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A High-Order, Adaptive, Discontinuous Galerkin Finite Element Method

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A High-Order, Adaptive, Discontinuous Galerkin FiniteElement Method for the Reynolds-Averaged Navier-StokesEquationsbyTodd A. OliverS.M., Massachusetts Institute of Technology (2004)S.B., Massachusetts Institute of Technolog y (2002)Submitted to the Department of Aeronautics and Astronauticsin partial fulfillment of the requirements for the degree ofDoctor of Philosophyat theMASSACHUSETTS INSTITUTE OF TECHNOLOGYSeptember 2 008c Massachusetts Institute of Technology 2008. All rights reserved.Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of Aeronautics and AstronauticsJuly 3, 2008Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .David L. DarmofalAsso ciate Pro fessor of Aeronautics and AstronauticsThesis SupervisorCertified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mark DrelaTerry J. Kohler Professor of Fluid DynamicsThesis CommitteeCertified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Jaime Pera ireProfessor of Aeronautics and AstronauticsThesis CommitteeAccepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .David L. DarmofalAsso ciate Pro fessor of Aeronautics and AstronauticsChairman, Department Committee o n Graduate Students2A High-Order, Adaptive, Discontinuous Galerkin Finite Element Methodfor the Reynolds-Averaged Navier-Stokes EquationsbyTodd A. OliverSubmitted to the Department of Aeronautics and Astronauticson July 3, 2008, in partial fulfillment of th erequirements for the d egree ofDo ctor of PhilosophyAbstractThis thesis presents high-order, discontinuous Galerkin (DG) discretizations of the Reynolds-Averaged Navier-Stokes (RANS) equations and an output-based err or estimation and meshadaptation algorithm for these discretizations. In particular, DG discretizations of theRANS equations with the Sp alart-Allmaras (SA) turbulence model are examined. Thedual consistency of multiple DG discretizations of the RANS-SA system is analyzed. Theapproach of simply weighting gradient dependent source terms by a test function and inte-grating is shown to be dual inconsistent. A dual consistency correction for this discretizationis derived. The analysis also demonstrates that discretizations based on the popular mixedformulation, where dependence on the state gradient is handled by introducing additionalstate variables, are generally asymptotically dual consistent. Numerical results are pre-sented to confirm the results of the analysis.The output error estimation and output-based adaptation algorithms used here areextensions of methods previously developed in the finite volume and finite element com-munities. In particular, the methods are extended for application on the curved, highlyanisotropic meshes required for boundary conforming, high-order RANS simulations. Twomethods for generating such curved meshes are demonstrated. One relies on a u s er-definedglobal mapping of the physical domain to a s tr aight meshing domain. The other uses a lin-ear elasticity node movement scheme to add curvature to an initially linear mesh. Finally,to improve the robustness of the adaptation process, an “unsteady” algorithm, where th emesh is adapted at each time step, is presented. The goal of the unsteady procedure is toallow mesh adaptation prior to converging a steady state solution, not to obtain a time-accurate solution of an unsteady problem. Thus, an estimate of the error due to spatialdiscretization in the output of interest averaged over the current time s tep is developed.This error estimate is then used to d rive an h-adaptation algorithm.Adaptation results demonstrate that the high-order discretizations are more efficientthan the second-order method in terms of degrees of freedom requ ir ed to achieve a desirederror tolerance. Furthermore, using the unsteady adaptation process, adaptive RANS sim-ulations may be started from extremely coarse meshes, significantly decreasing the meshgeneration burden to th e u ser.Thesis S upervisor: David L. DarmofalTitle: Associate Professor of Aeronautics and Astronautics34AcknowledgmentsTo begin, I would like to express my gratitude to my advisor, Prof. David Darmofal. With-out his guidance and encouragement throughout my time as a gradu ate student, this workwould not have been possible. His probing questions and insightful ideas contributed im-measurably to the research presented here, and his mentoring has been crucial to my de-velopment as a researcher.In addition, I would like to th an k my committee members, Prof. Mark Drela andProf. Jaime Peraire, for their criticism and feedback, which led to many improvementsin my research and this thesis. I would also like to thank Dr. Ralf Hartmann for hiscomments on the initial draft of the thesis, and I am particularly indebted to Dr. SteveAllmaras for his help with the SA model modifications and for his insightful comments onthe in itial thesis draft.Of course, none of this work would have taken place without the entire Project X team.I would particularly like to thank Garrett Barter, Krzysztof Fidkowski, Mike Park, RobertHaimes, Laslo Diosady, JM Modisette, and Josh Krakos for their contributions to the code aswell as their input throughout many discussions of research, DG methods, software practices,and Project X. In addition to those already mentioned, Project X wou ld not be what itis now without the contributions of Mathieu Serrano, Michael Brasher, James Lu, PaulNicholson, Er ic Liu, Eleanor Lin, Peter Whitney, Shannon Cheng, Jean-Baptise Brachet,Haufei Sun, and Masayuki Yano, an d I am looking forward to following the evolution of theproject in the capable hands of Laslo, JM, Josh, Haufei, and Masa.On a more personal note, I have sincerely enjoyed the many friendships I developedthrough both my undergraduate and graduate years at MIT. In particular, Dave Bennett,Garrett Barter, Mike Brasher, Mark Monroe, and Shana Diez have all helped to make MITmore f un than it might otherwise have been.I would also like to thank my family—Mom, Dad, Lee, and Lauren—for th eir


A High-Order, Adaptive, Discontinuous Galerkin Finite Element Method

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