fig. 10 centered was investigated in wind tunnel tests at the following subsonic flow conditions: V=25m/s to 75m/s (Ma=0.19 to 0.22); Re=1.55M to 1.75M; q/E≈ 5 · 10−8 and α ≈ −3.5◦ to +10.5◦ . The quantity q/E represents the ratio of dynamic pressure and Young’s modulus. It is directly linked to the loading of the wing, the deformation of which scales almost linearly with the q/E value. The results of the wind tunnel tests are presented in detail in [15]. These were compared to the numerical predictions of SOFIA in depth in [4, 16, 17]. Although the swept wing is constructed in a similar manner to the straight wing

Fig. 11 Static polars of the SFB 401 swept wing test case: Comparison of experimental and numerical spanwise bending deflection (left) and structural torsion (right) for various root angles of attack αW = αW,0 + αW,rel . (Flow conditions: V=75m/s, Ma=0.22, Re=1.73M, αW,0 =-1.36◦ )

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Fig. 12 Free vibration test case of the SFB 401 swept wing: Left: Initial setup before removing of the cord force. Top right: Time history of wing tip bending deflection. Bottom right: Time history of wing root bending moment. Each of the most right figures show a detail of the initial 0.2s of the corresponding neighbouring left figure. (Flow conditions: V=75m/s, Ma=0.22, Re=1.73M, αW =αW,0 =−1.36◦ )

(see Fig. 10 left), it exhibits a minor stiffness coupling between bending and torsion. This originates from ribs which are oriented parallel to the incoming flow direction. For an appropriate idealization of the spar-rib construction the beam formulation had to be enhanced from a single straight Timoshenko beam towards a framework of beams. Fig. 10 shows the employed framework projected onto the photograph of the wing placed in the wind tunnel test section. Details on the numerical idealization of the wind tunnel model are described in [4, 16, 17]. As plotted in Fig. 11, SOFIA correctly predicts the stationary deformation behaviour in terms of the spanwise bending deflection (left) and the torsional rotation (right) up to very high angles of attack. In Fig. 12 unsteady simulation and experimental results of free vibration tests are compared, here for an inflow velocity of V=75m/s. In the upper row the time history of the tip deflection is shown, with the right image containing a magnification of the time window from 0 to 0.2s. In the lower row the wing root bending moment is depicted. All diagrams document an excellent agreement between simulation and experiment. This is also the case for inflow velocities other than V=75m/s, as can be seen from the frequency shifts and aerodynamic dampings in Fig. 13.

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Fig. 13 Free vibration test case of the SFB 401 swept wing: Modal frequencies (left) and damping factors (right) evaluated experimentally and numerically from time histories of de√ caying free vibrations of the wing. (Damping factor D=Λ / 4π 2 + Λ 2 with Λ denoting the logarithmic decrement)

3.2 Static Deformation Effects in Wind Tunnel Experiments of Transport Aircraft During the HiReTT project (High Reynolds Number Tools and Techniques for Civil Aircraft Design), wind tunnel experiments were conducted in the European Transonic Wind Tunnel (ETW) in the transonic flow regime (Ma=0.70 bis 0.88) at high Reynolds numbers (Re > 30M). In this process extensive experimental data for a full and a half model of a wing fuselage configuration was gathered [18]. The wind tunnel half model is depicted in Fig. 10 on the right. High channel pressure, essential to achieve high Reynolds numbers, led to model deformation with significant influences on the aerodynamic characteristics of the investigated configuration. These wind tunnel tests provided data which was used to validate SOFIA for transonic static aeroelastic problems. Due to high Reynolds numbers, the employed multiblock structured flow grid featured a very fine resolution of the boundary layer. To ensure the quality of these grid cells close to the deforming wing surface, the grid deformation code MUGRIDO was developed and successfully applied (cf. Sec. 2.3). The comparison between computational results and measured data is discussed in detail in [4, 19, 20], and only some exemplary results can be presented here. It is well-known that for backward swept wings the coupling of bending deflections and local flow incidence angles leads to a considerably reduced lift coefficient with respect to the one of an undeformed configuration, the so called jig shape (cf. Fig. 14, left). The difference between the AEC and the jig shape becomes even more prominent if one regards the pitching moment coefficient (cf. Fig. 14, right). In both cases the aeroelastic simulation with SOFIA agrees almost perfectly with the wind tunnel measurement, in contrast to the simulations of the jig shape. For the reference

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Fig. 14 HiReTT test case: Global lift and pitching moment coefficients as a function of angle of attack. Aeroelastic simulations and simulations about the undeformed configuration (jig shape) are compared to experimental results from the test campaign in ETW. (Ma=0.85, Re=32.5M, q/E=0.5747·10−6 )

Fig. 15 HiReTT test case: Spanwise lift and pressure distribution of the experiment compared to results from an aeroelastic simulation and a simulation of the undeformed (rigid) configuration. (Ma=0.85, Re=32.5M, q/E=0.5747·10−6 , α =αre f )

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angle of attack defined in the context of HiReTT (see mark in Fig. 14) the spanwise distribution of the lift coefficient is additionally shown in Fig. 15. Also here, the reduction of the local lift due to elastic deformation becomes more pronounced with increasing span. In the outer part of the wing a strong shock exists if deformation is disregarded. This shock becomes a double-shock configuration at the tip in consequence of an aerodynamic twist of up to 2◦ , as can be seen from the plots of the pressure coefficient in the sections at 47%, 75% and 84% span. These significant changes of the pressure distribution are precisely captured by SOFIA.

4 Selected Results of the Final Funding Period – Validation for the HIRENASD Experiments In this chapter, selected results are shown which were achieved in the final SFB 401 funding period by applying SOFIA in its latest development status to the High Reynolds Number Aero-Structural Dynamics (HIRENASD) test case (see also Ballmann et al. in this volume). The HIRENASD experiment provides comprehensive data, in particular unsteady data, from aeroelastic experiments at high Reynolds numbers and transonic flow conditions which are scarce otherwise. The wind tunnel model used in these tests is shown in Fig. 16. Additional information about

Fig. 16 Setup of the HIRENASD test case

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the model geometry, its equipment with measuring technique and the test campaign can be found in [4, 21, 22]. During the test campaign the aeroelastic behaviour of the wing model was studied in two different ways: (i) in steady polars at slowly varying angle of attack; and (ii) in dynamic aeroelastic tests at fixed root angle of attack under vibration excitation in the wing root region. These experiments were performed at different Mach numbers, Reynolds numbers and model loading factors q/E. In Sections 4.1 and 4.2, the agreement between SOFIA’s predictions of the steady aeroelastic model behaviour and the measurements is discussed in terms of the changes of the global and the local lift coefficient resulting from (i) variations of the angle of attack; (ii) variations of the loading factor; and (iii) variations of the Mach number. In Section 4.3, the simulation results are compared to measurements of one series of dynamic aeroelastic HIRENASD experiments. It is studied how the unsteady aerodynamics is influenced by changes of the Mach number, while the wing vibrates in its second flap-bending dominated mode shape.

4.1 Dependence of the Lift Coefficient on the Angle of Attack Since backward-swept wings experience a spanwise reduction of local angles of attack when bent upwards, their integral lift coefficient in regular flow is always lower than for a wing without deformation (cf. Figs. 14 and 15). As to be expected, this is also the case with the HIRENASD wing, as can be seen in Fig. 17. The lift polars are plotted for Ma=0.80, Re=23.5M and two different loading factors, which are q/E=0.22 · 10−6 on the left and q/E=0.48 · 10−6 on the right. The lift loss with respect to simulations disregarding the wing deformation (dashed curves) is higher in the right part of the figure because of the increase of the wing loading factor q/E. With consideration of the model deformation, a far better prediction of lift coefficients is obtained, as is evidenced by the close agreement with the measured polars. The streamline patterns displayed in combination with c p distributions about AECs at α =5◦ and −2◦ show insights how the wing deformation interacts with the flow separation. The flow is detached in the two outer wing sections if a rigid wing is assumed. In case of the higher q/E level, the flow reattaches again in the outermost wing section if the aerodynamic twist is correctly captured. Nonlinearities of the cL α curves are only visible for the rigid case because of its stronger flow separations. Because zero lift is at about α =−1.3◦ for the wing, the impact of wing deformation on streamline patterns and c p distribution is hardly visible at α =−2◦ . At this angle of attack it is difficult to obtain a converged coupled solution. This is due to a strong shock occuring where the airfoil is thickened from originally 11% to 15% relative thickness in the inner section on the lower wing side. In addition, a separation bubble is present at negative α shortly behind the characteristic turning point in the nose region of the BAC-3/11 airfoil on the lower side, even for low Mach numbers. This behaviour may account for the differences between experiment and simulation for negative α . For α =+1◦ , +2◦ and +3◦ , denoted by I, II and III in Fig. 17, the spanwise lift distribution is extracted for both simulation variants and the experiment and compared to each other in Fig. 18. Although the agreement between aeroelastic simulation and

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Fig. 17 HIRENASD test case: Changes of lift coefficient with angle of attack α for Ma=0.80, Re=23.5M, q/E=0.22 · 10−6 (left) and q/E=0.48 · 10−6 (right)

Fig. 18 HIRENASD test case: Spanwise lift distribution for α =+1◦ , +2◦ and +3◦ , Ma=0.80, Re=23.5M, q/E=0.48 · 10−6

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measurement diminishes with increasing angles of attack, once more, the benefit of considering the model deformation is evident. To predict the spanwise lift distribution of the inner wing sections correctly, the influence of the fuselage substitute must be considered aerodynamically [23]. Since there is no mechanical contact between the fuselage and the wing, the fuselage structure does not have to be modelled. Nevertheless, the fuselage is considered with a dummy beam to preserve the smoothness of the aerodynamic surface during wing deformation. This is achieved by applying the blending techniques of ACM’s projection algorithm between the dummy beam and the actual wing beam, as is described in Sec. 2.4.1.

4.2 Lift Divergence Behaviour When the Mach number is increased, the integral lift coefficient diverges at some point due to the development of shocks and flow separation. This point of lift divergence is also visible in the upper left diagram of Fig. 19 for numerical predictions and experiments. The divergent behaviour of the experimental curves is well captured by the aeroelastic simulations in terms of the lift slope and the Mach number at divergence onset. Although the absolute lift value is predicted less accurately with increasing angle of attack, the Mach number from which the curves diverge is determined correctly for all angles of attack, even for α =5◦ . For this case the pressure distributions at Ma=0.75, 0.80, and 0.85 (Re=23.5 · 106, q/E=0.22 · 10−6) are compared between simulation and experiment in the 7 spanwise sections which were equipped with pressure sensors. Fig. 19 reveals that flow separation increasingly dominates the wing aerodynamics at higher Mach numbers. Overall, we obtain a very good agreement between simulation and experiment. Almost all shock positions, suction peaks and general pressure distributions along local chords are in very good agreement with the experiments. Slight deviations are present in the pressure level aft of the onset of shock-induced separation in the outer sections and in particular in both inner sections for Ma=0.85, where the computed shock position is slightly more upstream than in the experiment.

4.3 Prescribed Motion According to the Second Flap-Bending-Dominated Mode Shape In the dynamic HIRENASD experiments a wing excitation tuned to the resonance frequency of particular eigenmodes of the aeroelastic system was applied. Further details on the actual realization of the excitation mechanism are given in [22]. In the experiments, the mutual influence between the excited harmonic wing motion and the transient transonic aerodynamics was measured in terms of balance reactions, structural strains and accelerations, and the instantaneous pressure distribution. These experiments were simulated with SOFIA in two different ways. First, the wing vibration was excited numerically in the same manner as in the experiments. Starting from an initial AEC for the respective flow conditions, the numerical structural model, i.e. the Timoshenko beam idealization, was subjected to an equilibrium

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Fig. 19 HIRENASD test case: Divergence of the lift coefficient with increasing Mach number. In addition, the changes of the c p distribution for Ma=0.75 (I), 0.80 (II), and 0.85 (III). Section 1 is located close to the wing root whereas no. 7 is the outermost section. (Re=23.5M, q/E=0.22 · 10−6 ).

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Fig. 20 HIRENASD test case with prescribed motion of the wing according to the second flap-bending dominated mode shape: The mean values of the pressure fluctuations of the outermost wing section are plotted for Ma=0.80, 0.83, 0.85, and 0.88. (Re=23.5M, q/E=0.22 · 10−6 , α =1.5◦ )

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Fig. 21 HIRENASD test case with prescribed motion of the wing according to the second flap-bending dominated mode shape: The first harmonic of the pressure fluctuations of the outermost wing section are plotted for Ma=0.80, 0.83, 0.85, and 0.88 with respect to the amplitude of the outermost acceleration sensor acc15. (Re=23.5M, q/E=0.22 · 10−6 , α =1.5◦ )

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pair of bending moments which bends periodically the wing clamping part region. For this type of simulations, preliminary results can be found in [24]. In this paper, results for a second way of simulating the excitation experiments are presented. Rather then exciting the oscillatory wing motion, it was explicitly prescribed according to the selected numerical mode shape. The same values were chosen for the numerical oscillation frequency and amplitude as in the experiments. This study of the transient aerodynamics involved the dynamic HIRENASD tests no. 159, 161, 164 and 166. In these tests, the second flap-bending dominated mode was excited under the following wind-on conditions: Ma=0.80 (no. 159), 0.83 (no. 161), 0.85 (no. 164), 0.88 (no. 166); Re=7M; q/E=0.22 · 10−6; α =1.5◦ ; and fexc =78.9Hz. For these tests, Fig. 20 shows the mean pressure distributions of all simulations as derived from instantaneous pressure distributions. The mean pressure distribution is shown as a contour plot for the whole wing on the right. Although the fuselage is not displayed, it is considered in the computations aerdynamically. On the left, the pressure distribution is shown in the section which is closest to the wing tip (section 7). In addition, the pressure plots of this section are complemented by the predicted root mean square (RMS) of the instantaneous pressure. Due to the nonlinearity of the transonic flow, the mean pressure may differ from the pressure distribution of the respective AEC. This is evident in the pressure plot at Ma=0.88. Here, the mean shock position is slightly more upstream than for the AEC. In consideration of Fig. 20, the mean state of the wing vibrating about its AEC can be described as follows: At Ma=0.80, a shock is only present close to the wing root. It reduces its strength in spanwise direction and finally vanishes at the wing tip. The flow is still fully attached. With increasing Mach number, the shock strength increases and the shock moves upstream. Shock-induced flow separation starts to appear in the inner wing section and extends more and more towards the wing tip. In section 7 a weak shock emerges at Ma=0.83. In contrast to the behaviour of the shock in the inner wing section, this shock moves downstream with increasing Mach number. As an additional result of an increasing Mach number, the location of the maximum RMS of the pressure changes from the leading edge to the shock region. As the shock strength increases, the extent of the region with high pressure fluctuations becomes narrower. As a consequence of the increase of the Mach number, the RMS of the pressure also becomes larger in the region of the suction minimum on the lower side. Higher RMS values at the trailing edge might be attributed to additional crossflow (cf. streamlines in the right part of the figure) and a slightly enlarged trailing edge separation. In the following, the computed and measured pressure fluctuations are compared for section 7 for the same series of Mach numbers as before. To investigate the magnitudes and phase lags of the pressure fluctuations with respect to the actual wing motion, these pressure fluctuations were normalized with the mean acceleration amplitude at the wing tip. The location of sensor acc15 in the experiment is depicted in Fig. 16. In Fig. 21 the resulting relative data (cp /acc15) of experiment and simulation are plotted in terms of their real and imaginary parts. In the experiment, this data was obtained by Fourier analysis and elimination of any frequency content outside of a narrow band around the excitation frequency. Details about the band filtering

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technique applied to the experimental raw data are given in [21]. The resulting data of the simulation considers only the first harmonic of the pressure oscillations obtained also from Fourier analysis. In comparison with the experiment, the numerical predictions capture all principal trends observed from the measured behaviour with acceptable accuracy. The development of the weak shock between Ma=0.80 and 0.83 is predicted, which is evident from the emerging pronounced local minimum of the imaginary part and the sign change of the real part. The computational results also exhibit an increase of the shock strength and hence the increase of the pressure fluctuations. The maxima of the pressure oscillations move downstream with increasing Mach number due to the same motion of the mean shock location, which can be seen clearly from the imaginary part. The magnitudes of the imaginary part of the pressure oscillations are predicted significantly lower than observed in the experiments though. Apart from Ma=0.88, the phase shifts identified in the real part only occur in the shock region on the upper side. At Ma=0.88, the trend towards a shock formation on the lower side induces a phase shift on this side as well. The pressure fluctuations upstream and downstream of the shock are predominantly lower in the experiment than in the simulation, whereas the measured ones are higher directly at the shock. Since the turbulence model strongly affects the prediction of the transient aerodynamics and the shock-boundary layer interaction, its influence on the magnitude and phase lag of the pressure oscillations should be investigated more closely. The results shown here and the ones in the steady HIRENASD cases were obtained by employing the LEA k-ω turbulence model. Finally, the agreement between experimental and predicted pressure fluctuations can be regarded as satisfactory. This motivates the comparison with the experiments with higher loading factors and the excitation of the two remaining mode shapes, i.e. the excitation of the first flap-bending-dominated mode and the first torsion-dominated mode.

5 Conclusion This paper outlined the development of the aero-structural dynamics method SOFIA over the duration of the Collaborative Research Center SFB 401 from 1997 till 2008. Parallel to its implementation and extension, SOFIA was validated in this time for numerous aeroelastic test cases. It was validated for 3D wing models by comparing numerical predictions with the comprehensive data gathered by the SFB 401 in various wind tunnel tests. This validation started for a straight wing exposed to subsonic flow [12, 13]. In the subsequent study of the configuration shown in Sec. 3.1 the effects of wing sweep was included in the analysis. SOFIA’s validation proceeded with various aircraft-like configurations for the transonic flow regime. The predictions of SOFIA agreed excellently with the measurements of the HiReTT wing-fuselage configuration. This encouraged the application of SOFIA for preparing the HIRENASD wind tunnel tests. The dependence of lift on angle of attack, Mach number and loading factor observed in the steady tests were very accurately captured by SOFIA. The comparisons between SOFIA and unsteady HIRENASD test results were promising but have not been completed yet.

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The aeroelastic computations for such configurations like HIRENASD and HiReTT were not possible with SOFIA until its load and deformation projection algorithms were enhanced in the way presented in Sec. 2.4.1. Though not demonstrated in this paper, the enhancements also enabled the consideration of stationary deflected control surfaces [3, 4, 20]. During the funding of the SFB 401, SOFIA has been established as a versatile numerical method for the simulation of steady, unsteady, 2D and 3D aeroelastic problems. One of SOFIA’s further developments currently in progress aims at the inclusion of time-dependent control surface deflections [25].

References 1. Mok, D.P.: Partitionierte L¨osungsans¨atze in der Strukturdynamik und der Fluid-StrukturInteraktion. Doctoral thesis, University Stuttgart, Germany (2001) 2. Kroll, N., Fassbender, J.K.: MEGAFLOW — Numerical flow simulation for aircraft design. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 89. Springer, Heidelberg (2005) 3. Boucke, A.: Kopplungswerkzeuge f¨ur aeroelastische Simulationen. Doctoral thesis, RWTH Aachen University, Germany (2003) 4. Braun, C.: Ein modulares Verfahren f¨ur die numerische aeroelastische Analyse von Luftfahrzeugen. Doctoral thesis, RWTH Aachen University, Germany (2007) 5. Hesse, M.: Entwicklung eines automatischen Gitterdeformationsalgorithmus zur Str¨omungsberechnung um komplexe Konfigurationen auf Hexaeder-Netzen. Doctoral thesis, RWTH Aachen University, Germany (2006) 6. Farhat, C., Lesoinne, M., LeTallec, P.: Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity. Comput. Meth. Appl. Mech. Eng. 157, 95–114 (1998) 7. Samareh, J.: Discrete Data Transfer Technique for Fluid-Structure Interaction. AIAA paper 2007-4309 (2007) 8. Beckert, A.: Coupling fluid (CFD) and structural (FE) models using finite interpolation elements. Aerosp. Sci. Technol. 4, 13–22 (2000) 9. Badcock, K., Rampurawala, A., Richards, B.: Intergrid Transformation for Aircraft Aeroelastic Simulations. AIAA journal 42(9), 1936–1939 (2004) 10. Hurka, J., Ballmann, J.: Elastic Panels in Transonic Flow. AIAA paper 2001-2722 (2001) 11. Massjung, R.: Discrete conservation and coupling strategies in nonlinear aeroelasticity. Comput. Meth. Appl. Mech. Eng. 190, 91–102 (2006) 12. Britten, G., Braun, C., Hesse, M., Ballmann, J.: Computational Aeroelasticity with Reduced Structural Models. In: Ballmann, J. (ed.) Flow Modulation and Fluid-Structure Interaction at Airplane Wings – Research Results of the Collaborative Research Center SFB 401 at RWTH Aachen University. Notes on Numerical Fluid Mechanics, vol. 84, pp. 275–299. Springer, Heidelberg (2003) 13. Britten, G.: Numerische Aerostrukturdynamik von Tragfl¨ugeln großer Spannweite. Doctoral thesis, RWTH Aachen University, Germany (2003) 14. Dafnis, A., K¨ampchen, M., Reimerdes, H.G.: Aero-structural investigation on highly flexible wind tunnel wing models. In: Proc. of the International Forum on Aeroelasticity and Structural Dynamics IFASD 2001, Madrid, Spain, paper 093 (2001)

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15. K¨ampchen, M., Dafnis, A., Reimerdes, H.G., Britten, G., Ballmann, J.: Dynamic aerostructural response of an elastic wing model. J. Fluids and Structures 18, 63–77 (2003) 16. Reimer, L., Braun, C., Ballmann, J.: Computational Study of the Aeroelastic Equilibrium Configuration of a Swept Wind Tunnel Wing Model in Subsonic Flow. In: Nagel, W.E., J¨ager, W., Resch, M. (eds.) High Performance Computing in Science and Engineering 2006, pp. 421–434. Springer, Heidelberg (2006) 17. Reimer, L., Braun, C., Ballmann, J.: Analysis of the Static and Dynamic Aero-Structural Response of an Elastic Swept Wing Model by Direct Aeroelastic Simulation. In: Proc. of the International Council of the Aeronautical Sciences ICAS 2006, Hamburg, Germany, paper ICAS 2006-10.3.3 (2006) 18. Rolston, S.: Initial Achievements of the European High Reynolds Number Research Project HiReTT. AIAA paper 2002-0421 (2002) 19. Braun, C., Boucke, A., Hanke, M., Karavas, A., Ballmann, J.: Prediction of the Model Deformation of a High Speed Transport Aircraft Type Wing by Direct Aeroelastic Simulation. In: Krause, E., J¨ager, W., Resch, M. (eds.) High Performance Computing in Science and Engineering 2003, pp. 331–342. Springer, Heidelberg (2003) 20. Braun, C., Boucke, A., Ballmann, J.: Numerical Prediction of the Wing Deformation of a High-Speed Transport Aircraft Type Windtunnel Model by Direct Aeroelastic Simulation. In: Proc. of the International Forum on Aeroelasticity and Structural Dynamics IFASD 2005, Munich, Germany, paper IF-147 (2005) 21. Ballmann, J., Boucke, A., Dickopp, C., Reimer, L.: Results of Dynamics Experiments in the HIRENASD Project and Analysis of Observed Unsteady Processes. In: Proc. of the International Forum on Aeroelasticity and Structural Dynamics IFASD 2009, Seattle, USA, paper IFASD-2009-103 (2009) 22. Korsch, H., Dafnis, A., Reimerdes, H.G.: Dynamic qualification of the HIRENASD elastic wing model. Aerosp. Sci. Technol. 13(2-3), 130–138 (2009) 23. Reimer, L., Braun, C., Chen, B.H., Ballmann, J.: Computational Aeroelastic Design and Analysis of the HIRENASD Wind Tunnel Wing Model and Tests. In: Proc. of the International Forum on Aeroelasticity and Structural Dynamics IFASD 2007, Stockholm, Sweden, paper IF-077 (2007) 24. Reimer, L., Boucke, A., Ballmann, J., Behr, M.: Computational Analysis of High Reynolds Number Aero-Structural Dynamics (HIRENASD) Experiments. In: Proc. of the International Forum on Aeroelasticity and Structural Dynamics IFASD 2009, Seattle, USA, paper IFASD-2009-130 (2009) 25. Chen, B.H., Brakhage, K.H., Behr, M., Ballmann, J.: Numerical simulations for preparing new ASD experiments in ETW with a modified HIRENASD wing model. In: Proc. of the International Forum on Aeroelasticity and Structural Dynamics IFASD, Seattle, USA, paper IFASD-2009-131 (2009)

A Unified Approach to the Modeling of Airplane Wings and Numerical Grid Generation Using B-Spline Representations Karl-Heinz Brakhage, Wolfgang Dahmen, and Philipp Lamby

Abstract. In this article we summarize the development of a unified platform for treating the entire range of geometric preprocessing tasks that preceded the wind tunnel readings and the numerical simulations performed in the collaborative research center SFB 401. In particular, this includes the automated generation of the CAD models which were used for manufacturing multi-parted wing-fuselage configurations as well as the generation of the numerical grids for the corresponding, adaptive numerical simulations.

1 Introduction and Overview The numerical simulation of fluid-structure-interaction, especially when employing high level models such as the compressible Navier-Stokes equations on the fluid side, still pose enormous challenges that cannot be met solely by increased computing power. The processes are inherently nonstationary and the computational domain varies in time. Moreover, stiff components in the coupled fluid-structure problem require implicit time integration and hence the repeated solution of possible extremely large nonlinear systems of equations. To deal with problems of such complexity calls, on one hand, for adaptive spatial and temporal discretizations in the fluid and structure solvers, in order to keep the systems as small as possible in the first place. On the other hand, such adaptive meshes need to be frequently adapted to varying domain geometries. Therefore, a long term central objective has been the Karl-Heinz Brakhage · Wolfgang Dahmen Institut f¨ur Geometrie und Praktische Mathematik, RWTH Aachen, D-52056 Aachen, Germany e-mail: {brakhage,dahmen}@igpm.rwth-aachen.de Philipp Lamby University of South Carolina, Industrial Mathematics Institute, Columbia SC, 29208, USA e-mail: [email protected]

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 239–263. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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development of an integrated concept that closely intertwines adaptive discretizations with a tailormade grid generation in a way that the above tasks are supported to a possibly large extent. Moreover, manufacturing high quality models for corresponding wind tunnel experiments suggests taking related CAD tasks into account. The project B2 of the SFB 401 has been therefore concerned with the development of suitable grid generation concepts as well as with the generation of CAD models. More precisely, the work in this project can be divided into three major blocks: 1. In order to support adaptive discretizations based on hierarchies of nested grids, as needed by the multiscale algorithms designed in project A4, a parametric grid generation concept has been developed and integrated into the finite volume solver QUADFLOW. 2. A system, including a graphical interface, for generating parametric grids employing B-spline techniques has been implemented. In particular, properly adapted versions of many classical grid generation algorithms have been integrated into the B-spline concept as tools for generating suitable control nets for the B-spline mappings. Furthermore new methods, in particular for the generation of offset grids, have been invented. 3. In the context of the HiReNASD-project CAD models for the multi-parted wing-fuselage configuration have been generated, which are both highly parameterizable and high-quality, such that they can be used directly for both the manufacturing process and the grid generation. The central objective of project B2 has been to develop a unified framework for addressing all the geometric processing tasks related to the above three packages. In particular, geometry representations needed to be found that support both the manufacturing process and the grid generation already at the modeling stage. This goal has been achieved by extensive use of B-spline techniques. In the following three sections we shall recall the motivation behind this concept, sketch briefly the evolution of the project during the whole funding period of the research center and highlight some of the latest developments.

1.1 Parametric Grids Some basic features of QUADFLOW and corresponding demands on the underlying grid generation concept are indicated in Figure 1 which shows a transonic inviscid fluid flow around a BAC 3-11 profile. The computation starts with an initially very coarse grid. After some cycles of adaptive flow computation one arrives at a final adapted grid meeting a desired target accuracy. The grid adaptation is a quadtreetype h-refinement based on an initial block partition of the computational domain. Each block hosts a logically Cartesian grid. Grids on adjacent blocks need not match though which simplifies the process. The key task of the grid generation module is to provide hierarchies of grids at arbitrary levels of resolution that are structured blockwise, so that locally refined portions of such hierarchies can be efficiently activated by suitable adaptation criteria.

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Fig. 1 Initial coarse grid, final adapted grid and pressure distribution for flow around BAC 3-11 profile, Mach 0.85

The adaptation strategy employed in QUADFLOW requires the underlying grid hierarchy to be nested. This means that a cell V j,k on level j with index k is the union of cells V j+1,r , r ∈ M j,k on the next finer level so that the sum of the volumes of the fine grid cells is the same as the volume of the coarse grid cell: ∑r∈M j,k |V j+1,r | = |V j,k |. Here a problem arises because obviously in curvilinear domains standard polygonal grid hierarchies are not nested. In order to facilitate the efficient generation of nested grids of curvilinear type we do not use discrete grid models. Instead we provide as input for the flow solver parametric mappings that represent coordinate systems in the single blocks. By default these mappings are realized by tensor product B-splines which ensure fast point evaluation independent of the number of knots. Then grids at arbitrary level of resolution can be constructed easily by function evaluation and grid nestedness is automatically achieved if one considers a grid cell to be the geometric image of the corresponding cell in parameter space. Hence, within the frame of our grid generation concept a multi-block grid is a representation of the flow domain by B-spline tensor product patches. In 3D, of course, the blocks are trivariate B-spline tensor product volumes bounded by B-splines surfaces. The integration of the parametric grid generation concept into the QUADFLOW solver started at the beginning of the second funding period, when the first prototypes of the multiscale adaptation algorithm developed in project A4 and the finite volume solver implemented in project B3 became available. Since then the parametric system has been used and tested extensively. In the course of these investigations it turned out though, that in the curvilinear setting, special care has to be taken in order to fulfill the geometric conservation laws which can be considered as necessary consistency conditions for the accuracy and stability of the finite volume solver. For a detailed discussion of this topic we refer to [17].

1.2 B-Spline Grid Generation Of course, the concepts described above require a possibly automatized and robust generation of B-spline mappings to begin with. In principle this task can be related to classical block-structured grid generation by viewing the B-spline control nets, i.e. the coefficients in B-spline representations, as coarse versions of grids, since

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these control points indeed have a geometric significance. Hence, in principle, a host of algorithms and codes could be taken from the relevant literature to serve that purpose. The most notable ones are algebraic grid generation techniques using transfinite interpolation and elliptic grid generation systems. Therefore, already in the first funding period, these algorithms were thoroughly evaluated regarding their usability and, if approved, were added to the software package that was build in this project. As mentioned before these algorithms could be used to generate control nets, or to generate first classical grids and convert them subsequently into B-spline representations. This required the development of tailormade fast interpolation and approximation algorithms. A detailed documentation of this stage of the project can be found in [12]. On the other hand, the superior flexibility and accuracy of the B-spline representations motivated the invention of completely new grid generation methods, like for example algebraic-hyperbolic grid generation as described in [6], and later the generation of offset blocks, see [10]. Since complex grids cannot be represented by a single tensor-product mapping, a multiblock-grid data structure has to be defined. This became a major part of the work during the middle of the funding period of the research center. Finally a multiblock manager was implemented that mimics ideas from [21] and [19] and adapts them to the parametric setting, see [17]. At a later stage of the QUADFLOW development, when the solver became able to compute on time-varying grids, robust and efficient grid deformation algorithms became an important issue. A first set of such algorithms based on algebraic perturbation methods was presented in [11]. These methods were later expanded by a deformation algorithm based on radial basis functions, which will be discussed later in Section 4. Only at about this time, i.e. at a relatively late stage of the project, when the configurations and block-decompositions became too complex to be assembled based on sketches and scripts, a real, interactive graphical interface was implemented. In this context the need of revising some of the initial approximation algorithms became apparent since they did not perform well enough in a complex application. In particular, this was the case when for constructive reasons irregular B-spline knot sequences came into play. As a remedy at that stage of the work we chose to discretize an elliptic grid generation system by means of collocation instead by finite differences, which naturally ties into the B-spline representations and offers significantly more accuracy. These new results will be presented in Section 3.

1.3 Generation of Wing Models In the final stage of the SFB another problem came into focus. Usually the CAD-models which are used for the manufacturing of wind tunnel models are generated by commercial programs that provide hundreds of geometric entities to construct the various geometric features. Every program has its own way to represent and manipulate data. Conversion from one program to the other is accompanied by approximation processes which usually cause errors and loss of

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information, in particular, with regard to the topology. For instance, in CAD-models one finds very frequently trimmed surfaces, which destroy the logically Cartesian topology of the surface representation and therefore work very badly together with block-structured grids. Hence, after getting the CAD-data, the grid generator is usually committed to fix and repair the geometry before he can start with the genuine grid generation task. This geometry post-processing can take weeks of work for a complex configuration and is therefore a major bottleneck in the modeling and grid generation pipeline. In this project CAD-models for an airplane wing have been constructed that do not suffer from such deficiencies. The complete wing is constructed by exactly fitting, untrimmed B-spline patches. The wing itself corresponds to a three parted, back-swept BAC 3-11 aerofoil cruise configuration of scale 1:28 with rounded tip. To diminish wind tunnel influences a half-body is placed between the wing and the wind tunnel wall. Optionally the outer part of the wing can be replaced by a part with a winglet. This model is conveniently parameterizable and offers the choice to vary design parameters like bending radii, angles and top views easily. However, in order to fulfill several non-standard constraints, that stemmed from the design demands and the manufacturing needs, the algorithms for approximation and fairing, that can be found in literature, had to be extended, as will be explained in Section 2. Pure B-splines are ideal for communication between different software packages, because they are considered to be basic in CAD and are understood by basically every B-spline software. The basic data exchange between the modeling, grid generation and manufacturing software was carried out by IGES files. Concretely the milling machine employed hyperCAD/hyperMill from OpenMind, the inner technical constructions were planed with CATIA and for the visualization we used Rhino.

1.4 Outline of Paper In the remainder of this article we mostly concentrate on work, that has been done in the last funding period. First, in Section 2 we present in some detail the construction of the wing model that was used for the HiReNASD wind tunnel readings described somewhere else in this book. Section 3 sets up a B-spline collocation method to realize an elliptic grid generation system. In some sense, this section serves to demonstrate how classical grid generation theory and B-spline representation work together. Finally, in Section 4 we discuss an algorithm for grid deformation.

1.5 Notation Throughout this paper we write B-spline curves in the form x(t) = ∑Ni=0 pi Ni,p,T (t) where Ni,p,T (t) is the i-th normalized B-spline function of order p (degree p − 1) corresponding to the generally non-uniform knot vector T = (t0 ,t1 , . . . ,tN+p ) and pi are the control points that determine the curve. In fact, since the B-splines form a local partition of unity the location of the control points already conveys a good geometric information on the actual position of the corresponding curve. Moreover,

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(a properly defined notion of) oscillation of the sequence of control points and of the corresponding control polygon is known to control also the oscillation of the B-spline curve. We usually assume that T is clamped and all interior knots are of multiplicity one, i.e., t0 = . . . = t p−1 < t p < . . . < tN < tN+1 = . . . = tN+p . For the sake of simplicity we write Ni,p instead of Ni,p,T whenever it is clear from the context which knot vector is being referred to. Surfaces are represented by B-spline tensor products of the form N

M

∑ ∑ pi j Ni,p(u)N j,q (v).

(1)

i=0 j=0

We shall always adopt the following convention. If q is a point, then x(q), y(q), z(q) denote its x, y, or z-coordinate respectively. The orientation of our coordinate system is explained in Figures 4 and 5.

2 Geometry Description and Outline of the Modeling Process We start the technical part of this paper with a description of the construction of the CAD-model for the wing model including the mounting unit shown in Figure 2. Both parts are represented exclusively by untrimmed B-spline patches. More specifically, as pointed out later, the main part of the wing is defined by a set of cross-sections which are connected by ruled surfaces. Hereby the number of cross-sections can be varied on demand. Additionally the simplified half of a fuselage has been designed in order to reduce the influence of the wind tunnel walls, see figure 3. This fuselage is represented by a curvature continuous periodic B-spline surface.

2.1 Cross Sections and Wing Construction The first step of the modeling process is to find B-spline representations of the wing cross sections. Originally the reference cross section of the cruise configuration was described by 87 points given in the two ARGARD reports [18]. These points x j are scaled in such a way that the wing depth, i.e, the distance from the nose to the trailing edge, attains unit length. The tolerance relative to the wing depth is ε = 1.7 · 10−4 . Thus we have to guarantee that min x(t) − x j 2 ≤ ε t

∀ j.

The profile is represented by a closed curve with a discontinuous tangent only at the trailing edge which is located precisely at (1, 0). Moreover, the point of the profile corresponding to the nose of the wing is (0, 0) and has a vertical tangent there. Optionally the curvature of the cross section at this point can be prescribed as well. Since the commonly used approximation schemes do not allow one to meet such a variety of different approximation constraints, special approximation procedures had to be developed, which are described in [10].

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Fig. 2 View of the complete model with mounting unit

Fig. 3 Simplified fuselage – control points and knot-isolines

The relative thickness of the reference profile is rt = 11%. This thickness can be varied by scaling the control points of the spline in vertical direction. At the fuselage the thickness of the profile has to be 15%. This increase of thickness has to be achieved by changing the profile only in the lower part (see Figure 4). All these computations are done in 2D space because they require only varying cross sections. Therefore we can fix the knot vector after computing the smooth 11% reference cross-section and keep the same knot vector for the other profiles, in particular, for the profile at the fuselage. This will later lead to an easy representation of the airfoil sections. The result is shown in Figure 4. We have used order p = 4 and a parametrization according to chord length. The next step is to describe the top view sketch of the multi-parted back-swept wing. This can be done with the aid of an arbitrary 2D CAD program that can treat B-splines and export coordinates and lengths. We use WinCAG [3, 4]. For our purposes some special interfaces were added to that system. The only information we need from this step is the front position of the cross-sections Ai , their depth li and the relative position of R with respect to An (= A4 here), compare Figure 5). The scaled profiles are placed at the right position in 3D space. The corresponding 3D coordinates are generated from the 2D sketch. Using piecewise linear connections between the profiles, the multi-parted wing can be represented by tensor product B-spline patches of order (p × 2) which results in ruled surfaces between the cross sections Ai . The only exception is the lower part of the patch near the fuselage which consists of conics.

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design ordinates

smooth spline

smooth spline at fuselage

Fig. 4 Design ordinates, spline (rt = 11%) and spline at fuselage (rt = 15%). For later reference: the horizontal direction in this sketch corresponds to the y-coordinate of our 3Dcoordinate system. The vertical direction corresponds to the z-coordinate A1 A2 l1 l2

A3 A3

l3

l3 A4 l4

A’4 l4’

R

x0

a

l 5’

A’5 R’

Fig. 5 Sheer Plan of the multi-parted back-swept wing – winglet modifications on right hand side, see Section 2.4. For later reference: The coordinate in span direction (i.e. the horizontal direction in the sketch) is the x-coordinate, the vertical direction is described by the y-coordinate

The sensors and cables have to be placed inside the wing. The necessary shell thickness of the aerofoil is roughly known from stress and eigenfrequencies computations (FE shell model considering webs) and is of variable size. Therefore a variable inner offset surface of the wing was computed. All detail constructions for the interior equipment have to remain inside this surface. So far we have considered mainly 2D constructions. In the following sections we shall describe some genuinely 3D constructions.

2.2 Mounting Unit The mounting unit is given by its top and front view plans which also indicate the rounded angles needed for the milling process. Only the top view of the fillet is given

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in the 2D sketch. To avoid gaps, the blend is not computed as a trimmed surface. For this reason the B-spline representing the cross-section at the fuselage has to be split up into five parts. This is done by knot insertion. The fillet and the mounting unit are then computed as one block. The exact blending radii and their centers are exported separately to the milling software. This part was manufactured with a cylindrical milling cutter. Between the wing and the blend to the mounting unit a cylindrical continuation can be added (see Figure 6) to pass the (material of the) fuselage. The realization of GC1-continuity (tangent plane continuity) for the fillet is a little bit more difficult. Employing knot insertion we subdivide the profile spline of the wing into 5 parts (compare Figure 6). The partitions number 2 and 4 - the last one with inverted orientation - are the boundary curves of our fillet. Now we can apply the formulas from [15, Ch.7] to achieve the desired continuity properties. For the blending radii a pre-computation of the x-y-coordinates is done (fitting of the circular arc). We have still one degree of freedom left for the second row of control points. This can be used to design a fairly rapid transition from the wing to

r1

r3

r2

Fig. 6 Top and front view of mounting unit with continuation

Fig. 7 Plot of the fillet with mounting unit

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the inclined part. The same applies to the transition to the mounting unit. Figure 7 shows the four untrimmed surfaces forming the transition.

2.3 The Wing Tip The rounded wing tip is determined by the relative position of R to An (compare Figure 5) and the following construction. The engineers demand only GC1continuity at the crossover from the wing given by the control points pi j to the tip with control points qik . To achieve this with a possibly small number of control points we build a B-spline-B´ezier surface of order p × 3. The control point matrix (qik ) will consist of three rows (k = 0, 1, 2) of control points each with i = 0, . . . , N points. Here N + 1 is the number of control points in the spline representations of each cross-section. As first row of control points for the tip we take the one from the last cross section of the airfoil: qi0 = piN . We place the second row of control points on the intersection of the lines through pairs of two corresponding control points from the last two airfoil cross sections with the plane through the point R parallel to the cross-section A4 , i.e. qi1 = piN + α (piN − pi,N−1 ). Since the outer part of the wing is a conic with parallel cross-sections this construction guarantees the desired GC1continuity. For the third and last row we first project the points of the second row into the x-y-plane and then transform the y-coordinates of these points linearly onto the interval y(qi0 ), y(R) in order to round of the wing at the outer end of the leading edge: y(R) − y(q01 ) qi2 = (x(qi1 ), y(q01 ) + (y(qi1 ) − y(q01), 0), y(P) − y(q01) where P is the nose of the profile defined by the spline with control points qi1 . The surface defined by this procedure is a GC1 continuation of the wing and is automatically computed due to the above choice of parameters. In this form it is well suited for manufacturing, but for grid generation it has to be re-parameterized, see [10]. A plot of the wing tip and its control points is given in Figure 8.

Fig. 8 Wing tip – Control points (top), knot-isolines (bottom)

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2.4 Winglet Construction A future series of wind tunnel readings will be concerned with wing configurations where a winglet is added to the airfoil, see Figure 9. For this reason we have developed algorithms for automatic winglet constructions. In a first step the sheer plan (compare Figure 5) has to be extended. We mark the bending position x0 and introduce an additional dihedral angle a. From this the new positions A4 and A5 and the wing chords l4 and l5 are determined. Finally R takes over the role of R in the construction of the wing tip. The basic idea of the 3D construction is to do all computations directly on the control points. Remember that the sheer plan lies in the xy-plane and the xcoordinate corresponds to the span width. In this plane the control points of our surfaces have (xi j , yi j )-coordinates and certain heights zi j (positive or negative). In the following constructions the y values of the control points will not be changed. Therefore the bending process can be described by the sketch 10 which shows a projection in y-direction. The sketch shows, how the original middle axis of the straight wing, i.e. the plane y = 0 containing the leading and trailing edge of the wing, is transformed into a curved middle axis for the wing with winglet. The adjustable parameters of this construction are the bending position x0 , the radius r and the angle w. Given these quantities, the center C and the points x0 (= x0 ), x1 , x2 , x1 and x2 are determined according to the specifications in the sketch. We want to keep the wing span given by c. Thus we first have to prolong the straight wing from c to c . Since the original representation of the straight wing as piecewise linear spline prohibits the generation of a curved structure, we replace it by a spline of order pusing a degree raising algorithm. Then we determine the spline parameters v0 < v1 < v2 corresponding to the points x0 , x1 and x2 . Starting from a we want to keep the middle axis straight up to as closely as possible to x0 . Hence we insert some knots (< v1 ) around v0 . For the same reason we add some knots (> v1 ) around v2 . This

Fig. 9 Side view of winglet construction – bending angle 60◦

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c’

C w/2 x’2 d c"

c

x2

r

x’1 x1 d

x0

a

Fig. 10 Principle of winglet construction

Fig. 11 Winglet with dihedral angle a = 10◦ and bending angle w = 60◦

gives us a couple of control points lying between a and c to provide the necessary flexibility for modeling the bending region. The control points on the left of x2 are mapped by a rotation around center x1 onto the line c x2 . The control points between x2 and x0 are mapped onto the circular arc connecting x2 and x0 according to their distances. The control points on the right of x0 remain unchanged. Altogether this describes a piecewise defined function x = (x, 0) → x = (x , z ). With n = (nx , nz ) we can denote the unit normal to the image curve at the point (x , z ). Actually we are not interested primarily in the deformation of the middle axis but rather in the transformation of the surface points. To define this we assume that the cross sections are rigid and remain perpendicular to the middle axis, so that finally any surface control point p = (x, y, z) is mapped to (x + znx , y, z + znz ). Figure 11 shows the final result with parameters a = 10◦ , x0 = 1194mm (the whole wing span is 1312mm with, and 1286mm without tip) and r = 25mm. An IGES file containing the configuration that was actually manufactured and used for the wind tunnel experiments can be downloaded from [5].

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3 Elliptic Grid Generation In this section, which can be considered as a follow-up to [9], we show how an elliptic grid generation system can be integrated into our B-spline based system using collocation methods.

3.1 Spekreijse’s Grid Generation System Among the various elliptic grid generation methods that are described in the literature we have chosen Spekreijse’s approach which can be very briefly summarized as follows. Let xˆ (s) be a harmonic mapping from the d-dimensional parameter space P onto the physical domain D and s(ξ ) be a so-called control mapping from the computational domain C onto the parameter domain P. Then the composite mapping (2) x(ξ ) = xˆ (s(ξ )) : C −→ D fulfills a differential equation of the form d

L(x) =

∑

gi j

i, j=1

d ∂ 2x ∂x + ∑ Pk = 0. ∂ ξi ξ j k=1 ∂ ξk

(3)

Here, denoting by J = det x (ξ ) the Jacobian of the composite mapping, we have d

Pk =

∑

J 2 gi j Pikj ,

(4)

i, j=1

where the gi j and gi j are the covariant and contravariant metric tensors defined by gi j =

∂x ∂x · , ∂ ξi ∂ ξ j

d

∑ gik gk j = δi j ,

(5)

∂ 2s , ∂ ξi ∂ ξ j

(6)

k=1

the Pikj are the components of the vector Pi j = −T −1

and T = s (ξ ) is the Jacobian matrix of the control mapping. This PDE has to be solved numerically, and in the present context it is a natural idea to use a B-spline collocation method to do so because this way the solution will be readily represented in the desired format.

3.2 B-Spline Collocation The general idea of collocation is to seek a function that satisfies the differential equation at certain points, the collocation points. In a way collocation is similar

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to interpolation, but in contrast to interpolation we do not match function values but certain combination of function and derivative values. In order to simplify the notation we concentrate on the bivariate case from now on and denote the Cartesian coordinates of the computational domain, the unit square, with ξ = (u, v) and of the parameter domain with s = (s,t). Hence, we search a function of the form (1) which fulfills Lx(uˆi , vˆ j ) = 0, i = 1, . . . , N − 1, j = 1, . . . , M − 1, (7) at certain collocation points uˆi , vˆ j , yet to be chosen. Moreover, Dirichlet boundary conditions are imposed on the control points p0, j , pN, j , j = 0, . . . , M, and pi,0 , pi,M , i = 0, . . . , N. Among various available collocation schemes, we prefer a scheme that works for splines with arbitrary knot sequences and uses the Greville abscissae defined by 1 i+p (8) uˇi = ∑ uk , p k=i+1 as collocation points. This choice is motivated by the Schoenberg-Whitney Theorem, see [13], which says that the interpolation problem x(uˆi ) = fi is well posed if, and only if, every uˆi lies in the support of the i − th B-spline function, i.e., if Ni (uˆi ) > 0. As one can easily verify, the Greville abscissae always give a set of as many distinct points, as the spline has control points and they fulfill the conditions of the Schoenberg-Whitney Theorem. The Schoenberg-Whitney Theorem is also the reason why the strategy to use a standard finite difference code followed by an interpolation algorithm in order to generate elliptic B-spline grids sometimes fails to produce good results. In fact, a typical finite difference code is based on the assumption that the grid points xi, j are numerical approximations of regularly spaced values x(ihu , jhv ). However, depending on the structure of the underlying spline it could become necessary to work with unevenly spaced grid points in order to fulfill the stipulations of the SchoenbergWhitney Theorem during the interpolation process.

3.3 Application Example The afore-mentioned collocation scheme has been implemented and tested for planar grids, surfaces and volume grids. In order to solve the PDE we just follow the standard approach and use a fixed point iteration, freezing the metric coefficients in Equation (3) in order to get a linear system in every single iteration. Then we apply the collocation scheme to the linearized equations. As a first application we present the grid in a block that is taken from a grid for a dual-bell configuration, see Figure 12. The boundaries are approximatively parameterized by arc length, so that we can use the identity mapping as control mapping. Hence, all control functions Pikj are zero and the resulting grid mapping is harmonic. However, the spacing of the control points displayed in the upper plot is rather irregular. This irregularity stems from an adaptive B-spline approximation algorithm which tries to resolve the different features of the nozzle contour and from the necessity to mutually insert the knots which are not present in the representation

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Fig. 12 Control points and harmonic mesh for the dual bell

of the opposite boundary in order to build a tensor product. However, the resulting numerical grid, which is computed by evaluation of the B-spline function can be seen to have the desired smoothness properties. However, as is well known, harmonic grid generation systems have the tendency to push away the grid lines from concave boundaries and the grids are in general not orthogonal at the boundaries. This is clearly demonstrated in Figure 13.

3.4 Boundary Orthogonality There remains the problem to determine in Spekreijse’s approach suitable control functions in order to incorporate desired features into the grids. In order to find a control mapping that ensures boundary orthogonality Spekreijse proposes to proceed as follows. The main idea is to compose a harmonic mapping with a control mapping that are both orthogonal at the boundary. The inverse of the harmonic mapping s(x) ˆ we search for fulfills mixed Dirichlet and Neumann boundary conditions, in particular ∂ s/ ˆ ∂ n = 0 at the boundaries x(u, 0) and x(u, 1) and ∂ tˆ/∂ n = 0 at the boundaries x(0, v) and x(1, v) of the physical domain. At this point let us assume that a folding-free grid x(ξ ) is already available. This grid may be, for instance, a transfinite interpolant or the solution of the purely harmonic grid generation system. We use this grid to discretize the above problem on a uniform mesh. In the first step we use this given grid to transform the Laplace equation into the computational domain. The components (s,t) of the composed mapping

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Fig. 13 Discretization with 40 × 40 control points

s(ξ ) = sˆ(x(ξ )) fulfill individually a partial differential equation, which for brevity we only present for the s-component: div(A grad s) = 0 where A=J

11 12

1 g22 −g12 g g = . g12 g22 J −g12 g11

(9)

(10)

The Neumann boundary condition ∂ s/∂ n = 0 transforms to (grad s, An) = 0

(11)

at the corresponding boundary in C . The solution of this problem gives us a oneto-one mapping of the boundaries ∂ C −→ ∂ P. In the second step we complete this boundary mapping to a suitable control mapping that fulfills the orthogonality conditions ∂ t/∂ u = 0 along the boundaries u = 0 and u = 1 and ∂ s/∂ v = 0 along the boundaries v = 0 and v = 1 using an algebraic grid generation method. For the details of this method we refer to [20]. Whereas the discretization of Equation (3) by collocation is straightforward it is in this case equally convenient to discretize Equation (9) with a finite volume method. For this we observe that for any control volume Ω in the computational domain the equation ∂Ω

(grad s, An) d σ = 0

(12)

holds. Of course, as control volumes we will choose rectangles of the form [ui , ui+1 ] × [v j , v j+1 ]. Again we want to represent the control mapping as tensor

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product B-spline. Therefore, the integral over the boundary of the control volume is composed of segments of the form ui+1 ui+1

su 0 (grad s, An) d σ = dσ ,A sv 1 ui ui ui+1 u2 g12 g11 du + N j (v) du = ∑ pi j − N j (v) Ni (u) Ni (u) J J ui u1 i, j and

v j+1

su 1 dσ ,A (grad s, An) d σ = 0 sv vj vj v j+1 v j+1 g12 g22 dv + N du . = ∑ pi j − Ni (u) N j (v) N j (v) i (u) J J vj vj i, j

v j+1

The integrals in the brackets enter the matrix of the discretized problem and can cheaply be evaluated by quadrature formulas. In order to obtain as many equations as control points we center the control volumes around the Greville abscissae by choosing 1 ui = (uˇi−1 + uˇ i ), i = 1, . . . , N − 1, 2 1 vi = (vˇ j−1 + vˇ j ), i = 1, . . . , N − 1, 2

u0 = 0,

uN = 1,

v0 = 0,

vM = 1.

(13)

Due to Equation (11), the discretization at boundary grid points is also straightforward. Figure 14 shows the control map and the smooth evaluation of the resulting orthogonal grid. It can very well be observed how the new boundary parameterization of the grid serves to enforce orthogonality along the boundary. However, since we still do not have control over the boundary spacing, the grid is rather distorted. This will be addresses in the next section.

3.5 Complete Boundary Control In order to achieve control of both the angles and the grid spacing near the boundary we need to look for control mappings that do not only have the right boundary parameterization and fulfill the orthogonality conditions but also take prescribed values ∂ t/∂ v along the boundaries u = 0 and u = 1 and ∂ s/∂ u along the boundaries v = 0 and v = 1. In principle, it is possible to construct such grids with algebraic methods, namely with so-called cubic Coons-Patches. However, algebraic methods easily tend to produce grid-folding. Hence we take up again the ideas of Spekreijse who uses solutions of the biharmonic equation to incorporate these additional boundary conditions into the control mapping. To set up this method, we assume that we have computed already a grid x(u, v) which is orthogonal at the boundaries and its corresponding control mapping s(u, v).

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Fig. 14 Grid, orthogonal near the boundaries, and its control mapping

In a parametric grid the spacing of the first meshline in physical space is, within second order accuracy, proportional to cross derivatives, i.e., to

∂x ∂x (u, v) and (u, v) ∂u ∂v

on the boundaries u = 0, u = 1 or v = 0, v = 1 respectively. In the given grid x(u, v), these quantities can be evaluated and compared with the desired values, lets say f (u, v). Since we assume that the control mapping the s(u, v) is already orthogonal at the boundary and because the harmonic transformation does not change if we do not change the boundary distribution in the parameter space, we can conclude that the norm of the above cross derivatives are proportional to the cross derivatives of the control mapping. That means, to achieve the desired spacing in the physical domain, the cross derivatives of the improved control mapping s˜ = (s, ˜ t˜) should take the values f (u, v) ∂ t˜ ∂t (u, v) = (u, v) ∂ x . (14) ∂v ∂v || (u, v)|| ∂v

∂ s˜ ∂s f (u, v) (u, v) = (u, v) ∂ x ∂u ∂u || ∂ u (u, v)||

(15)

Hence, our new control mapping for complete boundary control is found by solving the biharmonic equation Δ Δ s˜ = 0 (16) subject to the Dirichlet boundary conditions s˜(u, v) = s(u, v)

(17)

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and the Neumann conditions (14) and (15). In addition, as before, ∂ t˜/∂ u = 0 has to hold along the boundaries u = 0 and u = 1 and ∂ s/ ˜ ∂ v = 0 along the boundaries v = 0 and v = 1. In this case, instead of employing a B-spline collocation method to solve the biharmonic equation, we apply a finite-difference algorithm, see [2]. The result is subsequently interpolated by a B-spline mapping. In this case there are no problems with irregular knot-sequences, because the control mapping does not enter the construction of the physical grid, so that we can just use uniform discretizations in the parameter space. Figure 15 shows the grid for the above test configuration which is generated in the way just described with control of the first spacing. In particular, notice the excellent smoothness of the control mapping compared to Figure 14.

Fig. 15 Final grid with complete boundary control

Generally, from our practical experiences we conclude that the collocation scheme proposed above offers a stable and reliable way to realize elliptic grid generation methods directly in terms of B-spline representations. Due to the higher approximation order of splines compared with finite differences, it might even be computationally more efficient than the standard discretization method.

4 Deforming Grids In general, when solving unsteady problems where the mesh has to conform to the instantaneous shape of a deforming body, the grid has to be updated in every time step. Therefore the grid deformation process should be cheap, ideally taking only a small fraction of the overall CPU time required by the flow solver. In practice it turns out that multi-block grids offer several advantages over unstructured methods when it comes to grid deformation. As long as the deformations

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are moderate, it is usually possible to keep the topology and the connectivity of the grid unchanged, whereas moving unstructured grids often require some kind of remeshing. Another point is that transfinite interpolation can be used to compute the displacement of the interior grid points in a block or face from previously computed displacements of the grid points on the boundaries. Therefore the main problem is the computation of the displacement of the vertices and the edges of the blocks, i.e., of the so-called framework. For this task more sophisticated and expensive methods have to be used. Fortunately this is feasible since the amount of points belonging to the framework is negligible compared to the overall number of grid points.

4.1 Deforming the Framework Our strategy for deforming the framework is based on interpolation with radial basis functions. This method is rather general and can be applied to arbitrary types of grids, both structured and unstructured. The main idea is to reduce the deformation problem to a scattered data interpolation problem. To this end, suppose that on the configuration surface and possibly on the far field boundaries the displacements dk of some specifically chosen points xk , k = 1, . . . , N, are prescribed. The objective is now to find a smooth function d(x) which assigns a displacement to each point in the physical space and interpolates the given data: d(xk ) = dk for all k. Once this problem has been solved, the function d(x) is used to determine the displacements of the grid elements constituting the multiblock framework, for the technical details see [17]. Typically the data sites xk are the vertices of the multiblock grid which lie on the configuration surface plus possibly a small number of additional specific points, for instance, the leading and trailing edge of a profile. Spekreijse et. al. [22] estimate that even for a complex fighter configuration no more than 100 data points are necessary to make the method described in the current section work. However, the data sites xk bear no regular structure. A standard numerical method to address such a scattered data interpolation problem is interpolation by radial basis functions (RBFs). The general ansatz is to interpolate the given data by a function of the form N

d(x) =

∑ ak Φ (x − xk) + p(x)

(18)

k=1

where p is a polynomial of a low, fixed maximum order ord(p) ≤ m and the function Φ actually depends only on the distance ||x − xk ||. That means there exists a univariate function φ : R+ −→ R such that Φ (x − y) = φ (||x − y||). The data sites xk are usually called centers. The coefficients ak and the polynomial p are determined by the interpolation conditions d(xk ) = dk , and the additional requirement

k = 1, . . . , N,

(19)

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N

∑ ak q(xk ) = 0

(20)

k=1

for all polynomials q of order ord(q) ≤ m. In this general ansatz we now have to choose a concrete radial basis function. One popular choice is to take the fundamental solutions of the biharmonic equation. In 2D the fundamental solution is the so-called thin plate spline

Φ (||x − y||) =

−1 ||x − y||2 log(||x − y||). 8π

(21)

In 3D the fundamental solution of the biharmonic equation is essentially just the distance function itself: ||x − y|| Φ (||x − y||) = . (22) 8π The theory of interpolation with radial basis functions, which can be found for example in [23], requires that the order of q must be at least m = 1, and indeed this minimal choice is the only one considered in [16] and [22]. However, we have a slight preference for the choice m = 2. This is motivated by the following lemma. Lemma 1. If m ≥ 2 then d(x) reproduces each affine transformation exactly, i.e., if dk = Rxk + v with R ∈ Rd×d and v ∈ Rd then d(x) = Rx + v for all x ∈ Rd . A proof can be found in [17]. Reproduction of affine functions has the useful implication, that if the configuration performs a rigid body transformation, a rotation or translation, this movement is exactly carried over to the whole grid.

4.2 Example: High Lift Configuration As a practical example we apply this technique to the high lift configuration. The construction of the blocking shown in Figure 16 follows the strategy outlined in [12]. There are offset areas around the single elements, and we have constructed

Fig. 16 Sample blocking for the three-element configuration

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a Cartesian bounding box around the configuration such that a Cartesian far field could easily be added. The slat and the flap can be rotated around the centers that are indicated in Figure 17.

CF

CS

Fig. 17 Three element high lift configuration. The dots mark the rotation centers for the slat and flap riggings

The offset areas are considered to be rigidly connected to their supporting element. On each offset curve 20 support points are chosen for the volume spline method. Furthermore on each edge of the bounding box 10 support points are defined. The bounding box, of course, is not allowed to move during the deformation. The edges of the block structure between the offset curves and the bounding box are moved according to the radial basis interpolation function. An overall view of a deformed block is given in Figure 18. Figure 19 shows scaled up sections of this plot near the flap and the slat. It turns out, that the most problematic part of the configuration is the area between the slat and the main element. Here highly distorted grids may arise if one reduces the slat

Fig. 18 Deformed configuration. Here the slat has been rotated 8◦ counter-clockwise and the flap 20◦ clockwise

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Fig. 19 Detailed viewed at the slat and flap. First row: grid around slat in initial configuration (left) and rotated 20◦ clock-wise. Second row: grid around flap in starting configuration (left and rotated counter-clockwise by 6◦ . Third row: same grid rotated 12 degrees counterclockwise with vertices rigidly fixed to the offset curves (left) and vertices allowed to move on the offset curve (right)

gap, i.e., if one rotates the slat clock-wise. The problem is that the angles between the offset curves and the edge connecting the offset curve around the main element with the offset area behind the slat become very large. The situation can be improved, if one allows the right hand vertex of this edge to slide freely on the offset curve.

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This possibility is a special feature of the grid generation technique presented here. With discrete grids a similar strategy is in general not available because the number of grid points in each block is determined by the position of such a supported vertex, and therefore reattaching a boundary curve to another vertex would at least require a complete remeshing of the affected blocks or even violate global combinatorial constraints. However, to use this feature in a nonstationary calculation, it would be necessary to have a flow solver, that could handle changes in the cell connectivity during one time step. Therefore at the moment, QUADFLOW can use this feature only for steady state parameter studies, if for example one wants to examine the effects of varying flap and slat riggings.

5 Conclusion In this paper we have considered several aspects of the B-spline based parametric grid generation system developed in project B2. Compared with classical blockstructured grid generation such a system offers more flexibility and better integration to CAD-software. However it is well known that the main disadvantage of blockstructured grid generation is the fact, that there are hardly any automatic methods to generate the block-decompositions. While some steps in this direction has been taken this problem has not really been addressed in a systematic way in the current work. It would therefore be one important objective of further research. Another direction would be to combine parametric representations with unstructured grid technology or with subdivision schemes as known from CAGD. In particular, the latter approach seems to be promising because of its close connections with both geometry and adaptivity.

References 1. Ballmann, J. (ed.): Flow Modulation and Fluid-Structure-Interaction at Airplane Wings. Numerical Notes on Fluid Mechanics, vol. 84. Springer, Heidelberg (2003) 2. Bjorstad, P.E.: Numerical Solution of the Biharmonic Equation. Ph.D. dissertation, Stanford University (1980) 3. Brakhage, K.-H.: Ein menugesteuertes, intelligentes System zur zwei- und dreidimensionalen Computergeometrie. VDI Reihe 20 CAD/CAM, Nr. 26 Edition. VDI Verlag (1990) 4. Brakhage, K.-H.: Wincag-education software for geometry. In: 11th International Conference on Engineering Computer Graphics and Descriptive Geometry, Guangzhou, China (2004) 5. Brakhage, K.-H.: http://www.igpm.rwth-aachen.de/brakhage/SFBmodel 6. Brakhage, K.-H., M¨uller, S.: Algebraic–hyperbolic grid generation with precise control of intersection of angles. J. Num. Meth. in Fluids 33(1), 89–123 (2000) 7. Brakhage, K.-H., Lamby, P.: Generating airplane wings for numerical simulation and manufacturing. In: Soni, B., H¨auser, J., Eiseman, P. (eds.) Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, California, USA (2005)

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8. Brakhage, K.-H., Lamby, P.: Modeling of Airplane Wings with Winglets. In: Soni, B., H¨auser, J., Eiseman, P. (eds.) Proceedings of the 10th International Conference on Numerical Grid Generation in Computational Field Simulations, FORTH, Crete, Greece (2007) 9. Lamby, P., Brakhage, K.-H.: Elliptic Grid Generation by B-Spline Collocation. In: Soni, B., H¨auser, J., Eiseman, P. (eds.) Proceedings of the 10th International Conference on Numerical Grid Generation in Computational Field Simulations, FORTH, Crete, Greece (2007) 10. Brakhage, K.-H., Lamby, P.: Application of B-Spline Techniques to the Modeling of Airplane Wings and Numerical Grid Generation. Computer Aided Geometric Design 25, 738–750 (2008) 11. Bramkamp, F., Lamby, P., M¨uller, S.: An adaptive multiscale finite volume solver for unsteady and steady state flow computations. Journal of Computational Physics 197, 460–490 (2004) 12. Bramkamp, F., Gottschlich-M¨uller, B., Hesse, M., Lamby, P., M¨uller, S., Ballmann, J., Brakhage, K.-H., Dahmen, W.: H-adaptive Multiscale Schemes for the Compressible Navier–Stokes Equations — Polyhedral Discretization, Data Compression and Mesh Generation. In: Ballmann, J. (ed.) Flow Modulation and Fluid-Structure-Interaction at Airplane Wings. Numerical Notes on Fluid Mechanics, vol. 84, pp. 125–204. Springer, Heidelberg (2003) 13. de Boor, C.: A Practical Guide To Splines. Springer, Heidelberg (1978) 14. Farin, G., Rein, G., Sapidis, N., Worsey, A.: Fairing cubic B-spline curves. Computer Aided Geometric Design 4, 91–103 (1987) 15. Hoschek, J., Lasser, D.: Grundlagen der geometrischen Datenverarbeitung, 2nd edn. Teubner Verlag, Stuttgart (1992) 16. Hounjet, M.H.L., Meijer, J.J.: Evaluation of elastomechanical and aerodynamic data transfer methods for non-planar configurations in computational aeroelastic analysis. In: Proceedings International Forum on Aeroelasticity and Structural Dynamics, Manchester, UK (1995) 17. Lamby, P.: Parametric Grid Generation and Application to Adaptive Flow Simulations. PhD thesis, RWTH Aachen University (2007) 18. Moir, I.: Measurements on a two-dimensional aerofoil with high lift devices. AGARDAR-303, vol. I+II, DRA, Farnborough (1994) 19. Runborg, O.: A Multiblock Mesh Manager. In: Computational Mesh Adaptation. Numerical Notes on Fluid Mechanics, vol. 69, European Commision, Directorate General XII, Science, Research and Development, Vieweg (1999) 20. Spekreijse, S.: Elliptic Grid Generation Based on Laplace-Equations and Algebraic Transformations. Journal of Computational Physics 118, 38–61 (1995) 21. Spekreijse, S., Boerstoel, J.W.: Multiblock Grid Generation. Part 2: Multiblock Aspects. In: Deconinck, H. (ed.) Computational Fluid Dynamics. VKI Lecture Series 1996-06, pp. 1–39. von Karman Institute for Fluid Dynamics (1996) 22. Spekreijse, S., Prananta, B., Kok, J.: A simple, robust and fast algorithm to compute deformations of multi-block structured grids. Tech. Rep. NLR-TP-2002-105, National Aerospace Laboratory NLR (2002) 23. Wendland, H.: Scattered Data Approximation. In: Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005)

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Parallel and Adaptive Methods for Fluid-Structure-Interactions Josef Ballmann, Marek Behr, Kolja Brix, Wolfgang Dahmen, Christoph Hohn, Ralf Massjung, Sorana Melian, Siegfried M¨uller, and Gero Schieffer

Abstract. The new flow solver Quadflow, developed within the SFB 401, has been designed for investigating flows around airfoils and simulating the interaction of the structural dynamics and aerodynamics. This article addresses the following issues arising in this context. After identifying proper coupling conditions and settling the well-posedness of the resulting coupled fluid-structure problem, suitable strategies for successively applying flow and structure solvers needed to be developed that give rise to a sufficiently close coupling of both media. Based on these findings the overall efficiency of numerical simulations hinges, for the current choice of structural models, on the efficiency of the flow solver. In addition to the multiscale-based grid adaptation concepts, proper parallelization concepts are needed to realize for such complex problems an acceptable computational performance on parallel architectures. Since the parallelization of dynamically varying adaptive discretizations is by far not straightforward we will mainly concentrate on this issue in connection with the above mentioned multiscale adaptivity concepts. In particular, we outline the way the multiscale library has been parallelized via MPI for distributed memory architectures. To ensure a proper scaling of the computational performance with respect to CPU time and memory, the load of data has to be well-balanced and Kolja Brix · Wolfgang Dahmen · Sorana Melian · Siegfried M¨uller Institut f¨ur Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen e-mail: {brix,dahmen,melian,mueller}@igpm.rwth-aachen.de Marek Behr · Gero Schieffer Lehrstuhl f¨ur Computergest¨utzte Analyse Technischer Systeme, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen e-mail: {behr,schieffer}@cats.rwth-aachen.de Josef Ballmann · Christoph Hohn Lehr- und Forschungsgebiet f¨ur Mechanik, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen e-mail: {ballmann,hohn}@lufmech.rwth-aachen.de

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 265–294. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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communication between processors has to be minimized. We point out how to meet these requirements by employing the concept of space-filling curves.

1 Introduction In this paper we present an overview of the work conducted in our research group concerning the numerical simulation for fluid-structure interaction problems in the context of aeroelasticity. There have been two major focus points, namely (i) the coupling of fluid and structure solvers, and (ii) parallelization of the fluid solver. As for (i), a typical approach to solving the coupled fluid-structure problem is the successive application of highly developed solvers for each separate task coupled through appropriate interface conditions. We shall first briefly recall the essence of our findings from an early stage of the research program concerning some foundational issues of coupling mutual fluid and structure response. In particular, this covers well posedness of a coupled problem formulation based on proper coupling conditions as well as the development of discretization concepts that ensure a correct energy balance at the interface. Needless to stress that the latter issue is essential for a time accurate integration of the inherently nonstationary processes. Once suitable coupling strategies for flow and structure solvers had been identified the focus of the work shifted towards improving the performance of the individual solvers. Here the primary demands arise on the fluid side which brings us to topic (ii). In order to be able to meet the requirements of highly accurate nonstationary simulations based on the full Navier Stokes equations as a fluid model a new fully adaptive solver Quadflow has been developed [6, 7]. More specifically, it has been designed to handle (i) unstructured grids composed of polygonal(2D)/polyhedral(3D) elements [5] and (ii) block-structured grids where in each block the grid is determined by local evaluation of B-Spline mappings [17]. While the solver can handle also grids provided by an external grid generator grid adaptation has been implemented only for block-structured grids where in each block the grid is locally refined using the concept of multiscale-based grid adaptation [19]. In order to treat both settings, two different data structures were developed for the flow solver and the grid adaptation, respectively. The time evolution is performed on one unsorted list of all cell averages not distinguishing between data that might correspond to different blocks. On the other hand, grid adaptation is carried out for each block separately sweeping through the different refinement levels. It turned out that hash maps are well suited for this purpose rather than tree structures. Therefore, in each adaptation step the data have to be transferred back and forth between unsorted lists and hash maps. Although multiscale-based grid adaptation leads to a significant reduction of the computational complexity (CPU time and memory) in comparison to computing on uniform meshes, this by itself is ultimately not sufficient to warrant an acceptable efficiency when dealing with realistic 3D computations for complex geometries. In addition, parallelization techniques are indispensible for further reducing the computational time to an affordable order of magnitude. In a first step, the unstructured finite volume solver was parallelized via MPI [12, 13] for distributed memory

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architectures. Here load-balancing was performed by graph partitioning techniques using the Metis software [16, 15] together with PETSc [2, 3, 4]. This parallelized flow solver was used together with the concept of adaptive, block-structured grids. However, in this case all data had to be transferred to one processor. Since this ruins the overall performance, we have parallelized the multiscale library, which realizes local grid refinement in each block by means of the multiscale-based grid adaptation concept. The performance of a parallelized code crucially depends on load-balancing and minimal interprocessor communication. Since due to hanging nodes, the underlying adaptive grids are unstructured this task is by no means trivial. In contrast to the flow solver, instead of employing graph partitioning methods, we use space-filling curves [25]. Here the basic idea is to map level-dependent multiindices identifying the cells in the grid hierarchy to a one dimensional line. The interval is then split into different parts each containing the same number of entries. For this mapping procedure we employ the same cell identifiers as in case of the hash maps. After briefly recalling in Section 2 the results concerning fluid-structure coupling, the main objective of the present work is to present the subsequent developments concerning the parallelization of multiscale-based grid adaptation. For this purpose, we first summarize in Section 3 the basic ingredients of the multiscale library: (i) the multiscale analysis of the discrete cell averages and grid adaptation, (ii) algorithms and (iii) data structures. The key issues of parallelization are load-balancing and interprocessor communication. These issues are addressed in Section 4. An optimal balancing of the computational work load can be realized using the concept of space-filling curves. Mainly in connection with local multiscale transformations we discuss the data transfer at processor boundaries. Finally in Section 6, we present some performance studies with the parallelized version of the multiscale-based grid adaptation and show first adaptive, parallel 3D computations of a Lamb-Oseen vortex.

2 Fluid-Structure Coupling This section is devoted to discretization issues concerning the numerical solution of fluid-structure interaction problems. Numerical solution methodologies for solving such problems are typically based on employing an already available fluid solver and a given structural solver. The task is then to incorporate them into a fluid-structure solver or, referring to our application background, into an aeroelastic solver. Similar to constructing efficient and reliable fluid solvers and structural solvers, by making those solvers obey certain discretization principles, one faces the question, how to configure a good fluid-structure solver from the given individual fluid and structure solvers, or expressed shortly, how to realize a good fluid-structure coupling. For this purpose we shall explain first the setting in which this question has been analyzed. Consider first the following physical model. The aeroelastic system consisting of an aircraft wing in transonic flow can develop the nonlinear vibration phenomenon of limit cycles which exhibit constant amplitude structural vibrations. It is important

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to know the parameter ranges, in which these unstable phenomena occur. The limit cycles are self-excited oscillations caused by the aerodynamic coupling of structural modes in linear theory. Among other effects, shock movements in the flow play a significant role in the destabilizing mechanisms which finally activate nonlinearities. These features can also be found in the panel flutter problem [9], which serves as our model problem. The panel flutter problem analyzed in this work consists of a plate (panel) over which a compressible fluid (air) flows at transonic speed. We assume that the panel is infinitely long in the spanwise direction (z–coordinate), so that a 2D flow in the x-y plane passes over a strip of the panel, compare Figure 1. The flow is modeled by the 2D Euler equations of gas dynamics and the structure by a strip of a von–Karman plate. The panel is supported at its ends in fixed hinges. It is placed on the x–axis between solid walls as drawn in Figure 1. Numerical results visualized in [18] show structural deflections and shock movements present in typical limit cycle oscillations for the panel flutter problem. Second, let us make more precise, what the fluid-structure coupling is about. Typically the fluid-structure solver is based on the following 5 ingredients. • • • • •

Fluid solver Structural solver Geometrical transfer (at nonconforming grid interface) Load transfer (at nonconforming grid interface) Coupling Scheme = rule to process above 4 steps in time

Here we assume that fluid and structural solvers are given. In the fluid, which is compressible in our case, we have chosen a Finite Volume MUSCL approach and for the structure we have chosen the Finite Element Method, which are currently the most popular methods in use in the respective disciplines. What remains, is to specify geometrical transfer, load transfer and coupling scheme. Together all 5 items define the discrete fluid-structure system with its intrinsic properties. In particular, the grid interface of fluid and structure are typically nonconforming, i.e. the structural nodes and the fluid nodes do not match, and the deflected structure and the deforming fluid grid do not coincide as spatial objects at the interface. As an illustration of this point see Figure 2, where the situation is shown for the panel flutter problem. Here, at least each fluid node can be fixed to a material point of the structure, a property we have employed in our discretization, and which can already be interpreted as the geometrical transfer. Note that the nonconformity is much more involved, when considering a flow around an aircraft wing, having in mind the way in which aircraft wings are modeled, and that deformations in 3D can be more complex. The load transfer determines how the load distribution for the structure is determined from the discrete fluid quantities. Finally as an example for a coupling scheme we show the simplest choice, namely the loose coupling illustrated in Figure 3. Here the fluid state, respectively the structural deflection, on time level n, are denoted by U n , respectively wn . One fluid-structure time step is performed by first applying the load transfer (Step 1), then advancing the structural solution in time (Step 2), then applying the geometrical transfer (Step 3), which means to deform the fluid grid according to the structural deformation, and finally by advancing the fluid in

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time (Step 4) on the grid that moves during the time step from its position at the beginning to that at the end of the time step. The property which we impose on the coupling is that the discrete fluid-structure system satisfies the same energy balance as the continuous system, see (6) below. Obviously this is an important property. A consistent energy transfer between fluid and structure is essential, in particular, when the objective is to accurately determine where unstable behaviour occurs in parameter space, or more generally, when determining bifurcations of the system. Using a Finite Volume Method in the fluid and a Finite Element Method in the structure, we have constructed in [18] a geometrical transfer and a load transfer such that the discrete fluid-structure system satisfies the same energy balance as the continuous system. In fact we have given a discretization of the equations (1)–(5), resulting in a discrete fluid-structure system, implicit in time, such that for each time step a coupled set of fluid-structure equations has to be solved. This is done approximately by iterating the loose coupling within each time step until a convergence criterion is met. In order to give an idea about the energy argument, we restrict ourselves to a description of the continuous model and its energy balance for the time-dependent interaction of a compressible inviscid and non-heatconducting fluid and a linear elastic plate with geometrical nonlinearity according to von-Karman for a plate with fixed ends in two space dimensions.

x2

flow ρ∞ ,u∞ ,p∞

p

h

? ? ?

x1

p∞ l Fig. 1 Geometry of the 2D panel flutter problem.

The fluid equations are ⎛

⎞ 0 ⎜ −p n1 ⎟ d ⎜ ⎟ U dx1 dx2 + U (v − x˙ )T n ds = ⎝ −p n2 ⎠ ds dt Ω (t) ∂ Ω (t) ∂ Ω (t) −p vT n

(1)

for arbitraryly moving domains Ω (t). The conservative state vector U = (ρ , ρ u, ρ v, p T 1 2 T 2 ρ |v| + γ −1 ) involves the density ρ , the velocity vector v = (u, v) and the presT sure p. On a point of the boundary ∂ Ω (t), n = (n1 , n2 ) is the outward unit normal of Ω (t) and x˙ is the velocity of a boundary point x(t).

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x2

→

U0

↓

fluid node

Step 1

w(xk ) structural deflection structural nodes

Step 4

U1

U2

↑

Step 3

→

Step 2 w0 w˙ 0

x1

w1 w˙ 1

w2 w˙ 2

Fig. 3 Loose coupling

Fig. 2 Representation of the interface in fluid and structure

For the formulation of the plate equation we introduce the space V := H 2 (0, l) ∩ find the plate deflection w(t, x) satisfying

H01 (0, l) . The variational formulation is to

D (wxx , ϕxx ) + N(w) · (wx , ϕx ) + m (w, ¨ ϕ ) = (p2 − p1 , ϕ )

(2)

for all test functions ϕ ∈ V , where the nonlinear term models a restoring force due to the mid-surface stretching with N(w) =

Eh 2l

l 0

w2x dx .

The constants in the panel equation are the stiffness D = Eh3 /12(1 − ν 2) , the mass per unit area m = ρs h , the panel thickness h and length l, the density ρs , Young’s modulus E and the Poisson ratio ν . Now the aeroelastic problem can be formulated. For convenience, all interface and boundary conditions are given in the strong sense. • The fluid domain at time t is defined as the space above the deflecting panel and solid walls and is denoted by ΩF (t), i.e. for each fluid interface point x(t) associated with a panel point ξ ∈ [0, l] , x(0) = (ξ , 0)T ,

x(t) = (ξ , w(t, ξ ))T .

(3)

• For all moving domains Ω (t) ⊆ ΩF (t) (1) holds. • (2) holds with the r.h.s. determined from the fluid pressure on top of the panel and p∞ on the bottom, p1 (t, ξ ) ≡ p(t, ξ , w(t, ξ )), p 2 ≡ p∞ .

ξ ∈ [0, l],

• At infinity we have ρ = ρ∞ , u = u∞ , v = 0 , p = p∞ .

(4)

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• On the remaining boundary of ΩF (t) the fluid velocity in normal direction n equals the velocity of the boundary in direction n, in particular on the fluid– structure interface, vT n = x˙ T n = w˙ n2 .

(5)

The energy equations for fluid and structure are given by

p d 1 2 ρv + −p vT n ds , EF + (v − x˙ )T n ds = dt γ −1 ∂ Ω F (t) 2 ∂ Ω F (t)

l d ES = (p2 − p1 ) w˙ dx , dt 0 where we have denoted the energy in fluid and structure by

EF (t) ≡

Ω F (t)

p 1 2 ρv + dx , 2 γ −1

2 l Eh 1 l 2 m 2 D 2 w˙ + wxx dx + ES (t) ≡ wx dx . 0

2

2

2l

2

0

The energy equation of the fluid is the fourth component of (1) and the energy equation of the structure is derived by plugging ϕ = w˙ into (2). Summation of the two energy equations yields the energy equation of the aeroelastic system. Here we restrict the fluid solution to a finite domain ΩF (t) with non– moving inflow and outflow boundary. Due to the boundary and coupling conditions (3), (4), (5), we end up with EF (tn+1 ) + ES (tn+1 ) − EF (tn ) − ES (tn ) =

tn+1 l pγ 1 2 T ρv + p∞ w˙ dx dt . − = v n ds + γ −1 0 tn ∂ Ω in/outflow 2

(6)

To illustrate the influence of the coupling on the accuracy when predicting bifurcations in the transonic regime, we show an example taken from [18], where further details are given. Increasing the dynamic pressure at a fixed Mach number of M∞ = 0.95 we consider the panel flutter problem for an aluminium panel for a nondimensional dynamic inflow pressure λ in the range 2000 ≤ λ ≤ 4000. A bifurcation in the system behaviour occurs in that range and we run calculations with several coupling schemes at various (fluid)-CFL numbers, to compare how the different schemes manage to track that bifurcation. The bifurcation points obtained are shown in Figure 4 and reveal a strong variation depending on the coupling scheme, when using moderate CFL numbers. Here a CFL number of 50 corresponds to 120 time-steps per limit cycle oscillation and shocks in the fluid moving half the grid size during a time-step. The dotted line shows the λ -value for which each coupling scheme predicts the bifurcation as Δ t → 0. The method denoted as FPI, solves in

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each time step the discrete system that satisfies (6), by a fixed-point iteration, which iterates the loose coupling until a convergence criterion is met. On average this happens after two iterations. In contrast the method denoted by LOOSE performs only one such iteration, i.e. it is the loose coupling. The method PRED performs a loose coupling but uses a prediction of the fluid load data from previous time steps, in order to obtain a better approximation of the equations that satisfy (6). The details of all methods are given in [18]. In summary, one can conclude that the validity of (6) was identified as an important design criterion for the fluid-structure coupling, and that it can be achieved in conjunction with standard solution methods available for fluids and structures.

Fig. 4 Determination of bifurcation point with various time step sizes

3 Multiscale Decompositions–Basic Ingredients In this section we summarise the ingredients to successively decompose a sequence of cell averages given on a uniform fine grid (reference grid) into a sequence of coarse-scale averages and details. The details describe the update between two discretisations on successive resolution levels corresponding to a nested grid hierarchy. They reveal insight in the local regularity of the underlying function. In particular, whenever the details become negligible small in certain locations this gives rise to data compression. From the remaining significant details, an adaptive grid, i.e., a locally refined grid with hanging nodes can be determined. In principle, the concept can be applied to any hierarchy of nested grids, no matter whether these grids are structured or unstructured. However, here we will confine ourselves to structured

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grids and uniform dyadic refinement,s where on each refinement level the grids can be determined by evaluating a grid mapping. This has been successfully realised in the multiscale library, see [19] for details, and incorporated into the Quadflow solver [6, 7].

3.1 Multiscale Analysis and Grid Adaptation Grid Mapping. The starting point is a smooth function x : R := [0, 1]d → Ω , which maps the parameter domain R onto the computational domain Ω (which is typically one of several blocks in a composite grid). The Jacobian is assumed to be regular, i.e., det (∂ x(ξ )/∂ ξ ) = 0, ξ ∈ R. In our applications we represent the grid function by B–splines, see [17]. This admits control of good local grid properties, e.g., orthogonality and smoothness of the grid, and a consistent boundary representation by a small number of control points depending on the configuration at hand. Nested Grid Hierarchy. A nested grid hierarchy is defined from the grid mapping by means of a sequence of nested uniform partitions of the parameter domain. To this end, we introduce the sets of multi–indices Il := ∏di=1 {0, . . . , Nl,i − 1} ⊂ Nd0 , l = 0, . . . , L, with Nl,i = 2 Nl−1,i initialised by some N0,i . Here l represents the refinement level where the coarsest partition is indicated by 0 and the finest by L. The product denotes the Cartesian product, i.e., ∏di=1 Ai := A1 × · · · × Ad . Then the nested sequence of parameter partitions Rl := {Rl,k }k∈Il , l = 0, . . . , L, is given by Rl,k := ∏di=1 [ki hl,i , (ki + 1) hl,i ], with hl,i := 1/Nl,i = hl−1,i /2, see Figure 5. Finally, a sequence of nested grids Gl := {Vl,k }k∈Il , l = 0, . . . , L, of the computational domain Ω is obtained by Vl,k := x(Rl,k ), see Figure 6 for an illustration. Each grid Gl

Fig. 5 Dyadic grid hierarchy in parameter space

Fig. 6 Transformation from parameter to computational domain

builds a partition of Ω , i.e., Ω = k∈Il Vl,k , and the cells of two neighbouring levels are nested, i.e., Vl,k = r∈M 0 Vl+1,r , k ∈ Il . Because of the dyadic refinement, the l,k

0 = {2k+ i ; i ∈ E := {0, 1}d } ⊂ I d refinement set is determined by Ml,k l+1 of 2 cells on level l + 1 resulting from the subdivision of the cell Vl,k .

Multiscale Decomposition. For any scalar, integrable function u ∈ L1 (Ω , R) we define the average ul,k := u, φl,k L2 (Ω ) as the inner product of u with the L1 – + normalised box function φl,k (x) := |Vl,k |−1 χ (x), x ∈ Ω , where |Vl,k | := Vl,k 1 dx denotes the cell volume and χ

Vl,k

Vl,k

the characteristic function on Vl,k . With each grid

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Gl we can associate then the sequence of averages uˆ l := {uˆl,k }k∈Il . The nestedness of the grids as well as the linearity of the integration functional imply the two–scale relation |Vl+1,r | l,0 uˆl,k = ∑ ml,0 (7) r,k uˆl+1,r , mr,k := |V | , l,k 0 r∈Ml,k

i.e., the coarse–grid average can be represented by a linear combination of the corresponding fine–grid averages. Consequently, starting on the finest level, the flow field represented by the corresponding array of cell averages can be successively downsampled by computing averages on coarser levels. Since information is, of course, lost by the averaging process, it is not possible to reverse (7). Therefore, we have to store the update between two successive refinement levels by additional coefficients, so-called details representing the fluctuation between two successive refinement levels. From the nestedness of the grid hierarchy we infer that the linear spaces Sl := span{φl,k ; k ∈ Il } are nested, i.e., Sl ⊂ Sl+1 . Hence there exist complement spaces Wl such that Sl+1 = Sl ⊕ Wl . These are spanned by some basis, i.e., Wl := span{ψl,k,e ; k ∈ Il , e ∈ E ∗ := E\{0}} whose elements are oscilatory. For the construction of an appropriate wavelet basis we refer to [19]. In analogy to the cell averages, the details can be encoded by inner products dl,k,e := u, ψl,k,e L2 (Ω ) of the function u now with the wavelet ψl,k,e . The rationale is that for us an ”appropriate” basis means that it is a Riesz basis in L2 say, and therefore has a companion Riesz basis consisting of elements ψ˜ l,k,e which form a biorthogonal system, i.e. ψl,k,e , ψ˜ l ,k ,e L2 (Ω ) = δ(l,k,e),(l ,k ,e ) . Therefore, the details are just the expansion coefficients of u with respect to the dual companion basis. Moreover, the choice of the companion basis determines, in particular, the order of vanishing moments of the ψl,k,e which, in turn, determine how small the details are when u is smooth on the respective wavelet support. In fact, a local Taylor argument readily shows that vanishing moments of order m imply that |dl,k,e | is of the order 2−ml when u has bounded derivatives of order m on the corresponding wavelet support. Since the box functions and the wavelets are linearly independent, there exists a two–scale relation for the details, i.e., dl,k,e =

∑ e

r∈Ml,k ⊂Il+1

ml,e r,k uˆl+1,r .

(8)

On the other hand, we deduce from the change of basis the existence of an inverse two–scale relation uˆl+1,k =

∑0

r∈Gl,k ⊂Il

gl,0 r,k uˆl,r +

∑∗ ∑e

e∈E r∈Gl,k ⊂Il

gl,e r,k dl,r,e .

(9)

l,e Note that the mask coefficients ml,e r,k and gr,k in (7), (8) and (9) do not depend on the data but only on the bases and their underlying geometric information.

Parallel and Adaptive Methods for Fluid-Structure-Interactions

Fig. 7 Pyramid scheme of multiscale transformation

275

Fig. 8 Locally refined grid

Grid Adaptation. The multiscale analysis outlined above allows us to generate a locally refined grid in the following way. First we perform the multiscale decomposition according to (7), (8), as illustrated in Figure 7. As indicated above, the details become small where the underlying function u is locally smooth. The basic idea is therefore to perform data compression on the vector of details using hard thresholding. This means we discard all detail coefficients dl,k,e whose absolute values fall below a level-dependent threshold value εl = 2(l−L)d ε and only the significant details identified by the index set DL,ε := (l, k) ; |dl,k,e | > εl , k ∈ Il , e ∈ E ∗ , l ∈ {0, . . . , L − 1} are retained. In order to account for the dynamics of a flow field, due to the time evolution, and to appropriately resolve all physical effects on the new time level, this set is to be inflated in such a way that the prediction set D˜ L,ε ⊃ DL,ε contains all significant details of the old and the new time level. Here one exploits roughly speaking the finite speed of information propagation in a (dominantly) hyperbolic problem. In a last step, we construct the locally refined grid, see Figure 8, and the corresponding cell averages. For this purpose, we proceed levelwise from coarse to fine, see Figure 7, and we check for all cells of a given level whether there exists a significant detail. Whenever we find one, we refine the respective cell, i.e., we replace the average of this cell by the averages of its children by locally applying the inverse multiscale transformation (9). The final grid is then determined by the index set G˜L,ε ⊂ Ll=0 {l} × Il such that (l,k)∈G˜L,ε Vl,k = Ω . Note that a locally refined grid always corresponds to a tree that represents the refinement history. The index sets resulting from the thresholding do not necessarily form a tree yet. Therefore we have to inflate the prediction set somewhat so as to form even a graded tree. This means that there is at most one hanging node at a cell edge which is not strictly necessary but greatly simplifies data management, see [19].

3.2 Algorithms In order to benefit from the reduced complexity offered by the fact that the cardinality of the set of significant details and hence of the locally refined grid is typically much smaller than the corresponding uniform grid on level L, all transformations have to be performed locally. In particular, we are not allowed to operate on the full arrays corresponding to the uniformly refined grids, i.e., the summation in the transformations (7), (8) and (9) have to be restricted to those indices which correspond to non–vanishing entries of the mask coefficients. Introducing the mask matrices l,e Ml,e = (ml,e r,k )r∈Il+1 ,k∈Il and Gl,e = (gk,r )r∈Il ,k∈Il+1 , these two-scale relations may be

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rewritten in terms of matrix-vector products as yT = xT A. Note that the mask matrices are sparse due to an appropriate choice of the wavelet basis. In order to confine the summation in the matrix-vector product only to non-vanishing entries of the matrices it is helpful to introduce the following notion the support of matrix columns and rows Ak := supp(A, k) := {r ; ar,k = 0} = support of kth column of A, Ak∗ := supp(AT , k) := {r ; ak,r = 0} = support of kth row of A. The support Ak of a column is comprised of all non–vanishing matrix elements that might yield a non–trivial contribution to the kth component of the matrix–vector product, i.e., yk . Therefore Ak can be interpreted as the domain of dependence for yk , i.e., the components xr which contribute to yk . The support Ak∗ of a row contains all non–vanishing matrix entries of the kth row that might yield a non–trivial contribution to the vector y of the matrix–vector product. Therefore Ak∗ can be interpreted as the range of influence, i.e., the components yr which are influenced by the component xk . With this notation in mind, we now can formulate efficient algorithms for locally performing the decoding and encoding processes as they have been realised in the multiscale library: Algorithm 1 (Encoding) Proceed levelwise from l = L − 1 downto 0: I. Computation of cell averages on level l: 1. For each active cell on level l + 1 determine its parent cell on level l: ∗,0 where Il+1,ε := {k ∈ Il+1 : (l + 1, k) ∈ G˜L,ε } Ul0 := r∈Il+1,ε Ml,r 2. Compute cell averages for parents on level l: 0 uˆl,k = ∑r∈M 0 ml,0 r,k uˆl+1,r , k ∈ Ul l,k

II. Computation of details on level l: 1. For each active cell on level l + 1 determine all cells on level l influencing their corresponding details: ∗,e Ule := r∈Il+1,ε Ml,r , e ∈ E∗ 2. For each detail on level l determine the cell averages on level l + 1 that are needed to compute the detail: e \I Pl+1 := e∈E ∗ k∈U e Ml,k l+1,ε l 3. Compute a prediction value for the cell averages on level l + 1 not available in adaptive grid: uˆl+1,k = ∑r∈G 0 gl,0 r,k uˆl,r , k ∈ Pl+1 l,k

4. Compute the details on level l: l,e e ∗ e m dl,k,e := ∑r∈Ml,k r,k uˆl+1,r , k ∈ Ul , e ∈ E

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Algorithm 2 (Decoding) Proceed levelwise from l = 0 to L − 1: I. Computation of cell averages on level l + 1: 1. Determine all cells on level l + 1 that are influenced by a detail on level l: + Il+1 := e∈E ∗ l∈Jl,ε Gl,l∗,e where Jl,ε := {k ∈ Il : (l, k) ∈ D˜ L,ε } 2. Compute cell averages for cells on level l + 1: l,e + uˆl+1,k = ∑ gl,0 ∑ gr,k dl,e,r , k ∈ Il+1 r,k uˆl,r + ∑ 0 r∈Gl,k

e e∈E ∗ r∈Gl,k

II. Remove refined cells on level l: 1. For each cell on level l + 1 determine its parent cell on level l: ∗,0 Il− := k∈I + Ml,k l+1 2. Remove the parent cells on level l from the adaptive grid: delete uˆl,k , k ∈ Il− , i.e., Il,ε := Il+ /Il− (Note: I0+ = I0 )

3.3 Data Structures In order to fully benefit from the above principal complexity reduction, we need appropriate supporting data structures. These have to be designed in such a way that the computational complexity (storage and CPU time) remains proportional to the cardinality of the adaptive grid and hence of the significant details, respectively. For this purpose, the C++-template class library igpm t lib [20, 24] has been developed. This library provides data structures that are tailored to the requirements set by Algorithms 1 and 2, from which the fundamental design criteria are deduced, namely, (i) dynamic memory operations and (ii) fast data access with respect to inserting, deleting and finding elements. Due to refinement and coarsening operations in the algorithm, memory operations are frequently performed and therefore should be very fast. This can be realised more efficiently by allocating a sufficiently large memory block and by managing the algorithm’s memory requirements with a specific data structure. In addition, since the overall memory demand can only be estimated, the data structure should provide dynamic extension of the memory. In order to facilitate an efficient memory management and a fast data access we use the well-known concept of hash maps, cf. [8], that is composed of two parts, namely, a vector of pointers, a so–called hash table, and a memory heap, see Figure 9. The hash table is connected to a hash function f : U → T , which maps a key, here (l, k), to the hash table of length #T , i.e., a number between 0 and #T − 1. Here the set U can be identified with all possible cells in the nested grid hierarchy (universe of keys), i.e., U = {(l, k) : k ∈ Il , l = 0, . . . , L}, and T corresponds to the keys of the dynamically changing adaptive grid, i.e., (l, k) ∈ G˜L,ε . The set of all possible keys is much larger than the length of the hash table, i.e., # T # U . Hence, the hash function cannot be injective. This leads to collisions in the hash table, i.e., different keys might be mapped to the same position by the hash function. As collision resolution we choose chaining: the corresponding values of these keys are linked to the list that starts at position f (key). Each element in the hash table is a pointer to a linked list whose elements are stored in the heap. Here

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each element of the list can be a complex data structure itself. It contains the key and usually additional data, the so–called value. In general, the value consists of the data corresponding to a cell. The performance of the hash map crucially depends on the number of collisions. In order to optimise the number of collisions, the length of the hash table # T and the number of collisions # {key ∈ U : f ({key}) = c} have to be well-balanced. Several strategies have been developed for the design of a hash function, see [23, 8]. For our purpose, choosing the modulo function and appropriate table lengths turned out to be sufficient, see [19]. Since the local multiscale transformations are performed level by level, see Algorithms 1 and 2, the hash map has to maintain the level information. For this purpose, the standard hash map is extended by a vector of length L. The idea is to have a linked list of all cells on level l: the lth component of the vector contains a pointer that points to the first element of level l put into the memory heap. Additionally, the value has to be internally extended by a pointer that points to the next element of level l. This is indicated in Figure 10. Then we can access all elements of level l by traversing the resulting singly linked list.

Fig. 9 Hashing (Courtesy of [25])

Fig. 10 Linked hash map

4 Parallelisation In the following we will present how we have parallelised the multiscale library by which we perform local grid adaptation in one block using multiscale techniques. For this purpose, we first outline the strategy of load-balancing using space-filling curves, see Section 4.1. In a second step, see Section 4.2, we explain how to perform the multiscale transformation in parallel. Here the crucial point is the handling of the cells on the partition boundary and interprocessor communication. Note that the multiscale-based grid adaptation consists of additional steps such as thresholding, prediction, grading and decoding. The parallelisation of these steps is in complete analogy and therefore not detailed here.

4.1 Load-Balancing via Space-Filling Curves Regarding parallelization, a first essential issue is the mesh partitioning or the loadbalancing problem. This is fairly straightforward for uniform grids. But having

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to deal with locally refined grids where not all cells on all levels of refinement are active, complicates matters significantly. A natural representation of a multilevel partition of a mesh is a global enumeration of the active cells. We need a method to do this at runtime, as the adaptive mesh is also created at runtime using the multiscale representation techniques sketched above. Such an enumeration can be based on space-filling curves (SFC) by mapping a higher-dimensional domain to a onedimensional curve. Specifically, the unit square or the unit cube is mapped to the unit interval. Using space-filling curves, each of the cells of the adaptive grid has a corresponding unique number on the curve. So, instead of assigning portions of the geometric domain to different processors, we only have to split the interval of numbers on the curve into parts that have roughly equal length. Each of these parts is mapped to a different processor, so that we obtain a well-balanced data partition. However, concerning the second major issue, we also have to pay the cost of interprocessor communications, because neighbours within a grid block may belong to different processors. We shall indicate below that the concept of space-filling-curves offers favorable features in this regard as well. Space-Filling Curve. Space-filling curves have been originally created for purely mathematical purposes , cf. [22]. Nowadays, these curves have found several applications, one of them being a good load-balancing for numerical simulations on parallel computer architectures. They can be used for data partitioning and, due to self-similarity features, multilevel partitions can also be constructed. In the mathematical definition, a space-filling curve is a surjective, continuous mapping of the unit interval [0, 1] to a compact d-dimensional domain Ω with positive measure. In our context, we restrict our attention to Ω being the unit square or the unit cube. In fact, as our grids have finite resolution, the iterates — so-called discrete space-filling curves — are applied, instead of the continuous space-filling curve. The construction of these curves is extremely inexpensive, as the SFC index for any cell in the grid can be computed using only local information, making it suitable for parallel computations. Hilbert Space-Filling Curve. One of the oldest space-filling curves, the Hilbert curve, can be defined geometrically, cf. [25]. The mapping is defined by the recursive subdivision of the interval I and the square Q. In 3D, the Hilbert curve is based on a subdivision into eight octants. The construction begins with a generator template, which defines the order in which the quadrants are visited. Then the template (identical, mirrored or rotated) is applied to each quadrant and, by connecting the loose ends of the curve, the next iterate of the space-filling curve is obtained. Actually, the mapping between the cells of the adaptive grid and the space-filling curve is realised using the finest iterate of the curve, which is constructed by recursively applying the template to the subquadrants (2D) and suboctants (3D) until the number of refinement levels is reached. Figures 11 and 12 show the first iterates of a 2D and 3D Hilbert SFC, respectively. A detailed discussion on the construction of the Hilbert space-filling curve is beyond the scope of this paper. For corresponding detailed expositions we refer the reader to [22, 25]. Here, we only summarise the

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procedure of the Hilbert curve construction and focus our attention on how the curve can contribute to an efficient parallelisation of the multiscale-based grid adaptation scheme.

Fig. 11 First 4 iterates of 2D Hilbert SFC

Fig. 12 1st and 2nd iterate of 3D Hilbert SFC (Courtesy of Gilbert [11])

Encoding the Hilbert SFC Ordering. By inverting the mapping induced by a discrete space-filling curve, multi-dimensional data can be mapped to a onedimensional interval. The basic idea is to map each cell of our adaptive grid to a point on the space-filling curve, so that we obtain a global enumeration of the grid cells. In the data structures, we use the key comprised of the cell’s refinement level and the cell’s multidimensional index on that level (l, k) to identify each cell. Since each cell of the adaptive grid is uniquely identified by this key, the aim is to use it in order to determine for each cell a corresponding number on the space-filling curve. Also, due to locality properties of the curves, each visited cell is directly connected to two face-neighbouring cells which remain face neighbours in the onedimensional space spanned by the curve. In this way, the cell’s children are sorted according to the SFC numbers and they will be nearest-neighbours on a contiguous segment of the SFC. Since we are dealing with multilevel adaptive rectangular grids, we restrict ourselves to recursively defined, self-similar SFC based on rectangular recursive decomposition of the domain. Encoding and decoding the Hilbert SFC order requires only local information, i.e., a cell’s 1D index can be constructed using only that cell’s integer coordinates in the d-dimensional space and the maximum number of refinement levels L that exist in the mesh. In a 2D space, consider a 2L × 2L square (0 ≤ X ≤ 2L−1 , 0 ≤ Y ≤ 2L−1 ) in a Cartesian coordinate system. Note that the variable L used for determining the Hilbert SFC order might be larger than the one introduced in Section 3.1 for the number of refinement levels, since N0,i > 1 in general and for the space-filling curve construction we should have a coarsest mesh with N0,i = 1. Any point can be expressed by its integer coordinates, (X,Y ), where X , Y are two sequences of L-bit binary numbers, as follows:

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Each sequence of two interleaved bits {xl , yl }l=1,...,L determines on each level l one of the four quadrants the cell belongs to, recursively. Thus by simply inspecting the cell’s integer coordinates and using a finite state machine, the cell’s location on a curve can easily be computed, cf. [25]. In the 3D case we proceed similarly. An important aspect that should be mentioned is that the construction of the space-filling curve on an adaptive mesh ensures that a parent cell (l, k) has exactly the same number on the SFC as one of its children (l + 1, 2k + e), e ∈ {0, 1}d , cf. [25]. This leads to the minimisation of the interprocessor communication in the case of a parallel MST using the Hilbert space-filling curve as partitioning scheme, because in most of the cases the parent cell should be computed on the same processor as its children. Load-Balancing. After computing the SFC indices for all the cells in the mesh, these indices are taken as sort keys and the mesh may be ordered along the curve using standard sorting routines, such as introsort in our case. Having all the cells sorted along the curve, the partition can be easily determined, just by choosing the number of cells that each processor should get. So the mesh cells are distributed to the different processors according to their index on the curve. Since the position of each cell on the curve can be computed very inexpensively at any time in the computation, there is no need to store all the keys. Instead, it is sufficient to store the separators between the elements of the partition of the interval, i.e., the first index on a processor, in order to determine for any cell’s multidimensional index the corresponding processor number. There are two options for handling the data partitioning and the load-balancing problem in the beginning of the computation, namely, (i) master-based partitioning and (ii) symmetric multiprocessing. In case of master-based partitioning, as its name says, the entire adaptive mesh is initialised on a master processor, according to the input file. Once the grid is initialised, the same master processor is also responsible for the entire partitioning procedure already described: the mapping of the cells to the SFC, the sorting of the keys, the load-balancing and separators’ determination. After having performed these steps once, the cells can be distributed to the corresponding processors. This approach is straight forward if the starting point is a running serial algorithm, as no data transfer and no barrier points are needed before the distribution of the data to processors actually begins. On the other hand, this implies that there is only one processor active during all the initialisation and sorting of the SFC procedures, while the others are idle, waiting to receive the data from the master for initialising their own data structures. The second possibility is symmetric multiprocessing: this avoids the use of a master processor, i.e., all processors should work in parallel, executing the same code and initialising only their corresponding part of the grid. For this, a set of initial separators on the space-filling curve has to be assumed, without knowing in advance anything about the structure of the adaptive grid. So the worst case has to be taken into account, when the grid would be uniformly refined, which is equivalent

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to the fact that, for each number on the discrete space-filling curve, there exists an active cell in the grid. In the case of a fully refined grid, the number of cells in the grid (2L × 2L and 2L × 2L × 2L , in 2D and 3D, respectively) corresponds to the last number on the SFC and the guess of the initial separators is straight forward. The main drawback of this second approach is that this initial guess might and is very likely to be far from the optimal choice, so the possibility of not having a remarkable performance improvement in the initialisation part is very high, also due to the interprocessor communication costs that arise. A rebalancing of the initial data is then required in order to obtain a well-balanced distribution of data among processors and a new set of separators is computed for the new partition. Applying either of these two strategies leads to a well-balanced data distribution as shown in Figure 13.

Fig. 13 Well-balanced distribution of a locally refined grid to 5 processors

Parallel Rebalancing. A reordering of the cells along the curve is also needed whenever the load-balancing is significantly spoiled due to the adaptivity of the grid. When this occurs, a new set of separators — that determine a new well balanced partition — has to be computed. There is no need to gather all cells on a master processor for the reordering, since all the cells on a processor p have smaller numbers on the SFC than the cells on processor p + 1 for all p = 0, . . . , nproc − 2. The parallel rebalancing is described in Algorithm 3. Algorithm 3 (Parallel Rebalancing) 1. Sort local cells along the SFC according to the old list of separators sepold [i] for i = 0, . . . , nproc − 1. 2. Compute the total workload from all processors: nproc −1 total workload = ∑ p=0 #cells(p) 3. Compute the positions of the new separators on the SFC:

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a) new positions[0] = 0 b) For i = 1, . . . , nproc − 1 do new positions[i] = new positions[i − 1] + i · (total workload/nproc) 4. For pos = 1, . . . , nproc − 1 do If new positions[pos] belongs to the local processor, then determine the new separator sepnew [pos] at position new positions[pos]. 5. Distribute new separators to all processors. 6. Redistribute data according to the new separators.

4.2 Parallel Grid Adaptation and Data Transfer Once load-balancing is achieved, each processor should perform the grid adaptation, see Section 3.1, on the local data. Special attention must be paid to the cells located at the processor’s boundary, i.e., the cells that have at least one neighbour belonging to another processor. Since these are the only ones that make the difference between serial and parallel algorithms and because they are similarly handled in all the steps of the grid adaptation, we shall address below only the parallelisation of the encoding step in some detail. More specifically, we shall mainly discuss the special treatment of the boundary cells. As described in Section 3.2, the encoding step consists of computing the cell averages on level l starting from data on level l + 1 and the computation of the details on level l. Since the approach to parallelising the coarsening is different from the one for computation of details, they will be discussed separately. Parallel Coarsening. To compute the parent’s cell average uˆl,k on the processor indicated by the parent’s position on the SFC and the separators between the elements 0 , should already be available on the of the partition, all its children uˆl+1,r , r ∈ Ml,k same processor. Due to the locality properties of the SFC and the compactness of each element of the partition, i.e., the ratio of an element’s volume and its surface is large, ensured by its construction, the children are nearest neighbours in the onedimensional space. This entails that only a few cells at the boundaries between partition elements have to be transferred between processors before computing the cell averages on level l, see Figure 14. When running through the active data on level l + 1 for constructing the set of parent cells that need to be computed, the processor the parent belongs is also determined and so a buffer is set up, containing the children cells that need to be transferred to a neighbour processor. The buffer is transferred to the corresponding processor before the computation of the averages on level l actually begins. Once the cell averages of all cells’ parents have been computed, the ghost cells transferred from other processors can be deleted from the local hash map.

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Algorithm 4 (Parallel Coarsening) Proceed levelwise from l = L − 1 downto 0: (cf. Algorithm 1, Step I) I. Computation of cell averages on level l: 1. For each active cell (l + 1, r), r ∈ Il+1,ε , determine a) the parent cell on level l; b) the processor p to which the parent belongs to. ∗,0 If p is the current processor then Ul0 = Ul0 ∪ Ml,r ; else transfer cell (l + 1, r) to processor p and there add it to the local hash map. 2. Compute cell averages for parents on level l: 0 uˆl,k = ∑r∈M 0 ml,0 r,k uˆl+1,r , k ∈ Ul l,k

3. Delete the cells received from other processors. Parallel Details Computation. In the case of the computation of details, more data from the neighbour processors have to be transferred, see Figure 15. For the parallel coarsening, a single step data transfer is made, since each processor can determine the cells on level l + 1 that are necessary for the neighbours to compute the averages on level l by itself. In case of the multiscale decomposition for computing the details a two step transfer has to be performed. In a first step, each processor has to send a request to the others for the cells that influence the details computation of the local cells, so we obtain a first pair of MPI Isend and MPI Recv calls. A new pair of such calls is needed to fulfil the requests, when actually all the data located at the geometrical boundary of the partitions has to be transferred to the neighbouring processors. Algorithm 5 (Parallel Details Computation) Proceed levelwise from l = L − 1 downto 0: (cf. Algorithm 1, Step II) I. Computation of details on level l: e = 0, / p = 0, . . . , n proc − 1 0. On each processor initialise the index sets Ul,p 1. For each active cells on level (l + 1, r), r ∈ Il+1,ε do a) determine all cells on level l influencing their corresponding details: ∗,e (l, k) ∈ Ml,r , e ∈ E ∗; ∗,e b) for each (l, k) ∈ Ml,r determine the processor p where the details of cell (l, k) should be computed: e := U e ∪ {k}; if p = ploc (current processor) then Ul,p l,p else transfer index (l, k) to processor p and there add it to the index set e := U e ∪ {k} and transfer cell (l + 1, r) to processor p and there add Ul,p l,p it to the local hash map. 2. For each detail on level l determine the cell averages on level l + 1 that are needed to compute the detail: e \I Pl+1 := e∈E ∗ k∈U e Ml,k l+1,ε l

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3. For all indices (l + 1, r) ∈ Pl+1 that belong to other processors p = ploc (current processor), p = 0, . . . , n proc − 1 do a) send requests to processor p to transfer the unavailable data; b) receive needed data from processor p. 4. Accept requests from other processors and send back the data to the other processors requested from the current processor. 5. Compute a prediction value for the cell averages on level l + 1 not available in the adaptive grid: uˆl+1,k = ∑r∈G 0 gl,0 r,k uˆl,r , k ∈ Pl+1 l,k

6. Compute the details on level l: l,e e ∗ e m dl,k,e := ∑r∈Ml,k r,k uˆl+1,r , k ∈ Ul , e ∈ E 7. Delete data received from other processors from the local hash map.

Fig. 14 Cells to be transferred for parallel coarsening

5 Embedding of Parallel Multiscale Library into the Quadflow Solver The finite volume solver Quadflow [6, 7] has been designed to handle (i) unstructured grids composed of polygonal(2D)/polyhedral(3D) elements [5] and (ii) blockstructured grids where in each block the grid is determined by local evaluation of B-Spline mappings [17]. Therefore the solver can handle grids provided by an external grid generator. However, grid adaptation is only available for block-structured grids, where in each block the grid is locally refined using the concept of multiscalebased grid adaptation [19]. Note that the flow solver and the multiscale-based grid adaptation pose totally different algorithmic requirements: on one hand, there is a finite volume scheme working on arbitrary, unstructured discretisations. On the other hand, there is the multiscale algorithm assuming the existence of hierarchies of structured meshes. The flow solver module is face-centred, since the central issue is the computation

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Fig. 15 Cells to be transferred for parallel details computations

Fig. 16 Parallel Transfer and Conversion

of the fluxes at the cell faces, while the adaptation module is cell-centred, analysing and manipulating cell averages. Moreover, the data structures used in the two parts are also different: while for the adaptation part a special implementation of hash maps is used, see Section 3.3, for the flow solver module the FORTRAN style data structures remained optimal. The link between these two modules is done by a data conversion algorithm, which organises all the data communication — the transfer of the conservative variables, volumes, cell centres, the registration of the knots, the construction of the faces and determination of their neighbouring cells and nodes — between the two modules in a connectivity list.

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In order to embed the parallelised version of the multiscale library into Quadflow, the transfer had to be adjusted: besides the special treatment required by some cells located at the physical boundary or at the far field boundary of the domain even in the serial case, cf. [17], in parallel special attention is needed also by the cells at the partition’s boundary. Since the adaptive mesh is determined at runtime, as well as the partition, and since the adaptive mesh and implicitly the shape of the elements of the partition could change at any time step, there is no way of knowing in advance which cells are located on a processor’s boundary. This entails that the partition boundary on each processor should be reconstructed each time the connectivity list is built, in order to transfer the cells on the boundary to the neighbour processors, see Figure 16. In this way, the boundary faces of the partition and the flux at these faces can be properly computed on each processor.

6 Numerical Results First of all, we investigate in Section 6.1 the performance of the parallelised multiscale library. For this purpose, we focus on the multiscale transformation (MST), i.e., the encoding of the data. Note that a complete cycle of grid adaptation also includes thresholding, prediction, grading and decoding. In Section 6.2 we then present numerical simulations of the behaviour of a Lamb-Oseen vortex using the parallelised Quadflow solver.

6.1 Performance Study for Multiscale Transformation The performance of the multiscale based grid adaptation has been investigated by means of a data set corresponding to a locally refined grid that consists of 437236 cells. The underlying grid hierarchy is determined by L = 10 refinement levels and a coarse grid discretisation of 8 × 8 cells. For this configuration, we performed the adaptation process using an increasing number of processors. The experiments were performed on a Sun Fire X4600 system, with 2 AMD Opteron 885 nodes having 8 sockets per node (a total of 16 processors), 32 GB memory and a high speed low latency network (InfiniBand) for parallel MPI and hybrid parallelised programs. Table 1 shows the results of these experiments with respect to the different number of processors mentioned in the first column. The second column contains the values for the initial workload, i.e. the number of cells on each processor. Columns 3 and 4 show the times measured when running one MST step and the time spent in the MPI Isend and MPI Recv routines, respectively. The number of cells sent by one processor during this MST step is shown in the fifth column, while the sixth column contains the number of cells received by the same processor. After the partitioning is done, each processor is getting a number of cells equal to the total number of cells in the adaptive grid over the number of processors. Note that the processor with the highest rank also takes the few cells that remain if the total number of entries cannot be divided by the number of processors. Better

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Table 1 Performance study for parallel multiscale transformation No. of Initial MST time Transfer time No. of cells No. of cells procs. workload [CPU s] [CPU s] sent received 1 437236 10.21580 0 0 0 2 218618 5.55433 0.066706 2020 2020 3 145746 3.45852 0.170083 5985 5975 4 109309 2.81636 0.125800 2048 2048 5 87448 2.13640 0.127011 10826 10706 6 72876 1.78998 0.126228 7240 7190 7 62464 1.55737 0.133888 9097 9078 8 54658 1.47147 0.103048 6899 6954 9 48588 1.33229 0.105228 7112 7068 10 43729 1.23401 0.135434 8856 8791

performance and good scaling may be observed in column 3 as the number of processors increases (see also Figure 17). When it comes to the transfer of cells between neighbour processors, the number of entries sent by one processor is different than the one received on most processor configurations chosen due to the adaptive grid: one processor might have more finer cells sitting on the partition boundary than its neighbours have on the other side of the boundary, which gives the difference in the number of ghost cells needed to be transferred from one processor to the other. The number of processors chosen also influences the shape of the elements of the partition created at runtime. Thus, having a symmetric adaptive grid on a single processor and choosing to run a parallel computation on 2 or 4 processors might lead to a symmetric partitioning of the grid on all refinement levels. This implies the minimisation of the interprocessor communication. The consequences of this symmetry can be observed in Table 1, where the minimum number of cells to be transferred is achieved on 2 processors. At small difference, the computation on 4 processors also gives few entries to be transferred in comparison with the rest of the configurations tested. The fact that on 2 and 4 processors a symmetric partitioning is obtained is also emphasised when inspecting the number of cells sent and received across the partition boundary: here the number of cells sent to the neighbours is equal to the number of cells received. The opposite is observed on a configuration of 5 processors, where the number of cells on the partition’s boundary reaches the maximum value and implicitly increases the interprocessor communication. A partitioning to 5 processors is shown in Figure 13, where the asymmetry of the partitions is obvious. Speedup. According to Amdahl’s law [1] for a given problem the maximal speedup p of a computation on p processors is bounded by smax ≤ 1+ f (p−1) , where f is the fraction of the runtime of the code that is not parallelised. We compared the scaling of our experiment with Amdahl’s law and therefore assumed that the maximum speedup was measured. Using a nonlinear least squares

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fit, we were able to estimate the fraction of the code that is not parallelised and obtained f = 0.0203 ± 0.0017, cf. Figure 21. Having a fraction of 2% of the program not parallelised, according to Amdahl’s law, the maximum speedup that could be reached for this fixed configuration would be 50. However, Amdahl’s law doesn’t take into account that the fraction of the serial parts can be reduced by scaling the problem to the number of processors. So,

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for a fixed configuration, infinite speedup cannot be achieved, but when the problem size grows, also speedup could be expected on an increasing number of processors.

6.2 Application The system of vortices in the wake of airplanes continues to exist for a long period of time in reality. It is possible to detect wake vortices as far as some 100 wing spans behind the airplane, which are a hazard to following airplanes. In the frame of the collaborative research centre SFB 401, the research aimed to induce instabilities into the system of vortices to accelerate their alleviation. The effects of different measures taken in order to to destabilise the vortices have been examined in a water tunnel. A model of a wing was mounted in a water tunnel and the velocity components in the area behind the wing were measured using particle image velocimetry. It was possible to conduct measurements over a length of 4 wing spans. The experimental analysis of a system of vortices far behind the wing poses great difficulties due to the size of the measuring system. Numerical simulations are not subject to such severe constraints and therefore Quadflow is used to examine the behaviour of vortices far behind the wing. To minimise the computational effort, the grid adaptation adjusts the refinement of the grid with the goal to resolve all important flow phenomena, while using as few cells as possible. In the present study instationary, quasi-incompressible, inviscid fluid flow described by the Euler equations for compressible low Mach number flow is considered. A first assessment is presented to validate the ability of Quadflow to simulate the behaviour of the wake of an airplane. A velocity field based on the experimental data from the water tunnel measurements is prescribed as boundary condition in the inflow plane. The circumferential part of the velocity distribution vΘ (r) is described by a Lamb-Oseen vortex according to 2 Γ − r/d0 . (10) 1−e vΘ (r) = 2π r The axial velocity component in the inflow direction is set to the constant inflow velocity of the water tunnel. The two parameters of the Lamb-Oseen vortex, circulation Γ and core radius d0 are chosen in such a way that the model fits the measured velocity field of the wing tip vortex as close as possible. The radius r is the distance from the centre of a boundary face in the inflow plane to the vortex core. Instead of water, which is used as fluid in the experiment, the computation relies on air as fluid. The inflow velocity in x-direction u∞ is computed to fulfil the condition that the Reynolds number in the computational test case is the same as in the experiment. The experimental conditions are a flow velocity uw = 1.1 ms and a Reynolds number Rew = 1.9 · 105. From the condition Reair = Rew the inflow velocity in x-direction can be determined as u∞ = 16.21 ms . For purpose of consistency the circumferential velocity vθ has also been multiplied by the factor uu∞w . The velocity of the initial solution is set to parallel, uniform flow u0 = u∞ , v0 = w0 = 0.0.

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The computation has been performed on 16 Intel Xeon E5450 processors running at 3 GHz clock speed. The computational domain matches the experimental setup which extents l = 6 m in x-direction, b = 1.5 m in y-direction and h = 1.1 m in zdirection. The boundaries parallel to the x-direction have been modelled as symmetry walls. This domain is discretised by a coarse grid with 40 cells in flow-direction, 14 cells in y-direction and 10 cells in z-direction, respectively. The number of refinement levels has been set to L = 6. With this setting the vortex can be resolved on the finest level by about 80 cells in the y-z-plane.

Fig. 22 Initial computational grid

Since Quadflow solves the compressible Euler equations a preconditioning for low Mach numbers has to be applied. The preconditioning is used in a dual-time framework wherein only the dual time-derivatives are preconditioned and used for the purposes of numerical discretisation and iterative solution, cf. [21]. The spatial discretisation of the convective fluxes is based on the AUSMDV(P) flux vector splitting method [10]. For time integration the implicit midpoint rule is applied. In each timestep the unsteady residual of the Newton iterations is reduced by four orders of magnitude. The physical timestep is set to Δ t = 5 · 10−5 s which corresponds to a maximum CFL-number of about CFLmax = 28.0 in the domain. The grid is adapted after each timestep. After every 100th timestep the load-balancing is repeated. To guarantee a sufficiently fine grid to resolve the vortex properly at the start of the computation, the grid on the inflow plane is refined to the maximum level, see Figure 22. Due to this procedure the first grid contains 384000 cells. When the information at the inlet has travelled through the first cell layer, the forced adaptation of the cells at the inlet is not necessary anymore. From there on the grid is only adapted due to the adaptation criterion based on the multiscale analysis. After 5466 timesteps, which corresponds to a computed real time of t = 0.27 s, the grid contains 787000 cells. Figure 23 shows six cross sections of the mesh, which are equally spaced in x-direction with distances Δ x = 1.0 m. In addition, the isosurface of the λ2 -criterion with the value λ2 = −3 is also presented. The λ2 criterion has been proposed by Jeong et al. [14] to detect vortices. A negative value of

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Fig. 23 Slices of the computational grid after 5466 timesteps at six different positions and the distribution of λ2 = −3

Fig. 24 Slices of the computational grid at two different positions in x-direction, the grid colour is consistent with the value of λ2 . Slice of the computational grid at x = 0.0 m (left) and x = 2.0 m (right)

λ2 identifies a vortex, whereas the smallest of these negative values marks the core of the vortex. As can be seen from Figure 23, the vortex is transported through the computational domain. The locally adapted grid exhibits high levels of refinement only in the vicinity of the vortex. A more detailed view of the grid for the cross sections at x = 0.0 m and x = 2.0 m is presented in Figure 24. It can be seen that only near the vortex core the grid is refined up to the maximum level. We conclude that Quadflow is well suited to examine the instationary behaviour of a vortex. In particular the complexity reduction due to the grid adaptation makes it possible to perform the computations in reasonable time.

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18. Massjung, R.: Discrete conservation and coupling strategies in nonlinear aeroelasticity. Comput. Methods Appl. Mech. Engrg. 196, 91–102 (2006) 19. M¨uller, S.: Adaptive Multiscale Schemes for Conservation Laws. Lecture Notes on Computational Science and Engineering, vol. 27. Springer, Heidelberg (2002) 20. M¨uller, S., Voss, A.: A Manual for the Template Class Library igpm_t_lib. IGPM– Report 197, RWTH Aachen (2000) 21. Pandya, S.A., Venkateswaran, S., Pulliam, T.H.: Implementation of preconditioned dualtime procedures in overflow. AIAA Paper 2003-0072 (2003) 22. Sagan, H.: Space-Filling Curves, 1st edn. Springer, New York (1994) 23. Stroustrup, B.: The C + + Programming Language. Addison-Wesley, Reading (1997) 24. Voss, A.: Notes on adaptive grids in 2d and 3d part i: Navigating through cell hierarchies using cell identifiers. IGPM–Report 268, RWTH Aachen (2006) 25. Zumbusch, G.: Parallel multilevel methods. Adaptive mesh refinement and loadbalancing. In: Advances in Numerical Mathematics. Teubner, Wiesbaden (2003)

Iterative Solvers for Discretized Stationary Euler Equations Bernhard Pollul and Arnold Reusken

Abstract. In this paper we treat subjects which are relevant in the context of iterative methods in implicit time integration for compressible flow simulations. We present a novel renumbering technique, some strategies for choosing the time step in the implicit time integration, and a novel implementation of a matrix-free evaluation for matrix-vector products. For the linearized compressible Euler equations, we present various comparative studies within the QUADFLOW package concerning preconditioning techniques, ordering methods, time stepping strategies, and different implementations of the matrix-vector product. The main goal is to improve efficiency and robustness of the iterative method used in the flow solver.

1 Introduction Large sparse non-linear system of equations resulting from a finite volume discretization of compressible Euler equations are considered in this paper. This discretization is based on adaptive wavelet techniques for local grid refinement, For an overview of the adaptivity concept and the finite volume discretization we refer to [8]. The methods are implemented in the QUADFLOW solver, cf. [7, 8]. In this paper we only consider iterative methods for solving the large non-linear systems of equations. The approach of a “pseudo-transient continuation” is followed. That is, an implicit time integration method is applied to the unsteady Euler equations so that the corresponding non-stationary solution converges to the stationary solution for time tending to infinity. This then yields a non-linear system of equations in each time step which is solved by a Newton-Krylov method. Therein one applies a linearization technique combined with a preconditioned Krylov subspace algorithm for Bernhard Pollul · Arnold Reusken Chair for Numerical Mathematics, RWTH Aachen University, Templergraben 55, D–52056 Aachen e-mail: {pollul,reusken}@igpm.rwth-aachen.de

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 295–323. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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solving the resulting linear problems. The computational work needed for solving the large sparse systems in the Newton-Krylov method determines to a large extent the total computing time. Because preconditioning is crucial for the convergence of the Krylov solver we investigate different so-called “point-block” preconditioners. Preconditioners usually strongly depend on the ordering of the (block) unknowns. We present a new renumbering technique that is based on a reduced matrix graph and that can significantly improve both the robustness and the efficiency of the iterative method. The selection of the time step size in the implicit time integration is crucial for the performance of the iterative solver. We investigate two known and two novel time step selection strategies. A further acceleration of the time integration can be achieved by the use of a second order accurate Jacobian. Because the stencils for second order methods are relatively large resulting in a complex Jacobian requiring much memory, we present a second order matrix-free method that uses automatic differentiation. Using the second order matrix-free evaluation of the matrix-vector product the corresponding computational time can be significantly decreased.

2 The Euler Equations Derived from the fundamental conservation laws of fluid dynamics, the timedependent Euler equations describe the motion for an inviscid, non-heat-conducting compressible gas. For an arbitrary control volume V ⊂ Ω ⊂ Rd with boundary ∂ V and outward unit normal vector n on the surface element dS ⊂ ∂ V they are given by V

∂u dV + ∂t

∂V

F(u)n dS = 0 .

(1)

The convective flux F(u) and the vector of unknown conserved quantities u containing the density ρ , the static pressure p, the velocity vector of the fluid v, and the total energy E are given by ⎞ ⎛ ⎞ ⎛ ρv ρ u = ⎝ ρv ⎠ , (2) F(u) = ⎝ ρ v ◦ v + pI ⎠ , Ev + pv E where ◦ denotes the dyadic product. The system is closed by suitable initial and boundary conditions and the equation of state for a perfect gas using the ratio of c specific heats γ = cvp : p 1 E= + ρ v2 . (3) γ −1 2 The equations (1) – (3) form the standard model that is considered in this paper.

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3 Test Problems We describe two classes of test problems which are used in the numerical experiments below.

3.1 Homogeneous Stationary Flow on the Unit Square In this simple test problem we consider Ω = [0, 1]2 on a uniform mesh Ωh = { (ih, jh) | 0 ≤ i, j ≤ n } , with nh = 1 and boundary conditions such that the stationary Euler equations have a constant solution. We apply the Van Leer flux vector-splitting scheme and use compatibility relations for the discretization of the boundary conditions. A lexicographic ordering of the grid points is applied. The discretization yields a non-linear system of equations F : R4N → R4N ,

F(U) = 0 .

(4)

The continuous constant solution (restricted to the grid) solves the discrete problem and thus the solution of the non-linear discrete problem, denoted by U∗ , is known a-priori. In Subsection 4.2 we investigate the behavior of different preconditioners when applied to a linear system of the form DF(U∗ )x = b. The matrix DF(U∗ ) has a point-block structure, that is, a regular block structure DF(U∗ ) = blockmatrix(Ai, j )0≤i, j≤N with Ai, j ∈ R4×4 for all i, j. Note that Ai, j = 0 can occur only if i = j or i and j correspond to neighboring grid points.

3.2 Stationary Flow around NACA-0012 Airfoil This problem is a standard test case for inviscid compressible flow solvers [25] in which the inviscid, transonic, stationary flow around the NACA0012 airfoil is considered. We present results for the three test cases given in Table 1 characterized by Mach number M∞ and angle of attack α . Table 1 Test cases 2A, 2B, 2C: Mach number M∞ and angle of attack α for NACA0012 airfoil NACA0012

M∞

α

test case 2A 0.80 1.25◦ test case 2B 0.95 0.00◦ test case 2C 1.20 0.00◦

The problems are discretized using a hierarchy of locally refined grids on which standard finite volumes are applied, cf. [8]. The steady-state solutions of test cases 2A, 2B, and 2C are evolved in a pseudo-transient continuation solving (1), starting on a coarse initial grid, and evolving a solution on an adaptively refined grid. We perform one inexact Newton iteration per time step. The corresponding Jacobian matrices are the system matrices of the occurring systems of linear equations. These

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systems are solved with a left-preconditioned BiCGSTAB method. Preconditioning will be explained in detail in Section 4. In the implicit time integration, the size of the time step is determined by a CFL number γ which is not limited by the Courant-Friedrichs-Levy (CFL) condition [16]. Initialized by γMIN on the coarse initial grid, the CFL number is increased by an evolution method as presented in Section 6 in every time step until an a-priori fixed upper bound γMAX is reached. Time integration is continued until a tolerance criterion for the residual is satisfied. Then a (local) grid refinement is performed and the procedure starts again with an initial CFL number equal to γMIN . 3.2.1

Grid Hierarchy

We show some first results for test cases 2A, 2B, and 2C, using the QUADFLOW solver with a standard time step selection strategy method (γk+1 = 1.1 · γk , γmin = γ0 = 1, and γmax = 1000), cf. Section 6. As in [8] we allow 8 maximum levels of refinement, 10 cycles of adaptations in the cases 2A and 2C and 13 cycles in case 2B. Table 2 Sequence of grids. Tabulated is the number of cells in nested grids for test cases 2A and 2C (10 adaptations performed) and for test case 2B (13 adaptations performed) Test case

Grid

1

2A 2B 2C

# cells # cells # cells

400 400 400

Grid

8

Test case 2A 2B 2C

2

3

4

5

6

7

1 384 2 947 3 805 4 636 5 689 6 817 1 600 4 264 7 006 11 827 15 634 21 841 1 600 4 864 10 189 16 885 23 290 30 598 9

10

11

12

13

14

# cells 7 753 9 028 9 523 9 874 only 10 adaptations carried out # cells 25 870 28 627 30 547 31 828 33 067 33 955 34 552 # cells 36 160 38 764 39 961 40 708 only 10 adaptations carried out

In Table 2 the sequence of nested grids for the four test cases is given. In a full simulation, the density residuals are decreased by a factor of 104 in the finest grid and by a factor of 102 on all coarser grids. Note that the finest grids contain up to 102 times as much cells as the initial grids. Therefore the focus in the following sections will be on reducing the time that is needed to achieve convergence on the finest grid. Therein, the main effort is the solution of the large, sparse linear equation systems that arise in Newton’s linearization method.

4 Point-Block Preconditioners In the Newton-Krylov approach the arising linear systems of equations are solved by a Krylov method. Therein, the choice of the preconditioner is crucial for the convergence process. Our main focus is on the incomplete LU-factorization (ILU) and

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Gauss-Seidel (GS) preconditioners that are widely-used in solvers in the numerical simulation of compressible flows [1, 6, 18, 31, 35, 40].

4.1 Methods In the test problems described in the previous section we have to solve large systems of linear equations. The matrices have a point-block structure in which the blocks correspond to the d + 2 unknowns in each of the N cells (finite volume method). Thus, we have linear systems of the form Ax = b ,

A = blockmatrix(Ai, j )1≤i, j≤N ,

Ai, j ∈ R(d+2)×(d+2) .

(5)

In the following we describe basic point-block iterative methods that are used as preconditioners in the iterative solver. For the right-hand side we use a block representation b = (b1 , . . . , bN )T , bi ∈ Rd+2 that corresponds to the block structure of A. The same is done for the iterands xk that approximate the solution of (5). For the description of the preconditioners the nonzero pattern P(A) corresponding to the point-blocks in the matrix A is important: P(A) = { (i, j) | Ai, j = 0 } 4.1.1

(6)

Point-Block-Gauss-Seidel Method

The point-block-Gauss-Seidel method (PBGS) is the standard block Gauss-Seidel method applied to (5). Let x0 be a given starting vector. For k ≥ 0 the iterand xk+1 = k+1 T (xk+1 1 , . . . , xN ) should satisfy i−1

= bi − ∑ Ai, j xk+1 − Ai,i xk+1 i j j=1

N

∑

Ai, j xkj ,

i = 1, . . . , N .

(7)

j=i+1

This method is well-defined if the (d + 2)× (d + 2) linear systems in (7) are uniquely solvable, that is, if the diagonal blocks Ai,i are nonsingular. In our applications this was always satisfied. This elementary method is very easy to implement and needs no additional storage. The algorithm is available in the PETSc library [2]. 4.1.2

Point-Block-ILU(0) Method

We consider the point-block version of the standard point ILU(0) algorithm, denoted by PBILU(0). For the PBILU(0) preconditioner a preprocessing phase is needed in which the incomplete factorization is computed. Furthermore, additional storage similar to the storage requirements for the matrix A is needed. One can consider variants of this algorithm, e.g. PBILU(p), p = 1, 2, . . .. This produces additional storage requirements and additional arithmetic costs. Both, the PBILU(0) algorithm and such variants, are available in the PETSc library [2].

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Point-Block Sparse Approximate Inverse

The SPAI method [20] can be modified to its point-block formulation in the same way as Gauss-Seidel and ILU. In its point-block version, denoted by PBSPAI(0), we use M = blockmatrix(Mi, j )1≤i, j≤N , Mi, j ∈ R(d+2)×(d+2) and denote the set of admissible approximate inverses by M := { M ∈ R(d+2)N×(d+2)N | P(M) ⊆ P(A) }. A sparse approximate inverse M is determined by minimization over this set: ˜ − IF AM − IF = min AM ˜ M∈M

(8)

The choice for the Frobenius norm allows a splitting of this minimization problem leading to multiple low dimensional least squares problems that can be solved by standard methods in parallel. The application of the PBSPAI(0) preconditioner requires a sparse matrix-vector product computation which also has a high parallelization potential. As for the PBILU(0) preconditioner a preprocessing phase is needed in which the PBSPAI(0) preconditioner M is computed. Additional storage similar to the storage requirements for the matrix A is needed. We also implemented the row-variant of SPAI, denoted by PBSPAIrow (0). As for the ILU preconditioner, there exist variants in which additional fill-in is allowed, cf. [20].

4.2 Numerical Experiments We present results of numerical experiments. Our goal is to illustrate and to compare the behavior of the different preconditioners presented above for both test problems. In test problem 1, the Jacobian is evaluated at the discrete solution U∗ . The solution is trivial, namely, constant. The solution is a complex flow field in test problem 2. In the latter linear systems with matrices as in (5) arise in the solver used in the QUADFLOW package. In all experiments below we use a left preconditioned BiCGSTAB method. For test problem 1, the discretization routines, methods for the construction of the Jacobian matrices and the preconditioners (PBGS, PBILU(0) and PBSPAI(0)) are implemented in MATLAB. For the other test problems the approximate Jacobian matrices are computed in QUADFLOW using PETSc [2]. More results are presented in [32]. 4.2.1

Arithmetic Costs

To measure the quality of the preconditioners we present the number of iterations that is needed to satisfy a certain tolerance criterion. We briefly comment on the arithmetic work needed for the construction of the preconditioner and the arithmetic costs of one application of the preconditioner. As unit of arithmetic work we take the costs of one matrix-vector multiplication with the matrix A, denoted by 1 matvec. For the PBGS method we have no construction costs. The arithmetic work per application of the PBGS preconditioner is about 0.7 matvec. In our experiments the costs for constructing the PBILU(0) preconditioner are between 2 and 4 matvecs. We typically need 1.2–1.6 matvecs per application of the PBILU(0) preconditioner. The costs for constructing the PBSPAI(0) preconditioner are much higher. Typical

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values (depending on P(A)) in our experiments are 20–50 matvecs. We typically need 1.2–1.5 matvecs per application of the PBSPAI(0) preconditioner. 4.2.2

Stationary 2D Euler

We consider the discretized stationary Euler equations as described in Paragraph 3.1 with mesh size h = 0.02. We vary the Mach number in x1 -direction, which is denoted by Mx : 0.05 ≤ Mx ≤ 1.25. For the Mach number in x2 -direction, denoted by My , we take My = 32 Mx . The BiCGSTAB iteration, initialized by the all-zero starting vector, is stopped if the relative residual is below 10−6, measured in the 2-norm. The results are presented in Figure 1. In the supersonic case (Mx > 1), due to the downwind numbering, the upper block-diagonal part of the Jacobian is zero and thus both the PBILU(0) method and PBGS are exact solvers. The PBSPAI(0) preconditioner does not have this property, due to the fact that M is a sparse approximation of A−1 , which is a dense block lower triangular matrix. For Mx < 1 with PBGS preconditioning we need about 1 to 4 times as much iterations as with PBILU(0) preconditioning. Both preconditioners show a clear tendency, namely that the convergence becomes faster if Mx is increased. For Mx < 1 the PBSPAI(0) preconditioners show an undesirable very irregular behavior, therefore we do not apply this preconditioner for test problem 2. Test case 1 300

Iterations BiCGSTAB

250

200

150

100 PBGS PBILU(0) PBSPAI(0) PBSPAI (0)

50

row

0

0.2

0.4

0.6 0.8 Mach number M

1

1.2

x

Fig. 1 Test problem 1: Iteration count for different Mach numbers

4.2.3

Stationary Flow around NACA0012 Airfoil

In the computations with the three standard NACA0012 airfoil test cases, the choice of the time step is based on an exponential strategy as already described in Paragraph 3.2.1. The linear systems with the approximate Jacobians are solved until the

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relative residual is smaller than 10−2 . In Table 3 the averaged number of preconditioned BiCGSTAB iterations for the two finest grids is given. Note that applying PBGS we need about 2–3 times as much iterations as when using PBILU(0). With PBILU(2) we save between 25% and 54% on the average iteration count compared with PBILU(0). Taking the arithmetic work per iteration into account, cf. Paragraph 4.2.1, we conclude that PBGS and PBILU(0) have comparable efficiency, whereas the PBILU(p), p = 1, 2, preconditioners are (much) less efficient due to the high memory requirements. In Section 5 we will see that an adequate renumbering technique significantly improves the situation for PBGS. Table 3 Test problem 2: Average iteration count on two finest grids test case Grid PBGS PBILU(0) PBILU(1) PBILU(2)

2A 10 11 2.89 1.33 1.04 1.00

27.0 9.83 6.07 5.09

2B 13 14 14.9 6.17 4.24 3.52

18.6 8.37 4.65 3.83

2C 10 11 6.50 2.69 1.81 1.60

20.7 6.21 2.41 3.42

4.3 Concluding Remarks We summarize the main conclusions of this section. Already for our relatively simple model problems the PBSPAI(0) method has turned out to be a poor preconditioner. This method should not be used in a Newton-Krylov method for solving compressible Euler equations. Both for model problems and a realistic application (QUADFLOW solver, test problem 2) the efficiency of the PBGS preconditioner and the PBILU(0) method are comparable. For our applications the PBILU(1) and PBILU(2) preconditioners are less efficient than the PBILU(0) preconditioner.

5 Renumbering Techniques In this section we present ordering algorithms for the PBGS preconditioner. We do not know of any literature in the context of linearized Euler equations dealing with ordering techniques for Gauss-Seidel preconditioners. The presented ordering algorithms consist of three steps. In the first step we construct a weighted directed graph in which every vertex corresponds to a block unknown and the weights correspond to the magnitude of the fluxes. This graph is usually very complex making it almost impossible to work with standard ordering techniques. Therefore, we use an approach that is very similar to coarsening techniques used in algebraic multigrid methods [37]: At first we reduce the complex graph by deleting edges with relatively small weights. Then we consider three different algorithms to determine the renumbering of the vertices of the reduced graph.

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5.1 Methods ILU and Gauss-Seidel preconditioners depend on the ordering of the cells [5, 21, 38]. This holds for their point-block variants, too. Many studies on numbering techniques for ILU preconditioners appear in the literature, cf., e.g., [18, 36] and references therein. For ILU methods, in many applications, a reverse Cuthill-McKee ordering algorithm [17] provides good results [6, 29, 34, 35]. The PBGS preconditioner can be significantly improved by reordering techniques that should be such that one approximately follows the directions in which information is propagated. In this section we introduce three renumbering methods that aim at realizing this. All three algorithms are completely matrix-based, that is, only the blockstructured matrix from (5) is needed as input. We distinguish the following three steps: 1. Construct a weighted directed matrix graph in which • every vertex corresponds to a block unknown (= cell) • every edge corresponds to a nonzero off-diagonal block of the given matrix A 2. Construct a reduced weighted directed matrix graph by • deleting edges with relatively small weights 3. Determine a renumbering of the vertices While for all three algorithms presented below steps 1 and 2 are identical, they differ in the methods used in the third step. We explain the first two steps in Paragraphs 5.1.1 and 5.1.2. In Paragraphs 5.1.3 – 5.1.5 we give the three different methods that are used in step 3 to determine the reordering. 5.1.1

Construction of Weighted Directed Matrix Graph G (A)

We introduce standard notation related to matrix graphs. Let V = {1, . . . , N} be a vertex set such that each vertex corresponds to a discretization cell. The set of edges E contains all directed edges and the mapping ω : E → (0, ∞) assigns to every directed edge (i, j) ∈ E a weight ωi j : E = {(i, j) ∈ V × V | Ai, j = 0, i = j}

,

ωi j := ω (i, j) := Ai, j F .

(9)

We take the Frobenius-norm because it is easy to compute and all entries in a block Ai, j are weighted equally. This yields a weighted, directed matrix graph G (A) := (V , E , ω ). Opposite to the commonly used definition we call an edge (i, j) ∈ E an inflow edge of vertex i ∈ V and an outflow edge of vertex j ∈ V . This is motivated by the following: In our applications, an edge (i, j) in the graph corresponds to a flow from cell j into cell i in the underlying physical problem. Consequently, for (i, j) ∈ E we call j a predecessor of i and i a successor of j. The set of predecessors of vertex i ∈ V is denoted by Ii := { j ∈ V | (i, j) ∈ E } .

(10)

In the construction of G (A) one only has to compute the weights ωi j in (9). For storage of this information we use a sparse matrix format. Note that the size of the

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sparse matrix corresponding to G (A) is N × N (and not N(d + 2) × N(d + 2), as for A). Hence, the costs both for the computation and for the storage of G (A) are low. 5.1.2

Construction of Reduced Matrix Graph Gˆ

Based on reduction techniques from algebraic multigrid methods in which strong couplings and weak couplings are distinguished [37], we separate strong edges from weak edges. For every vertex i ∈ V we neglect all inflow edges (i, j) ∈ E with a weight smaller than τ -times the average of the weights of all inflow edges of vertex i. Thus we obtain a reduced set of strong edges Eˆ and a corresponding reduced (weighted, directed) graph Gˆ(A) := (V , Eˆ , ω|Eˆ ):

σi :=

1 |Ii |

∑ ωi j

,

Eˆ := {(i, j) ∈ E | ωi j ≥ τ · σi }

(11)

j∈Ii

This simple construction of a reduced matrix graph Gˆ(A) can be realized with low computational costs. Moreover, we can overwrite G (A) with Gˆ(A). The use of graph reduction is essential for the performance of the reordering techniques discussed below. Note that the parameter τ controls the size of the reduced graph: for τ = 0 there is no reduction of the original graph, whereas for τ → ∞ the reduced graph contains only vertices and no edges. The choice of an appropriate value for the parameter τ is discussed in Subsection 5.2. In particular it will be shown that the performance of the numbering techniques is not very sensitive with respect to perturbations of the parameter value. We call τ “graph reduction parameter” below. 5.1.3

Downwind Numbering Based on (V , Eˆ ) (Bey and Wittum)

The downwind numbering algorithm due to Bey and Wittum [5], denoted by “BW”, is presented in Figure 2. This ordering is used in multigrid methods for scalar convection-diffusion problems for the construction of so-called “robust smoothers”. To apply this algorithm for our class of problems we need the reduced directed graph (V , Eˆ ) as input. Note that although they have been used to compute the reduced graph (V , Eˆ ), the weights ωi j are not used in the ordering algorithm. Remark 1. In the loop over P ∈ V in algorithm BW the ordering of the blockunknowns (cells) corresponding to the input matrix A is used. In the procedure SetF(P) a vertex is assigned the next number if all its predecessors have already been numbered. Hence, the first number is assigned to a vertex that has no inflow edges. Note that in the procedure SetF(P) there is freedom in the order in which the successors Q are processed. In our implementation we again use the ordering induced by the given matrix A. The BW numbering is applied to the reduced matrix graph. If that graph is cycle-free, the algorithm returns a renumbering that is optimal in the sense that this reordering applied to the matrix corresponding to Gˆ(A) results in a lower triangular matrix. However, in our problem class the reduced graphs in general contain cycles. In that case, after algorithm BW has finished, there still are vertices P ∈ V with Index(P)= −1, that is, there are N − nF > 0 vertices that have

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for all P ∈ V : Index(P) := −1 ; nF := 1 ; for P ∈ V if (Index(P) < 0 ) SetF(P) ; end P procedure SetF(P) if (all predecessors B of P have Index(B) > 0 ) Index(P) := nF ; nF := nF + 1 ; for Q successor of P if (Index(Q) < 0) SetF(Q) ; end Q end if Fig. 2 Downwind numbering algorithm BW

no (new) number. The numbers nF , . . . , N are assigned to these remaining vertices in the order induced by the input matrix ordering. The two variants of BW that are treated below in general have less of such “remaining” vertices. Note that in the BW algorithm there are logical operations and assignments but no arithmetic operations. 5.1.4

Down- and Upwind Numbering Based on (V , Eˆ ) (Hackbusch)

In Figure 3 we present an ordering algorithm, referred to as down- and upwind numbering and denoted by “HB”, that is due to Hackbusch [21]. As input for this algorithm one needs the reduced directed graph (V , Eˆ ). The routine “SetF” is the same as in the BW algorithm in Figure 2.

for all P ∈ V : Index(P) := −1 ; nF := 1 ; nL := N ; for P ∈ V if (Index(P) < 0 ) SetF(P) ; if (Index(P) < 0 ) SetL(P) ; end P procedure SetL(P) if (all successors B of P have Index(B) > 0 ) Index(P) := nL ; nL := nL − 1 ; for Q predecessor of P if (Index(Q) < 0) SetL(Q) ; end Q end if Fig. 3 Down- and upwind numbering algorithm HB

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Remark 2. While in the BW algorithm the vertices are ordered in one direction, namely “downwind”, that is, in the direction of the flow, the algorithm due to Hackbusch uses two directions: “downwind” (SetF) and “upwind” (SetL). The computational cost of algorithm HB is comparable to that of BW. 5.1.5

Weighted Reduced Graph Numbering Based on (V , Eˆ , ω|Eˆ )

The performance of the BW and HB numbering depend on the ordering of the input graph. We present an algorithm that uses the weights of the reduced graph to avoid the dependence on the initial ordering. The algorithm, denoted by “WRG”, is presented in Figure 4.

for all P ∈ V : Index(P) := −1 ; nF := 1 ; nL := N ; /* (i) apply SetF and SetL to starting vertices */ do in an outflow-ordered list , S(Σi ωip , P): for P ∈ V (12) if (Index(P) < 0 ) SetF(P, 1) ; end P (13) do in an inflow-ordered list , S(Σ j ω p j , P): for P ∈ V if (Index(P) < 0 ) SetL(P) ; end P /* (ii) number remaining vertices */ do in an outflow-ordered list , S(Σi ωip , P): for P ∈ V if (Index(P) < 0 ) SetF(P, 0) ; end P

(14)

procedure SetF(P, s) if (all predecessors B of P have Index(B) > 0 ) or (s = 0) Index(P) := nF ; nF := nF + 1 ; do in an outflow-ordered list , S(Σi ωiq , Q): for Q successor of P if (Index(Q) < 0) SetF(Q, 1) ; end Q end if

(15)

procedure SetL(P) if (all successors B of P have Index(B) > 0 ) Index(P) := nL ; nL := nL − 1 ; do in an inflow-ordered list , S(Σ j ωq j , Q): for Q predecessor of P if (Index(Q) < 0) SetL(Q) ; end Q end if

(16)

: p denotes the index of the vertex P of the input graph. S(Σi ωip , P) sorts the vertices P descending in the corresponding values Σ i ωip (similar for S(Σ j ω p j , P)).

Fig. 4 Weighted reduced graph numbering algorithm WRG

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There are two important differences to the algorithms HB and BW. The first difference is related to the arbitrariness of the order in which the vertices are handled in the loops in HB and BW, cf. Remark 1. If there are different possibilities for which vertex is to be handled next we now use the weights ωi j of the reduced graph to make a decision. This decision is guided by the principle that edges with larger weights are declared to be more important than those with relatively small weights. A weight based sorting occurs at several places, namely in (12) – (16). In (12) the vertices with no inflow edges (“starting” vertices) are sorted using the sum of the weights of the outflow edges at each vertex. Similarly, in (13) the vertices with no outflow edges are sorted. The “remaining” vertices, that is, all vertices that have inflow and outflow edges, are finally sorted based on the sum of the outflow edges at each vertex in (14). In all three cases the number of vertices to be sorted is much smaller than N and thus the time for sorting is acceptable. Sorting is also used in (15) and (16) to determine the order in which successors and predecessors are handled. In SetF(·, ·) the successors Q of the current P are sorted using the sum over the weights of all outflow edges for each Q. This is done similarly in SetL(·) for all predecessors of the current P. The second difference is that the loop over the numbering routine SetF is called two times. The first call SetF(P, 1) in part (i) of algorithm WRG is similar to the call of SetF(P) in the algorithms BW and HB but now with an ordering procedure used in SetF. The second call SetF(P, 0) (in part (ii) in WRG) is introduced to handle the remaining vertices that still have index value −1. In this call we do not consider the status of inflow edges and continue numbering in downwind direction (SetF(·,0)). The inner call SetF(Q, 1) to number the successors still requires that all predecessors have been numbered. After part (ii) of the algorithm is finished the only possibly not yet numbered vertices are trivial ones, in the sense that these are vertices that have no edges to other vertices. Note that although the first part of this numbering (cf. (i) in Figure 4) can be also obtained by applying HB to an a-priori sorted graph, the second step (ii) of WRG does neither have a counterpart in HB nor in BW. Remark 3. In all three algorithms the computational time that is needed and the storage requirements are modest compared with other components of the iterative solver. Moreover, since the Jacobian matrices of consecutive time steps are in some sense similar we apply the reordering not in each iteration but only “now and then” and keep it for the subsequent time steps, cf. Subsection 5.2. Because of the infrequent application of the numbering the total execution time for the reordering routines is very small compared with the total time needed.

5.2 Numerical Experiments We illustrate the behavior of four different numberings for a few test problems. The BW, HB and WRG methods have been explained above. The fourth numbering, denoted by QN, is induced by multiscale analysis that is used for error estimation and generates local refinement leading to a hierarchy of locally refined grids. In the

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QN numbering the cells are numbered level-wise from the coarsest to the finest level resulting in a sort of hierarchical block-structure of the matrix. For efficiency reasons we do not apply the renumbering method (steps 1–3) to every new Jacobian but use the known renumbering as computed in the first time step. All three numbering techniques are sensitive with respect to the choice of the value for the parameter τ . In our sub- and supersonic problems τ = 1.25 turned out to be a good default value. In highly transonic problems (M∞ ≈ 1) the performance can often be improved by taking a somewhat large τ -value (e.g., τ = 2.00). Table 4 shows the average iteration count on the finest level for the different orderings. The average is taken over all time steps that are needed to achieve convergence on the finest discretization level for test cases 2A, 2B, and 2C. The savings compared with the QN ordering are displayed in the last rows of Table 4. In all numerical experiments the reduced matrix graph was constructed with τ = 1.25. For test case 2C we give the graph G (A) and the corresponding renumbered reduced graph of a typical Jacobian matrix in Figure 5. Table 4 Test problems 2A, 2B, and 2C: Average iteration count on finest level

Numbering

Test case 2A QN BW

HB

WRG

Average iteration count 32.0 30.6 28.6 23.0 Saving 0% 4.4% 10.6% 28.1%

Numbering

Test case 2B QN BW

HB

WRG

Average iteration count 20.2 20.1 18.2 18.4 Saving 0% 0.5% 9.9% 8.9%

Numbering

Test case 2C QN BW

HB

WRG

Average iteration count 24.2 12.5 12.6 10.9 Saving 0% 48.3% 47.9% 55.0%

Using the WRG renumbering method we save between 9% and 55% of PBGSpreconditioned BiCGSTAB iterations on the finest level compared with the original numbering QN. Since the renumbering has to be computed only once, the additional computational costs for WRG are negligible. The improvement is strongest for case 2C, which is due to the fact that in this case the flow is almost supersonic and thus there is a main stream in which information is transported. For test case 2C we illustrate the dependence of the iteration count on the graph reduction parameter τ . In Figure 6 the results for τ = 0.25 · k, k = 0, 1, . . . , 12 are given. The dashed line (right y-axis) shows that the number of edges in the corresponding reduced graph of the Jacobian is decreasing monotonically if the value of τ is increased. Table 5 shows how many vertices are renumbered in each of the steps (i) and (ii) in the WRG algorithm, cf. Figure 4. For values τ ≤ 0.75 the reduced

Iterative Solvers for Discretized Stationary Euler Equations Test case 2C − Graph

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Fig. 5 Test problem 2C: Graph G (A) (left) and renumbered reduced graph (right) of Jacobian matrix on finest grid, τ = 1.25

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Fig. 6 Test case 2C, finest computational grid: Experiment with different values for the graph reduction parameter τ . Average iteration count using WRG numbering (dashed, right axis) and number of edges of the corresponding reduced graph (solid, left axis) Table 5 Test problem 2C, finest computational grid: Different values for τ . Number of cells that were numbered in steps (i) and (ii) in WRG algorithm, cf. Figure 4 Step of WRG

τ τ τ τ

≤ 0.75 = 1.00 = 1.25 = 1.50

(i) (ii) 0 40 213 11 153 29 060 39 869 344 40 205 8

Step of WRG

τ τ τ τ

= 1.75 = 2.00 = 2.25 ≥ 2.50

(i) (ii) 40 207 40 211 40 211 40 213

6 2 2 0

graph is too complex so that in the first step, cf. (i) in Figure 4, none of the 40 213 vertices is given a new number. On the other hand with τ = 1.50, 98% of the vertices are given a new number in the first step (i), so that a further increase of the value for τ would be counterproductive. The dashed line in Figure 6 representing the performance of the preconditioned BiCGSTAB method indicates that the choice

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of the value for τ is not very sensitive. For this test case values 0.75 ≤ τ ≤ 2.00 all give quite good results. We obtain the best results for τ = 1.25; in this case the reduced graph Gˆ(A) can be reordered so that it is nearly a lower-diagonal graph as shown in the right subplot of Figure 5. The effect of the reordering is that the dominant entries of the reordered Jacobian lie mostly in the lower triangular part. It should be noted that such reordering techniques can only be effective for point-block matrices in which there is a significant difference between Ai, j F and A j,i F for most i, j with Ai, j F = 0. Further results are presented in [33].

5.3 Concluding Remarks We have presented ordering techniques for the PBGS method that use ideas from algebraic multigrid methods. Except for the (critical) graph reduction parameter τ in (11), the ordering methods are “black-box”. In most test cases a good choice for this grid-reduction parameter has turned out to be τ = 1.25. Only one reordering per adaptation level has been needed neglecting the additional costs of the ordering algorithm. Using the WRG reordering one can improve the robustness of and the efficiency of the linear solver.

6 Time Integration In the pseudo-transient continuation [26], large time steps in an implicit time discretization method are preferred to achieve fast convergence. On every level of adaptation we start with an initial CFL number which determines the first time step. The local time step Δ ti for the i-th cell is given by

Δ ti = γ

|Ω i | , λic

λic =

∂ Ωi

(|vn| + c) dS ,

(17)

where γ is the CFL number [16] and λic is the maximum eigenvalue of the Euler equations averaged over the bounding surface of the control volume Ωi , cf. [7]. During the time integration the CFL number is varied by one of the three strategies described in Subsection 6.1. In every time step a non-linear system of equations has to be solved. Note that the Jacobian has a structure and thus in general a smaller time step will improve the condition number of the approximated Jacobian in the Newton-Krylov method. |Ωi | ∂ R(u) + . J(u) = diag Δ ti ∂u

(18)

6.1 CFL Evolution Strategies Implicit time integration methods in principle allow large time steps (γ > 1). For steady flows the CFL number γk = γ (k) at a time step k is usually varied in a

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prescribed interval γk ∈ [γmin , γmax ]. With small CFL numbers γk one has to perform many time steps in order to achieve convergence. Choosing the CFL number γk too large may result in a breakdown of the iteration process. 6.1.1

Exponential Progression (EXP)

The exponential law (EXP)

γk+1 = γ0 · (γEXP )k , k ∈ N0

(19)

increases the CFL number in a regular manner, also used, e.g., in [24, 39]. The control parameters γ0 and γEXP completely determine a sequence of CFL numbers. Appropriate values for the control parameters are problem-dependent and in general not known a priori. Typical values are γ0 = γmin (= 1.0), and γEXP ∈ [1.05, 1.5]. 6.1.2

Switched Evolution Relaxation (SER)

The Switched Evolution Relaxation (SER) [30] method often appears in the context of compressible flow, cf., e.g., [7, 15, 27, 39]. The norm of the density residual, denoted by Rk , is directly coupled with the CFL number of the following time step:

αSER R0 γk+1 = γSER · , k ∈ N0 (20) Rk This way, the sequence of CFL numbers selected by the SER method is not only determined by the choice of the control parameters γSER and αSER but also depends on the particular flow problem at hand. In this study we use αSER ∈ [1.0, 5.0] and γSER = γmin (= 1.0) as in [7]. A coupling of the CFL number with the residual is also used in the RDM strategy: 6.1.3

Residual Difference Method (RDM)

If the solutions seem to stagnate while evolving to a steady-state solution one may increase the size of the time step to check if the solution remains stable. Since in QUADFLOW the residual is taken as an estimate for the error, we suggest the following evolution strategy, referred to as Residual Difference Method (RDM), as an alternative to the SER method: ⎧ ⎪ γ if k < k0 , ⎪ ⎨ min αRDM

γk+1 = (21) k ∈ N0 1 ⎪ ⎪ ⎩γRDM · if k ≥ k0 , |Rk − Rk−1| where k0 ≥ 1 denotes the first index satisfying Rk0 ≤ Rk0 −1 − εRDM . The control parameters are γRDM and αRDM , and –like in the EXP and SER strategies– they must be selected carefully in order to obtain rapid convergenceand to avoid breakdowns. In this study, we set εRDM = 10−2 and choose γRDM ∈ {1, 2, 5} and αRDM ∈ [0.6, 6.0].

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6.2 Numerical Experiments We present results of numerical experiments using the different CFL evolution strategies described in Subsection 6.1, where the CFL numbers are allowed to vary in the interval [γmin , γmax ] = [1, 105 ]. We also investigate a test case mimicking a nonadaptive scheme, called 2n A, in which the calculation on the finest adaptation level of test case 2A is initialized with free instream conditions rather than an interpolated solution of the previous adaptation level. The point-block-ILU(0) method is the preconditioner chosen for all tests in this section. More results are presented in [10]. 6.2.1

Parameter Study on the CFL Control Parameters

We present the results of a parameter study on the CFL control parameters for the three evolution strategies. Here, the parameters γEXP , αSER , αRDM , and γRDM are varied, while γ0 and γSER are assigned the fixed value 1.0. Table 6 Test cases 2A, 2B, and 2C: Time steps needed for convergence on finest grid and corresponding CPU times in seconds for different values for the control parameters Test Case 2A SER

EXP

RDM

γEXP # ts CPU αSER # ts CPU αRDM γRDM # ts CPU 1.1 10 15 20 50

124 40 37 38 35

80.8 36.0 31.6 32.4 33.4

1.1 4.0 4.5 5.0 10.0

172 39 37 34 †

121.7 32.9 27.2 28.5 –

1.0 3.0 4.0 5.0 6.0

Test Case 2B SER

EXP

1 1 1 1 1

90 42 39 36 36

67.2 34.5 29.3 28.4 29.0

RDM

γEXP # ts CPU αSER # ts CPU αRDM γRDM # ts CPU 1.1 3.0 5.0 10 15

115 56 46 49 41

204.1 123.7 101.9 115.3 125.3

1.1 271 406.2 3.0 58 135.7 4.0 51 117.8 4.5 41 105.6 5.0 47 105.1

1.0 2.0 3.0 5.0 6.0

Test Case 2C SER

EXP

1 5 1 1 1

117 51 49 50 50

216.5 112.9 113.7 112.3 111.7

RDM

γEXP # ts CPU αSER # ts CPU αRDM γRDM # ts CPU 1.1 10 15 20 100

125 70 69 69 69

292.8 1.1 211.4 3.5 204.8 4.0 207.8 5.0 212.9 10.0

120 71 70 69 68

285.5 211.5 209.8 211.1 199.1

1.0 3.0 4.0 5.0 10.0

1 1 1 1 1

85 71 70 70 70

237.7 208.3 205.5 210.5 208.7

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The results for test case 2A are displayed in the upper tabular of Table 6, showing, for varying parameter values, the number of time steps needed to achieve convergence, denoted by “# ts”, as well as the actual CPU time, given in seconds. All timing results are obtained on an Intel Xeon processor running at 3 GHz clock speed. It turns out that, for all three CFL evolution strategies, the choice of the control parameters has a great impact on the number of time steps needed for convergence and the total execution time. For each CFL evolution strategy, the “best” parameter values are given in bold. Results for test cases 2B and 2C are given in the other tables of Table 6. The results show that using other than the “best” values for the control parameters may lead to a doubling of time steps needed and corresponding CPU time. Because the results are equally efficient for all methods, for these test cases, the choice of the CFL evolution strategy is not the crucial factor. 6.2.2

Results Mimicking a Non-adaptive Scheme

The situation is, however, different in test case 2n A. The results of a corresponding parameter study are presented in Table 7. It seems more difficult to find feasible values for the control parameters because values assumed to be appropriate for test case 2A may not even yield a converging iteration process. Such a divergence in the iteration process is indicated by “†” and “–” in the table. Although EXP with γEXP = 1.09 yields the fastest convergence process, when increasing the parameter γEXP to 1.1 or higher the iteration process diverges. Compared to EXP, the SER strategy leads to a much slower convergence. The reason for this slow convergence of SER is that it chooses only relatively small CFL numbers γk in the first 1200 iterations. For a feasible pair of values, RDM is significantly faster than SER. Table 7 Test case 2n A: Time steps needed for convergence on finest grid and corresponding CPU times in seconds EXP

SER

RDM

γEXP # ts CPU αSER # ts CPU αRDM γRDM # ts CPU 1.05 1.08 1.09 1.10 1.11

189 116.3 134 93.0 124 87.6 † – † –

2.6 1354 322.6 2.7 † – 2.8 1329 314.0 2.9 1320 314.3 3.0 † –

0.80 0.94 0.95 0.98 1.00

1 1 1 1 1

290 151.0 186 120.8 † – 168 100.5 † –

We conclude from this experiment that SER is not suitable for non-adaptive schemes. Therefore, we rather recommend using an exponential law (EXP) or RDM.

6.3 Locally Optimal CFL Numbers Since there is no clear winner among the three basic strategies, a different approach is presented in this subsection. Reconsider that the relative density residual is used in the stopping criterion in QUADFLOW. For each time step k we define a function

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−3

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Fig. 7 Relative residual of density Rk for different values γk for test case 2A on the finest grid. The residuals Rk for time steps k = 13 to k = 15 and for k = 16 to k = 18 are shown in the left and right subplot, respectively. The CFL numbers γ1 , . . . , γk−1 for the first k − 1 time steps are selected by the LOC strategy, that is, by approximating (23)

Rk : R+ → R+ ,

γk → Rk (γk )

(22)

which maps a CFL number γk to the norm of the density residual Rk that is obtained after performing this iteration using γk . Some typical plots of the function (22) are given in Figure 7. Apparently the shape of the functions does not change much from one time step to the following time step. The idea is to find the best CFL number in every iteration, that is, we determine a value for γk such that the residual gets as small as possible: Rk (γk ) = min Rk (γ ). γ ∈R+

(23)

Note that the residual Rk in the k-th iteration depends not only on γk , but also on the CFL numbers γ1 , . . . , γk−1 used in the previous iterations. In order to test this approach, we implemented a heuristic search strategy approximating γk , denoted by LOC in the sequel, where in each iteration several trial steps using different values for CFLare carried out. In the neighborhood of the CFL value yielding a minimal residual further trial steps are performed. From the set of CFL numbers tested during this heuristic search, the best CFL number, that is, the value γk that yields the smallest residual, is then employed to perform the actual iteration. This method is (very) expensive and therefore only of theoretical concern. This approach can be used in a faster strategy using derivative information of the function (22) so that no additional trial steps have to be carried out, cf. Remark 4. The picked CFL numbers γk to actually perform time step k is the one that corresponds to the smallest residual Rk . As indicated in Figure 7, for time steps 13 − 18 a clear decrease of the residuals Rk can be observed in every time step.

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Test case 2A

3

Test case 2A

10

EXP LOC(2) LOC(20) LOC(40) LOC(80)

10

EXP LOC(2) LOC(20) LOC(40) LOC(80)

−2

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Fig. 8 Test case 2A: CFL numbers γk selected by EXP and LOC(nLOC ), for various nLOC (left), and corresponding residual history Rk (right)

However, it turns out that this method does not necessarily decrease the total number of iterations. In fact, the total number of iterations needed for convergence is typically larger than if any of the other methods described in section Subsection 6.1 were used. Results for the “pure” EXP strategy and switches after nLOC − 1 time steps to the LOC strategy, denoted by LOC(nLOC ), reveal that, as soon as the LOC strategy is initiated, the relative density residual decreases quite fast. In subsequent iterations, the rate of decrease of the residual gets smaller such that almost no progress can be observed. In the long run, the pure EXP strategy yields faster convergence although the residual actually increases during several iterations. This result is presented in Figure 8. Remark 4. In the selected example, test case 2A, a closer look at the convergence behavior corresponding to Figure 8 shows that the position of the shock is slightly moving during the time integration. This can be interpreted as a “coarse grid effect”. Avoiding this effect can be achieved by a more precise solution on the previous grid leading to a better initial solution on the finest grid. A decrease of the tolerance on the next coarser grid to ε1 = 10−3 yields a fast convergence for the LOC(2) strategy, denoted by LOC, for test case 2A. Because the LOC strategy is very expensive, we compare the results not only with the EXP strategy but also with an approximation of the LOC strategy, denoted by ADL. This approximation is feasible thanks to the similar shapes of the functions (22), cf. Figure 7. The ADL strategy uses two derivatives of the function (22) obtained by automatic differentiation and approximates γk → Rk (γk ) by a quadratic polynomial.

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This approach works fine in some of the test cases, however, it does not perform satisfactory for test case 2C. Figure 9 shows the residual history for the three strategies for test cases 2A (left subplot) and 2C (right subplot). Remark 5. Reconsidering the right subplot of Figure 8, it seems that a fast iteration process must allow an increase of the residual Rk . We have noted in Paragraph 6.2.2 that this fact results in a slow convergence behavior when using the SER method. A similar effect eventually yields very small CFL numbers when using the LOC(nLOC ) strategy. In test case 2n A, the calculation on the finest adaptation level of test case 2A was initialized with free instream conditions. Apparently the initial residual vanishes in most of the cells. As reported in [26] an increase of the residuals is natural. Even in an adaptive computation, the same effect can occur. To better understand the impact of the CFL numbers on the residuals, a sensitivity analysis has been carried out in [10] using automatic differentiation for evaluating sensitivities without additional truncation error. The analysis has confirmed that CFL control is a subtle issue and that the three basic strategies have comparable sensitivities.

6.4 Concluding Remarks The results have shown that the best strategy does not have to locally minimize the density residuals as much as possible in every time step, and that even an increase of the residual must be accepted in order to achieve rapid overall convergence. A new CFL evolution strategy, called RDM, has been introduced and compared with the existing strategies EXP and SER. For the residual-based strategies SER and RDM, RDM has turned out to be faster than SER. Currently, application-specific knowledge, intuition, and trial and error are still needed in order to determine appropriate Test case 2A 10 EXP LOC ADL 10

−2

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Test case 2C

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−3

−4

20

40

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80 100 Time step k

120

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Fig. 9 Test cases 2A (left) and 2C (right): Residual histories on finest grid when solving on coarser levels to the smaller tolerance ε1 = 10−3 , EXP, LOC, and ADL strategies

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values for the CFL control parameters. Using all CFL evolution strategies within an expert system like advocated in [39] may improve this situation.

7 Matrix-Free Methods for Second Order Jacobians Matrix-free evaluations of matrix-vector products are popular because the system matrix does not have to be stored and thus, one can simulate problems with larger stencils or that would not fit into memory when explicitly building the Jacobian. The Krylov subspace method does not require the system matrix J in (18) explicitly but needs the evaluation of the product of J with some given vector x ∈ R(d+2)N . Hence, an evaluation of the Jacobian-vector product can be realized without actually storing the Jacobian matrix. In contrast to the approach of using divided differences to approximate the Jacobian-vector product we use AD that does not produce any additional truncation error. Thus, the derivatives can be computed more accurately than any approach using divided differences. The computational effort to compute a Jacobian-vector product by the forward mode of AD is typically similar to the computational cost of a first order divided difference approximation. For additional information on automatic differentiation and the actual implementation we refer to the article by Arno Rasch within this issue of the book.

7.1 Numerical Experiments It is known from experience, as reported in, e.g., [29, 31], that a benefit from second order methods can only be expected after a certain number of time steps have been elapsed. Usually in the early iterations a first order implementation of a matrixvector product is faster and more robust. In our numerical experiments we therefore usually switch at a certain threshold ν for the density residual between the different methods. A similar approach is followed in [6, 29]. In the actual implementation we do not switch back from the higher order method to the lower-order method if the residual increases again during the computation. If kν denotes the first time step satisfying Rkν −1 ≤ ν the Jacobian approximations are as follows: , Jlow Δ u = −Rhigh , k < kν (24) Jhigh Δ u = −Rhigh , k ≥ kν In the following subsection we also use a variant in which we switch at an a-priori prescribed time step kν . This approach is also used, e.g., in [31]. If not stated otherwise, we use the PBILU(0) preconditioner for the first order methods and a PBILU(2) preconditioner, based on the first order Jacobian, for the matrix-free second order method. Other results are presented in the article by Gero Schieffer within this issue of the book and in [9].

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7.1.1

Newton Convergence

We investigate the impact of the CFL number on the performance of Newton’s method using the first and second order methods by carrying out a fixed number of time steps using the first order matrix-based method. In this simulation the corresponding value for kν was chosen to be kν = 20 (upper row of Figure 10) and kν = 80 (lower row of Figure 10). Thereafter we perform 20 Newton steps with the matrix-based first order method (left subplots) and matrix-free second order method (right subplots). The plots show the Newton iteration history for different CFL numbers γ = 100 , 101 , . . . , 104 . The first order method shows linear or slower than linear convergence for all tested values for γ in both time steps (cf. left plots in Figure 10). The larger the CFL number is selected, the slower the convergence of Newton’s method is. This is expected because for larger time step sizes the non-linearity of the corresponding

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non-linear system of equation is increasing. Note that Newton’s method converges only locally and the corresponding initial guesses have to be in the region for that the non-linear method converges. In time step 20 the flow is still in its startup phase, that is, most flow features (such as position of shocks) are not resolved, and thus even a a divergence of Newton’s method may occur. Nevertheless, all 20 Newton iterations can be performed without a breakdown (cf. Subsection 6.1). As expected, the convergence for Newton’s method is better in time step 80. Although the exact Newton method has the advantage of local quadratic convergence, it may be counterproductive to use it in the start-up phase of the computation. This is demonstrated by the fact that two graphs in the upper right subplot of Figure 10 show that the iteration process does not converge for some higher values for γ when using the second order method. This divergence occurs if the initial guess for Newton’s method is too far away from the corresponding solution or the Krylov solver diverges when solving the corresponding linear system of equations. However, typical values at time step 20 are γ ∈ [1, 10] and it can be observed that for γ = 1 and γ = 10 the corresponding convergence of Newton’s method is significantly faster (than for the first order method). The second order method does not show any divergence and the corresponding convergence is significantly faster in time step 80. Although the convergence of Newton’s method is faster for smaller values of γ in any case —which is related to the fact that the non-linearity of (18) increases with larger time step sizes— in the computation with kν = 80, the second order method shows a significant faster convergence for all —and especially for large— CFL numbers than the first order method, as shown in the two plots in the lower row of Figure 10. This is a major benefit of the second order method: Towards the end of the computation a significant acceleration of the time integration process can be achieved also due to the selection of larger CFL numbers γ . 7.1.2

Acceleration of Time Integration

We study the impact of ν in order to reduce the execution time of the time integration. As for all stationary test cases in the previous sections, only one Newton step is performed per time step. In Figure 11 numerical results using an exponential CFL strategy with γEXP = 1.5, γMAX = 106 are presented. We plotted the residual history in terms of iteration number (left subplot) and the corresponding execution time (right subplot) for the matrix-based first order method and the second order method. We give results for different switch tolerances ν . It can be observed that the second order methods need in general less iterations. But the effect of the selection of ν is crucial: ν has to be selected small enough so that not most of the iterations are performed by the —potentially slower— first order method. On the other hand, if the switch tolerance ν is chosen too small the number of iterations is only insignificant smaller resulting in a bad CPU behavior. We also show a plot for the first order matrix-free variant.

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7.2 Concluding Remarks The convergence of Newton’s method is significantly better for the second order method than for the first order method, especially when using larger time step sizes. This allows a clear reduction in the time steps needed for achieving convergence. However, in the test problems for the stationary two-dimensional Euler equations, a benefit from the second order method can only be achieved in the final iteration process We have shown that switching at the “right” time, i.e. ν = 10−5, from the first order method to the second order method can speed up the overall computation process compared with the “pure” first order matrix-based method.

8 Outlook We conclude with some remarks on topics that could be considered in future work in this research area. The first two aspects address preconditioning, the last item suggests an expert system for time stepping and switching between first order matrixbased and second order matrix-free methods. Parallel Preconditioners The preconditioner and the corresponding ordering routines have to be adapted and optimized for a fully parallel version of QUADFLOW. The use of parallel ILU-type preconditioners is possible if a multi-color ordering or subdomain preconditioning is used. These approaches are investigated and compared in, e.g., [4]. For a parallel preconditioner a Newton-Krylov-Schwarz algorithm [19] using overlapping domains could be used. Such additive or multiplicative Schwarz preconditioners can be viewed as an overlapping block-Jacobi or block-Gauss-Seidel preconditioner, respectively [3]. This technique is widely-used in the context of partial differential

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equations as reported in many articles in the proceedings on the annually conferences on domain decomposition methods. A cheap and fast variant, the Restricted Additive Schwarz Method (RAS, RASM) by Cai and Sarkis [13], that is also integrated in the PETSc library [2], has been successfully combined with a NewtonKrylov method within the context of flow solvers [12, 19, 22, 23]. As demonstrated in [19] the RAS method can also be combined with the PBILU(0) algorithm for the subdomains. While the RAS method is used as preconditioner for the linear systems in [13], Schwarz preconditioners can also be used as a preconditioner for the non-linear systems of equations as presented in [11]. This non-linear technique was applied to a one-dimensional compressible flow, denoted by “additive Schwarz preconditioned inexact Newton method” (ASPIN), in [12] and has also been successfully used in [23]. ASPIN is a non-linear block-Jacobi iteration followed by a Newton linearization. This non-linear Schwarz preconditioner could significantly enlarge the region for that the non-linear solver converges compared with Newton’s method. Matrix-Free Preconditioners The matrix-based preconditioner PBGS, in which Jlow is used for the computation of the preconditioner, can be replaced by some kind of matrix-free preconditioner in the second order matrix-free implementation of the matrix-vector product. A general approach for building a matrix-free preconditioner can be found in [14]. The use of a second order matrix-free preconditioner could certainly reduce the storage requirements for the first order matrix-based preconditioner. One could also implement a symmetric variant of PBGS, such as the matrix-free LU-SGS preconditioner which is proposed in [28]. A symmetric PBGS-type preconditioner can also be used with the described WRG ordering. In principle the renumbering technique works in a matrix-free context because only the relatively small reduced graph has to be stored. Expert Systems for Time Integration The implicit time integration process may be automated by some kind of advanced expert system leading to a kind of “black-box” CFL evolution strategy. A basic expert system is proposed in [39]. One can think of a complex expert system including all basic strategies, the ADL strategy, plausibility checks, a breakdown control, as well as repetitions of time steps or the use of multiple Newton steps. In a more advanced expert system different switches between the CFL evolution strategies and the first and second order methods can be realized including also switches between different preconditioners and Krylov methods.

Acknowledgment The research for this article has been performed with funding by the Deutsche Forschungsgemeinschaft (DFG) in the Collaborative Research Center SFB 401 “Flow Modulation and Fluid-Structure Interaction at Airplane Wings” of RWTH Aachen University. We acknowledge the fruitful collaboration with several members of the QUADFLOW research group.

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20. Grote, M.J., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM Journal on Scientific Computing 18(3), 838–853 (1997) 21. Hackbusch, W.: On the feedback vertex set for a planar graph. Computing 58, 129–155 (1997) 22. Hovland, P.D., McInnes, L.C.: Parallel simulation of compressible flow using automatic differentiation and PETSc. Parallel Computing 27(4), 503–519 (2001) 23. Hwang, F.-N., Cai, X.-C.: A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations. Journal of Computational Physics 204(2), 666–691 (2005) 24. Issman, E., Degrez, G., Deconinck, H.: Implicit upwind residual-distribution Euler and Navier-Stokes solver on unstructured meshes. AIAA Journal 34(10), 2021–2028 (1996) 25. Jones, D.J.: Reference test cases and contributors. In: AGARD–AR–211: Test Cases for Inviscid Flow Field Methods. Advisory Group for Aerospace Research & Development. Neuilly-sur-Seine, France (1986) 26. Kelley, C.T., Keyes, D.E.: Convergence analysis of pseudo-transient continuation. SIAM Journal on Numerical Analysis 35(2), 508–523 (1998) 27. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications. Journal of Computational Physics 193(2), 357–397 (2004) 28. Luo, H., Sharov, D., Baum, J.D., L¨ohner, R.: Parallel unstructured grid GMRES+LUSGS method for turbulent flows. AIAA Paper 2003–0273 (2003) 29. Manzano, L., Lassaline, J.V., Wong, P., Zingg, D.W.: A Newton-Krylov algorithm for the Euler equations using unstructured grids. AIAA Paper 2003–0274 (2003) 30. Mulder, W.A., van Leer, B.: Experiments with implicit upwind methods for the Euler equations. Journal of Computational Physics 59(2), 232–246 (1985) 31. Nejat, A., Ollivier-Gooch, C.: Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations. Journal of Computational Physics 227(4), 2366–2386 (2008) 32. Pollul, B.: Preconditioners for linearized discrete compressible Euler equations. In: Wesseling, P., O˜nate, E., P´eriaux, J. (eds.) Proceedings of the European Conference on Computational Fluid Dynamics ECCOMAS, Egmond aan Zee, The Netherlands (2006) 33. Pollul, B., Reusken, A.: Numbering techniques for preconditioners in iterative solvers for compressible flows. International Journal for Numerical Methods in Fluids 55(3), 241–261 (2007) 34. Pueyo, A., Zingg, D.W.: Efficient Newton-Krylov solver for aerodynamic computations. AIAA Journal 36(11), 1991–1997 (1998) 35. Saad, Y.: Preconditioned Krylov subspace methods for CFD applications. In: Habashi, W. (ed.) Solution techniques for Large-Scale CFD-Problems, pp. 139–158. John Wiley & Sons, New York (1995) 36. Saad, Y.: Iterative methods for sparse linear systems, 2nd edn. SIAM, Philadelphia (2003) 37. St¨uben, K.: An introduction in algebraic multigrid. In: Trottenberg, U., Osterlee, C., Sch¨uller, A. (eds.) Multigrid, pp. 413–532. Academic Press, London (2001) 38. Turek, S.: On ordering strategies in a multigrid algorithm. In: Hackbusch, W., Wittum, G. (eds.) Proc. 8th GAMM-Seminar, Kiel. Notes on Numerical Fluid Mechanics, vol. 41. Vieweg, Braunschweig (1997) 39. Vanderstraeten, D., Cs´ık, A., Rose, D.: An expert-system to control the CFL number of implicit upwind methods. Technical Report TM 304, Universiteit Leuven, Belgium (2000) 40. Wong, P., Zingg, D.W.: Three-dimensional aerodynamic computations on unstructured grids using a Newton-Krylov approach. Computers & Fluids 37(2), 107–120 (2008)

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Unsteady Transonic Fluid – Structure – Interaction at the BAC 3-11 High Aspect Ratio Swept Wing P.C. Steimle, W. Schröder, and M. Klaas

*

Abstract. A comprehensive analysis of experimental results from several wind tunnel test campaigns is presented. The experiments were conducted in the subproject B6 of the Collaborative Research Center SFB 401. The main focus of this research is the interaction of unsteady aerodynamic phenomena in transonic flow over a supercritical BAC 3-11/RES/30/21 high aspect ratio swept wing configuration in the context of aeroelastic instabilities occurring at modern transport type wing configurations. The fluid-structure-interaction is simulated using a simplified aeroelastic test setup with harmonic forcing in the bending and torsional wing degree of freedom. The interaction between shock wave and turbulent separation is the most essential feature in the unsteady wing flow leading to a distinct selfinduced oscillation of the flow field. Flow cases with incipient separation and with full scale shock induced separation at Mach numbers 0.86 and 0.92 are compared showing the interference effects of self-induced shock oscillations with the harmonically oscillating wind tunnel model. The results demonstrate the self-limiting nature of the unsteady flow with incipient separation and the aeroelastic coupling in the presence of shock induced separation.

1 Introduction The highly non-linear character of the transonic flow field is a major reason for the occurrence of aeroelastic instabilities at transport type wing configurations under cruise flight conditions. The unsteady nature of the viscous interaction of a shock wave with the wing surface boundary layer, in some cases involving separation, induces a time-dependent load distribution on the wing structure. This may result in a distinctive structural response to the fluctuating pressure distribution. Modern transport aircraft wings with sweep, taper, and high aspect ratio typically exhibit the structural response in the first coupled bending – torsion mode. Although the high aspect ratio swept wing is generally susceptible to destructive P.C. Steimle · W. Schröder · M. Klaas Institute of Aerodynamics, RWTH Aachen University, Wüllnerstraße 5a, 52062 Aachen, Germany e-mail: [email protected] *

W. Schröder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 325–361. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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bending – torsion flutter [1], the energy transfer rate from the flow to the wing structure is limited by the aerodynamic non-linearity arising from the shock dynamics to reduce the flutter response to a limit cycle oscillation (LCO) with moderate amplitude. Transonic limit cycle flutter has been subject to numerous experimental investigations and numerical studies, mainly of the two-dimensional flow case [2]-[8]. Although these research activities focus on a variety of aspects of the aeroelastic phenomenon, there is a general agreement on the mechanism limiting the energy transfer to the wing structure, thus preventing a destructive flutter amplitude growth to occur. The transonic flow field itself has the ability to provide a certain throttling of the structural excitation through a change from harmonic shock motion (type A in [9]) to interrupted shock motion (type B) [7]. The viscous shock interaction with the boundary layer is speculated by Schewe et al. to have a similar effect due to the occurrence of a trailing edge separation of intermittent character, which causes a damping of oscillation amplitudes in the two-dimensional flow case by a shock induced and trailing edge separation growing together in some part of the oscillation cycle [4]. The intention of this research project is to extend the study of viscous twodimensional aeroelastic interaction, which was conducted by Hillenherms et al. [10]-[12] with the BAC 3-11 aerofoil to the three-dimensional transonic flow domain. At the swept wing, the structural washout, i. e., the bending – torsion coupling due to the swept elastic wing axis, has an additionally stabilizing effect on the flutter response [13]. Nevertheless, aeroelastic instabilities can be observed on swept wing configurations [14], because the reduced damping in transonic flow gives rise to a dynamic structural response to and interaction with disturbances in the surrounding flow field. Several experimental test campaigns have been conducted at the Institute of Aerodynamics with special attention directed to the dynamic shock-boundary layer interaction in the context of dynamic aeroelastic behavior. To understand transonic flutter on the aerodynamic side, the highly complex system of aero-structural action and reaction is simplified to a predefined harmonic motion of the wing. The structural response to the aerodynamic forces is suppressed by the high bending and torsional stiffness of the wing model. To further reduce the aeroelastic system, the structural unsteadiness of the swept wing, which is typically a coupled bending-torsion mode, is separated into a pure bending response simulated by harmonic heave oscillations [15], [16] and pure torsional response simulated by harmonic pitch oscillations about the averaged quarter chord line of the wing model [17]. Wind tunnel experiments have been performed with a variation of the between 0.78 and 0.92 and Reynolds number refreestream Mach number lated to the mean aerodynamic chord = (106) at angles of attack at the wing root between -2 and +3°. The harmonic wing oscillation was introduced with between 0.1% and 0.5% of the wing half span and pitch heave amplitudes amplitudes between 0.1 and 1.1° at reduced frequencies between 0.025 and 0.125 calculated with the wing aerodynamic mean chord and the incident flow velocity .

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This paper presents an overview of the steady and unsteady data collected for the three-dimensional wing flow. The dynamic fluid-structure interaction at the wing undergoing harmonic oscillations is analyzed on the basis of the steady wing flow, which in itself exhibits a high degree of unsteadiness due to a small amplitude shock oscillation and the presence of a trailing edge separation. Section 2 gives a brief description of the experimental setup including the methods of flow analysis used in the experiments, followed by the outline of the steady wing flow properties in section 3, and the analysis of harmonic wing oscillations in section 4. The results are discussed in section 5.

2 Experimental Setup 2.1 Wind Tunnel Facility The experimental investigation has been conducted in the Trisonic Wind Tunnel of the RWTH Aachen University. This facility is an intermittently working vacuum storage tunnel capable of producing flows with Mach numbers between 0.4 and 3. For transonic flows with freestream Mach numbers below 1, the tunnel is equipped with a 0.4m x 0.4m two-dimensional adaptive test section consisting of parallel side walls and flexible upper and bottom walls to simulate unconfined flow conditions ([18], Fig. 1). The wall contours are calculated by the one-step method solving the Cauchy integral based on the time-averaged pressure distribution measured along the centre line of the flexible walls [19]. The twodimensional wall adaptation is assumed to be adequate due to the small size of the three dimensional wing configuration used in the experiments. The tunnel total pressure and temperature are equal to the ambient conditions. Therefore, the Reynolds number depends on the Mach number and ambient temperature of each test ranging from 1.3 to 1.6 x 107m-1 in the present experiments. The relative humidity of the airflow is always kept well below 4% at total temperatures around 293K to exclude any influence on the shock wave position [20]. The acoustic environment in the wind tunnel is of major interest in the experimental simulation of dynamic fluid – structure interaction processes. For this reason, the adaptive test section also equipped with several dynamic pressure transducers distributed along center line of the upper and lower wall. Figure 2 displays a summary of frequency spectra measured in the empty test section for different flow conditions. The freestream chamber located downstream of the test section is identified as the main source of acoustic disturbances in the test section, since spectral powers always peak in the transducer locations closest to this area. One example of the streamwise development of spectral powers is given in Fig. 2a for a Mach number of = 0.86. Depending on the incident Mach number, the acoustic disturbances contain three predominant frequencies, most likely evolving from different acoustic modes in the freestream chamber. The fluctuation power contained in the modes is also depending on the Mach number (Fig. 2b). The results from experiments with the BAC 3-11 swept wing model will be analyzed regarding the influence of the test section acoustics on the basis of the empty test section acoustic analysis.

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2.2 Swept Wing Model The wind tunnel model under consideration in the experiments is a backswept semi-span wing with the supercritical airfoil BAC 3-11/RES/30/21 [21]. The wing geometry consists of two segments with a constant leading edge sweep angle of 34° and has a semi span = 280mm, aspect ratio of 7.54, and mean

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aerodynamic chord of wing root (Fig. 3).

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Fig. 3 Plan view of the supercritical BAC 3-11/RES/30/21 swept wing model, dimensions given in mm. Pressure orifices are located in one wing section at a relative wing span = 0.286 and 0.714 for the pressure sensitive paint experiments. The wing upper side contains markers with a diameter of 5mm for a videogrammetric wing deformation measurement

The wing structure is made of an orthotropic ultra-high modulus carbon fiber laminate sandwich shell. The precise simulation of fluid-structure interaction by forced harmonic oscillations introduced at the wing root requires a rigid structure or at least an ultimately high stiffness of the wing model, since rigidity cannot be achieved with a one-sided wing suspension. The time-dependent bending deformation resulting from the aerodynamic pressure distribution around the wing is the major source of changes in the local flow conditions, since it directly causes a change in the local angle of attack due to the bending-torsion coupling of the swept wing. This structural washout effect can be successfully reduced by maximizing the bending stiffness of the fiber laminate. This adjustment means reducing the torsional stiffness at the same, since the shell thickness is generally limited by the requirement of a low area-related weight to keep the inertia forces and

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associated additional deformation small, when inducing the forced oscillation to the structure. To incorporate measurement equipment into the extremely slender wing structure, the sandwich shell furthermore had to be designed as the major structural component without any supporting ribs or spars. Hence, the major part of the carbon fiber inlays was arranged parallel to the elastic axis, which is inclined to the incoming flow by approximately 29°. This structural design provides the necessary stiffness for the forced harmonic oscillation experiments up to a relative wing half span ≈ 0.75. In the outer region the wing stiffness inadvertently degrades due to the decreasing stiffness caused by the small wing chord and shell height near the wing tip. The wing model aeroelastic deformation can be measured by the tracking of 24 white markers distributed in 8 semi-span positions on the black wing upper surface (Table 1, sec. 2.3). The markers have a diameter of 5mm and a thickness of 80μm. Their effect on the local flow behavior is negligible. The wing model is equipped with 27 subminiature pressure transducers Entran EP-I and Kulite XCQ-080 incorporated along the wing section at = 0.286 of the length = 82.71mm. Due to the slenderness of the BAC 3-11 trailing edge, the pressure measuring chord is limited to 0.051 ≤ ≤ 0.785 on the upper and ≤ 0.648 on the lower side. Table 2 shows the chord-wise pressure 0.085 ≤ tap positions. The dynamic transducer signals are recorded and digitized with the data acquisition system of the Trisonic wind tunnel at a sample rate of = 20kHz with anti-aliasing filter at 10kHz. Each transducer is installed in closest proximity to the corresponding pressure orifice to minimize the damping and phase shift of the measured pressure signal against the actual signal on the wing surface. Fig. 4a shows the dynamic response of the sensor installation (Fig. 4b) plotted against the reduced frequency at a Mach number between 0.8 and 1.4, which corresponds to the bandwidth of the local Mach number in the flow field. The response function was calculated with the theory developed by Bergh and Tijdeman for the propagation of small harmonic pressure perturbations through tube-transducer systems [22]. It has a minor influence on pressure fluctuations in the interesting reduced frequency range of the harmonic wing oscillations, which is 0.025 ≤ ≤ 0.125. The laminar-turbulent transition of the boundary layer is a significant parameter to determine the shock – boundary layer interaction. To limit the number of variables and to simulate a flow that in most respects resembles a realistic high Reynolds number flow, the boundary layer transition is fixed at a line of 5% chord with a 117μm zic-zac shaped transition strip on the lower side and trip dots of 1mm in diameter and heights between 101.6μm at the wing root and 52.0μm at the tip with a constant spacing of 2mm on the wing upper side. The variable trip dot height is just sufficient to trigger transition to avoid an over fixation of the boundary layer on the upper surface, which was assessed by a visualization of the boundary layer state with shear-sensitive liquid crystals.

Unsteady Transonic Fluid – Structure – Interaction

Table 1 Spanwise optical target positions on the upper wing surface

0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.971

0.127 0.141 0.159 0.180 0.195 0.227 0.259 0.291

0.507 0.507 0.508 0.507 0.489 0.509 0.510 0.511

0.886 0.874 0.858 0.833 0.784 0.790 0.761 0.732

331

Table 2 Chordwise pressure orifice positions, spanwise station = 0.286 Upper side

Lower side

0.051 0.150 0.250 0.351 0.417 0.484 0.518 0.551 0.585 0.618 0.652 0.685 0.718 0.751 0.785

0.085 0.150 0.217 0.283 0.383 0.416 0.482 0.516 0.549 0.582 0.614 0.648

1

Gain

0.8 0.6 0.4

Phase

0.2 0 1.5

1

0.8 1.0 1.2 1.4 05

1

15

2

25

1.5

2

2.5

0.8 1.0 1.2 1.4

0.5 0 0

0.5

1

ω∗

a)

b)

Fig. 4 Pressure transducer installation in the wing model. a) Transfer function over the reduced frequency for relevant local Mach numbers 0.8 ≤ ≤ 1.4. b) Sensor installation, dimensions in drawing given in mm

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P.C. Steimle, W. Schröder, and M. Klaas

2.3 Videogrammetric Model Deformation Measurement Setup The time-dependent surface motion resulting from the deformation of the wing structure is the major influencing variable in the aeroelastic system under consideration. Small surface oscillations can induce significant changes in the flow dynamics due to the inherently nonlinear character of the transonic flow field. Hence, the precise time-resolved measurement of wing deflection is a critical item to evaluate aerodynamic and structural effects during the aero-structural interaction process. The dynamic aeroelastic deformation of the wind tunnel model is measured with a single view videogrammetric setup. The deformation is determined by the time-dependent position of optical marker targets (Fig. 3, Table 1) in a digital image of the wing upper surface. Image time series were recorded simultaneously to the dynamic pressure measurement with a single high speed charge coupled device camera Photron 1024-PCI with Nikon 50mm f/1.2 objective lens at a rate of 3kHz. This recording frequency is the maximum frame rate of the camera at a resolution of 768x432 pixels, which in this optical setup was optimal for the pixel resolution of markers within the recorded images. The synchronization with the pressure measurement system was obtained using the camera start signal recorded with the DAQ system. The camera position is on the side of the adaptive wind tunnel test section at a viewing angle of about 25° relative to the wing model upplane and perpendicular to the flow direction in the per surface in the plane. The camera position and angle in relation to the objective lens satisfy the Scheimpflug condition to focus the entire model from the wing tip to the wing root. The Scheimpflug angle between the image plane and the lens plane is about 5° (Fig. 5). The light source is a 1kW halogen lamp positioned downstream of the test section directed towards the white upper test section wall to provide a diffuse lighting of the wing markers.

Fig. 5 Videogrammetric model deformation measurement setup, view in the upstream direction

Unsteady Transonic Fluid – Structure – Interaction

333

This setup is a simplified version of the single view videogrammetric deformation measurement described by Burner et al. [23] and Graves et al. [24]. The right angle between the camera viewing plane and the flow direction allows determining each marker position without an elaborate mapping of the two-dimensional image to the three-dimensional object space. The target displacements during the recorded image sequence are determined by a particle tracking algorithm with sub-pixel resolution. The displacements in the physical length scale can easily be calculated by calibrating the measurement system with a known marker displacement for a fixed camera position. This calibration is done by moving the wing upwards by a known length and comparing the displaced marker positions with the zero positions. The result is the length sensitivity of each marker in the pixel domain taking into account the perspective reduction of lengths from the wing tip to the root.

2.4 Time-Resolved Pressure-Sensitive Paint Visualization Setup Measuring dynamic pressure distributions around the BAC 3-11 transport wing is limited to one spanwise station in = 0.286 due to the small wing model dimensions to fit into the Trisonic Wind Tunnel test section. Of course, this setup alone does not allow investigating three-dimensional effects in the dynamic flow field. Furthermore, the use of additional flow visualization techniques like oil-flow and shear-sensitive liquid crystalline coatings, which were also used to gather additional information about the wing aerodynamics, only provides a time-averaged flow image. For this reason, a coating of pyrene-based pressure-sensitive paint (PSP) on porous anodic aluminum binder (AA-PSP) [25] was used to visualize the instantaneous pressure distribution on the upper surface with a frequency resolution being high enough to accurately display the dominant events in the dynamic wing flow. The PSP measurement technique is based on quenching of photo-chemically excited organic molecules by the interaction with oxygen, which allows a measurement of the local oxygen partial pressure by a change in luminescence intensity. A complete review can be found, e. g. in [26] and [27]. The oxygen permeability of the binder containing the luminophore determines the response time of the PSP luminescence to changes in the local pressure. Very short response times between 35 and 100μs can be realized by using a porous anodic aluminum binder [28], [29], where the photo-chemically active molecule is adsorbed to the surface to allow direct correspondence with oxygen in the surrounding flow. A 47μm aluminum A1050 foil was attached to the carbon fiber wind tunnel model using double sided adhesive tape of 70μm thickness. The model surface was then anodized in 1mol dilute sulphuric acid following the procedure of Kameda et al. [30]. An anodizing voltage of 20V at a current of 15mA/cm2 and constant solution temperature of 18°C resulted in a porous surface with pore diameters between 25 and 40nm (Fig. 6). Among the different luminophores, which have already been used by other groups, 1-pyrene-sulfonic acid sodium salt (PSA) was chosen in the current experiments (Fig. 7). Molecules with sulphuric or carboxyl group can be adsorbed

334

P.C. Steimle, W. Schröder, and M. Klaas

easily onto anodic aluminum surfaces through a dehydrate ester reaction resulting in a chemical bonding between luminophore and aluminum in the micro pores [31]. The PSA compound has a high acidity and forms a strong link to the surface due to its high polarity. For this reason, it showed significantly less photodegradation in similar experiments reported by Merienne et al. with PSA on an aluminum tape [32].

Fig. 6 Scanning electron microscope image of aluminum foil sample anodized with a voltage of 20V, area-related current 15mA/cm2 at a temperature of 18°C

Fig. 7 Chemical composition of 1-pyrene-sulfonic acid sodium salt (PSA), CAS no. 59323-54-5, used as luminophore

Nonetheless, the hydrophilic nature of the porous surface causes a long term influence of air humidity on the pressure sensitivity of AA-PSP [33]. To avoid any sensitivity degradation during the test campaign, the wing was always kept in dry air conditions after the aluminum foil had been anodized. The wing was either stored under pure nitrogen atmosphere or installed in the dry test section of the wind tunnel with a relative humidity below 1%. The optical setup for the PSP experiments is shown in Fig. 8. The light source for the excitation of the pyrene luminophore was a flicker-free mercury vapor lamp Osram HBO 500W combined with the band pass filter Schott UG-11 and a focusing lens to concentrate the light on the wing surface. Images were recorded with a Photron Fastcam 1024 PCI CMOS camera with a Leica Noctilux M 1:1/50mm lens and a UV-blocking filter Schott KV-408 combined with the bandpass Lee filter V28 Blueberry 8. The widest aperture 1:1 was chosen in the experiments to maximize the light intensity on the camera chip, thus to improve the signal-to-noise ratio in the recorded images sampled at different acquisition rates between 1500 and 2000Hz. The UV lamp and the camera was in a position next to the wind tunnel side wall which ensured optimal lighting as well as optimal viewing of the model surface for a given image resolution of 896 x 576pixels on the CMOS chip. The

Unsteady Transonic Fluid – Structure – Interaction

335

camera position is on the side of the adaptive wind tunnel test section at a viewing angle of about 25° relative to the wing model upper surface in the y-z plane and perpendicular to the flow direction in the x-y plane. The camera position and angle in relation to the lens satisfy the Scheimpflug condition to focus the entire model from the wing tip to the wing root. The Scheimpflug angle between the image plane and the lens plane is about 5° (Fig. 5).

CMOS camera filter

UV-lamp

wing model

Fig. 8 AA-PSP visualization setup at the wind tunnel side wall

The image analysis procedure generally follows [25]. The calculation of the luminescence ratio has to account for the deformation of the wind-tunnel model, which is time-dependent and of large quantity in comparison to other PSP experiments in the literature. The calculation of the intensity ratio used for the local pressure measurement requires a mapping of each measurement image to the reference image. This image registration was performed using the b-spline algorithm described by Rueckert et al.[34], which combines the advantages of voxel-based similarity measures like mutual information [35] with a non-rigid transformation model [36], [37]. The algorithm lays out a grid of b-spline control points guiding the transformation of an input image. The control points are moved by a quasi Newton scheme such that the registration error between the input and the reference image is limited by a defined boundary.

3 Steady Wing Flow Properties 3.1 Time-Averaged Flow Topology The unsteady aerodynamic flow field around the swept wing is determined by quasi-steady flow phenomena, which are accessible to time-averaged flow measurement tools. The steady wing aerodynamics has been investigated for a wide

336

P.C. Steimle, W. Schröder, and M. Klaas

cp

variety of angles of attack measured at the wing root between = -3° and +3°, freestream Mach numbers between = 0.78 and 0.92 and Reynolds numbers around 106, calculated with the mean aerodynamic chord . The main feature of the transonic swept wing flow is a weak compression shock terminating the supersonic bubble in the wing flow field. Figure 9 displays the Mach number effect on the time-averaged pressure distributions in = 0.286 for a constant mean angle of attack = 0°. The steep pressure rise at = 0.84 and above indicates a weak shock wave on the wing upper side. On the lower surface, the shock does = 0.86 due to the lower local Mach numbers. The supernot develop below sonic flow area grows with the incident Mach number leading to a stronger shock, which appears closer to the trailing edge. Above = 0.9 a weak oblique shock is followed by a short supersonic expansion and a strong shock on the wing upper side. Since the leading shock is caused by a local deflection of the streamlines most likely in the vicinity of the outer boundary layer region, the appearance of this shock system marks the change from a mild to a severe interaction with the boundary layer. −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0

0.778 0.802 0.822 0.840 0.861 0.882 0.903 0.919

upper side 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c1

cp

BAC 3-11 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0

lower side

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c1

Fig. 9 Mach number effect on time averaged pressure distributions at ≤ 0.02°, 0.78 ≤ ≤ 0.92 angle of 0.01°≤

= 0.286 for a fixed

Instantaneous three-dimensional pressure distributions on the wing upper side = 0.86 (Fig. measured with the AA-PSP coating are exemplary presented for 10a) and = 0.92 (Fig. 10b) [38]. Due to the almost elliptical lift distribution, which is a typical attribute of back-swept wings, both distributions show the

Unsteady Transonic Fluid – Structure – Interaction

337

weakening of the supersonic flow towards the wing tip, yielding the disappearance of the shock wave in the outer half-span region. 1

0.5

0

-0.5

-1

-1.5

a) 1

0.5

0

-0.5

-1

-1.5

b) Fig. 10 Instantaneous pressure distribution on the upper wing surface measured with apriori calibrated AA-PSP, image acquisition rate = 1500Hz, a) = 0°, = 0.86. = 0°, = 0.92 b)

Besides the presence of a shock wave, the wing flow field features a separation of the turbulent boundary layer in the trailing edge region. The separation occurs in the area of a positive pressure gradient, first at sub-critical conditions. Then, it grows upstream to finally coincide with the time-averaged shock position as soon as the shock is strong enough to induce separation. This flow development resembles the type B3 flow in the phenomenological description given by Pearcey et al. [39]. The oil flow visualization shows the development of the skin friction lines

338

P.C. Steimle, W. Schröder, and M. Klaas

along the wing surface. Fig. 11 displays the surface flow pattern for = 0.86 and 0.92 at = 0°, thereby demonstrating the three-dimensionality of the wing flow. In the area of the positive pressure gradient a distinctive deflection of the skin friction lines towards the wing tip can be observed, indicating the influence of the transverse pressure gradient as well as a considerable thickening of the boundary layer and skewing of the viscous velocity profile that eventually cause a threedimensional separation [40]. In this flow case the boundary layer amplification due to the adverse pressure gradient is naturally much stronger than in a turbulent high Reynolds number flow [41]. At the BAC 3-11 wing section the shock appears immediately behind the pressure minimum. Hence, it is located, where the skin friction line starts to be deflected towards the wing tip. At low shock strength, a weak shock – boundary layer interaction occurs on the wing upper surface. The flow exhibits a pronounced cross flow component and separates close to the trail= 0.86 the shock induced separation is incipient (Fig. 11a). A ing edge. At further increase in the shock strength results in a separation at the shock foot merging with the trailing edge separation, which is associated with the first lift divergence (Fig. 12). The appearance of the lambda shock system at higher Mach numbers 0.9 and 0.92 corresponds with a full scale separation involving back flow in the trailing edge region (Fig. 11b) and marks the performance boundary for this aerodynamic configuration due to the breakdown of lift (Fig. 12). The flow behavior on the lower side is similar, albeit with much weaker pressure gradients. A = 0.92 with reattachment close to shock induced separation appears only at the trailing edge.

a)

b)

= 0.01°. a) Flow with a single shock Fig. 11 Surface flow pattern on the wing upper surface, = 0.859. b) Flow with a wave and shock induced separation near the trailing edge at = 0.921 lambda shock combination and shock induced separation involving backflow at

Unsteady Transonic Fluid – Structure – Interaction

339

0.08 Lambda shock test case, = 0.92

cL

0.06 0.04 0.02

Normal shock = 0.86 test case,

0 0.75

0.8

0.85

0.9

0.95

1

M∞

Fig. 12 Lift coefficient measured at the wing root and definition of test cases,

= 0°

Although flutter characteristics cannot be determined in the Trisonic Wind Tunnel facility, the single shock wave flow at = 0°, = 0.86 is suspected to be close to the minimum of the stability boundary curve, commonly referred to as “transonic dip” [9], while the lambda shock flow with full scale trailing edge sepa= 0°, = 0.92 is on the increasing branch of the stability limit. In ration at the following, these two different flows are used as test cases for the unsteady wing experiments. The steady data from these representative flow problems serve as references to investigate differences in their dynamic behavior and the impact on aeroelastic wing responses simulated by forced harmonic oscillations. The flow downstream of the shock experiences significant shearing between the viscous rear flow and the external flow, since the skin friction lines are much more deflected towards the tip than the external stream lines. Nevertheless, the amount of shear stress acting on the wing surface is significantly reduced in this flow region. Images of a shear sensitive chiral-nematic liquid crystal coating [42]-[44] Hallcrest CN/R2 clearly show extended regions of low shear stress by the discrete change in color from light blue to dark red close to the trailing edge behind the shock wave. The distinctiveness of the color change is increasing from the flow with mild shock = 0.86 to the one with a strong shock at = 0.92 (Fig. 13), strength at thereby demonstrating a difference between these two flow types, which is very important for the unsteady flow behavior. The flow behind the weaker shock has the ability to dynamically exchange information with the upstream flow, especially with the boundary layer in the shock foot region and can be much more sensitive to upstream disturbances. On the other hand, the separation behind the stronger lambda shock system lacks this ability due to the presence of a quasi-steady separation line, which isolates the separated flow near the trailing edge from the attached flow upstream. This quasi-steady isolation line coincides with the shock foot and therefore reduces the sensitivity of the lambda shock to the flow dynamics. Figure 14 shows the angle-of-attack effect on the time-averaged pressure distributions demonstrating the distinct shock stall behavior of the flow. In general, angle of attack and Mach number effect are comparable. An increase in the angle of attack also causes a mild strengthening of the shock wave, although without the appearance a lambda shock combination.

340

P.C. Steimle, W. Schröder, and M. Klaas

a)

b)

Fig. 13 Comparison of steady pressure distributions and qualitative visualization of the wall by a chiral-nematic liquid crystal coating on the wing upper side shear stress modulus = 0.86 (a), and 0.92 at = 0°(b). The light blue reflective color indicates a high for shear stress region, dark red low shear stress −0.8

upper wing surface −0.6

-2.02◦ -1.00◦◦ +0.00 +0.99◦ +2.00◦ +3.01◦

cp

−0.4

cp,crit −0.2 0

lower wing surface

0.2 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

[x/c]1

Fig. 14 Angle of attack effect on time averaged pressure distributions at = 0.286 for = 0.86, = 1.1x106, -2° ≤ ≤ 3° on the wing upper (●, ■, ♦, ▼, ▲, ►) and lower side (○, □, ◊, V, U, Z)

3.2 Dynamic Shock-Boundary Layer Interaction The results discussed above demonstrate the influence of the trailing-edge separation on the time-averaged flow topology. However, the interaction between shock wave and boundary layer is also a very important source of flow unsteadiness around the BAC 3-11 swept wing. In the single shock flow case with incipient separation the time-resolved pressure distribution shows a chord-wise motion of the shock wave, which is coupled to a local subsonic acceleration in the cross flow region behind the shock. The origin of this behavior is most likely the thickening and skewing of the boundary layer profile at the line of incipient separation in the shock foot, which might react to disturbances contained in the incoming turbulent boundary layer and acoustic disturbances from the trailing edge. The local boundary layer unsteadiness leads to a shock oscillation with amplitude around 4% of the local wing chord.

Unsteady Transonic Fluid – Structure – Interaction

341

The spectral analysis of the wing flow unsteadiness reveals several reduced frequencies ! ¤ = 2¼f c¹=u1 incorporated in the flow dynamics of the single shock test case. The highest pressure fluctuation levels for all frequencies can be observed in the shock – boundary layer interaction region (Fig. 15a, b). The fluctuation of the shock properties on the upper surface (Fig. 15c - f) essentially contains = 0.43 including its first harmonic and 0.73. In this the reduced frequencies analysis the shock deflection angle and shock angle were calculated from the time-dependent pressure distribution using the oblique shock relationships and the Rankine-Hugoniot equations taking into account the local oblique shock sweep angle (Table 3), which was determined by a close-up analysis of the oil flow visualization following [45] and [46]. Power spectra of intensity fluctuations in the pressure-sensitive paint layer, which directly correspond to pressure fluctuations, are presented in Fig. 16 for the single shock flow. The spectral analysis shows a wide band unsteadiness contained in the three-dimensional wing flow, as expected from the pressure transducer measurements. The prominent reduced frequencies are 0.5 and 0.72. The reduced frequency = 0.73 originating from the acoustic feedback of the shock motion with the separated boundary at the trailing edge determines all semi-span positions and is most significant in = 0.572 close to the trailing edge kink. Obviously, the acoustic [x/c]1

cp

−0.6 −0.4 −0.2 0 0.2

−2

−2 −3

log10 (|(p2 /p1 ) |2 /fs ) [1/Hz]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −1 cp upper side cp lower side −0.8 cp,crit

log10(|xs /c1 |2 /fs ) [1/Hz]

a)

c)

−4 −5 −6 −7 −8 −9 −10 −11 −12 0

M∞ = 0.86

0.5

1

ω∗

1.5

2

= 0.286

e)

−2 −3 −4 −5 −6 −7 −8

0.2

0.4

0.6

0.8

−7 −8 −9 −10 −11 0.5

1

0.5

1

ω∗

1.5

2

2.5

1.5

2

2.5

f)

−2 −3 −4 −5 −6 −7 −8 −9

0.5

1

ω

0

−6

−1

−1

−9 0

d)

−5

0

log10(|θ |2 /fs ) [1/Hz]

log10 (|β |2 /fs ) [1/Hz]

b)

−4

−12 0

2.5

1 0

−3

∗

1.5

2

2.5

−10 0

ω∗

1

Fig. 15 Close-up analysis of the single shock test case, = 0.02°, = 0.861, = 1.08x106, wing upper surface, = 0.286. a) Time-averaged pressure distribution. b) Approximated wall streamlines in the area of = 0.286 visualized with the oil flow technique. c) Power spectrum of the time dependent shock position . d) Power spectrum . e) Power spectrum of the shock angle . f) Power spectrum of of the shock strength the deflection angle

342

P.C. Steimle, W. Schröder, and M. Klaas

Table 3 Local shock sweep angle = 0.286, wing upper side

[°] -2 -1 0 1 2 3

[°] in the

plane of the wing coordinate system in

0.84

0.86

0.88

0.90

0.92

19.12 -

16.42 18.91 19.17 19.84 21.27 25.05

21.87 -

18.89 -

15.88 -

source, which is responsible for pressure fluctuations with this reduced frequency, is very strong in this area. Here, the sudden change of the trailing edge sweep angle induces a discontinuity in the shear layer, which is interacting with the wing trailing edge forming a three-dimensional source of acoustic disturbance. −4

0.8 0.6 0.4 0.2 0 0

0.5

ω∗

1

x 10

0.6 0.4 0.2

0.5

ω∗ b)

a) x 10

−4

0.8 0.6 0.4 0.2 0 0

1

0.5

d)

ω∗

1

1.5

0.6 0.4 0.2 0 0

1.5

0.5

ω∗ c)

1

1.5

0.5

ω∗

1

1.5

−4

x 10

1

0.8 0.6 0.4 0.2 0 0

x 10

0.8

−4

1

|I /I0 |2 /fs [1/Hz]

|I /I0 |2 /fs [1/Hz]

1

1

0.8

0 0

1.5

−4

−4

|I /I0 |2 /fs [1/Hz]

1

|I /I0 |2 /fs [1/Hz]

x 10

|I /I0 |2 /fs [1/Hz]

|I /I0 |2 /fs [1/Hz]

1

0.5

ω∗ e)

1

1.5

x 10

0.8 0.6 0.4 0.2 0 0

f)

Fig. 16 Power spectra of time-dependent intensity ratios measured with AA-PSP on the = 0.6, = 1500Hz, single shock flow. a) Relative wing half span wing upper side in = 0.148, b) = 0.286, c) = 0.429, d) = 0.572, e) = 0.714, f) = 0.858

Unsteady Transonic Fluid – Structure – Interaction

343

A comparison of these spectrograms with resonance frequencies measured in the empty test section (Fig. 2b) clearly identifies wind tunnel resonance to be the = 0.43 or 0.5. However, there is no acoustic influorigin of fluctuations with ence at = 0.73, suggesting that this frequency originates from the wing flow unsteadiness and the local fluid – structure interaction. The onset of periodic shock oscillations has been investigated in several experimental and numerical flow analyses. Brunet et al. describe a “pulsation” of the separated area to be the origin of buffet oscillations on the OAT15A supercritical airfoil with a thickness to chord ratio of 12.5% [47]. Lee’s widely accepted shock buffet model [48] describes the inviscid shock interaction with upstream propagating sound waves generated by the impingement of large scale turbulent eddies on the sharp trailing edge [49] forming a feedback loop with disturbances convected downstream to be the main buffet mechanism. Shock buffet has mainly been investigated in two-dimensional flows. Finke and Lee measured shock buffet fre= 0.5 to 2.0 on the 12% thick NACA 63-012 and around 0.5 on quencies of the 11.8% BGK No. 1 airfoil [48], [50]. A reduced buffet frequency of = 0.55 was measured by Schewe et al. for the NLR-7301 airfoil [51], the BAC 3-11 air= 0.53 to 0.58 in the two-dimensional flow foil exhibits a buffet frequency of case [12]. Hence, the observed reduced buffet frequency of = 0.73 on the BAC 3-11 wing is in the range of other supercritical airfoils with this thickness ratio and comparable shock position. The close resemblance of the frequencies may also be a sign of a wide similarity between the buffet mechanisms in two and threedimensional flow. Dor et al. also speculated that the local buffet mechanisms might be similar [52], but currently there is no clear analysis in the literature to the authors’ knowledge. Nevertheless, the shock buffet phenomenon observed on the BAC 3-11 swept wing has a different characteristic than shock buffet flows described in the literature. In the case of classical buffet the shock is strong enough to cause a fully separated flow between the shock boundary-layer interaction area and the trailing edge. The formation of the acoustic feed back loop can be observed at high Mach numbers and high angles of attack. In the BAC 3-11 three-dimensional buffet flow considered here, the cross-flow region with skewed boundary layer profile developing into a trailing edge separation is unsteady and leads to an oscillatory shock motion only due to the local interference in the shock foot. Acoustic disturbances in this case originate mainly in an area where shear layers undergo an almost abrupt change like in the trailing edge kink region. This oscillatory shock motion is only present in the range of M∞ = 0.84 to 0.88 and changes to a flow with significantly lower unsteadiness at M∞ = 0.90 and 0.92. Since the time-averaged shock foot position shifts downstream at rising Mach number, it coincides with the separation line thereby replacing the dynamic interaction between shock foot and the cross-flow with a rather steady separation line. Hence, the flow at M∞ = 0.92 can be considered as a flow, where almost no shock buffet occurs (Fig. 17, 18).

344

P.C. Steimle, W. Schröder, and M. Klaas [x/c]1

−0.4 −0.2 0

−3 −4

log10 (|(p2 /p1 ) |2 /fs ) [1/Hz]

log10(|xs /c1 |2 /fs ) [1/Hz]

cp

−0.6

0.2

−2

−2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −1 cp upper side cp lower side −0.8 cp,crit

a)

c)

−5 −6 −7 −8 −9 −10 −11 −12 0

M∞ = 0.92

0.5

1

ω∗

1.5

2

= 0.286

e)

−2 −3 −4 −5 −6 −7 −8

0.2

0.4

0.6

0.8

−7 −8 −9 −10 −11 0.5

1

0.5

1

ω∗

1.5

2

2.5

1.5

2

2.5

f)

−2 −3 −4 −5 −6 −7 −8 −9

0.5

1

ω

0

−6

−1

−1

−9 0

−5

0

log10 (|θ | 2 /fs ) [1/Hz]

log10(|β |2 /fs ) [1/Hz]

b)

d)

−4

−12 0

2.5

1 0

−3

∗

1.5

2

−10 0

2.5

ω∗

1

Fig. 17 Close-up analysis of the single shock test case, = 0.02°, = 0.919, = 1.11x106, wing upper surface, = 0.286. a) Time-averaged pressure distribution. b) Approximated wall streamlines in the area of = 0.286 visualized with the oil flow technique. c) Power spectrum of the time dependent shock position . d) Power spectrum of the shock strength . e) Power spectrum of the shock angle . f) Power spectrum of the deflection angle x 10

−4

1

|I /I0 |2 /fs [1/Hz]

|I /I0 |2 /fs [1/Hz]

1 0.8 0.6 0.4 0.2

0 0

0.5

1

ω∗

1.5

x 10

−4

0.8 0.6 0.4 0.2 0 0

2

0.5

a) 1

0.8 0.6 0.4 0.2 0 0

0.5

1

ω∗

c)

1

1.5

2

1

1.5

2

b)

−4

|I /I0 |2 /fs [1/Hz]

|I /I0 |2 /fs [1/Hz]

1

x 10

ω∗

1.5

2

x 10

−4

0.8 0.6 0.4 0.2 0 0

0.5

ω∗

d)

Fig. 18 Power spectra of time-dependent intensity ratios measured with AA-PSP on the = 0.6, = 2000Hz, lambda shock flow. a) Relative wing half span wing upper side in = 0.148, b) = 0.286, c) = 0.429, d) = 0.572

Unsteady Transonic Fluid – Structure – Interaction

345

The fluctuation power is solely contained in the frequency of the acoustic = 0.72 in this flow case, but the unsteadiness in the shock feedback, which is position, strength, and angle is greatly reduced at the frequency of the acoustic feedback compared to = 0.86. The local deflection angle shows a more discrete fluctuation (Fig. 17f) due to the pulsation of the separated boundary layer, in comparison to the single shock flow. At = 0.92 the nearest reduced tunnel resonance frequency is = 0.69. However, the difference between the main wing aerodynamic frequency and the resonance frequency is almost 50Hz in this flow case, making an interference of the test section acoustics with this aerodynamic unsteadiness rather unlikely. The existence of a steady flow case at higher Mach numbers than in the unsteady flow is also described by Xiao on a circular arc airfoil of 18% thickness [53] and by Geissler for the NLR 7301 airfoil in the context of numerically simulating limit cycle oscillations [54].

3.3 Wing Model Deformation The wing model deformation resulting from the dynamic fluid-structure interaction was measured using the videogrammetric deformation measurement setup. An accurate determination of the model deformation is a critical item to identify the sources of local flow unsteadiness in the dynamic system. Despite the high rigidity of the wind tunnel model a certain structural response to aerodynamic loads is inevitable. Fig. 19 shows the relative heave motion h/s and pitch motion α over time. In comparison to the wing root the difference in the time-averaged angle of attack is around 0.02° in η = 0.286 due to the structural washout effect. 0.2

α0 = 0.02 deg

α(t) [deg]

h/s [%]

0.1 0 −0.1 −0.2

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

Time [s]

Fig. 19 Wing section angle of attack α and z-position h/s in η = 0.286 as a result of torsional and bending deformation of the wing model under fluctuating flow pressures as a function of time

The dynamic wing response, however, is not a coupled bending-torsion mode. The heave and pitch motion along the wing span generally are not in phase, although some of the frequencies contained in these degrees of freedom coincide

346

P.C. Steimle, W. Schröder, and M. Klaas

(Fig. 20b, c). The main reduced pitch frequency in 0.286 relative half-span is 0.42, which is relevant in the pressure fluctuation spectrum, especially in the shock region on the upper surface (Fig. 20a). This is a clear indication for a fluid-structure interaction at this frequency, which is limited to the pitching degree of freedom. This interaction does not happen due to a wing model response in the first Eigenmode, which is dominated by bending at a much higher frequency, but rather due to a structural reaction to disturbances contained in the flow. This is especially = 0.73 in the single true for the main reduced shock oscillation frequency shock flow and 0.72 in the lambda shock flow, which is present in the torsional as well as bending deformation of the wing model. Nevertheless, the dynamic wing model deformation remains very small with a twist angle below 0.1° and bending deformation below 0.15% half-span in = 0.286.

a)

b)

0 −2 −4 −6 −8 −10 0 −4

05

1

15

2

25

05

1

15

2

25

0.5

1

1.5

2

2.5

−5 −6 −7 −8 0

c)

−8 −10 −12 −14 0

ω∗

Fig. 20 Comparison of power spectra of pressure fluctuations in = 0.286 with the spectrum of the time-dependent wing deformation. a) Pressure fluctuations in the shock region = 0.685, single shock test case. b) Fluctuations in the local wing angle of attack. c) Local plunging of the wing section

4 Forced Harmonic Wing Oscillation Experiments During the dynamic experiments the wing model undergoes either forced harmonic heave or pitch oscillations around the averaged quarter chord line resulting in a pitching axis of 32° inclination to the incident flow. The excitation mechanism uses an adjustable eccentricity transforming the shaft rotation into a harmonic motion, which is directly connected to an oscillating carriage in case of

Unsteady Transonic Fluid – Structure – Interaction

347

forced heaving (Fig. 21a) or to a rotational shaft bearing the wing model by a lever arm to induce pitching (Fig. 21b). The harmonic motion of the wing suspension is detected by a laser triangulator, the signal of which is synchronously measured with the pressure distributions. This assembly is driven by a 5.5kW asynchronous motor with a maximum excitation frequency of 100Hz. The oscillation frequency was restricted to 80Hz as a measure of precaution. Due to dynamic deformation effects in the excitation assembly the forced oscillation amplitude unfortunately depends on the frequency, making a clear determination of the frequency effect in the forced wing experiments unfeasible. Therefore, the following presentation of the data focuses on the amplitude effect on the flow. Unsteady pressure distributions from the section at = 0.286 measured synof chronously to the model motion at the wing root at pitching amplitudes about 0.1, 0.5, and 0.9° and heave amplitudes of 0.1, 0.3, and 0.5% are presented for a mean angle of attack = 0° and reduced excitation frequency 0.075.

a)

b)

Fig. 21 Close-up view of oscillation mechanic installed at the wind tunnel side wall. a) Heave oscillation mechanic. b) Pitch oscillation mechanic

4.1 Bending Degree of Freedom The spectral analysis of the wing surface motion measured during the forced oscillation experiments shows that the wing model oscillation is indeed dominated by the forced heave motion introduced at the wing root. Other frequency components, which were contained in the deformation spectrum of the steady wing, reduce significantly even at outer half-span positions. Fig. 22a shows the spectrogram of the wing surface motion in the pressure measurement section = 0.286 for the smallest heave amplitude. Even in this case the harmonic heave oscillation is much stronger than any other frequency content of the wing deformation induced by the flow unsteadiness. The time-dependent angle of attack is also determined primar(Fig. ily by the disturbance velocity from the instantaneous heave motion 22b). Thus, the experiments simulate a harmonic wing bending oscillation with negligible torsion coupling.

348

P.C. Steimle, W. Schröder, and M. Klaas

The interesting effect of the harmonic heave oscillation of the wing is the amplification of the flow unsteadiness in the single shock flow test case. The selfinduced shock oscillation is intensively modulated by the wing unsteadiness depending on the frequency as well as amplitude of the wing motion (Fig. 23a). Some parameter combinations induce a significant increase of the pressure fluctuation strength, especially at the medium heave amplitude. In general, the strength of fluctuations from the oscillating shock wave decreases with increasing heaving amplitude. This sensitivity to the wing heave motion reduces immensely in the lambda shock flow, since the mean pressure fluctuation power level reduces by an approximate factor of 5 in comparison to the single shock flow (Fig. 23b). The aerodynamic response function to harmonic oscillations of the wing structure can be assessed using the first harmonic unsteady pressure distribution calcuand the heave lated with the Fourier coefficients of the pressure fluctuation motion at the fundamental frequency Hcp ; h =

Cp (i!h ) . H(i!h )

(1)

wc/4 /s = 0.073%, α = 0.044 , ωα = 0, test 3007, Δω =0

−5

harmonic heave oscillation steady wing

−6 −7 −8 −9 −10 −11 −12 −13 −14 −15 0

0.5

1

ω∗

a)

1.5

2

2.5

log10 (|α (η = 0.286)| 2 /fs ) [deg2 /Hz]

log10 (|h /s(η = 0.286)|2 /fs ) [1/Hz]

This frequency domain representation of the unsteady flow only contains the fun. Due to the broad undamental frequency of the forced wing motion steadiness the flow field is also determined by other frequencies! The first harmonic pressure distribution for the single shock test case is displayed in Fig. 24. No clear tendency can be established from the amplitude variation, since the harmonic flow answer to the heave motion reduces from the smallest heave amplitude of 0.073% to the medium amplitude of 0.269%, to increase again at an amplitude of 0.515% at a fixed reduced have frequency. wc/4 /s = 0.098%, α = 0.054◦ , ωh∗ =0.099, test 4074, Δω ∗ =

−3

harmonic heave oscillation steady wing

−4 −5 −6 −7 −8 −9 0

0.5

1

ω∗

1.5

2

2.5

b)

Fig. 22 Forced heave excitation effect on the wing deformation at = 0.286 displayed for = 0.098% and reduced frequency = 0.1, the smallest amplitude heave amplitude single shock flow. a) Wing section heave motion. b) Wing section angle of attack

Unsteady Transonic Fluid – Structure – Interaction −3

x 10

6

|cp |2 /fs [1/Hz]

|cp |2 /fs [1/Hz]

x 10

349

4 2 0 0.1 0.05

ωh∗

0

0.2

0.1

0

0.3

0.5

0.4

0.6

−3

6 4 2 0 0.1 0.05

ωh∗

h1 /s [%]

0

a)

0.2

0.1

0

0.3

0.4

0.6

0.5

h1 /s [%]

b)

Fig. 23 Comparison of pressure fluctuation spectral power in the shock region as a function of harmonic wing oscillation amplitude and reduced frequency . a) Single shock flow with main reduced unsteadiness frequency 0.73. b) Lambda shock flow case with main reduced unsteadiness frequency 0.72

|Hcp,h |

12

16

a)

h1 /s = 0.073% h1 /s = 0.269% h1 /s = 0.515%

|Hcp,h |

16

a)

8 4

1.5

01

02

03

04

05

06

07

8

h1 /s = 0.073% h1 /s = 0.269% h1 /s = 0.515%

−1.5 0.1

0.2

0.3

0.4

0 0 3

0

0

−3 0

h1 /s = 0.117% h1 /s = 0.313% h1 /s = 0.518%

4

ϕ(Hcp,h )

ϕ(Hcp,h )

0 0 3

12

0.5

0.6

0.7

1.5

0.1

0.2

0.3

−3 0

0.8

|Hcp,h |

6 4

8

01

02

03

04

05

06

07

6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

05

06

07

0

h1 /s = 0.117% h1 /s = 0.313% h1 /s = 0.518%

4

−1.5 0.1

0.2

0.3

0.4

0 0 3

0

h1 /s = 0.073% h1 /s = 0.269% h1 /s = 0.515%

0

−3 0

0.

2

ϕ(Hcp,h )

ϕ(Hcp,h )

1.5

10

b)

2 0 0 3

0.7

x/c1 h1 /s = 0.073% h1 /s = 0.269% h1 /s = 0.515%

8

0.6

−1.5

|Hcp,h |

10

0.5

0

x/c1

b)

0.4

x/c1 h1 /s = 0.117% h1 /s = 0.313% h1 /s = 0.518%

0.5

0.6

0.7

0.8

x/c1

Fig. 24 Modulus and phase of the 1st unsteady pressure distribution in = 0.286 = 0.075. for the single shock flow, a) Wing upper side. b) Wing lower side

01

02

03

04

h1 /s = 0.117% h1 /s = 0.313% h1 /s = 0.518%

1.5 0 −1.5 −3 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c1

Fig. 25 Modulus and phase of the 1st unsteady pressure distribution in = 0.286 = 0.075. for lambda shock flow, a) Wing upper side. b) Wing lower side

350

P.C. Steimle, W. Schröder, and M. Klaas

The flow oscillation around the wing is generally in phase with the wing motion with a phase angle of around on the upper and around 0 on the lower surface. The cross flow and separated regions near the trailing edge ≥ 0.65 on the upper surface and in the cove region on the lower surface on the other hand show a reversed behavior. The development of the lambda shock flow field is slightly different. At small heave amplitude a distinct harmonic flow response is measured in the region of the downstream shock on the upper side, which is accompanied by a significant phase change in the local flow oscillation (Fig. 25a). The shock wave on the lower side also shows a distinct harmonic answer to the heave motion, here at all considered amplitudes. The increase in heave amplitude from 0.117% to 0.313% also yields a remarkable phase change in the entire upper flow field. The phase angle is almost reversed from approximately – /2 to + . The following anti-phase flow motion relative to the lower side (Fig. 25b) resembles the single shock flow behavior in this respect.

4.2 Torsional Degree of Freedom

wc/4 /s = 0.088%, α = 0.165 , ωα =0.101, test 5009, Δω =

−5

harmonic pitch oscillation steady wing

−6 −7 −8 −9 −10 −11 −12 −13 −14 −15 0

0.5

1

ω∗

a)

1.5

2

2.5

log10 (|α (η = 0.286)|2 /fs ) [deg2 /Hz]

log10 (|h /s(η = 0.286)|2 /fs ) [1/Hz]

The measurements considering a pure torsional oscillation around the wing quarter line are intended to complete the simplified aeroelastic analysis in the relevant degrees of freedom of the swept wing deformation. Here, the time-dependent motion of the wing structure is dominated by the forced pitch oscillation. The harmonic pitching is the primary content of the local time-dependent angle of attack at = 0.286, while the heave motion of this wing section remains almost unaffected at small pitch amplitudes (Fig. 26). wc/4 /s = 0.088%, α = 0.165◦ , ωα∗ =0.101, test 5009, Δω∗ =

−2

harmonic pitch oscillation steady wing

−3 −4 −5 −6 −7 −8 −9 0

0.5

1

ω∗

1.5

2

2.5

b)

Fig. 26 Forced pitch excitation effect on the wing deformation at = 0.286 displayed for = 0.165° and reduced frequency = 0.1, single the smallest amplitude pitch amplitude shock flow. a) Wing section heave motion. b) Wing section angle of attack

Unsteady Transonic Fluid – Structure – Interaction

351

0

0

−1

−1

log10 (|cp |2 /fs ) [1/Hz]

log10 (|cp |2 /fs ) [1/Hz]

The pressure fluctuations in the supersonic flow region are dominated by the reduced fundamental oscillation frequency. Especially the spectra of pressure (Fig. 27a). fluctuations in the supersonic field show a significant maximum at The main shock oscillation frequency, which was identified in the steady wing flow, is also strong in the unsteady pressure field around the pitching wing, demonstrating that the acoustic mechanism responsible for shock buffet remains unaffected by the oscillatory motion of the wing. In the single shock flow the fundamental frequency almost vanishes from the spectrum in the interaction region between shock wave and boundary layer separation at = 0.652 (Fig. 27b). This behavior cannot be observed in the lambda shock flow. The strength of the pressure oscillations on the wing, i. e., the power contained in the shock buffet process, is sensitive to certain combinations of pitching amplitude and frequency. Fig. 28 shows an overview containing the power spectral = 0.73 at the location of the incipient density of the buffet reduced frequency separation displayed for the oscillation parameter field. This is the position, where the strongest fluctuations occur. While the single shock test case generally exhibits stronger fluctuations (Fig. 28a) in comparison to the lambda shock flow = 0.12° with (Fig. 28b), the combination of a very small pitching amplitude of a reduced frequency of = 0.1 triggers a very strong answer of the flow field in the buffet frequency. Similar although weaker peaks can also be found for other oscillation parameter combinations in this flow case. The normal shock flow is obviously much more susceptible to disturbances from the wing oscillations than the lambda shock flow. The existence of certain trigger points is most likely a

−2 −3 −4 −5 −6 −7 −8

−2 −3 −4 −5 −6 −7 −8

−9

−9

−10 0

−10 0

0.5

1

ω∗

a)

1.5

2

2.5

0.5

1

ω∗

1.5

2

2.5

b)

Fig. 27 Pressure fluctuation power spectral density in the single shock flow, = 0.568°. a) Supersonic flow, = 0.25. b) Shock region, = 0.652

= 0.075,

352

P.C. Steimle, W. Schröder, and M. Klaas

feature caused by the inherent non-linearity of the flow, although due to the lack of reports about similar interference effects between an oscillating shock wave and an unsteady wing structure in the literature, no further explanation for this behavior can be given yet. For the single shock flow over the wing pitching at = 0.075, the first harmonic pressure distribution is displayed in Fig. 29 for different pitching amplitudes . Generally, the flow exhibits a rather distinct periodic motion on both sides with a maximum at the quarter chord and in the shock – boundary layer interaction region. The flow on the wing upper side advances the pitch oscillation by an almost constant phase angle of approximately 2 /3. With increasing pitch amplitude the phase changes significantly in the shock region to lag behind the wing oscillation. The periodic flow answer in the shock region also reduces at increasing pitch amplitude . This trend results in the development of limit cycle oscillations as the regular aeroelastic instability process in this flow case. There is no such development for the modulus on the lower side leading to the assumption that the skewed boundary layer profile and trailing edge separation on the upper side has a damping effect, which is increasing at the pitch amplitude. The aspect of a frequency selective change in flow behavior which became apparent in Fig. 28 is also evident in the harmonic pressure distribution at the smallest pitching amplitude. When the reduced oscillation frequency is changed, the periodic answer in the shock region is much smaller than at 0.075. It is suspected that the unsteady flow is more susceptible to the = 0.73 is close reduced frequency 0.075, since the reduced buffet frequency to a multiple of it.

−3

−3

x 10

|cp |2 /fs [1/Hz]

|cp |2 /fs [1/Hz]

x 10 6 4 2 0 0.1

0.05

ωα∗

0

0.4

0

0.8

α1 [deg]

a)

1.2

1.6

6 4 2 0 0.1 0.05

ωα∗

0

0.4

0

0.8

1.2

1.6

α1 [deg]

b)

Fig. 28 Comparison of pressure fluctuation spectral power in the shock region as a function and reduced frequency , a) Single shock of harmonic wing oscillation amplitude flow, b) Lambda shock flow case

Unsteady Transonic Fluid – Structure – Interaction

|Hcp,α |

8

10

a)

α1 = 0.169◦ α1 = 0.568◦ α1 = 1.004◦

8

|Hcp,α |

10

a)

353

6 4

02

03

04

05

06

07

0

−3 0

4

α1 = 0.169◦ α1 = 0.568◦ α1 = 1.004◦ 0.1

0.2

0.3

0 0 3

08

ϕ(Hcp,α )

ϕ(Hcp,α )

01

1.5

−1.5

6

2

2 0 0 3

α1 = 0.173◦ α1 = 0.421◦ α1 = 1.1◦

0.4

0.5

0.6

0.7

01

02

03

|Hcp,α |

8

−1.5 −3 0

0.8

0.1

0.2

0.3

0.5

0.6

0.7

0.8

α1 = 0.173◦ α1 = 0.421◦ α1 = 1.1◦

6 4 2

01

02

03

04

05

06

07

α1 = α1 = α1 = 1.004

0 −1.5 0.1

0.2

0.3

0 0 3

08

0.169◦ 0.568◦ ◦

ϕ(Hcp,α )

ϕ(Hcp,α )

0.4

α1 = 0.173◦ α1 = 0.421◦ α1 = 1.1◦

8

2

−3 0

08

10

b)

α1 = 0.169◦ α1 = 0.568◦ α1 = 1.004◦

4

1.5

07

x/c1

6

0 0 3

06

0

|Hcp,α |

10

05

1.5

x/c1

b)

04

0.4

0.5

0.6

0.7

0.8

x/c1

Fig. 29 Modulus and phase of the 1st unsteady pressure distribution in = 0.286 = 0.075. for the single shock flow, a) Upper side. b) Lower side

1.5

01

02

03

04

05

06

07

08

0.4

0.5

0.6

0.7

0.8

α1 = 0.173◦ α1 = 0.421◦ α1 = 1.1◦

0 −1.5 −3 0

0.1

0.2

0.3

x/c1

Fig. 30 Modulus and phase of the 1st unsteady pressure distribution in = 0.286 = 0.075. for lambda shock flow, a) Upper side. b) Lower side

The comparison of the unsteady pressure distributions of the normal shock and lambda shock flows demonstrates another effect which buffet has on harmonic wing oscillations in the pure torsional degree of freedom. Compared to the unsteady single shock flow the lambda shock flow is much more sensitive to disturbances from the wing structure and exhibits a rather strong harmonic oscillation with the fundamental frequency in anti-phase to the rest of the flow field (Fig. 30a). The observation is valid for all amplitudes and frequencies of the harmonic wing oscillations investigated. The effect is, however, most significant at very small pitch oscillations. Obviously, the shock buffet mechanism in the single shock flow has a stabilizing influence on the fluid – structure interaction, since the wing oscillation cannot change the acoustic feedback loop being the origin of the buffet oscillation. On the other hand, the lambda shock pattern quickly adjusts to changes in the local boundary layer, especially to the position of the trailing edge separation line, which is depending on the instantaneous local angle of attack.

354

P.C. Steimle, W. Schröder, and M. Klaas

Hence, it locally amplifies the structural oscillation (Fig. 31) and the buffet oscillation is not strong enough to further dominate the flow field.

log10 (|xs /c1 |2 /fs ) [1/Hz]

−2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 0

0.5

1

ω∗

1.5

2

2.5

Fig. 31 Power spectrum of the downstream shock position in = 0.286 for the lambda = 0.075 and = 1.1° shock flow, forced pitch oscillation with

4.3 Fluid-Structure Energy Exchange The probably most important aspect in the analysis of the dynamic pressure distribution is the energy exchange between the transonic flow field and the wing structure, since it determines the development of aeroelastic instabilities potentially occurring in cruise flight condition. The first harmonic pressure distributions presented in Figs. 24, 25, 29 and 30 so far do not contain enough information for an adequate judgment, because the linearization implemented herein does not represent the entire flow unsteadiness, which is highly non-linear. A better criterion taking the non-linearities into account can be the averaged local energy exchange on the unsteady wing based on the method discussed by Dietz et al. [8]. The fluiddescribing structure energy exchange is quantified by the local work coefficient the local work exerted by the pressure fluctuating quantity on a surface element with the corresponding normal vector and instantaneous local velocity of the wing surface in the chord-wise position summed over the entire measuring time T and averaged by the number of oscillation periods : .

(2)

The step between two subsequent data points in time is the reciprocal value of the sampling rate . A positive leading sign of the work coefficient corresponds to work exerted on the wing structure by the flow, thus to an excitation of the wing by the flow in η = 0.286. This methodology can, of course, only be applied to locations on the wing surface, where the instantaneous pressure distribution is a known quantity. Since the result from the PSP measurements still contain a significant amount of optical noise, only the pressure distribution at η = 0.286 is used in this analysis. However, it should be possible to extrapolate the local analysis to

Unsteady Transonic Fluid – Structure – Interaction

355

10

10

8

8

6

6

4

4

2

2

cw

cw

the entire wing surface based on the observation in the PSP measurements that pressure fluctuations on the surface develop synchronously along the wing span. The work coefficient distributions show a pronounced exciting effect in the single shock test case with a maximum in the shock – boundary layer interaction zone for both forced heaving and forced pitching motion of the wing model (Fig. 32, Fig. 34). The harmonic heave oscillation enables a significant energy flux to the wing model in the supersonic flow region, which is increasing progressively with the heave amplitude on both sides (Fig. 32). The region of the shockboundary layer interaction is the source of a strong energy production, although the shock oscillation yields much higher frequencies than the harmonic wing motion. This amplitude effect would have the potential to drive a destructive flutter amplitude increase, if pure bending motion were the main structural unsteadiness.

0 −2

−4

−4 −6 −8 −10 0

0 −2

steady wing h1 /s = 0.073% h1 /s = 0.269% h1 /s = 0.515% 0.1

0.2

0.3

0.4

−6 −8 0.5

0.6

0.7

−10 0

0.8

steady wing h1 /s = 0.117% h1 /s = 0.313% h1 /s = 0.518% 0.1

0.2

0.3

8

8

6

6

4

4

2

2

0 −2

−8

0.6

0.7

0.8

0.5

0.6

0.7

0.8

0 −2

−4

−10 0

0.5

a) 10

cw

cw

a) 10

−6

0.4

x/c1

x/c1

−4 steady wing h1 /s = 0.073% h1 /s = 0.269% h1 /s = 0.515% 0.1

0.2

0.3

0.4

−6 −8 0.5

0.6

0.7

0.8

−10 0

steady wing h1 /s = 0.117% h1 /s = 0.313% h1 /s = 0.518% 0.1

0.2

0.3

0.4

x/c1

x/c1

b)

b)

Fig. 32 Time-averaged work coefficient in η = 0.286 for the single shock flow, forced heave oscillation with = 0.075. a) Upper side. b) Lower side

Fig. 33 Time-averaged work coefficient in η = 0.286 for the lambda shock flow, = 0.075. forced pitch oscillation with a) Upper side. b) Lower side

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The behavior of the aeroelastic system is reversed for the harmonic pitching motion about the mean quarter chord line of the wing model (Fig. 34). The structural excitation beneath the supersonic flow bubble reduces with increasing oscillation amplitude and even has a damping effect at the highest pitching amplitude = 1.004°. The cross-flow region on the other hand shows a loss of its damping capability with increasing pitch amplitude (Fig. 34a). This effect can be observed in the pitch as well as heave motion experiments and can be attributed to increased dynamics in the separation line yielding a higher degree of unsteadiness in the cross-flow and separated flow area near the trailing edge. The amplitude effect in the case of wing pitching motion demonstrates the self-regulating mechanism in the aeroelastic system for the single shock flow case, which is near the minimum of the transonic dip in the aeroelastic stability curve. In this flow case wing unsteadiness may develop into a limit cycle oscillation, if the pitching degree of freedom is primarily involved. This conclusion is consistent with the analysis of swept wing transonic limit cycle flutter given in [13] and [55] attributing the positive influence of the bending-torsion coupling and structural washout on the limitation of flutter amplitudes. A significant difference in the energy exchange can be observed in the lambda shock flow case (Fig. 33, 35), which is in the region of the rising branch of the stability boundary. During the harmonic heave motion the wing still experiences an increasing excitation in the supersonic bubble, but with progressively increasing energy consumption by the lambda shock system, which is coupled to the fundamental frequency. This damping is generally effective, when the lambda shock is present. The full scale trailing edge separation, which characterizes this flow case, exhibits a much higher work coefficient, due to a strong pulsation of the separated flow. This observation is in qualitatively good agreement with the energy exchange described for the NLR 7301 profile as test case TL4 by Dietz et al. [8]. The reverse in the energy transfer behavior, which was a feature of the single shock flow, when changing the pure harmonic heave motion into a pitch motion, does not occur in the lambda shock flow. Here, the fluctuations are generally much weaker yielding a lower level of energy exchange between flow and structure. Nevertheless, a structural excitation can be induced by large pitch amplitudes with the lambda shock damping a significant amount of it (Fig. 35a). On the other hand, the shock wave on the lower side contributes to the structural excitation (Fig. 35b). The arising structural excitation in the supersonic flow field can be attributed to a lack of intrinsic stability, which is present in the single shock flow due to the pronounced shock buffet at much higher reduced frequencies. This behavior of the aeroelastic system however does not occur in the region of the rising branch of the transonic dip, since the amplitude of structural oscillations stay at a lower level due to generally much weaker fluctuations of the flow. Here, the shock motion on the wing upper side itself absorbs energy from the structural oscillation, especially in the case of very small amplitudes.

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Fig. 34 Time-averaged work coefficient in = 0.286 for the single shock flow, = 0.075. forced pitch oscillation with a) Upper side. b) Lower side

Fig. 35 Time-averaged work coefficient in = 0.286 for the lambda shock flow, = 0.075. forced pitch oscillation with a) Upper side. b) Lower side

5 Conclusion Results from three experimental test campaigns with a supercritical high aspect ratio swept wing have been presented to analyze unsteady flow in the context of aeroelastic instabilities developing in the transonic flight regime. The dynamic fluid-structure interaction involving bending-torsion coupling, which is characteristic for the swept wing geometry, was simulated by forced harmonic oscillations of a rigid wing model separately in the bending and torsional degree of freedom. Two different supercritical flow types, one with a single shock wave with incipient boundary layer separation close to the trailing edge, the other with a lambda

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shock system induced by a full scale trailing edge separation, were analyzed regarding their interaction with the harmonically oscillating wing. The flow with incipient shock induced separation exhibits a high level of unsteadiness with the Lee acoustic feedback mechanism causing a periodic oscillation in the position and properties of the shock wave. This shock unsteadiness generates local pressure fluctuations, thus further exciting an existing structural instability. On the other hand, it also introduces a dynamic stability into the flow field, such that the shock wave is less sensitive to disturbances from the oscillating wing structure. The lambda shock flow shows a much lower level of unsteadiness, although the buffet feedback loop is also active. In this flow the steady character of the trailing edge separation line significantly reduces the local flow dynamics. Only the pulsation of the separated flow is a source of unsteadiness mainly affecting the flow deflection angle over the shock wave. However, the experiments with the simulated pure bending and torsional flutter demonstrate yet another special feature of the trailing edge separation. The separation line position depends primarily on the actual flow angle of attack, which is harmonically oscillating during the forced oscillation of the wing, thus inducing a harmonic oscillation of the lambda shock wave. The shock wave position couples with the wing oscillation, thereby consuming kinetic energy from the wing structure. The appearance of the trailing edge separation coincides with increasing stability in the flow, as long as the separation line position has a steady character. Otherwise a large shock motion might develop. At pure heave oscillations, the energy transfer to the structure increases dramatically with the oscillation amplitude showing the potential for destructive flutter developing from this single degree of freedom oscillation. On the other hand, the torsion coupling introduces a stabilizing effect on the structural unsteadiness to enable the development of less severe limit-cycle flutter. The modulation of power contained in pressure fluctuations in the shock interaction region revealed in the forced oscillation experiments does, however, not appear to be linked to a destabilization of the wing structure motion.

References [1] Bendiksen, O.O.: Energy Approach to Flutter Suppression and Aeroelastic Control. J. Guidance, Control, and Dynamics 24(1), 176–184 (2001) [2] Lee, B.H.K., Price, S.J., Wong, Y.S.: Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos. Progr. Aerosp. Sci. 35, 205–334 (1999) [3] Thomas, J.P., Dowell, E.H., Hall, K.C.: Non-linear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit Cycle Oscillations. AIAA J. 40(4), 638–646 (2002) [4] Schewe, G., Mai, H., Dietz, G.: Nonlinear Effects in Transonic Flutter with Emphasis on Manifestations of Limit Cycle Oscillations. J. Fluids Structures 18, 3–22 (2003) [5] Dietz, G., Schewe, G., Mai, H.: Experiments on Heave/Pitch Limit Cycle Oscillations of a Supercritical Airfoil Close to the Transonic Dip. J. Fluids Structures 19, 1–16 (2004)

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[6] Thomas, J.P., Dowell, E.H., Hall, K.C.: Transonic limit cycle oscillation analysis using reduced order aerodynamic models. J. Fluids Structures 19, 17–27 (2004) [7] Bendiksen, O.O.: Transonic Limit Cycle Flutter/LCO. In: 45th AIAA/ASME/ASCE/ AHS/ ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, California, April 19 - 22, AIAA-Paper 2004-1694 (2004) [8] Dietz, G., Schewe, G., Mai, H.: Amplification and amplitude limitation of heave/pitch limit cycle oscillations close to the transonic dip. J. Fluids Structures 22, 505–527 (2006) [9] Tijdeman, H.: Investigation on the Transonic Flow around Oscillating Airfoils. PhD thesis, NLR TR 77090 U, TU Delft, The Netherlands (1977) [10] Hillenherms, C., Schröder, W., Limberg, W.: Experiments on Transonic Aerodynamics about Elastically Suspended Airfoils. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 84, pp. 351–376 (2003) [11] Hillenherms, C., Schröder, W., Limberg, W.: Experimental investigation of a pitching airfoil in transonic flow. Aerosp. Sci. Technol. 8(7), 569–671 (2004) [12] Hillenherms, H.C.: Experimental Investigation of a Supercritical Airfoil Oscillating in Pitch at Transonic Flow. Doctoral thesis D82, RWTH Aachen University, Shaker, Aachen, Germany (2003) [13] Bendiksen, O.O.: Transonic Limit Cycle Flutter of High-Aspect-Ratio Swept Wings. J. Aircraft 45(5), 1522–1533 (2008) [14] Dietz, G., Schewe, G., Kießling, F., Sinapius, M.: Limit-Cycle-Oscillation Experiments At a Transport Aircraft Wing Model. In: CEAS/AIAA/NWL Int. Forum on Aeroelasticity and Structural Dynamics (IFASD), Amsterdam, The Netherlands (2003) [15] Steimle, P.C., Schröder, W., Althaus, W.: Forced Oscillations of a Supercritical Swept Wing in Transonic Flow. In: 13th Intl. Conf. on Methods of Aerophysical Research (ICMAR), Novosibirsk, Russian Federation (2007) [16] Steimle, P.C., Schröder, W., Limberg, W.: Transonic shock boundary-layer interaction on an oscillating high aspect ratio swept wing. In: AIAA/CEAS/KTH Intl. Forum on Aeroelasticity and Structural Dynamics, Stockholm, Sweden (2007) [17] Steimle, P.C., Schröder, W., Klaas, M.: Transonic Shock Buffet Interference of an Oscillating High Aspect Ratio Swept Wing. In: 26th AIAA App. Aerodyn. Conf., Honolulu, Hawaii, USA, AIAA-Paper 2008-6908 (2008) [18] Romberg, H.-J.: Two-dimensional wall adaption in the transonic wind tunnel of the AIA. J. Aircraft 38(4), 177–180 (1990) [19] Amecke, J.: Direkte Berechnung von Wandinterferenzen und Wandadaption bei zweidimensionaler Strömung in Windkanälen mit geschlossenen Wänden (in German). DFVLR-FB 85-62 (1985) [20] Binion, T.W.: Potentials for Pseudo-Reynolds Number Effects. Reynolds Number Effects in Transonic Flow. AGARDograph No. 303 (1988) [21] Moir, I.R.M.: Measurements on a two-dimensional aerofoil with high-lift devices. AGARD Advisory Report 303, DRA, Farnborough, United Kingdom (1994) [22] Tijdeman, H.: Theoretical and experimental results for the dynamic response of pressure measuring systems. NLR-TR F.238 (1965) [23] Burner, A.W., Liu, T.: Videogrammetric Model Deformation Measurement Technique. J. Aircraft 38(4), 745–754 (2001) [24] Graves, S.S., Burner, A.W., Edwards, J.W., Schuster, D.M.: Dynamic Deformation Measurements of an Aeroelastic Semispan Model. AIAA-Paper 2001-2454

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[25] Asai, K., Kanda, H., Cunningham, C.T., Erausquin, R., Sullivan, J.P.: Surface pressure measurements in a cryogenic wind tunnel by using luminescent coatings. In: 17th Int. Congr. Instrumentation in Aerospace Simulation Facilities (ICIASF), September 28 - October 2 (1997) [26] Bell, J.H., Schairer, E.T., Hand, L.A., Mehta, R.D.: Surface Pressure Measurements Using Luminescent Coatings. Ann. Rev. Fluid Mech. 33, 155–206 (2001) [27] Klein, C.: Application of Pressure Sensitive Paint (PSP) for the determination of the instantaneous pressure field of models in a wind tunnel. Aerosp. Sci. Technol 4, 103– 109 (2000) [28] Sakaue, H., Gregory, J.W., Sullivan, J.P.: Porous Pressure-Sensitive Paint for Characterizing Unsteady Flow Fields. AIAA J. 40(6), 1094–1098 (2002) [29] Sakaue, H., Sullivan, P.: Time Response of Anodized Aluminum Pressure-Sensitive Paint. AIAA J. 39(10), 1944–1949 (2001) [30] Kameda, M., Tezuka, N., Hangai, T., Asai, K., Nakakita, K., Amao, Y.: Adsorptive pressure-sensitive coatings on porous anodized aluminium. Meas. Sci. Technol. 15, 489–500 (2004) [31] Amao, Y., Asai, K., Okura, I.: Photoluminiscent oxygen sensing using palladium tetrakis(4-carboxyphenyl)porphyrin self-assembled membrane on alumina. Anal. Commun. 36, 179–180 (1999) [32] Merienne, M.-C., Le Sant, Y., Ancelle, J., Soulevant, D.: Unsteady pressure measurement instrumentation using anodized aluminium PSP applied in a transonic wind tunnel. Meas. Sci. Technol. 15, 2349–2360 (2004) [33] Kawakami, T., Tabei, T., Kameda, M., Nakakita, K., Asai, K.: Unsteady pressurefield measurements by a fast-responding PSP on porous anodized aluminium. In: Proc. 11th Int. Symp. on Flow Visualization, Paper No. 217 (2004) [34] Rueckert, D., Sonda, L.I., Hayes, C., Hill, D.L.G., Leach, M.O., Hawkes, D.J.: Nonrigid Registration Using Free-Form Deformations: Application to Breast MR Images. IEEE Transactions on Medical Images 18(8) (1999) [35] Meyer, C.R., Boes, J.L., Kim, B., Bland, P.H., Zasaduy, K.R., Kison, P.V., Koral, K., Frey, K.A., Wahl, R.L.: Demonstration of accuracy and clinical versatility of mutual information for automatic multimodality image fusion using affine and thin-plate spline warped geometric deformations. Medical Image Analysis 1(3), 195–207 (1997) [36] Christensen, G.E., Müller, M.I., Mars, J.L., Vaunier, M.W.: Automatic analysis of medical images using a deformable textbook. In: Computer Assisted Radiology, pp. 146–151. Springer, Berlin (1995) [37] Bro-Nielsen, M.: Fast fluid registration of medical images. In: Proc. 4th Int. Conf. Visualization in Biomedical Computing (VBC), pp. 267–276 (1996) [38] Steimle, P.C., Karhoff, D.-C., Nakata, S., Schröder, W.: Unsteady Anodized Aluminum Pressure-Sensitive Paint Measurements on a High Aspect Ratio Swept Wing in Transonic Flow. In: Intl. Forum on Aeroelasticity and Structural Dynamics (IFASD), Seattle, Washington, USA, June 21-25 (2009) [39] Pearcey, H.H., Osborne, J., Haines, A.B.: The Interaction between Local Effects at the Shock and Rear Separation - A Source of Significant Scale Effects in WindTunnel Tests on Aerofoils and Wings. AGARD-CP-35, 11-1 – 11-23 (1968) [40] Délery, J., Marvin, J.G.: Shock-Wave Boundary Layer Interactions. AGARDograph (280), 90–108 (1986) [41] Délery, J.M.: Shock wave / turbulent boundary layer interaction and its control. Progr. Aerosp. Sci. 22, 209–280 (1985)

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[42] Naughton, J.W., Sheplak, M.: Modern Developments in Shear-stress Measurement. Progr. Aerosp. Sci. 38, 515–570 (2002) [43] Meijering, A., Schröder, W., Limberg, W.: Aerodynamic Design of an Adaptive Airfoil at Transonic Speeds. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 84, pp. 85–104 (2003) [44] Ting, C.-C., Henze, A., Schröder, W.: Heat Transfer Measurements in Supersonic Flow Using the Liquid Crystal Display Technique. AIAA-Paper 2002-5200 (2002) [45] Inger, G.R.: Analytical Investigation of Swept Shock-Turbulent Boundary Layer Interaction in Supersonic Flow. In: AIAA 17th Fluid Dynamics, Plasma Dynamics, and Laser Conference, June 25-27, AIAA-Paper 1984-1555 (1984) [46] Stanewsky, E.: Shock boundary layer interaction. In: Boundary Layer Simulation and Control in Wind Tunnels, AGARD-AR-224, pp. 271–305 (1988) [47] Brunet, V., Deck, S., Jacquin, L., Molton, P.: Transonic Buffet Investigations Using Experimental and DES Techniques. In: 7th ONERA-DLR Aerospace Symposium ODAS 2006, Toulouse, France. ONERA-TP-2006-165 (2006) [48] Lee, B.H.K.: Self-sustained shock oscillations on airfoils at transonic speeds. Progr. Aerosp. Sci. 37, 147–196 (2001) [49] Deck, S.: Numerical Simulation of Transonic Buffet over a Supercritical Airfoil. AIAA J. 43(7), 1556–1566 (2005) [50] Finke, K.: Stoßschwingungen in schallnahen Strömungen (in German). VDIForschungsheft 580, VDI-Verlag, Düsseldorf, Germany (1977) [51] Schewe, G., Knipfer, A., Mai, H., Dietz, G.: Experimental and numerical investigation of nonlinear effects in transonic flutter. German Aerospace Center internal report DLR-IB 232-2002 J 01 (2002) [52] Dor, J.B., Mignosi, A., Seraudie, A., Benoit, B.: Wind Tunnel Studies of Natural Shock Wave – Separation Instabilities for Transonic Airfoil Tests. In: Zierep, J., Oertel, H. (eds.) Symposium Transonicum III, IUTAM Symposium Göttingen, Germany (1988) [53] Xiao, Q., Tsai, H.M.: Numerical Study of Transonic Buffet on a Supercritical Airfoil. AIAA J. 44(3), 620–628 (2006) [54] Geissler, W.: Numerical study of buffet and transonic flutter on the NLR 7301 airfoil. Aerosp. Sci. Technol. 7(77), 540–550 (2003) [55] Bendiksen, O.O.: Effect of Wing Deformations and Sweep on Transonic Limit Cycle Flutter of Flexible Wings. In: AIAA/CEAS/KTH Int. Forum on Aeroelasticity and Structural Dynamics, Stockholm, Sweden (2007)

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Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity J.A. Kengmogne Tchakam and H.-G. Reimerdes

Abstract. In the present contribution concepts of reduced structural models for Computational Aero-Elastic simulation (CAE) on aircraft wings are presented. Here the idealization approach relies on analytical methods with the aim to shorten in comparison to a typical finite element method computational cost and time, by preserving nearly the same accuracy. Prior to more detailed investigations using higher order models, these simplified models allow an earlier access of insight regarding the aeroelastic and structural behavior of the wing at the very beginning of the design process. At first a one-dimensional idealization that extends the Timoshenko beam by taking into account additional effects due to warpings is developed. To better describe the influence of swept, a three dimensional idealization is derived. Both idealizations yield good agreements in results concerning the global static deformation and the modal behavior of the wing.

1 Introduction To investigate aeroelastic effects on high capacity aircrafts, computational methods are employed. These procedures enhance the capability to predict the static and the dynamic aeroelastic response of the aircraft. It becomes therefore possible to provide detailed information concerning the aeroelastic behavior of the structure during the preliminary design phase. The present study was conducted within the framework of the collaborative research center SFB401 at the RWTH-Aachen university. A numerical method that combines Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) to investigate transonic aeroelastic phenomena has been developed [2, 14]. This tool uses partitioned algorithms and staggered coupling to solve the differential equations of the fluid-structure-interaction J.A. Kengmogne Tchakam · H.-G. Reimerdes Department of Aerospace Structures, RWTH Aachen University, W¨ullnerstrasse 7, D-52062 Aachen Germany e-mail: {kengmogne,reimerdes}@ilb.rwth-aachen.de

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 363–387. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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problem in aeroelasticity. The fluid flow is modeled by means of compressible Navier-Stokes equations with associated turbulent RANS models, the structure by means of the Timoshenko beam model. For more complex configurations, finite element models are available as well. Partitioned algorithms belong to the field of domain decomposition methods since they combine procedures from the fluid and structural mechanics to address the aeroelastic problem. Over the last decades partitioned algorithms in aeroelasticity have widespread and merged to solve growing issues in the framework of Multidisciplinary Design Optimization (MDO). An example here is the aerostructural optimization [1]. Aerodynamic shape optimization and structural optimization are linked in Computational Aero-Elasticity (CAE). One concern at applying partitioned algorithms for aeroelastic purposes is to minimize computational time. Although great progress has been achieved toward developing efficient aeroelastic tools by including preconditioning techniques [15], multigrid methods [16] and multiprocessing through parallel algorithms, there is still a necessity to improve the computational time since a non-negligible number of iterations are run until a convergent state is available. Unlike the structural discretization, the flow field discretization is the most demanding in terms of time consumption as higher order fluid models are employed, and a tremendous amount of grid points are needed. Therefore a risk against loss of accuracy is greater for the flow field discretization if time saving is sought at alleviating the mesh size. Simplified structural models are preferred in the preliminary design stage since they enable a rapid access of useful results, thus helping to accelerate the design process. This approach can also be considered within a numerical aeroelastic environment. By doing so, the preprocessing time of the computational method also benefits, in the sense that the structural preprocessing is minimized. With this respect, concepts based on reduced structural models in computational aeroelasticity were worked out. The objectives behind these concepts have been to develop analytical methods that are capable through simplified models to deliver a high-fidelity structural behavior of the wing, with nearly the same accuracy as their finite element model counterparts. The natural advantage which arises from this approach is that more control on the design parameters is allowed. A systematical variation of the wing planform, and structural properties at fast model generation is then enabled and parametric studies for sensitivity analysis can be easily performed. The state of the art practices in analytical idealization of aircraft wings include one-dimensional structural models using modified box-beam [17], plate-beam theories [7], plate models [8, 9] and three dimensional models based on partial differential equations or the shearing theory [12]. The first work of the research program presented here has focused on a one-dimensional structure using an extended beam theory. So was developed the so called membrane-panel model [3]. This idealization extends the Timoshenko beam by considering warping due to transverse shear and torsion. Though the model can predict the wing deformation quite well, the stress distribution in case of swept wing suffers from shortcomings at the root. In the second work a three-dimensional -cell- model was developed [18]. For its formulation, the wing is discretized using cell elements that are made of panels and truss beams.

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For panel elements, continuum equations of the thin plate theory are derived. In order to avoid complicated analytical solution procedures a simplification regarding the stress flow is met. Anisotropic material properties are considered in the models. By taking the anisotropy into account, the benefits of its directional stiffness properties and stiffness coupling can be investigated in the framework of aeroelastic tailoring [11].

2 Structural Design of Wing Box The Structural layout of conventional aircraft wing structures is complex. The main issue in their design is to ensure sufficient torsion as well as bending stiffness and strength to resist the wing loading. The purpose of this section is to give the reader substantial aspects that were considered for the analytical idealization as well as some aspects of generic wing structural design.

2.1 Geometry Configuration The load carrying structure of a wing is typically a single cell box consisting of webs, parabolic skins, ribs and stringers as depicted in Fig. 1. The wing cross section is unsymmetrical and it shows spar booms at each web corner (see for example Fig. 6 in section 3). Different wing planforms are obtained from the sweep angle, including multiple sweep and the wing taper. The structural chord length of the wing box is chosen such that enough bending-torsion stiffness is ensured. The arrangement of the above-specified elements has to be considered in the idealization of the wing structure.

skin stringers ribs webs

Fig. 1 Wing box geometry and structural components

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2.2 Rib Arrangements In the wing box design philosophy one generally distinguishes between orthogonal or nearly orthogonal and oblique arrangement of ribs as shown in Fig. 2. While the orthogonal arrangement has advantages from the structural point of view, the oblique arrangement is more interesting from the aerodynamic point of view. It is suspected that in orthogonal systems, the wing loading might drive to deformations that are likely to disturb the smoothness of the aerodynamic surface [20].

s

s

rear spar

Fig. 2 Rib arrangement types: conventional arrangement (left) and arrangement parallel to the flight path (right). Courtesy of Niu [20]

Ribs have an influence on the torsional stiffness property of the wing. The rib spacing is adopted such that enough support of bending material (spar caps and skin sheets) against buckling is provided. In the structural idealizations introduced here, ribs are arranged parallel to the root rib plane. Furthermore, they are assumed to be rigid in plane and flexible out of plane.

2.3 Stringer Arrangements In typical wing box designs, stringers are arranged either parallel to the beam axis or parallel to the front spar or parallel to the rear spar (see Fig. 3). In the structural idealizations stringers are assumed to be continuously distributed on the skin sheets. Their arrangement happens to be at equal fraction of the wing chord (see Fig. 1). They are idealized on the skins either smeared or lumped. Spar and web booms are treated as lumped stringers.

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Fig. 3 Stringer arrangements: Parallel to beam axis, parallel to front spar and parallel to rear spar. Courtesy of Dugas [25]

2.4 Sweep and Taper Effects One effect of the wing sweep on the wing deformation is the kinematic coupling between bending and torsion. From a strength point of view it will also result in stress coupling between shear and normal stress since the rib arrangement produces skewed skin sheets. Stress concentration is located at the wing root trailing edge if the wing is swept backward or at the leading edge if the wing is forward-swept. This area is therefore critical for strength and stability. An idealization using beam models cannot capture the above-mentioned stress concentration because the ”equivalent axis” approach is used in those formulations. The wing taper mainly affects the longitudinal strength of the wing. If the stringers are aligned as shown in Fig. 3 then run-outs are created at the leading edge or the trailing edge of the skin area. These run-outs cause severe fatigue problems. If stringers are aligned at equal fraction of the structural chord (which is the case in the here derived idealizations), their spacing becomes narrower in each cross section, leading to an overestimation in the dimensioning, if the stringer cross sections are not adjusted.

2.5 Warping Effects To accurately determine the stress distribution in the wing box it is important beside bending and twisting loads to account also loads resulting from restrained warping. The torsion and shear related warping displacement functions on wing boxes with parabolic skins are shown in Fig. 4.

2.6 Material Anisotropy Materials with anisotropic behavior have proved some advantages in the design of aerospace structures. Among others, the weight saving can be enforced at preserved

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Fig. 4 Torsion and shear related warping functions for a wing box with parabolic skins

structural stiffness. Using composite materials, the structural and aerodynamic performance may be enhanced if the wing is tailored [5]. Shirk has summarized the beneficial effects of aeroelastic tailoring in Fig. 5. While the wash-in effect which is the ”bend-up” and ”twist-up” of the wing helps to prevent flutter, wing washout which is opposite is suitable for raising the divergence velocity. The aeroelastic

maneuver drag reduction maneuver load relief lift effectiveness

U¥ divergence prevention

control effectiveness flutter prevention

wash out wash in

primary stiffness direction structural reference axis

Fig. 5 Main effects of an anisotropic wing. Courtesy of Shirk [10]

tailoring gains most effectiveness if structural optimization schemes are employed [6]. For static and dynamic aeroelasticity of composite wings, the idealizations presented here can be integrated in a computational environment to investigate the effect of anisotropic skin sheets on divergence and flutter.

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3 Structural Idealization In this section the structural idealizations are presented. These idealizations ought to supply the structural solver of the numerical aeroelastic code for simulations on realistic aircraft wing configurations. At first a one-dimensional idealization is presented. In this idealization the classical Timoshenko beam is expanded by introducing the already mentioned restrained warping effects in the kinematic behavior and the internal loading of the structure. Secondly a three-dimensional idealization is discussed. Unlike the one-dimensional idealization that may lack accurate information on the stresses at the root for swept wings, the three-dimensional idealization is sufficiently consistent to better address this issue.

3.1 One-Dimensional Idealization In the one-dimensional structural idealization of wings, simplified models based on modified beam theories are widely used [6]. One of these extended beam theories has been developed in the framework of the collaborative research center SFB 401. A brief overview of this idealization is given here. For detailed information concerning the formulation of the underlined method refer to [3, 4]. The aim of developing the theory has been to provide a structural model which describes the static and dynamic behavior of aircraft wings with few degrees of freedom. Secondly the model should include additional effects typical to wing box structures such as warpings due to torsion and transversal forces. A load carrying wing geometry configuration with unsymmetrical cross section as shown in Fig. 6 is considered. In this idealization -also called membrane-panel idealization- the longitudinal forces, bending

tupper skin

E SB, A SB 2

E upper skin ESB, ASB3

a1 E Web 1

EWeb 2 tWeb 2 ESB, ASB 4

b/2

h1 tWeb 1

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E lower skin

z h2 dz

tlower skin

Fig. 6 Cross section of the idealized wing box

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moments and transversal forces are carried by the wing membrane shell built from stringer and rib stiffened skin sheets. The following assumptions hold: • The normal and shear stresses are constant over the skin and web thickness. • The ribs of the wing are assumed to be parallel to the root section and have infinite shear stiffness in their own plane, while they are flexible out-of-plane. • The stringers are continuously distributed along the cover sheets while discrete spar booms are considered.The stringers, spar- and rib booms carry only normal forces. 3.1.1

Effects of Restrained Warping

The considered warping deformations that occur in the wing have been shown in Fig. 4. Both warping shapes are approximated using a separation ansatz. For torsion related warping the displacement is approximated as: uT = ψT (x) u0T (y, z)

(1)

uS = ψS (x) u0S (y, z)

(2)

and for shear related warping:

where ψT and ψS are the generalized coordinates, while u0T and u0S are the warping functions of the cross section. They are evaluated according to the theory of restrained torsion of thin walled prismatic shell beams. The displacement in spanwise direction is expanded by means of these warping functions as: u(x, y, z) = uN (x) + ϕy (x) z − ϕz (x) y + ψT (x) u0T (y, z) + ψS (x) u0S (y, z) and the normal stress as:

σx (x, y, z) = E(y, z)

My Nx Mz WT 0 WS 0 + z − y+ uT + uS . EN A Ey Iy Ez Iz ET CT ES CS

(3)

(4)

It is obvious from equation (4) that the normal stress is also expanded by additional warping moments WT and WS . 3.1.2

Effects of Wing Taper

In beam models taper is achieved by variation of the cross sections, where the geometry of each section itself is constant. As an immediate consequence, a stiffness variation occurs in each cross section. From the strength point of view a part of the bending moments are carried by shear forces, while the normal stresses become larger.

Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity

3.1.3

371

General Remarks

Material anisotropy is integrated in the theory according to the classical lamination theory [19]. For anisotropic stiffness equivalent Young’s and shear moduli are used. A dynamic model for the computation of eigenfrequencies and eigenmodes, in which inertia warping forces and moments are neglected is also available.

3.2 Three-Dimensional Idealization Using the membrane panel model, the prediction of the global deformation of the wing with a good accuracy is possible. For a stress calculation on swept wings, this one dimensional idealization may rather exhibit some shortcomings that are generally inherent to beam theories. Here the accuracy of results concerning the stress distribution is difficult to achieve, especially at the wing root area where coupling effects become dominant. Analytical methods for computing the stress in a swept wing box can be confined to two categories: continuous methods where a regular arrangement of load carrying components is assumed and discontinuous methods based on shearing theory. The developed theory uses a combination of continuous and discontinuous method similar to the approach published in [13] and presented in more detail in [18]. 3.2.1

Discretization

The load carrying structure in Fig. 1 is discretized in spanwise direction using cell elements. In each cell, discrete truss elements for stringers and panel elements for skins are used. An example of this discretized wing is shown in Fig. 7 for a swept configuration. In this typical case a cell element is made of eight panels and eight truss elements. If one cell element is unfolded, the cell layout reveals skewed panels showing more complexity in this view than in the planform or the perspective view of the cell. Each panel has its own oblique coordinate denoted by the local parallelogram angle βi . 3.2.2

Panel Element

In the following, the equations of the linear theory of elasticity for a parallelogram element are derived. The parallelogram angle describes the oblique coordinate system in which equations of thin plate theory are formulated. These continuum equations are derived from the fundamentals of linear theory of elasticity. Kinematics From Fig. 8 the displacement-strain relations of a thin plate in oblique coordinate system write: ∂ u¯ ∂ v¯ ε¯x¯ = (5) + sin (β ) d x¯ d x¯

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discretized wing

skewed panels cell element

z y x

z

wing root Fig. 7 Cell element discretization of a swept wing and the resulting skewed panels

∂ v¯ ∂ u¯ + sin (β ) d y¯ d y¯

(6)

∂ v¯ ∂ u¯ ∂ u¯ ∂ v¯ + sin (β ) + + sin (β ) . d x¯ d x¯ d y¯ d y¯

(7)

ε¯y¯ = γ¯x¯y¯ = Equilibrium Condition

From the stress flows in Fig. 9 the equilibrium of forces and moments can be written as follow:

∑ F¯x¯ = 0 : ∑ F¯y¯ = 0 : ∑ M¯ z¯ = 0 :

∂ n¯ x¯y¯ ∂ n¯ x¯ + =0 ∂ x¯ ∂ y¯ ∂ n¯ y¯ ∂ n¯ y¯x¯ + =0 ∂ y¯ ∂ x¯ n¯ x¯y¯ = n¯ y¯x¯ .

(8) (9) (10)

Constitutive Law Using the expressions from the equations (5), (6) and (7) the Hooke’s law writes:

∂ (u¯ + v¯ sin(β )) = s¯11 n¯ x¯ + s¯12 n¯ y¯ + s¯16 n¯ x¯y¯ ∂ x¯

(11)

Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity

y V u dy sinb Vy

y

b V v dy Vy

V v dy VVy

dy b

373

b V v dy sinb VVy

V u dy VVy V u dx sinb Vx V v dx Vx

b

V v dx Vx b x,x

V u dx Vx Vx

dx

V v dx sinb Vx

Fig. 8 Kinematic relations at parallelogram element in oblique coordinates

y

ny dx + V ny dy dx V Vy

y

dx nx dy

dy

nyx dx + V nyx dy dx V Vy nxy dy + V nxy dx dy V Vx Vn nx dy + x dx dy V Vx

nxy dy b

nyx dx

ny dx

x,x

Fig. 9 Applied forces at parallelogram element in oblique coordinates

∂ (u¯ sin(β ) + v) ¯ = s¯21 n¯ x¯ + s¯22 n¯ y¯ + s¯26 n¯ x¯y¯ ∂ y¯ ∂ ∂ (u¯ sin(β ) + v) ( u¯ + v¯ sin(β ) ) ¯ =− ∂ x¯ ∂ y¯ + s¯61 n¯ x¯ + s¯62 n¯ y¯ + s¯66 n¯ x¯y¯ .

(12)

(13)

374

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J.A. Kengmogne Tchakam and H.-G. Reimerdes

Mathematical Model of the Stress Problem

The most commonly used equation for the plane stress problem is based on the Airy stress function as described in [24]. An analytical solution of this equation for general configurations can be cumbersome because of the high sensitivity of the solution against load boundary conditions. Based on the hypothesis that the shear stress is carried by rigid ribs (which means that no shear deformation occurs in the ribs) and imposing a geometrical compatibility between rib and panel in circumferential direction (equal strain between rib and panel is fulfilled), a simplification upon the circumferential normal stress flow is introduced. The normal stress flow in circumferential direction is assumed to be zero. n¯ y¯(x, ¯ y) ¯ =0. (14) From equation (9) the shear stress flow becomes constant in x-direction ¯ and is denoted as n¯ x¯y¯0 . The remaining equations (8), (11), (12) and (13) are then transformed to ∂ n¯ x¯y¯0 (y) ¯ ∂ n¯ x¯ (x, ¯ y) ¯ + =0 (15) ∂ x¯ ∂ y¯ and

∂ 2 n¯ x¯y¯0 (y) ¯ ∂ u¯ ∂ v¯ − μ 2 n¯ x¯y¯0 (y) , ) ¯ = g(β , u, ¯ v, ¯ ∂ 2 y¯ ∂ y¯ ∂ y¯

where

μ 2 := 12

2 s¯6,6 s¯1,1 − s¯1,6 2 l2 s¯1,1

(16)

(17)

and s¯1,6 − s¯1,1 sin(β ) s¯1,6 sin(β ) − s¯1,1 (u¯l − u¯0 ) + 12 (v¯l − v¯0 ) 2 l3 2 l3 s¯1,1 s¯1,1

∂ u¯l ∂ u¯0 ∂ v¯l ∂ v¯0 6 − + − sin( β ) . − s¯1,1 l 2 ∂ y¯ ∂ y¯ ∂ y¯ ∂ y¯

g := 12

(18)

Equation (16) is a second order linear differential equation. The general solution of equation (16) is computed according to the theory of ordinary differential equations [23] using the superposition of a homogeneous solution and a particular one. 3.2.4

Boundary Conditions

The elementary solution in one complete cell is found by coupling elementary solutions from each panel. The coupling is achieved using dynamic and kinematic boundary conditions at each interface between two panels and one stringer. Thus the equilibrium of forces on stringer element k from Fig. 10 is stated as follows: n¯ x¯y¯i+1 = n¯ x¯y¯i −

N¯ x¯ = l − N¯ x= ¯ 0 l

(19)

Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity

375

i+1

i+1

panel i+1

i+1

i+1 i

i+1

i+1 i+1

k

i+1

i+1

stringer k i i i

panel i

i i

i

i i

Fig. 10 Coupling between stringer and neighboring panels

and the geometrical compatibility between stringer and panels:

ε¯x¯k = ε¯x¯i = ε¯x¯i+1 . 3.2.5

(20)

Approximation

According to the right hand side of the differential equation (16) the uniqueness of the solution is achieved if the complete structure is kinematically definite. To ensure this, the global boundary displacement field of the cell will be computed. The displacement field of each cell is approximated by means of panel boundary collocation. For this the boundary displacement function of one panel reads: ¯ y) ¯ = uˆgi ∗ ug1,i (x, ¯ y) ¯ . ugi (x,

(21)

The trial functions ug1,i include linear, quadratic and cubic shape functions of displacements perpendicular to the rib plane and the in-plane rigid body motions of the rib. This results in 3n + 3 degrees of freedom for a cell, where n is the total number of panels at each cell. 3.2.6

Element Stiffness Matrix

The element stiffness matrix is obtained from the displacement method as described in [21]. Using the approximation from equation (21) as well as the elementary solutions obtained in section 3.2.4 the principle of virtual work is applied:

δ (U − Wa ) = 0

(22)

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where U denotes the deformation energy and Wa is the potential of external loads. The element stiffness matrix writes: nS

K=

∑

j=1

¯+ ε¯1Tj σ¯ 1j dV

kmax

∑

k=1

ε¯1Tk σ¯ 1k dV .

(23)

The second term of the element stiffness matrix K in equation (23) considers the deformation energy of discrete stringers. Using stress flow terms the coefficients of the stiffness matrix become: ⎞ ⎛ n¯ x,1 ¯ j,ι nS kmax ⎜ 0 ⎟ ¯ ¯ ¯ d A + ki,ι = ∑ ε¯x,1 , ε , γ ⎠ ⎝ ¯ j,i y,1 ¯ j,i x¯y,1 ¯ j,i ¯ k,i N1,Str k,ι d x¯ ∑ εx,1 j=1

n¯ x¯y,1 ¯ j,ι

k=1

(24) where the index “ 1“ in the subscript list reminds the computed elementary solutions of stress flows.

3.3 Structural Dynamics For the dynamic of the wing the equation of motion reads in the form: Mu¨ + Ku = F(t) .

(25)

The mass matrix M is built by taking the mass contribution from displacements in spanwise direction as well as rigid body motions into account. For displacements in spanwise direction a consistent mass matrix is first determined as described in [22]. Since only the lowest eigenfrequencies are of interest, a lumped mass matrix is sufficient. 3.3.1

Displacements in Spanwise Direction

The mass contributions for displacements in spanwise direction are measured at discrete stringer nodes as illustrated in Fig. 11. 3.3.2

Rigid Body Motions

Unlike displacements in spanwise direction, in-plane rigid body movements of ribs are measured from a reference axis located at the front spar web foot point as shown in Fig. 12, where acceleration components for mass and moment of inertia are assigned. The rigid body motions are coupled since the mass and the reference axis do

Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity

377

panel stringer rib

Fig. 11 Mass contribution from displacements in spanwise direction at stringer node

.. -z j .. rj W .. w zRi

.. j

W

.. w

.. v

dm

r .. j r=1

.. yj

z .. v

yRi

y

Fig. 12 Rigid body acceleration components of a mass element dm as a result of reference axis accelerations. The radial component of acceleration ϕ˙ 2 r is neglected

not coincide. From the Newton and Euler axioms, the inertia forces and moments thus write:

Fy = v¨ Fz = w¨ M = w¨

dm − ϕ¨ x

dm + ϕ¨ x y dm − v¨

z dm

(26)

y dm

z dm + ϕ¨ x

(27)

r2 dm .

(28)

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J.A. Kengmogne Tchakam and H.-G. Reimerdes

Mass moments and moments of inertia resulting from the rotation about the reference axis are: ϕ¨x z dm, ϕ¨x y dm, ϕ¨x r2 dm . Added masses can be used to cover the mass distribution of secondary masses such as flaps, fuel and engine masses. 3.3.3

General Remark

In fact the procedure to derive the stiffness matrix of the cell element is very close to that of the displacement method from the finite element analysis. The slight difference lies in the shortcut that can be taken using elementary stress solutions of the ordinary differential equation. They are directly applied to build the coefficients of the stiffness matrix.

4 Numerical Results In this section some numerical results are displayed. Results of the one-dimensional idealization have been published and broadly commented in [3, 4], so that an emphasis is put here on the results of the three-dimensional idealization. The first example aims at showing sweep effects which the described theory accounts for. Then a validation of stresses against the finite element method follows. Thereafter a dynamic simulation based on the modal analysis of a generic wing box will be carried out. Finally the possibility to use the cell model for the static and the vibration tailoring is studied, with the objectives to gain some insights toward the ability to control flutter and divergence of composite wings during aeroelastic simulations.

4.1 Sweep Effects In this example the effect of sweep on the stress distribution is investigated at two straight wing configurations with parabolic skin sheets. The first wing is unswept and the second is backward swept at 35o . Both wings have the following dimensions: Halfspan width = 30 m, chord length = 6 m, box height = 1.25 m. The wings are clamped at the root and loaded at the tip with a single force of 100kN applied upward in the shear center. As expected for the unswept wing the normal stress (Fig. 13) varies linearly over the wing span and the shear stress (see Fig. 14) remains constant. For the swept wing a critical area with both normal and shear stress concentration can be found at the wing root (see also Fig. 13 and Fig. 14).

4.2 Warping Effects To illustrate warping effects, the stresses in spar booms are plotted in Fig. 15. It is obvious that an unsteady variation of the normal stress occurs in the vicinity of

Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity

379

Fig. 13 Normal stress distribution at upper skin (left) and lower skin (right)

Fig. 14 Shear stress distribution at upper skin (left) and lower skin (right)

the wing root, proving the existence of warping stresses. These warping stresses are less dominant in the straight wing where an almost linear variation from the wing

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500

500 boom 1 boom 2 boom 6 boom 7 cross section

400

300

200

200

100

100 σx [MPa]

σx [MPa]

300

boom 1 boom 2 boom 6 boom 7 cross section

400

0

-100

0

-100

-200

1

-200

6

-300

1

6

-300

-400

2

-400

7

-500

2

7

-500 0

5

10

15 20 span width [m]

25

30

0

5

10

15 20 span width [m]

25

30

Fig. 15 Boom stresses: Straight wing (left), swept wing (right)

root to the wing tip is observed, rather than the swept wing counterpart where the linear decrease starts away from the wing root. In the latter, the stress and warping intensities are amplified by coupling effects.

4.3 FE-Validation of Stress To show that reliable results can be obtained from the cell model on realistic configurations a comparison with finite element model is carried out. The wing configuration has the following data: halfspan width: 30 m chord length (root): 5 m box height (root): 1.27 m wing sweep: 35o taper: 9.5 This wing is loaded with a distributed force of 1 kN m applied upward at the front spar foot point. The results of normal stresses are compared in Fig. 16 and they show a satisfactory agreement between finite element analysis and cell model.

4.4 Modal Analysis To perform the modal analysis a symmetric generic straight wing as shown in Fig. 17 is used . The dimensions of the cantilever wing are: L = 10 m, B = 1m, H = 0.3 m. Aluminium is used here as material (isotropic, ρ = 2700 mkg3 ). In Fig. 18 the first four mode shapes and their eigenfrequencies are plotted. These are the first bending in y-direction , the first bending in z-direction, the second bending in y-direction and

Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity

381

FEM

CM

Fig. 16 Normal stress distribution at upper skin (left) and lower skin (right): FEM (top) Cell Model (CM) bottom Z

X

Y

0 2 4 6 8

pan g s n i w

[m]

10 Fig. 17 Unswept generic wing box

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J.A. Kengmogne Tchakam and H.-G. Reimerdes

Fig. 18 Mode shapes and eigenfrequencies of the unswept generic wing box

Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity

383

the first torsion. A finite element verification shows good agreement in the prediction of eigenfrequencies.

4.5 Static Tailoring The generic wing of the last section has been used to conduct static tailoring. The upper and the lower skin of the wing box are provided in a first configuration with orthotropic laminate layup while in the second configuration the orthotropic axis is rotated about 10o . The spanwise loading is a distributed force of 2 kN m applied in the elastic axis of the wing. The measured deformation values are the spanwise twist and deflection (see Fig. 19). It can be seen in Fig. 20 that no structural torsion is produced at orthotropic stacking sequence, whereas at anisotropic stacking sequence a wing wash-in is created. The chosen laminate ply layup controls the wing twist and leads to an increase of the wing deflection (see also Fig. 20). Finite element comparison shows very good agreements concerning the wing deformation.

4.6 Vibration Tailoring The generic wing is now modified by raising the structural chord length to 2m and the height to 0.5m. For the skin sheets a symmetrical and balanced laminate is used,

f

w

Fig. 19 Measures of wing deformation

-0.3

0.8

material: CFRP laminate (T300) number of plies: 8 thickness: 1 mm

material: CFRP laminate (T300) number of plies: 8 thickness: 1 mm

0.6 -0.2

w [m]

φ [°]

0.4 -0.1

C.M. anisotropic FEM anisotropic C.M. orthotropic FEM orthotropic

0

0.2

C.M. anisotropic FEM anisotropic C.M. orthotropic FEM orthotropic orthotropic: [0°, +45°, -45°, 0°]s

0 orthotropic: [0°, +45°, -45°, 0°]s 0.1

anisotropic: [β, β+45°, β-45°, β]s, β = 10° 0

2

4 6 Span width [m]

8

1

-0.2

anisotropic: [β, β+45°, β-45°, β]s, β = 10° 0

2

4 6 Span width [m]

Fig. 20 Wing spanwise twist and deflection

8

10

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with 70 % of fibers orientated at β = 0o at the beginning, 20% at ±45o and the remaining 10% at 90o . During the modal analysis the angle β is varied between −90o and 90o . 4.6.1

Modal Frequency

Without static unbalance the first four modes exhibit symmetrical behavior in their eigenfrequencies in Fig. 21, unlike with static unbalance where an unsymmetrical eigenfrequency characteristic is seen. There, a modal coupling between the third and the fourth mode takes place at −30o ply angle, where a coalescence of eigenfrequencies is noticeable. While the first mode remains symmetrical to the zero axis the second mode is slightly shifted. Both the first and the second mode are uncoupled and do not show any sensitivity toward the excentricity of the center of gravity.

50

40

1.mode 2.mode 3.mode 4.mode

30

35

modal frequency [Hz]

modal frequency [Hz]

35

1.mode 2.mode 3.mode 4.mode

45

30 25 20 15

25 20 15 10

10 5 5 0

-90

-45

0

β[°]

45

90

0

-90

-45

0

45

90

β[°]

Fig. 21 Modal frequency of the tailored generic wing without static unbalance (left) and with static unbalance (right)

4.6.2

Modal Amplitude

To study the effect of ply orientation on eigenmodes, the modal shapes of the third and the fourth mode are considered at β = −30o and β = 0o (Fig. 22). From β = 0o to β = −30o a modal transmutation is observed. This modal transmutation is characterized in the third mode by an increase of the bending coupling and for the fourth mode an increase of torsional coupling. These observations show that modal activities upon the variation of the angle β take place. This is useful in the mode selection prior to dynamic aeroelastic analysis, because then a mode shape can be updated with respect to its damping characteristic.

Structural Idealization of Flexible Generic Wings in Computational Aeroelasticity

3.mode β = 0°

0.4

modal amplitude

modal amplitude

0.2

0

-0.2

0

-0.2

w φ

-0.4

0

2

w φ

-0.4

4 6 span width [m]

8

10

0

3.mode β = -30°

0.4

2

4 6 span width [m]

8

10

4 6 span width [m]

8

10

4.mode β = -30°

0.4

0.2

modal amplitude

0.2

modal amplitude

4.mode β = 0°

0.4

0.2

385

0

-0.2

0

-0.2

w φ

-0.4

0

2

w φ

-0.4

4 6 span width [m]

8

10

0

2

Fig. 22 Third and fourth beam mode shapes for β = 0o and β = −30o

5 Conclusions In computational aeroelasticity, partitioned methods are widely employed to simulate the fluid-structure interaction using staggered coupling. In the framework of the collaborative research center SFB 401, a computational code has been developed using partitioned algorithms to predict the aeroelastic behavior of aircraft wings in transonic flow. To supply this numerical aeroelastic method with simplified structural models, analytical procedures in structural idealization are employed. At first a one-dimensional idealization is described. The so called membranepanel model is an extension of the Timoshenko beam that takes into account additional effects due to restrained warpings. Thereafter, a three-dimensional idealization, so called cell model, is presented. In this three-dimensional idealization the wing structure is discretized in spanwise direction using cell elements that are composed of truss and panel elements. In case of swept wings, panels are skewed. For the skewed panels continuum equations of the linear theory of elasticity are

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derived in oblique coordinate system. A hypothesis on the normal stress flow in circumferential direction is defined to condense the plane stress problem in an ordinary differential equation. Results obtained from the cell model reveal considerable improvements in comparison to the membrane-panel model, in the prediction of the stress distribution on swept wings. Indeed sweep effects characterized by stress concentrations at the root could be found. A realistic configuration was also investigated and good agreement against the finite element method concerning the normal stress distribution has been achieved. Eigenfrequencies and mode shapes have been validated on a generic straight wing. To show that the three-dimensional idealization is suitable for tailoring purposes, the static and modal behavior of a composite wing configuration has been investigated. In the static case, a wash-in of the wing was produced using a suitable stacking sequence of the ply layup, whereas in the dynamic case an eigenfrequency coalescence and a transmutation of mode shapes with respect to ply angle variation were observed. The cell model thus succeeded to deliver information about the beneficial application of composite material in aircraft wing structural design. From the aeroelastic point of view the static tailoring as well as the flutter tailoring can be studied. It is therefore intended to develop a multidisciplinary environment which combines the numerical aeroelastic method and a structural optimization tool.

References 1. Reuther, J.J., Alonso, J.J., Joaquim, R.R.A., Smith, S.C.: A Coupled Aero-Structural Optimization Method for Complete Aircraft Configurations. In: AIAA 37th Aerospace Sciences Meeting (1999) 2. Britten, G., Braun, C., Hesse, M., Ballmann, J.: Computational Aeroelasticity with Reduced Structural Models. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 84 (2003) 3. Jung, W., Reimerdes, H.-G.: Concepts for Reduced Structural Models of Airplane Wings in Aeroelasticity. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 84 (2003) 4. Jung, W., Reimerdes, H.-G.: Ein Beitrag zur aeroelastischen Untersuchung mit idealisierten Tragf¨ugeln. Report, DGLR-Jahrestagung, Berlin (1999) 5. Schneider, G., Zimmermann, H.: Static Aeroelastic Effects on High-Performance Aircraft. AGARD Report No. 725 (1986) 6. Guo, S., Cheng, W., Cui, D.: Aeroelastic Tailoring of Composite Wing Structure by Laminate Layup Optimization. AIAA Journal 44(12) (December 2006) 7. Gern, F.H., Librescu, L.: Static and Dynamic Aeroelasticity of Advanced Aircraft Wings Carrying External Stores. AIAA Journal 36(7) (July 1998) 8. Giles, G.L.: Equivalent Plate Modeling for Conceptual Design of Aircraft Wing Structures. AIAA Journal 95-3945, 1–17 (1995) 9. Kapania, R.K., Castel, F.: A Simple Element for Aeroelastic Analysis of Undamaged and Damaged Wings. AIAA Journal 28(2) (February 1990) 10. Shirk, M.H., Hertz, T.J., Weisshaar, T.A.: Aeroelastic Tailoring-Theory, Practice, and Promise. Journal of Aircraft 23(1), 6–17 (1985)

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11. Weisshaar, T.A.: Aeroelastic Tailoring creative Uses of Unusual Materials. AIAA Paper 87-0976-cp (1987) 12. Ebner, H.: Die Beanspruchung d¨unnwandiger Kastentr¨ager auf Drillung bei behinderter Querschnittsverw¨olbung. Zeitschrift f¨ur Flugtechnik und Motorluft-Schiffahrt 24, 645– 692 (1933) 13. Benthem, J.P.: Analysis of a symmetrical swept-back Box Beam with non-swept Centre Part. Reports and Transactions 28, 71–101 (1964) 14. Braun, C.: Ein modulares Verfahren f¨ur die numerische aeroelastische Analyse von Luftfahrzeugen. PhD Thesis, RWTH-Aachen University (2007) 15. Saad, Y.: Iterative Methods for Sparse Linear Systems. Cambridge University Press, Cambridge (2003) 16. Hirsch, C.: Numerical Computation of Internal and External Flows. Butterworth Heinemann (2007) 17. Duan, S., Piening, M.: Investigation of the torsion-related Warping Behavior of anisotropic Boxbeam Structure. DLR Bericht, IB-131-95/33 (1995) 18. Jung, W.: Analyse des Strukturverhaltens gepfeilter Tragfl¨gel. PhD Thesis RWTHAachen University, Germany (2003) 19. Sch¨urmann, H.: Konstruierem mit Faser-Kunststoff-Verbunden. Springer, Heidelberg (2007) 20. Niu, M.C.Y.: Airframe Structural Design. Conmilit Press LTD (1996) 21. Dieker, S., Reimerdes, H.-G.: Elementare Festigkeitslehre im Leichtbau. Donat Verlag (2005) 22. Bathe, K.-J.: Finite-Elemente-Methoden. Springer, Heidelberg (1990) 23. Collatz, L.: Differentialgleichungen. Teubner (1990) 24. Altenbach, H., Altenbach, J., Naumenko, K.: Ebene Fl¨achentragwerke. Grundlagen der Modellierung und Berechnung von Scheiben und Platte. Springer, Heidelberg (1998) 25. Dugas, M.: Ein Beitrag zur Auslegung von Faserverbundtrafl¨ugeln im Vorentwurf. PhD Thesis, Stuttgart University (2002)

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Aero-structural Dynamics Experiments at High Reynolds Numbers Josef Ballmann, Athanasios Dafnis, Arndt Baars, Alexander Boucke, Karl-Heinz Brakhage, Carsten Braun, Christian Buxel, Bae-Hong Chen, Christian Dickopp, Manuel K¨ampchen, Helge Korsch, Herbert Olivier, Saurya Ray, Lars Reimer, and Hans-G¨unther Reimerdes

Abstract. The elastic wing model, its excitation and comprehensive high frequency measuring equipment for the High Reynolds Number Aero-Structural Dynamics (HIRENASD) tests in the European Transonic Windtunnel (ETW) are shortly described. Some of the stationary polars are presented in terms of wing deformation, as well as aerodynamic coefficients and pressure distributions. Then unsteady processes observed in the measurements of static aerodynamic coefficients, are regarded with focus on small amplitude pressure waves travelling upstream from the trailing edge and triggering periodically break-down and redeployment of the local supersonic domains with transonic shock waves to run upstream and to disappear. Another focus is on stochastic vibrations excitation while moving forward during nominally static experiments. Emphasis is put on measured variations of pressure distribution on the wing surface caused by defined vibration excitation applying internal force couples at the wing root, whereby the exciter frequencies were chosen Josef Ballmann · Arndt Baars · Alexander Boucke · Carsten Braun · Saurya Ray LFM, Lehr- und Forschungsgebiet f¨ur Mechanik, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen e-mail: [email protected] Karl-Heinz Brakhage · Christian Dickopp IGPM, Institut f¨ur Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen Athanasios Dafnis · Christian Buxel · Manuel K¨ampchen · Helge Korsch Hans-G¨unther Reimerdes ilb, Institut f¨ur Leichtbau, RWTH Aachen University, W¨ullnerstraße 7, 52062 Aachen e-mail: [email protected] Herbert Olivier SWL, Stosswellenlabor, RWTH Aachen University, Schurzelter Straße 35, 52056 Aachen Bae-Hong Chen · Lars Reimer CATS, Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen

W. Schr¨oder (Ed.): Flow Modulation & Fluid-Structure Interaction, NNFM 109, pp. 389–424. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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close to natural frequencies of the wing model. Phase and magnitude of measured local lift fluctuations as well as real and imaginary parts of pressure distributions are presented.

1 Introduction The High Reynolds Number Aero-Structural Dynamics (HIRENASD) project is devoted to the analysis of static aeroelasticity and dynamical aeroelastic processes observed in experiments with a supercritical elastic wing model in the transonic regime at Reynolds numbers which are realistic for large passenger aircraft in cruise flight. Experiments were performed in the European Transonic Windtunnel (ETW) under cryogenic conditions in nitrogen gas. Usually dynamical wind tunnel experiments with elastic wing models have been conducted mainly at Reynolds numbers which are about one order of magnitude less than in cruise flight of large aircraft [1]. Only for a rigid airfoil, oscillating with prescribed frequency, experiments in transonic flow at high Reynolds numbers had been made earlier at NASA LaRC [2] for buffet analysis, whereby cryogenic wind tunnel conditions made the high Reynolds numbers feasible too. Principle aims of the HIRENASD project are to enhance understanding of aeroelastic phenomena in transonic flow at high Reynolds numbers and to gather experimental data for validation of methods for multidisciplinary airplane design and Computational Aero-Structural Dynamics (CASD) simulation of airplanes in flight. The project was funded by the German Research Foundation (DFG) through the Collaborative Research Centre ”Modulation of Flow and Fluid-Structure Interaction at Airplane Wings” ( SFB 401 at RWTH Aachen University, Aachen, Germany). The very expensive experimental project HIRENASD is a special central project which has been initiated after seven years of successful research work of SFB 401. An overview of this research is presented in [3]. The HIRENASD project has been conducted in strong cooperation of four member chairs LFM/CATS, ILB, IGPM and SWL of SFB 401. It included the layout of the model and the experiments, the design and construction of the model and the vibration excitation mechanism, the design and construction of a new windtunnel balance, the manufacturing of all this and equipment with 5 different measuring techniques for measuring pressure distribution, forces, strains, displacements and accelerations. For checking the suitability of the pressure sensors with respect to cryogenic temperatures in the wind tunnel, an airfoil with the profile of the HIRENASD wing model was manufactured, equipped in one section with 41 pressure sensors and tested successfully in KRG G¨ottingen [4, 5, 6, 7]. Considering the safety of model and wind tunnel installations in the tests at the strongest wind tunnel condition which is characterised by the highest value of the load factor q/E=0.70 · 10−6, representing the ratio of dynamic pressure and Young’s modulus of the model material, a resistant construction with high toughness had to be realised. As a consequence, the model became very stiff which rendered its excitation for vibration difficult. For this purpose a piezoelectric mechanism was

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constructed to excite the whole wing model by strong interior force couples acting in the wing clamping zone at the wing root. After first CASD simulations of the planned ASD experiments in ETW [4] and the dynamic qualification of the model under laboratory conditions [8, 9] it was decided to choose the frequencies of excitation close to resonance, defined on the basis of an online decision during the tests at the respective wind tunnel conditions, i.e. total temperature of the gas, value of q/E, total pressure, Mach number and Reynolds number, in order to achieve significant deflection amplitudes and pressure variations during excited vibration. First results of the HIRENASD project were presented in [10, 11], where emphasis was mainly put on static aero-elastic measurements. More detailed analysis of the dynamical experiments has been presented in [12]. Predictions that had been made for these experiments [13, 14, 15] using the SOFIA code package, which was developed in SFB 401 [3, 16] and is still being enhanced [17, 18], have been confirmed by the experimental results very well [19]. In chapter 2 of the paper the wind tunnel ETW, the wing model and its measuring equipment are shortly described. For more details about the project the reader is referred to the cited literature and to the HIRENASD homepage http://www.lufmech.rwth-aachen.de/HIRENASD/ where all geometrical and physical properties of the model, including all sensor positions and wind tunnel conditions as well as a complete list of the experiments, are presented. Furthermore, there are computational grids for the wing structure and the flow field provided for downloading which can considerably facilitate numerical simulations of the ASD experiments by interested researchers. In chapter 3 the focus is on stationary results and chapter 4 to 6 emphasize on observed unsteady processes during nominally stationary polars in the HIRENASD tests and ASD results of the dynamic tests with vibration excitation.

2 Experimental Setup and Windtunnel Conditions 2.1 Wind Tunnel Model in the European Transonic Windtunnel ETW is the European cryogenic transonic facility, which is similar to the NTF in the US at NASA LaRC but a little smaller in cross-section. It has a closed circuit (see Fig. 1), and the test gas is nitrogen. The dimensions of the test section are 2.0m height and 2.4m width. Controlled liquid nitrogen injection and gaseous nitrogen blow-off maintain pressure and temperature of the test gas at the chosen conditions for Mach number, Reynolds number and load factor q/E. A two-stage 50MW compressor provides Mach numbers Ma=0.15 up to Ma=1.3, as shown in Fig. 2. At lowest temperature and highest static pressure condition the Reynolds number, with the aerodynamic mean chord as reference length, achieves about 80 millions in half-model testing. For half-model testing, the wing model is fixed at the wind tunnel balance above the turntable which is located on the wind tunnel ceiling. The HIRENASD wing model corresponds to the SFB 401 clean wing reference configuration, which has a horizontal projection as characteristic for modern passenger aircraft, with a leading edge backward sweep of 34o [20]. The wing

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Fig. 1 The European Transonic Windtunnel circuit

Fig. 2 Test envelope of the European Transonic Windtunnel in comparison with other European windtunnels

configuration consists of three sections. The profile of the two outboard sections corresponds to the BAC 3-11/RES/30/21 originally published in [21] as a reference case for multi-element high lift configuration analysis. For the HIRENASD project the elements have been arranged as for cruise flight to a supercritical profile, which has an 11% thickness. The inboard section is linearly thickened from 11% at 285.71mm span up to 15% at the wing root. This thickening is completely realised on the wing lower, i.e. the pressure side of the wing. The suction side (also called upper wing side in the paper) has the same profile everywhere. The wing model alone has a half span of 1.28571m and its aerodynamic mean chord and horizontal projection area are cre f =0.3445m and Are f =0.39255m2. Figure 3 displays the placement of the model assembly in the test section of ETW on the wind tunnel ceiling as for half model testing [22]. The height 90mm of the fuselage substitute, which is mechanically separated from the wing model itself by a non-contact labyrinth sealing brings the span of the

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Fig. 3 HIRENASD wing model assembly. Dimensions, profile and placement at the wind tunnel ceiling

complete assembly which is wetted by the wind tunnel flow to about 1.36m. The wing model is composed of two separate parts, the suction side part and the main pressure side part. The suction side part contains the leading edge and the trailing edge, including the rear part of the pressure side. Both parts are connected on formlocking contact surfaces (see Fig. 5 for details) and bolted together by more than 150 bolts. For CASD simulations it is suggested to model the wing as one single body, although in the statically indeterminate form-locking contact surface dry friction and unilateral contact problems might be important somewhere in the large test parameter space.

2.2 Measuring Equipment of the Wing Model The measurement equipment contains outside the wing model a 6 components balance and an optical system for surface deformation measurement, recording the displacement distribution via surface pattern tracking (SPT) by means of markers on the pressure side of the wing. Figure 4 shows on a photo a view upon the wetted part of the model assembly mounted in the wind tunnel, where the markers for SPT can be seen. For monitoring acceleration during the tests several acceleration sensors were placed in the

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Fig. 4 Photo of the HIRENASD wind model in the wind tunnel, equipped with markers for SPT on its pressure side (left), and inside view of its suction side part with coordinates of accelerometer positions (right)

Fig. 5 Positions of strain gauges inside of the wing model

assembly, including the balance and the excitation mechanism. 11 from these accelerometers were implemented inside the wing model, all at the upper part of the wing model as shown in Fig. 4 on the right.

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The windtunnel model assembly was equipped with 28 strain gauges of which 6 were placed in the wing clamping at the excitation force transmitters and the other 22 were distributed inside the wing model as shown in Fig. 5. The origin of the coordinate system 0, to which the coordinates of the sensors are referred to in Fig. 4 and 5, is situated at the leading edge in the wing root profile-section, see Fig. 5. The exposure of the adhesive of the strain gauges inside the wing model to the large temperature range from 100K to about 300K caused detachment and drift of many strain gauges. But a certain number of the sensors worked correctly and made a rough reconstruction of wing deformation possible. A number of 259 pressure sensors for cryogenic conditions (Kulite miniature and ultra-miniature sensors) are implemented in 7 pressure sections from relative wing span η =0.14 to η =0.95 as shown in Fig. 6. The numbers of sensors in the different pressure sections varies from 43 in the innermost section to 31 in the outermost section.

Fig. 6 Sensor equipment of the HIRENASD wing model summarized

As always in real experimental situations, not all pressure sensors were active during the experiments and some were recording wrong pressure signals. Figure 7 shows the actively measuring pressure sensors in the different sections, whereby the five sensors marked with red crosses (see sections 2, 5 and 6) did not register correct signals. For steady polars within which α -sweeps were conducted at an angular sweep velocity of the turntable of 0.2o /second, data was simultaneously recorded from all implemented sensors for pressure, acceleration, strain and balance forces at a frequency of 4kHz. During vibration excitation data were recorded with frequencies suited to the vibration excitation frequency with 128 signals per cycle, i.e. for

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Fig. 7 Distribution of pressure sensors giving correct signals in the seven pressure sections

exciting frequency of 26Hz the acquisition rate would be 3328Hz and for 260Hz ten times this rate. The balance for measuring forces is designed for high frequency measurements. It has been constructed and manufactured within the frame of the HIRENASD project as well as the piezoelectric mechanism for exciting vibration. Figure 8 shows on top the mounting flange, which is fixed to the balance adapter inside a heated enclosure which is attached to the turntable outside on top of the wind tunnel. Four piezoelectric force sensors measure the forces and moments transmitted in all three space directions from the wing via the wing clamping and the wing adapter. More then 180 experiments/polars at different conditions were conducted in total, among them about 50 static tests and about 130 dynamic tests with defined vibration excitation to analyse nonstationary pressure distributions and vibrations as well as displacement fields of static equilibrium configurations.

2.3 Internal Forced Vibration Excitation Vibration excitation is realized by four high voltage piezoelectric stacks, two of them on the pressure side and two on the suction side of the wing model, which are acting in anti-phase, such that two equal force couples, resulting in a bending moment about the wing root chord, are applied through force transmitters on the prominent noses at the wing root (marked as solid blocks in Fig. 8). The force couples act as inner forces in the wing clamping block and its prolongation by the containment for the piezoelectric stacks producing the forces due to controlled timeharmonic voltage changes. Because the swept wing model has no axis or plane of symmetry every mode shape of the model can be excited predominantly by choosing the exciting frequency equal or as close as possible to the corresponding natural mode frequency. As mentioned already, the wing model was designed very stiff for

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Fig. 8 Piezoelectric six components balance (upper part) and wing clamping with integrated piezoelectric vibration excitation mechanism at wing root (lower part)

safety reasons. Therefore, it suggested oneself, also for achieving significant and measurable vibration amplitudes and pressure fluctuations, to excite the model in the dynamic experiments with single natural frequencies. The dynamic experiments with excitation were concentrated on three eigenfunctions of the wing model. These were the two lowest modes with natural frequencies about 27Hz and 80Hz, which both are dominated by flap bending, and the 5th mode with natural frequency about 268Hz which is predominantly torsion. Due to large differences of temperature in the experimental program and the different flow conditions the natural mode frequencies changed within a range of some Hz.

2.4 Wind Tunnel Conditions Test Mach numbers were chosen Ma=0.70, 0.75, 0.8, 0.83, 0.85, 0.88, and Reynolds number was varied from 7 to 73 millions. The main conditions were those for Mach number Ma = 0.8, for which the test envelope is presented in Fig. 9. The encircled number indicate the sequence of the performed test series. During each series the values of q/E and Reynolds number are fixed. Series 1 to 3 with Re=7 to 14 million were conducted with artificial transition to turbulent flow by means of transition bands fixed at 12% chord on the body side section and at 15% chord on the two outer sections of the suction side, and continuously at 5% chord on the pressure side. Thereafter for series no. 4 and higher, the transition bands were removed. According to the experience at ETW with other wing models in former experiments, free transition can be assumed at values of Reynolds number Re=23.5 · 106 and

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higher already at the leading edge. Thus the flow can be assumed fully turbulent in all experiments which have been conducted within the HIRENASD project in series 5 to 11. During the test runs it was decided to concentrate mainly on three lower levels, corresponding to q/E = 0.22 · 10−6, 0.34 · 10−6 and 0.48 · 10−6. With increasing values of q/E the wing deformation increases and consequently the changes in the pressure distribution grow for fixed Mach and Reynolds numbers. This can be analysed looking at the results of series 6, 10 and 5 or 11. For Reynolds number effects, series 2, 10, 7 or 3, 5, 8 can be studied. Similar test polars as for Ma = 0.8 were used for the other Mach numbers mentioned in this paragraph to analyse the effect of Mach number changes.

Fig. 9 Windtunnel test envelope for Mach number Ma=0.80

The Reynolds number variation is made possible by variation of the total pressure between 140kPa and 450kPa on the one hand and by variation of total temperature from 300K down to about 120K on the other hand. Thus the load factor of the wing model was chosen in that context between q/E=0.22 · 10−6 and 0.70 · 10−6 which produced total loads up to 2.5tons onto the wing model.

3 Selected Experimental Results from Static Tests In each series, first the static polars were measured for all Mach numbers of the respective series at angles of attack from about α = −2o to positive angles corresponding to the load limits admitted by ETW for the model assembly, e.g. α = +4.2o in

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series 1, 2, 6 and 7. The angle of incidence has been changed continuously with an angular velocity of about 0.2 degrees per second as mentioned already in Sect. 2.2. This value corresponds to a reduced frequency of the order of 10−6 which is really quasi-stationary. Data has been recorded with 4kHz in most static experiments, except the lower frequency of SPT for deformation measurement, for which an exemplary result is depicted in Fig. 10. It contains the change of aerodynamic twist over the angle of incidence for three levels of q/E taken from the test series 1, 2, 3 (symbols) in comparison to results from numerical simulations using SOFIA taken from [15]. The aerodynamic twist reached values between 0.7 and 1.2 degrees in the experiments which is mainly due to wing bending in combination with sweep. The contribution of structural torsion to the twist is essentially less.

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The raw data of static tests measured for pressure and strain have been smoothed out by low pass filtering with 5Hz. Using these strain data and suitable boundary conditions at the mounting flange of the balance, the bending deformation could be roughly reconstructed. The reconstruction is plotted exemplarily together with raw data and filtered data of the SPT measurements of test no. 196 in Fig. 11. Results for Ma = 0.80 and Re = 23.5 million, for which the flow is fully turbulent almost everywhere, are depicted for three levels of q/E in Fig. 12. It can be observed that the lift decreases with increasing q/E due to the aerodynamic twist by deformation which diminishes the local angle of attack. This diminution increases in span-wise direction. The pressure distribution for one root angle of attack is presented in Fig. 13 for two levels of q/E=0.22 · 10−6 and 0.48 · 10−6.

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Fig. 11 Displacement along a generating line of the wing where strain gauges are located

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The influence of varying Mach number for fixed q/E = 0.34 · 10−6 and Re = 23.5 million is displayed in Figs. 14 and 15, where lift over α and lift over drag polars are presented in Fig. 14 and pressure distribution in two wing sections in Fig. 15 for four

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Fig. 17 Influence of varying Reynolds number on pressure coefficient c p distribution in two wing sections at 14% (left) and 95% (right), α = 2o , Ma=0.80, q/E=0.48 · 10−6

Mach numbers at α = 2o root angle of incidence. As can be seen lift deteroriates due to compressibility effects at Ma = 0.88. The changes of shock strength and position due to variations of the Mach number can be seen very clearly in Fig. 15. For Ma = 0.70 no shock is present at all. Figure 16 and 17 show exemplarily the measured influence of varying Reynolds number for Mach number fixed at Ma=0.8 and load factor q/E fixed at q/E=0.48 · 10−6 . Results are taken from polars of test series 3, 5 and 8. Contrary to the lift decrease when increasing q/E, the lift rises slightly with ascending Reynolds number, and drag over α descends with increasing Reynolds number as is displayed in Fig. 16. Pressure distributions in wing sections 1 and 7 are presented in Fig. 17 for α = 2o . The lift contribution of the suction side seems to be a little higher for Re=23.5 million than for Re=50.0 million, where in both cases the flow is fully turbulent and no transition band is present. In case of Re=14.0 million transition bands

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are applied which may have some differing influence by themselves due to their dimension.

4 Wave Processes during the Tests for Steady Polars 4.1 Upstream Travelling Waves The pressure sensors indicate that small amplitude pressure waves start at the trailing edge and move upstream versus the leading edge. The experimental observation from the raw data of the pressure sensors is depicted in Fig. 18 exemplarily in pressure section 3 for experiment number 249 with oncoming flow at Ma=0.75 for pressure section 3. In the case of pure subsonic flow with Ma=0.75 and angle of attack α =−1.5o these waves have very small amplitudes as can been seen in the upper left diagram of Fig. 18. For angle of attack α =0o the flow speed exceeds locally very little Ma=1.0. The amplitudes of the waves in that case are a little higher than α=-1.5°

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Experiment 249 pressure section 3 Ma=0.75, Re=23.5e+6 q/E=0.48e-6, α=3° cp∞*=-0.591

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for α =−1.5o and the changes with time become locally steeper. For α =1.5o the supersonic regions have greater extent, and the waves starting at the trailing edge get steeper and steeper with increasing angle of attack, whereby the Mach number exceeds Ma=1.0 periodically but not steadily with time, see the lower right picture of Fig. 18. In Fig. 19 the space time behavior of pressure in section 3 is used to show where in space and when in time the flow is locally sub- or supersonic. In the black marked domains the flow is supersonic, elsewhere it is subsonic. Obviously local transient supersonic domains develop and collapse periodically. Drawing in mind suitable lines t=const. in Fig. 19 one can easily recognize that in x-direction alternately one single and two subsequent supersonic domains are crossed by the thought line.

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As well-known, in stationary flow supersonic regions are normally closed on the downstream side by a shock. Here the flow is non-stationary and obviously pressure waves or even weak shocks are moving in upstream direction as could already be seen in Fig. 19, but more in detail in Fig. 20 for the time interval from t=29.01s to t=29.02s in steps of two milliseconds. The solid line belongs to the instant t=29.010s when only one single supersonic domain is present. The flow is supersonic from the leading edge to about 42% chord. Through the shock the pressure coefficient increases by a jump of about Δ c p =0.4, and behind this jump, at this instant the flow is subsonic over the rest of the pressure section. The weak shock moves upstream, gets weaker and transforms into a compression wave, which arrives after 4ms at about 28% chord and is immediately followed by a rarefaction wave. The loss of total pressure through this compression wave is very small such that the flow has regained by the following rarefaction farther downstream supersonic speed at the same time. This second transient supersonic domain is closed by a strong pressure wave which has steepened to a shock at time instant t=24.016s whilst the compression wave running on ahead has reached 10% chord at this instant and will vanish

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running further upstream. The play is periodical with a frequency of about f =112Hz as can be seen in the pressure curves for the following time steps, where the pressure distribution at t=29.018s corresponds approximately to the one at t=29.01s and so on. The frequency 112Hz corresponds to a reduced frequency ω ∗ =1.17, where ω ∗ =2π f cre f /Uinf. The behavior of the pressure coefficient behind the final shock seems not to be heavily alternated for the different shock situations. Apparently only weak flow separation occurs, although the wave pattern appears to be driven by disturbances in the boundary layer arriving from the trailing edge. Therefore the observed phenomenon does not appear as classical shock buffet. In the experiment no. 250 the free-stream Mach number is Ma=0.80. Here the wave pattern in the outermost pressure section is studied (section 7). In this section the flow on the front part of the suction side remains supersonic until almost 25% chord over time. The supersonic region varies with time between 25% and 60% Experiment 250: Ma=0.80, Re=23.5e+6, q/E=0.48e-6, α=3° 29 29.005

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chord. Again, two distinct supersonic regions are possible in this case as can be seen from Fig. 21 and 22. The foremost supersonic region which extends from the leading edge is closed by an upstream moving shock, which again gets weakened to a small amplitude wave while moving upstream. It decelerates the flow only little such that the velocity remains supersonic. In the case when a second supersonic region appears, this has a smaller extent and the flow is only weakly supersonic such that a smooth compression behind it decelerates the flow to subsonic, see curve at instant time t=29.012s in Fig. 23. Due to the convexity of the suction side of the wing, flow accelerates again behind this weak wave and may get finally decelerated smoothly to subsonic, see time t=29.014s. During the time interval from t=29.014s to t=29.016s the smooth compression wave steepens to a weak shock-wave. Thereafter the periodic wave pattern starts again at time between t=29.018s and t=29.020s, similarly to that observed in experiment no. 249 in pressure section 3, were at Ma=0.80 only small amplitude shock motion occurs in place of the wave pattern at Ma=0.75. The reason for the different behaviour in the two pressure sections in that the elastic twist of the wing reduces the local angle of attack at pressure section in test 250 to about the value at section 3 in test 249.

5 Stochastic Processes during the Tests 5.1 Frequency and Damping Analysis Dynamic analysis of quasi-stationary, i.e. tests without using the vibration excitation mechanism for defined excitation, reveals the presence of stochastic aerodynamic disturbances which excite the wing model for vibrations. That is shown exemplarily in this section where the data recorded during experiment no. 318 are discussed. That is a static polar at Mach number Ma=0.85, Reynolds number Re=23.5 · 106 , load factor q/E = 0.22 · 10−6, total temperature Tt =120K, total pressure pt =141kPa within which the angle of incidence α was changed at a reduced frequency in the order of magnitude 10−6 , see Sect. 3. The data registered by the implemented acceleration sensors (see Fig. 4) during the observed stochastic excitation has been used for modal analysis in terms of mode shapes, mode frequencies and damping factors. Results from the evaluation of HIRENASD experiment no. 318 with quasistationary change of α are presented in fig 24. The shapes of the 1st, 2nd and 3rd mode with about 29Hz, 80Hz and 173Hz frequencies indicate clearly flap bending. The 4th mode shape with about 242Hz frequency exhibits still predominantly flap bending, but it contains also visible torsion effects whilst the 5th mode with frequency about 268Hz is dominated by torsion but contains also small effects of bending. The higher modes include also bending and torsion, but one type of deformation is always dominating, the 6th mode with about 350Hz frequency contains e.g. predominantly bending as visible in Fig. 24.

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Fig. 24 Mode shapes and characteristic parameters of vibration of the HIRENASD wing model during the static polar exp. no.318. All presented modes are predominantly flap bending, except the 5th mode with frequency 268.2Hz which is dominated by torsion. Ma=0.85, Re=23.5 · 106 , q/E=0.22 · 10−6

5.2 Analysis of Normal Force Caused by Stochastic Aerodynamic Disturbances For evaluating the static aerodynamic coefficients lift and drag over angle of incidence α , the force data measured by the piezoelectric wind tunnel balance has been low pass filtered with 5Hz. After subtracting the low pass filtered data from the raw data of the force component BFy normal to the wing plane over the time of more than 40s needed for the α -sweep, this difference has been analysed w. r. t. its frequency contents in terms of the mode frequencies of the model. Applying band filtering on this most relevant balance force component around these frequencies using discrete Fourier transform as described later in Sect. 6.1 for dynamic tests, it was found that the first 7 modes contain almost all stochastic influences. The normal force contributions by three of these modes will be now regarded more closely. One observes that the amplitudes of the first mode can reach maximum amplitudes up to about 1kN over the complete sweep from α =−2.9o up to α =5o (see Fig. 25, upper two rows of diagrams), whereas the second mode (see Fig. 25, middle two rows of diagrams) gets excited first weaker with maximum amplitudes of 0.5kN of the force component BFy, but after about 19s which corresponds to α =1o the stochastic excitation yields beyond this angle of incidence increasing force amplitudes which can reach even more than 1kN (see time 35s). The first two modes

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Fig. 25 Stochastic disturbances during quasi-stationary HIRENASD exp. no. 318 (steady polar). Band filtered dynamic parts of normal force component BFy for first, second and fifth mode over time for α sweep from −2.9o to +5o in 41 seconds and with higher time resolution in a subinterval of time from 25.0s to25.5s

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contribute the main part to the balance force of induced stochastic excitation. For the fifth mode amplitudes remained less then 0.1kN until the angle of incidence reached 2.5o at time about 28s (see Fig. 25, lower two rows of diagrams). At further increasing angle of incidence amplitudes start growing up to finally possible values of 0.3kN. For the third, fourth, sixth and seventh mode, results are not presented by figures, for sake of brevity. For these it was found that stochastic force amplitudes did not exceed 0.1kN for α between −2.9o and 2.5o , but then the values increased and could achieve finally at α = 5o up to 0.3kN except for the fourth mode which achieved at the end of the α -sweep up to 0.8kN. Because of the stochastic excitation and consequently stochastic relative phase angles of modes, for the maximum significant disturbance of the regarded balance force component all amplitudes of the different modes have to be added. That may result in a remarkable percentage of the balance force compared with the respective force amplitudes occurring during defined excitation (compare with results for BFy displayed in Fig. 26). Evaluation of experiments with defined excitation of the normal modes 1, 2 or 5 gave as results that the stochastic disturbances occurred with the same intensity before, during and after the time interval of defined excitation, i.e. without being affected by the defined excitation of one of the three separately excited mode shapes.

6 Selected Experimental Results from Dynamic Tests 6.1 Processing of Measured Data as for Dynamic Tests The recorded measurement data exhibit considerable noise and disturbances possibly caused by aerodynamic effects. The results of measurements presented in this chapter represent outcomes of a band filtering process based on Fourier analysis. Discrete Fourier transform has been applied to the raw data, followed up by back transformation within frequency bands around the frequencies which are of particular interest. In case of vibration excitation with a defined frequency, an appropriate frequency band width has been elaborated using the data record of the control voltage in the piezo-stacks as a reference. The control voltage is very clean from disturbances and must not be degraded applying the filtering process on it. After finding out the maximum of admissible band width which fulfils this requirement, the upper limit of the filter band width has been defined. As an example, Fig. 26 shows the comparison of the unfiltered and filtered data record of the control voltage in the two upper diagrams on the right of the figure and the unfiltered and filtered data of the measured balance force in normal direction to the wing plane. The results presented in this figure belong to the experiment no. 346 within which predominantly the 2nd mode was being excited at a frequency of 83.3Hz. Mach and Reynolds numbers were Ma=0.85 and Re=23.5 · 106. The angle of attack corresponds to no lift under stationary conditions and was measured α =−1.33o at the turntable. Figure 27 shows for experiment no. 346 a result of the pressure distribution in pressure section 7 during about 4 periods in the fully excited vibration state which

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Fig. 26 Experiment no. 346. Example for processing of measured data applying band filtering based on Fourier analysis. Angle of attack α =−1.33o (no lift condition). Excitation with second mode frequency 83.3Hz

would exhibit pressure distributions varying periodically and smoothly at constant amplitudes in an ideal case. In the experimental reality, that does not apply due to disturbances. The recorded measurement data behave non-smoothly as can be seen in the upper pictures of Fig. 27 where the unfiltered measurement data is presented for the suction side of the wing on the left and for the pressure side on the right as recorded in the already mentioned experiment no. 346. At Mach number Ma=0.85 and angle of attack α =−1.33o three supersonic regions, which are each closed by a shock on their downstream side, are present in the wing section and changing in position and strength during the vibration. One shock is present on the upper side of the wing in the pressure section referred to, while two shocks are established on the lower side, where one results from the special shape of the BAC 3-11 profile which exhibits a very strong positive curvature near the leading edge at the wing lower side which is followed by a small interval with negative curvature until the flow reaches again a convex region with positive curvature. There the thickness maximum is situated at about 37% chord. Behind that the second shock on the lower side closes the second supersonic region. The filtering process using a bandwidth of 11Hz yielded the results presented in the pictures on the bottom of Fig. 27. All shocks are well represented including their changes in amplitude and position. The loss of magnitude in the pressure variation compared to the unfiltered data is actually not only caused by noise in the usual sense, its main part seems to be caused by aerodynamic disturbances occurring in the experiments as discussed in Sect. 5.2.

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Fig. 27 Experiment no. 346. Ma=0.85, Re=23.5 · 106 , q/E=0.22 · 10−6 , α =−1.33o (no lift), frequency of excitation 83.3Hz. Raw data (pictures on top) and band filtered data (pictures beneath) in pressure section 7 on the suction side and on the pressure side of the wing model

6.2 Unsteady Polars with Defined Vibration Excitation 6.2.1

Local Lift in Pressure Sections during Excited Vibration

Experiments with defined vibration excitation as described in Sect. 2.1 have been performed exciting separately with the frequencies of the first, second and fifth mode which were determined online during the quasi-stationary experiments with slow α sweeps, as described in Sect. 5.1, acceleration data from the stochastic excitation at the respective wind tunnel conditions. Data recording started a short interval of time before the excitation and ended a short time interval after stopping the excitation. The diagrams for the exciting piezo-voltage and the band filtered local lift depicted in Fig. 28 contain the complete recording time and show that the excitation voltage is zero outside the time interval from 11s to 17s, contrary to the measured local lift which exhibits remarkable amplitudes outside this interval which are due to the stochastic disturbances discussed in Sect. 5.2. One observes strongly varying amplitudes over the complete recording time. Those were detected by all sensors, i.e. by the acceleration sensors, the strain gauges and the balance as well. Their mean over the time periods without excitation is zero. Therefore, it is useful to represent the results of the period with excitation also by their means over this time interval. The time histories of the signals exhibited a certain coherence which justified the definition of measured quantities, here the acceleration at wing tip measured by sensor no. 15/1 (see Fig. 4) as a reference for the other signals. The mean amplitude of this signal over the period of fully excited vibration has been used for normalising

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Fig. 28 Experiment no. 457. Excitation of the first eigenmode with frequency 29.4Hz at the flow conditions Ma=0.8, Re=23.5 · 106 , q/E=0.34 · 10−6 , Tt =162K, pt =216kPa. Voltage in the exciting piezo-stacks (upper diagram), local lift in pressure section 7 after band filtering around the excitation frequency (diagram at bottom left) and local lift over wing span represented by the static means and the phase lags and means of normalised amplitudes in all pressure sections (bottom right)

the other measured data as for example the mean amplitude and phase lag relative to the reference quantity of the local pressure-based lift cl p in section 7, depicted in the lower right diagram Fig. 28. This way the number of data for e.g. validation purposes can not only be reduced efficiently but also improved for practical use and better understanding of the influence of defined excitation because of a certain cleaning from stochastically induced effects. But first of course, one has to see for oneself that the coherence of reference data is ensured. Particularly, for measured data during excitation of the torsion dominated mode the difference of the data measured by the acceleration sensors 14/1 and 14/2 (see Fig. 4) might be useful. Nevertheless the processed data (see Sect. 6.1) of the acceleration measured by sensor 15/1 which is positioned close to the leading edge at wing tip, has been used for all normalised results presented in this report. In the experiments with defined excitation the parameters load factor q/E, Mach number Ma, Reynolds number Re, angle of incidence α and excitation frequency have been varied independently. For reasons of limited space, only a small selection of results can be presented, whereby the parameters q/E and α are varied in Fig. 29 for fixed Mach number Ma=0.8 and excitation of the first mode and the same in Fig. 30 for fixed Ma=0.85 and excitation in the first mode. The distributions of stationary means of lift and the normalised mean amplitudes represented by upper

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Fig. 29 Normalised local lift in pressure section 7 represented by the static means and the phase lags and means of normalised amplitudes in all pressure sections for experiments no. 340, 457, 260 (all α =−1.33o ) and 261 (α =1.5o ). Excitation of the first eigenmode with frequency corresponding to the flow conditions Ma=0.8, Re=23.5 · 106 and varied value of q/E=0.22 · 10−6 with Tt =120K and pt =141kPa (exp. 340, diagram top left), q/E=0.34 · 10−6 with Tt =162K and pt =216kPa (exp. 457, diagram bottom left), q/E=0.48·10−6 with Tt =205K and pt =302kPa (Exp. 260 (diagram top right) and 261 (diagram bottom right)

and lower values about the stationary means of local lift over span are presented together with the phase lag relative to the reference quantity in the diagrams of the two figures. Three of the diagrams in each of the Figs. 29 and 30 belong to dynamic experiments with no lift under quasi-stationary conditions. Regarding these diagrams in Fig. 29 for Ma=0.8 where the flow still remains subsonic in pressure section 7 (no shock), it can be recognised that the mean amplitudes decrease with increasing values of q/E, as expected. For α =1.5o (diagram on bottom right for exp. no. 261) a qualitative change of shape of the static mean of the local lift distribution is observed. It is caused by the supersonic region now present for Ma=0.8 on the suction side, which is closed by a shock. Nevertheless, the mean amplitudes of pressure variations caused by the excited vibration are comparable to their values in the diagram above it, where q/E has the same value.

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Fig. 30 Normalised local lift in pressure section 7 represented by the static means and the phase lags and means of normalised amplitudes in all pressure sections for experiments no. 342, 461, 264 (all α =−1.33o ) and 265 (α =1.5o ). Excitation of first eigenmode with frequencies corresponding to the flow conditions Ma=0.85, Re=23.5 · 106 and varied value of q/E=0.22 · 10−6 with Tt =120K and pt =141kPa (exp. 342, diagram top left), q/E=0.34 · 10−6 with Tt =162K and pt =216kPa (exp. 461, diagram bottom left), q/E=0.48·10−6 with Tt =205K and pt =302kPa (Exp. 264 (diagram top right) and 265 (diagram bottom right)

6.2.2

Comparison of Time Histories Represented by Filtered Data versus Reduced Measuring Data

As mentioned in Sect. 6.2.1 efficient measurement data reduction, which is useful at the same time for code validation, is possible by declaring one characteristic quantity as reference magnitude for normalising other measured quantities. In order to verify exemplarily the justification of this procedure and the chosen reference quantity, band filtered signals of data over time for local lift are confronted to time histories reconstructed from normalised data with acceleration at wing tip (sensor no 15/1 in Fig. 4) as reference quantity. The comparisons presented in Figs. 31 and 32 for the local lift in pressure section 7 show for two different Mach numbers and angles of incidence α that the reconstructed data reproduce almost all variations of amplitudes over time, which are present in the band-filtered data. The adulteration

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Fig. 31 Time history of local lift in pressure section 7 as represented by band filtered data (left) and reconstructed from normalised data for experiment no. 459; Ma=0.85, Re=23.5 · 106 , q/E=0.34 · 10−6 , excitation with 29.4Hz (close to resonance), α =2.5o resulting in a supersonic region with shock on the suction side

of data is not too heavy. Thus, in this case the mean amplitudes of local lift referred to the mean amplitudes of acceleration at wing tip during the excitation time interval could be successfully used for code validation. The direct use of band filtered time history of the signals recorded in the dynamic HIRENASD experiments for code validation is not recommendable because of the strong stochastic disturbances, particulary in experiments like the one presented in Fig. 32, where three shocks are present on the wing surface which can move with small amplitudes and change their strength. The foremost shock which closes the first of two subsequent supersonic domains on the wing pressure side in the case of Ma=0.85 is the strongest at stationary no-lift angle of attack, and it varies from vanishing completely to very strong during vibration.

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Fig. 32 Time history of local lift in pressure section 7 as represented by band filtered data (left) and reconstructed from normalised data for experiment no. 461; Ma=0.85, Re=23.5 · 106 , q/E=0.34 · 10−6 , excitation with 29.4Hz, α =−1.33o resulting in one supersonic region with shock on the suction side and two supersonic regions with shocks on the pressure side

6.3 Influence of Parameters in Dynamic Tests on Chord-Wise Pressure Distribution The wind tunnel conditions in ETW can be varied independently w.r.t. the three parameters load factor q/E, Mach number Ma and Reynolds number Re. In the hierarchy of their importance with respect to elastic deformation of the wing model under wind on conditions, the parameter q/E is leading, followed by Ma and thereafter by Re. Another important and independent parameter in this respect is the angle of incidence α . The elastic deformation due to structural torsion and/or flap bending of the backward swept wing model result in an aerodynamic twist which increases in span-wise direction. For positive lift, an increasing loss of effective angle of incidence with span is observed. All results of this section are concerning measured pressure distributions after band-filtering which are presented in terms of real and imaginary parts using mean amplitudes of pressure data and their phase angles relative to acceleration data at wing tip. The focus is on the parameters Ma and α .

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HIRENASD Test no.457 vibration excitation 29.4 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered Top side Bottom side

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HIRENASD Test no.462 vibration excitation 26.6 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered Top side Bottom side

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HIRENASD Test no.461 vibration excitation 26.6 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered

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Fig. 33 Real and imaginary parts of normalised pressure variations in section 7 due to excited vibration with frequency close to resonance with the first natural frequency at the respective wind tunnel conditions in experiments, q/E=0.48 · 10−6 , Re=23.5 · 106

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HIRENASD Test no.158 vibration excitation 78.9 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered Top side Bottom side

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HIRENASD Test no.159 vibration excitation 78.9 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered Top side Bottom side 20 0

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HIRENASD Test no.163 vibration excitation 78.7 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered

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HIRENASD Test no.474 vibration excitation 269 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered Top side Bottom side

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HIRENASD Test no.478 vibration excitation 270.8 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered Top side Bottom side

20

15

10

10

5 0 -5 -10 -15

0.4

x

0.6

0.8

1

Re = 23.5 mio w/oTB Ma = 0.85 q = 66173Pa, q/E=0.34e-06 Filter Band Width = 24.8Hz

Top side Bottom side

20

15

Im(Δcp norm.)

Re(Δcp norm.)

x

-15

-20

5 0 -5 -10 -15

-20 -25

0.4

20

-15

Ma=0.85 α =2.5o

1

-20 0

20

-25

0.8

Top side Bottom side

HIRENASD Test no.477 vibration excitation 270.8 Hz real and imaginary parts of normalized Δcp in Pressure Section 7 ( η = 0.95), Data Filtered

Ma=0.85 αno−li f t

0.6

-15

-20 -25

x

20

Im(Δcp norm.)

Ma=0.80 α =2.5o

Re(Δcp norm.)

20

0.4

Re = 23.5 mio w/oTB Ma = 0.8 q = 63495Pa, q/E=0.34e-06 Filter Band Width = 24.8Hz

-20 0

0.2

0.4

x

0.6

0.8

1

-25

0

0.2

0.4

x

0.6

0.8

1

Fig. 35 Real and imaginary parts of normalised pressure variations in section 7 due to excited vibration with frequency close to resonance with the fifth natural frequency at the respective wind tunnel conditions in experiments, q/E=0.34 · 10−6 , Re=23.5 · 106

Aero-structural Dynamics Experiments at High Reynolds Numbers

421

Figure 33 shows results for pressure section 7 from experiments at different Mach numbers Ma=0.8 and 0.85 at two different angles of attack, αno−li f t (first and third diagram from top) and α =2.5o (second and fourth diagram from top), also with excitation frequencies close to the lowest respective natural frequencies. The value of the angle of attack for zero lift is in all cases approximately αno−li f t =−1.33o. Only for Ma=0.8 and αno−li f t (uppermost diagram in Fig. 33) no shock is present in section 7. Obviously, the shocks occurring in the other cases of Fig. 33 were responsible for strong variations of the real and imaginary parts of pressure in the zones where shocks are observed. That is on the pressure side in the third diagram from top for Ma=0.85 at αno−li f t , and on the suction side behind the thickness maximum. But additional strong variations of pressure and of phase as well can be detected near the leading edge which are caused by the strong pressure gradient due to the suction peak occurring at incidence angle α =2.5o . Real- and imaginary parts of pressure variation during excited vibration with the second natural frequency of the wing model (about 79Hz) is shown in Fig. 34. The displayed results belong to experiments of series 1, where Re=7.0 · 106 and turbulence was provoked artificially using transition bands as described in Sect. 2.4. In this case the load factor is q/E=0.22 · 10−6. For Ma=0.80 no shocks occur and correspondingly the Real- and Imaginary parts exhibit no peaks (uppermost diagram of Fig. 34). Also for α =1.5o and Ma=0.80 (second diagram from top) only very weak peaks are observed, contrarily to the case of α =2.5o and Ma=0.80 in Fig. 33, where due to the higher angle of attack stronger peaks can be seen. The two lower diagrams of Fig. 34 belong to Ma=0.85 and αno−li f t (second from bottom of Fig. 34) and α =2.5o (bottom). In these cases peaks in the real and imaginary parts can be seen at shock locations on the suction side of the pressure section 7 for both angles of attack α , and for α =2.5o while no peak is present on the pressure side. Finally, some results are presented for one of the higher excitation frequencies in Fig. 35. Here the fifth natural frequency corresponding to the lowest mode shape which is dominated by torsion. Its natural frequency is about 270Hz. The other parameters are the same as in the case discussed before. Also pressure section 7 is regarded. The amplitudes of the pressure variations due to excited vibration are significantly smaller than in the case of the lowest bending dominated natural frequency. That is because of the higher stiffness of the model against torsion. The excitation mechanism had no power in reserve, it was operated in all dynamic experiments near its limit of capacity.

7 Conclusions Selected experimental ASD results from experiments with an elastic wing model in the HIRENASD project have been presented and critically discussed. The raw data recorded with high frequency in quasi-stationary experiments, that exhibited stochastic excitation effects, and dynamic experiments with defined vibration excitation has been worked up based on Fourier analysis to distillate the physical effects

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to be understood and to achieve reference data for code validation. In the dynamic experiments data was recorded a certain period of time before and after the excitation time period. This data has been included in the evaluation. The signal to noise ratio, in terms of the band-filtered data for an experiment with defined excitation related to the raw data, came out to low in the experiments for practical use. But for the relation of band-filtered data during defined excitation time period to band-filtered data in the time period without defined excitation, the signal to disturbance ratio reaches values of about 3 for the experiments with defined excitation of the first and the fifth mode and values of about 10 for the second mode, which could be therefore preferably a first candidate being used in simulations for code validation. HIRENASD data have been needed already by several authors for code validation, e.g. [23, 24, 15, 17, 19, 25] and also software developers from outside Germany.

Acknowledgments Authors are very grateful to the German Research Foundation (DFG) for funding the HIRENASD project. Furthermore, they want to thank the referees of DFG for their review and recommendation as well as colleagues in the field for making useful suggestions and finally the institutions having supported the project, in particular Airbus Germany for providing financial support for the development of the piezoelectric wind tunnel balance, ETW for providing wind tunnel adaptations for dynamic measurements and giving continuously advice during the model preparation and conduction of the experiments, DLR for advice and support concerning data acquisition.

References 1. Cole, S.R., Noll, T.E., Perry III, B.: Transonic Dynamics Tunnel Aeroelastic Testing in Support of Aircraft Development. Journal of Aircraft 40(5), 820–832 (2003) 2. Bartels, R.E., Edwards, J.W.: Cryogenic Tunnel Pressure Measurements on a Supercritical Airfoil for Several Shock Buffet Conditions, NASA TM-110272 (1997) 3. Ballmann, J. (ed.): Flow Modulation and Fluid-Structure Interaction at Airplane Wings. Notes On Numerical Fluid Mechanics And Multidisciplinary Design, vol. 84, pp. 105– 122. Springer, Heidelberg (2003) 4. Ballmann, J., Dafnis, A., Braun, C., Korsch, H., Reimerdes, H.-G., Olivier, H.: The HIRENASD Project: High Reynolds Number Aerostructural Dynamics Experiments in the European Transonic Wind Tunnel (ETW). In: Proceedings of the International Council of the Aeronautical Sciences (ICAS) Congress, Hamburg, Paper-No. 2006-5.11.2 (2006) 5. Klioutchnikov, I., Ballmann, J.: DNS of Transitional Transonic Flow about a Supercritical BAC3-11 Airfoil using High-Order Shock Capturing Schemes. Direct and Large Eddy Simulation VI, pp. 737–744. Springer, Netherlands (2006)

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6. Hermes, V., Klioutchnikov, I., Alshabu, A., Olivier, H.: Investigation of Unsteady Transonic Airfoil Flow. In: 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 7-10, AIAA-2008-627 (2008) 7. Alshabu, A., Olivier, H.: Unsteady Wave Phenomena on Supercritical Airfoil. AIAA Journal 46(8), 2066–2073 (2008) 8. Korsch, H., Dafnis, A., Reimerdes, H.-G.: Dynamic Qualification of the HIRENASD Elastic Wing Model. Aerospace Science and Technology 13(2-3), 130–138 (2009) 9. Dafnis, A., Korsch, H., Buxel, C., Reimerdes, H.-G.: Dynamic Response of the HIRENASD Elastic Wing Model under Wind-Off and Wind-On Conditions. In: International Forum of Aeroelasticity and Structural Dynamics (IFASD), Stockholm, Sweden, Paper IF-073 (2007) 10. Ballmann, J., et al.: First Results of the High Reynolds Number Aero-Structural Dynamics (HIRENASD) Experiments in ETW. In: Keynote lecture at International Forum of Aeroelasticity and Structural Dynamics (IFASD), Stockholm, Sweden (2007) 11. Ballmann, J., Dafnis, A., Korsch, H., Buxel, C., Reimerdes, H.-G., Brakhage, K.-H., Olivier, H., Braun, C., Baars, A., Bouke, A.: Experimental Analysis of High Reynolds Number Aero-Structural Dynamics in ETW. In: 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, January 7-10, AIAA Paper 2008-841 (2008) 12. Ballmann, J., Boucke, A., Reimer, L., Dickopp, C.: Results of Dynamic Experiments in the HIRENASD Project and Analysis of Observed Unsteady Processes. In: IFASD2009-103, Seattle, June 22-24 (2009) 13. Ballmann, J., Braun, C., Dafnis, A., Korsch, H., Reimerdes, H.-G., Brakhage, K.-H., Olivier, H.: Numerically Predicted and Expected Experimental Results of the HIRENASD Project, DGLR-2006-117, DGLR-Jahrestagung, Braunschweig (2006) 14. Braun, C.: Ein modulares Verfahren f¨ur die numerische aeroelastische Analyse von Luftfahrzeugen. Doctoral thesis, RWTH Aachen, Germany 15. Reimer, L., Braun, C., Chen, B.-H., Ballmann, J.: Computational Aeroelastic Design and Analysis of the HIRENASD Wind Tunnel Wing Model and Tests. In: International Forum on Aeroelasticity and Structural Dynamics (IFASD), Stockholm, Sweden, paper IF-077 (2007) 16. Braun, C., Boucke, A., Ballmann, J.: Numerical Prediction of the Wing Deformation of a High-Speed Transport Aircraft Type Windtunnel Model by Direct Aeroelastic Simulation. In: International Forum on Aeroelasticity and Structural Dynamics (IFASD), Munich, Germany, paper IF-147 (2005) 17. Reimer, L., Ballmann, J., Behr, M.: Computational Analysis of High Reynolds Number Aerostructural Dynamics (HIRENASD) Experiments. In: IFASD-2009-130, Seattle, June 22-24 (2009) 18. Chen, B.-H., Brakhage, K.-H., Behr, M., Ballmann, J.: Numerical simulations for preparing new ASD experiments in ETW with a modified HIRENASD wing model. In: IFASD2009-131, Seattle, June 22-24 (2009) 19. Reimer, L., Braun, C., Ballmann, J., Behr, M.: Development of a Modular Method for Computational Aero-Structural Analysis of Aircraft (2009) ¨ 20. Ozger, E., Schell, I., Jacob, D.: On the Structure and Attenuation of an Aircraft Wake. AIAA Journal of Aircraft 38(5), 878–887 (2001) 21. Moir, I.-R.M.: Measurements on a two-dimensional aerofoil with high-lift devices. AGARD-AR-303 II, 58–59 (1994) 22. Wright, M.C.N.: Half Model Testing at ETW, Technical Memorandum ETW/TM/ 2000028 (2000)

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23. Nitzsche, J.: A Numerical Study on Aerodynamic Resonance in Transonic Seperated Flow. In: IFASD-2009-126, Seattle, June 22-24 (2009) 24. Neumann, J., Ritter, M.: Steady and Unsteady Aeroelastic Simulations of the HIRENASD Wind Tunnel Experiment. In: IFASD-2009-132, Seattle, June 22-24 (2009) 25. Schieffer, G., Ray, S., Bramkamp, F.D., Behr, M., Ballmann, J.: An Adaptive Implicit Finite Volume Scheme for Compressible Turbulent Flows around Elastic Configurations (2009)

Author Index

Alshabu, Atef

137

Lamby, Philipp

Baars, Arndt 389 Ballmann, Josef 25, 205, 265, 389 Behr, Marek 25, 205, 265 Bischof, Christian H. 153 Boucke, Alexander 389 Brakhage, Karl-Heinz 239, 389 Bramkamp, Frank Dieter 25 Braun, Carsten 205, 389 Brix, Kolja 265 B¨ ucker, H. Martin 153 Buxel, Christian 389 Chen, Bae-Hong Dafnis, Athanasios Dahmen, Wolfgang Dickopp, Christian Fares, Ehab

239

Massjung, Ralf 265 Meinke, Matthias 105 Melian, Sorana 265 M¨ uller, Siegfried 77, 265 Neuwerth, G. 181 Neuwerth, G¨ unther 1 Noelle, Sebastian 53 Olivier, Herbert

137, 389

Pollul, Bernhard

295

389 389 77, 239, 265 389

105

Henke, R. 181 Henke, Rolf 1 Hermes, Viktor 137 Hohn, Christoph 265 H¨ ornschemeyer, R. 181 Hovhannisyan, Nune 77 Huppertz, Guido 105

Rasch, Arno 153 Ray, Saurya 25, 389 Reimer, Lars 205, 389 Reimerdes, H.-G. 363 Reimerdes, Hans-G¨ unther Reusken, Arnold 295 Schieﬀer, Gero 25, 265 Sch¨ oll, Robert 1 Schr¨ oder, W. 325 Schr¨ oder, Wolfgang 105 Steimle, P.C. 325 Steiner, Christina 53 Tchakam, J.A. Kengmogne

K¨ ampchen, Manuel 389 Klaas, M. 325 Klioutchnikov, Igor 137 Korsch, Helge 389

389

Wellmer, Georg

205

Zurheide, Frank T.

105

363

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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