Discontinuous Galerkin Spectral Element Approximations for CFD

Transcription

1 Discontinuous Galerkin Spectral Element Approimations for CFD D.A. Kopriva Florida State Universit Tallahassee, FL 3236 G.B. Jacobs San Diego State Universit San Diego, CA September 3, Code description 1.1 Discretization The discontinuous Galerkin approimation generates a framework for developing solvers for conservation laws of the form q t + f = such as the Euler Equations of gas-dnamics ρ q = ρ u, f i = ρe f = f i + f v ρ u ρ u u + pi ρuh, f v = and the compressible Navier-Stokes Equations f v = τ τ u + k T The three basic characteristics are: Approimate the solution and flues b polnomials within elements q Q P N, f F P M on E Professor, Department of Mathematics Associate Professor, Department of Aerospace Engineering 1

2 Start with a weak form for the equations E (Q t + F ) φ = and require no continuit on φ P N between elements. 1.2 Relevant solvers Within these broad constraints one has a huge number of choices: 1. Use Quad/He or Tri/Tet elements? 2. Use a Nodal or a modal basis? 3. What basis polnomials? 4. Approimate boundaries with different orders?. Approimate solution and flues with different orders? 6. Eact integrals or quadrature? 7. Ineact or eact quadrature? 8. Integrate b parts once or twice? We will present solutions to the workshop benchmark problems using a Classical spectral element approimation: Quadrilateral/ Heahedral elements for efficient tensor product bases A Nodal basis for Eas for nonlinear/variable coefficient/general comple geometr problems. All approimations at same polnomial order, which Simplifies coding Legendre Polnomial basis for spectral accurac and p-refinement Legendre-Gauss quadrature Isoparametric element boundaries These choices lead to a spectrall accurate, unstructured mesh approimation that is nevertheless simple to code. Within each element one updates the solution b dq i,j dt + {[ F (1, η j ) l i(1) w (ξ) i {[ + G (ξ i, 1) l j(1) w (η) j ] F ( 1, η j ) l i( 1) w (ξ) i ] G (ξ i, 1) l j( 1) w (η) j + + N k= N k= F k,j ˆD(ξ) ik G i,k ˆD(η) jk } } =, 2

3 where F represents the Riemann Flu. Time integrators include eplicit Runge-Kutta, orders one to four, including low dissipation and dispersion error versions, Implicit Runge-Kutta up to order four and BDF up to order four with Newton-Raphson with GMRES or BICGSTAB linear solvers. 1.3 High-order capabilit An order depending on polnomial order approimation, N, within element. Spectrall accurate. 1.4 Parallel capabilit Full parallel capabilit, tested on 1k processors. 1. Post-processing Turbulence statistics. Tecplot, Matlab, Fieldview 1.6 Other features used for case (e.g. adaptivit) Non-conformal, adaptive grid, using mortar approach. 2 Case summar We solve the flat plate, bump flow and NACA12 problems posed in the workshop case director. We will determine the residual tolerances and other convergence criteria that were requested on parallel machines at Florida State Universit and San Diego State Universit. 2.1 Meshes Meshes were created with an in-house mesher with high-order (isoparametric) boundar fitting. The mesh for the NACA test problems are shown in Figures 1 and 2. The mesh used for the flat plate computation is shown in Figure a. 3

4 Figure 1: The mesh for M =. flow over a NACA12 airfoil at 2deg. angle of attack Figure 2: The mesh in the neighborhood of the airfoil for M =. flow over a NACA12 airfoil at 2deg. angle of attack. 4

5 3 Results 3.1 Problem C.1.1: Inviscid Flow over a Bump in a Channel Fig. 3 shows pressure contours for the M=. inviscid flow over a smooth bump in a channel using the DGSEM and fifth order polnomial approimations Figure 3: Pressure contours for a fifth order polnomial approimation to flow over a smooth bump. 3.2 Problem C.1.3: Flow over a NACA12 Airfoil The DGSEM is geometricall fleible. In Fig. 4 we show N = order solutions to the M =. flow over a NACA12 airfoil at 2 deg angle of attack Figure 4: Mach contours for M =. flow over a NACA12 airfoil at 2deg. angle of attack.

6 3.3 Problem C.1.4: Flat Plate Boundar Laer The DGSEM is also applicable to viscous compressible problems. Fig. shows a flat plate computation of the tpe requested at M =.6 and RE = 1. A different mesh would be used for the requested RE = 16 computation. Fig. compares the horizontal velocit profile as a function of height from the plate at = Figure : Mesh and temperature contours for flow over a flat plate Horizontal Velocit at = u Figure 6: Horizontal velocit for the boundar laer flow at = 12 6