Woodgate, Mark A. (2008) Fast prediction of transonic aeroelasticity using computational fluid dynamics. PhD thesis. http://theses.gla.ac.uk/923/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Glasgow Theses Service http://theses.gla.ac.uk/ [email protected] Fast Prediction of Transonic Aeroelasticity Using Computational Fluid Dynamics by Mark Woodgate BSc. A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Glasgow Department of Aerospace Engineering September 2008 © 2008 Mark Woodgate Declaration I hereby declare that this dissertation is a record of work carried out in the Department of Aerospace Engineering at the University of Glasgow during the pe- riod from October 1999 to September 2008. The dissertation is original in content except where otherwise indicated. September 2008 ....................................................... (Mark Andrew Woodgate) ii Abstract The exploitation of computational fluid dynamics for non linear aeroelastic simulations is mainly based on time domain simulations of the Euler and Navier- Stokes equations coupled with structural models. Current industrial practice relies heavily on linear methods which can lead to conservative design and flight envelope restrictions. The significant aeroelastic effects caused by nonlinear aerodynamics include the transonic flutter dip and limit cycle oscillations. An intensive research effort is underway to account for aerodynamic nonlinearity at a practical computa- tional cost. To achieve this a large reduction in the numbers of degrees of freedoms is required and leads to the construction of reduced order models which provide compared with CFD simulations an accurate description of the dynamical system at much lower cost. In this thesis we consider limit cycle oscillations as local bifurcations of equi- libria which are associated with degenerate behaviour of a system of linearised aeroelastic equations. This extra information can be used to formulate a method for the augmented solve of the onset point of instability - the flutter point. This method contains all the fidelity of the original aeroelastic equations at much lower cost as the stability calculation has been reduced from multiple unsteady computations to a single steady state one. Once the flutter point has been found, the centre mani- fold theory is used to reduce the full order system to two degrees of freedom. The thesis describes three methods for finding stability boundaries, the calculation of a reduced order models for damping and for limit cycle oscillations predictions. Re- sults are shown for aerofoils, and the AGARD, Goland, and a supercritical transport wing. It is shown that the methods presented allow results comparable to the full order system predictions to be obtained with CPU time reductions of between one and three orders of magnitude. iii Acknowledgements I am grateful to BAE SYSTEMS, Engineering and Physical Sciences Re- search Council, MoD and DERA for funding this work as part of the programme of the Partnership for Unsteady Methods in Aerodynamics (PUMA) Defence and Aerospace Research Partnership (DARP). I would like to thank my supervisor Professor Ken Badcock for this support, encouragement, guidance and patience over the past 8 years. I would also like to thank all the members of the CFD Lab, past and present, for creating a stimulating working environment which is has been a privilege to work at over the years. I especially like to thank Professor Bryan Richards for this support during my early years at Glasgow University and Dr George Barakos for motivating me to get this work written up. I am grateful to Professor Michael Henshaw and all the aerodynamicists at BAE SYSTEM Brough that made my years secondment there so productive and opened my eyes between practises used in the worlds of academia and business in the field of aeroelastics. iv List of Most Relevant Publications M.A. Woodgate and K.J. Badcock. Fast prediction of transonic aeroelastic stability and limit cycles. AIAA Journal, vol 45(6):1370-1381, 2007. M.A. Woodgate and K.J Badcock. A reduced order model for damping derived from CFD based aeroelastic simulations. In 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, Rhode Island, 1-4 May 2006. AIAA-2006-2021. K.J. Badcock and M.A. Woodgate. Aeroelastic damping model derived from discrete Euler equations. AIAA Journal, vol 44(11):2601-2611, 2006. M.A. Woodgate, K.J. Badcock, A.M. Rampurawala, B.E. Richards, D. Nardini, and M.J. Henshaw. Aeroelastic calculations for the Hawk aircraft using the Euler equations. Journal of Aircraft, vol 42(4):1005-1012, 2005. K.J. Badcock, M.A. Woodgate, and B.E. Richards. Direct aeroelastic bifurcation analysis of a symmetric wing based on the Euler equations. Journal of Aircraft, vol 42(3):731-737, 2005. K.J. Badcock, M.A. Woodgate, and B.E. Richards. Hopf bifurcation calculations for a symmetric airfoil in transonic flow. AIAA Journal, vol 42(5):883-892, 2004. G.S.L. Goura, K.J. Badcock, M.A. Woodgate, and B.E. Richards. Extrapolation effects on coupled computational fluid dynamics/computational structural dynamics simulations. AIAA Journal, vol 41(2):312-314, 2003. G.S.L. Goura, K.J. Badcock, M.A. Woodgate, and B.E. Richards. Implicit method for the time marching analysis of flutter. Aeronautical Journal, vol 105:199-214, 2001. G.S.L. Goura, K.J. Badcock, M.A. Woodgate, and B.E. Richards. Evaluation of methods for the time marching analysis of transonic aeroelasticity. In 19th AIAA Applied Aerodynamics Conference, Anaheim, CA, 11-14 June 11-14 2001. AIAA-2001-2457. K.J. Badcock, B.E. Richards, and M.A. Woodgate. Elements of computational fluid dynamics on block structured grids using implicit solvers. Progress in Aerospace Sciences, vol 36:351-392, 2000. M.A. Woodgate, K.J. Badcock, B.E. Richards, and J. Anderson. Towards the direct calculation of non-linear transonic flutter characteristics. RAeSoc Aerody- namics Conference 2000, London, 17-18 April 2000. v Contents Abstract iii Acknowledgements iv List of Most Relevant Publications v Table of Contents vi List of Figures ix List of Tables xii Nomenclature xiii 1 Introduction 1 1.1 AeroelasticPrediction. 3 1.2 Computational Aeroelasticity . 4 1.3 ReducedOrderModelling . 7 1.3.1 The Eigenmode Methodology . 7 1.3.2 Proper Orthogonal Decomposition . 9 1.3.3 Harmonic Balance Method . 11 1.4 DynamicalSystemsBasedMethods . 13 1.4.1 Numerical Analysis of Bifurcations Points . 13 1.4.2 Calculation of Bifurcation Points . 14 1.4.3 Normal Forms for Bifurcations . 15 1.5 ThesisOutline............................. 15 2 Calculation of Hopf Bifurcation Points 17 2.1 Introduction.............................. 17 2.2 One Parameter Bifurcation Equilibria . 18 2.3 ClassesofHopfBifurcation. 19 2.4 Numerical Methods for Calculating Equilibrium Solutions ..... 22 2.4.1 Newton’s Method . 22 2.4.2 Relaxed Newton’s Method . 23 2.4.3 Modified Newton’s Methods . 23 2.5 Numerical Methods for Calculating Hopf Bifurcations . 25 2.5.1 Indirect Calculation . 25 vi 2.5.2 Directcalculation. 27 2.5.3 Evaluation .......................... 28 2.6 ModelProblem ............................ 29 2.7 Conclusions.............................. 35 3 Model Reduction 36 3.1 Background.............................. 36 3.2 Centre Manifold Theorems . 36 3.3 ChangeofCoordinates . 38 3.4 MethodofProjection . 40 3.5 Centre manifolds with one parameter dependent systems . 43 3.6 Computational Cost of the Method of Projection . 44 3.7 ModelProblem ............................ 45 3.8 Conclusions.............................. 46 4 Two Degree of Freedom Aeroelastic System 50 4.1 Aerodynamic and Structural Simulations . 51 4.2 Formulation of Augmented System . 53 4.3 Calculation of the Jacobian Matrix . 55 4.4 Solution of the Linear System . 58 4.5 Iteration scheme for flutter boundaries . 62 4.6 Results for Symmetric Problem . 65 4.7 Conclusions.............................. 68 5 Aeroelastic Stability Prediction for Wings 77 5.1 Introduction.............................. 77 5.2 Aerodynamic and Structural Simulations . 77 5.2.1 Aerodynamics ........................ 77 5.2.2 Structural Dynamics, Inter-grid Transformation and Mesh Movement .......................... 78 5.3 Formulation of Augmented Solver . 81 5.4 Results for Symmetric Problem . 82 5.4.1 TestCase ........................... 82 5.4.2 Time Marching Solutions . 83 5.4.3 Augmented Solver Results . 85 5.5 Formulation of a Dedicated Linear Solver . 86 5.5.1 Generalized Conjugate Residual . 88 5.5.2 Block Incomplete Lower Upper Factorisation . 89 5.5.3 Real and Complex Variable Formulations . 90 5.5.4 Results ............................ 91 5.6 Symmetric case: AGARD Wing . 93 5.7 Asymmetriccase:MDOWing . 93 5.8 Conclusions.............................. 94 vii 6 Prediction of Aeroelastic Limit Cycle Oscillations 108 6.1 Introduction..............................108 6.2 Model Reduction for LCO Calculation . 109 6.3 Calculation of First, Second and Third Jacobians