Curved grid generation and DG computation for the DLR-F11 high lift configuration

Transcription

1 ECCOMAS Congress 2016, Crete Island, Greece, 5-10 June 2016 Curved grid generation and DG computation for the DLR-F11 high lift configuration Ralf Hartmann 1, Harlan McMorris 2, Tobias Leicht 1 1 DLR (German Aerospace Center), Institute of Aerodynamics and Flow Technology, Braunschweig 2 CentaurSoft, Austin, TX 10. June / 22

2 HioCFD-4 Computational/Meshing Challenge DLR-F11 Meshing challenge: Quadratic hybrid/mixed-element meshes by the CENTAUR grid generator Computational challenge: Stabilized DG discretization and implicit solver in the DLR-PADGE code DLR-F11 high-lift configuration with slat tracks and flap track fairings hybrid/mixed-element 3 rd -order grid 3 rd -order flow solution 2 / 22

3 Linear and quadratic mesh generation process (CENTAUR) (1) Generate a coarse, linear hybrid grid. (2) Insert an additional midpoint for every grid edge and quadrilateral face. (3) Use the CAD information to map each new boundary point onto the underlying CAD surface. (4) Adjust the position of the interior points based on the mapped position of the boundary edge midpoints, in order to prevent self-intersecting grid elements and to ensure grid validity. (5) Quadratic interpolation of points and interior points for a curved element representation. (1) (2) (3) (4) (5) 3 / 22

4 Linear and quadratic mesh generation process (CENTAUR) For the grid generation of a particularly coarse linear mesh a lower analytic curvature clustering, and a larger maximum element size are used than for linear meshes typically generated for 2nd order schemes. After curving of the mesh the grid quality is evaluated using the volumes, and the the volume ratios of the complete elements and sub-elements. 4 / 22

5 Linear and quadratic mesh generation process: Examples linear mesh quadratic mesh linear mesh quadratic mesh 5 / 22

6 Linear and quadratic mesh generation: Application to DLR-F11 Coarse mesh with prisms pyramids tetrahedra elements Outboard flap track fairing surface mesh: prism layers around the slat and leading edge of the main linear mesh quadratic mesh 6 / 22

7 Linear and quadratic mesh generation: Application to DLR-F11 Coarse mesh with prisms pyramids tetrahedra elements Hybrid mesh cuts in the slat region: linear mesh quadratic mesh 7 / 22

8 The DG discretization of the compressible Euler equations The problem: u F c (u) = 0 in Ω R 2 + u h h, κ with u = (ϱ, ϱv 1, ϱv 2, ϱe) T. The DG(p) discretization: Find u h in V p h such that N h (u h, v h ) { κ T h κ } F c (u h ) : v h dx + ĥ(u + h, u h, n) v+ h ds κ\γ + ĥ Γ (u + h, n) v+ h ds = 0 v h V p h, Γ Numerical flux function ĥ: Roe flux, local Lax-Friedrichs flux,... Numerical flux function ĥ Γ at the boundary: ĥ Γ (u + h, n) = n F c (u Γ (u + h )) allows for an adjoint consistent discretization ĥ Γ (u + h, n) = ĥ(u+ h, u Γ (u+ h ), n) is, to our experience, significantly more stable 8 / 22

9 Numerical flux function at the boundary Use the same numerical flux on the boundary like on interior faces 1 2 ĥ Γ,h = ĥγ(u + h, n) = ĥ(u+ h, u Γ (u+ h ), n), where ( the boundary exterior state u Γ (u+ h ) is computed from u + h + u Γ (u+ h )) = u Γ (u + h ) with n v Γ = 0, the discretization is given by F c (u h ) : h v h dx+ ĥ h v + h ds+ ĥ(u + h, u Γ (u+ h ), n) v+ h ds = 0. Ω κ T κ\γ Γ h This discretization is adjoint inconsistent in combination with both J h (u h ) = p ( u Γ (u + h )) n ψ ds = J(u Γ (u + h )), Γ W and J(u h ) = p(u h ) n ψ ds. Γ W 9 / 22

10 Adjoint consistent discretization of force coefficients Consider the discretization: find u h in V p h such that F c (u h ) : v h dx + ĥ h v + h ds + ĥ Γ,h v + h ds = 0 Ω κ T κ\γ Γ h v h V p h (1) Consider the target quantity and its discretization J(u) = p(u) n ψ ds, J h (u h ) = ĥ Γ,h ψ ds, (2) Γ W Γ W with ψ = (0, ψ 1, ψ 2, 0) for ψ = (ψ 1, ψ 2 ), and ĥ Γ,h = ĥ Γ (u + h, n) = ĥ(u+ h, u Γ (u+ h ), n), It can be shown 1, that (1) in combination with (2) is adjoint consistent. - 1 R. Hartmann and T. Leicht. Generalized adjoint consistent treatment of wall boundary conditions for compressible flows, J. Comput. Phys., 300: , / 22

11 Example: Inviscid flow around the NACA0012 airfoil M = 0.5, α = 2, numerical flux at the wall boundary: ĥ Γ,h = ĥ(u + h, u Γ (u+ h ), n) z 1 isolines of the discrete adjoint solution z h for C dp : y 0.02 y x J h (u h ) = J(u Γ (u + h )) adjoint inconsistent x J h (u h ) = Γ W ĥ Γ,h ψ ds adjoint consistent 11 / 22

12 Extension to viscous flows Ω ( F c (u h ) + F v (u h, h u h )) : h v h dx (ĥγ,h ˆσ Γ,h n) v h ds is adjoint consistent in combination with J(u) = (p n τ n) ψ ds, J h (u h ) = Γ W with ψ = (0, ψ 1, ψ 2, 0) for ψ = (ψ 1, ψ 2 ). Γ W Γ (ĥγ,h ˆσ Γ,h n) ψ ds, 12 / 22

13 Adjoint consistent treatment of integral and local quantities Discretization of flow equations: ( F c (u h ) + F v (u h, h u h )) : h v h dx (ĥγ,h ˆσ Γ,h n) v h ds Ω Adjoint consistent discretization of integral quantities (drag, lift coefficients): J(u) = (p n τ n) ψ ds, J h (u h ) = (ĥγ,h ˆσ Γ,h n) ψ ds. Γ W Adjoint consistent discretization of local quantities (surface pressure, skin friction): c p (u) = p(u) p 1 2 ρ v 2 Γ W, c p,h (u h ) = ĥγ,h ñ p 1 2 ρ v 2 c f (u, u) = τ W (u, u) 1 2 ρ, c v 2 f,h (u h, u h ) = with ñ = (0, n 1, n 2, 0) for the normal vector n = (n 1, n 2 ), and t = (0, t 1, t 2, 0) for the tangential vector t = (t 1, t 2 ). Γ (ˆσΓ,h n ) t 1 2 ρ, v 2, 13 / 22

14 Example: L1T2 high-lift configuration RANS & Wilcox k-ω model, M = 0.197, α = 20.18, Re = ĥ Γ,h = ĥγ(u + h, n) = ĥ(u+ h, u Γ (u+ h ), n). c p- and c f -distributions of the p = 1 flow solution 14 / 22

15 Regularity check of a curved mesh Discrete regularity check: Consider the mapping σ from the reference element ˆκ to the element κ in physical space. The determinant of its Jacobian matrix dσ dx is required in numerical quadrature: κ = det(j) dx det(j(ˆx q ))w q ˆκ q Check whether the Jacobian determinants det(j(ˆx q )) are positive in each of the quadrature points. Note, that this discrete regularity check checks necessary conditions only, whereas the check based on Bézier functions 1 would give a necessary and sufficient condition for validity. - 1 A. Johnen, J.-F. Remacle, and C. Geuzaine. Geometrical validity of curvilinear finite elements. J. Comput. Phys., 233: , / 22

16 Regularity check of the quadratic DLR-F11 mesh κ = det(j) dx ˆκ q det(j(x q ))w q Discrete regularity check: det(j(x q )) > 0 for all quadrature points x q? Discrete regularity check of the quadratic mesh around the DLR-F11 mesh with curved elements (prisms, pyramids and tetrahedras): degree # DoFs/eqn # quadrature # irregular worst # neg. points/elem. elements Jac./element / / /64 16 / 22

17 Application to the DLR-F11 (Config 4): The solution process RANS & Wilcox k-ω model, flow conditions: M = 0.175, Re = , α = 7 Flow solver: Fully implicit (Backward-Euler) solver Convergence criterion: nonlinear residual below Linear systems solved with GMRes and ILU per process The CFL number is increased as the nonlinear residual decreases Diverging steps are recomputed with CFL/2. Numerical boundary flux: ĥγ = ĥ(u+ h, u Γ (u+ h ), n). degree # DoFs/eqn res tol start CFL end CFL steps lost 0 3.5e / e / / e / / 22

18 Application to the DLR-F11 (Config 4): The solution process RANS & Wilcox k-ω model, flow conditions: M = 0.175, Re = , α = 7 Flow solver: Fully implicit (Backward-Euler) solver Convergence criterion: nonlinear residual below Linear systems solved with GMRes and ILU per process The CFL number is increased as the nonlinear residual decreases legend numerical flux recompute diverging steps with CFL/2 p=1 (1) ĥ(u + h, u Γ (u+ h ), n) yes p=1 (2) n F c (u Γ (u + h )) yes p=1 (3) n F c (u Γ (u + h )) no 18 / 22

19 DLR-F11 high lift configuration: The flow solution RANS & Wilcox k-ω model, flow conditions: M = 0.175, Re = , α = 7. 3 rd -order solution on the quadratic (3 rd -order) mesh: 19 / 22

20 DLR-F11 high lift configuration: The flow solution RANS & Wilcox k-ω model, flow conditions: M = 0.175, Re = , α = 7. 3 rd -order solution on the quadratic (3 rd -order) mesh: separation above flap track consider pressure tab locations 01, 04, 06, 10 fairings and at wing tip 20 / 22

21 DLR-F11 high lift configuration: The flow solution RANS & Wilcox k-ω model M = Re = α = 7 PS01(η=0.150) PS04(η=0.449) 3 rd -order solution on 3 rd -order mesh PS06(η=0.681) PS10(η=0.891) C L C D C M Exp TAU % +4.1% -4.8% DG-O % +2.1% -5.8% 21 / 22

22 Computational/Meshing Challenge DLR-F11: Summary Quadratic hybrid/mixed-element mesh by the CENTAUR grid generator. Stabilized DG discretization and solver in the DLR-PADGE code: For stability use interior flux also on the boundary. For adjoint consistency use the same flux also for the evaluation of integral quantities (force coefficients), and for an according evaluation of local quantities (c p- and c f -distributions) Recompute diverging solver steps with CFL/2. DLR-F11 high-lift configuration with slat tracks and flap track fairings hybrid/mixed-element 3 rd -order grid 3 rd -order flow solution 22 / 22