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



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A High Order Adaptive Discontinuous Galerkin Finite Element Method for the Reynolds Averaged Navier Stokes Equations by Todd A Oliver S M Massachusetts Institute of Technology 2004 S B Massachusetts Institute of Technology 2002 Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2008 c Massachusetts Institute of Technology 2008 All rights reserved Author Department of Aeronautics and Astronautics July 3 2008 Certified by David L Darmofal Associate Professor of Aeronautics and Astronautics Thesis Supervisor Certified by Mark Drela Terry J Kohler Professor of Fluid Dynamics Thesis Committee Certified by Jaime Peraire Professor of Aeronautics and Astronautics Thesis Committee Accepted by David L Darmofal Associate Professor of Aeronautics and Astronautics Chairman Department Committee on Graduate Students 2 A High Order Adaptive Discontinuous Galerkin Finite Element Method for the Reynolds Averaged Navier Stokes Equations by Todd A Oliver Submitted to the Department of Aeronautics and Astronautics on July 3 2008 in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis presents high order discontinuous Galerkin DG discretizations of the ReynoldsAveraged Navier Stokes RANS equations and an output based error estimation and mesh adaptation algorithm for these discretizations In particular DG discretizations of the RANS equations with the Spalart Allmaras SA turbulence model are examined The dual consistency of multiple DG discretizations of the RANS SA system is analyzed The approach of simply weighting gradient dependent source terms by a test function and integrating is shown to be dual inconsistent A dual consistency correction for this discretization is derived The analysis also demonstrates that discretizations based on the popular mixed formulation where dependence on the state



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