DG Methods for Aerodynamic Flows: Higher Order, Error Estimation and Adaptive Mesh Refinement

HONOM 2011 in Trento DG Methods for Aerodynamic Flows: Higher Order, Error Estimation and Adaptive Mesh Refinement Institute of Aerodynamics and Flow Technology DLR Braunschweig 11. April 2011 1 / 35

Research group working on discontinuous Galerkin methods for aerodynamic flows at DLR The current group members are: Dr. Ralf Hartmann Tobias Leicht (PhD student) Stefan Schoenawa (PhD student) Marcel Wallraff (PhD student) former group member were: Dr. Joachim Held Florian Prill (PhD student) Numerical results are based on: The DLR-PADGE code which is based on a modified version of deal.ii. 2 / 35

Overview Higher-order discontinuous Galerkin methods Error estimation and adaptive mesh refinement for force coefficients Residual-based mesh refinement Numerical results for aerodynamic test cases considered in the EU-project ADIGMA turbulent flow around the 3-element L1T2 high-lift configuration turbulent flow around the DLR-F6 wing-body configuration considered in the EU-project IDIHOM subsonic turbulent flow around the VFE-2 delta wing configuration transonic turbulent flow around the VFE-2 delta wing configuration 3 / 35

DG discretization of the RANS-kω equations RANS and Wilcox k-ω turbulence model equations: (F c (u) F v (u, u) ) = S(u, u) Discontinuous Galerkin discretization of order p + 1: Find u h V p h such that R(u h, v h ) R(u h ) v h dx + r(u h ) v + h + ρ(u h) : v + h ds κ T h with the element residual, Ω κ\γ + r Γ (u h ) v + h + ρ (u Γ h) : v + h ds = 0 v h V p h, Γ R(u h ) = S(u h, u h ) F c (u h ) + F v (u h, u h ), and face and boundary residuals r(u h ), ρ(u h ) and r Γ (u h ), ρ Γ (u h ). 4 / 35

Error estimation with respect to target quantities Target quantities J(u) of interest are the drag, lift and moment coefficients pressure induced and viscous stress induced parts of the force coefficients We want to quantity the error of the discrete function u h in terms of a target quantity J( ), i.e. we want to quantity the error Here, J(u) J(u h ) J(u h ) is the computed force coefficient, and J(u) is the exact (but unknown) value of the force coefficient 5 / 35

Error estimation for single target quantities Given a discretization: find u h V h,p such that N (u h, v h ) = 0 v h V h,p. and a target quantity J. Using a duality argument we obtain an error representation wrt. J( ): J(u) J(u h ) = R(u h, z) := N (u h, z) R(u h, z h ) = κ η κ. where z h is the solution to the discrete adjoint problem: find z h V h,p such that N [u h ](w h, z h ) = J [u h ](w h ) w h V h,p, and η κ are adjoint-based indicators which are particularly suited for the accurate and efficient approximation of the target quantity J(u). 6 / 35

Residual-based mesh refinement The DG discretization: Find u h V p h such that R(u h, v h ) R(u h ) v h dx + r(u h ) v + h + ρ(u h) : v + h ds κ T h Error representation: Residual-based indicators: Ω κ\γ + r Γ (u h ) v + h + ρ (u Γ h) : v + h ds = 0 v h V p h, Γ J(u) J(u h ) = R(u h, z) ( ) 1/2 J(u) J(u h ) (ηκ res ) 2 κ T h ηκ res = h κ R(u h ) κ + hκ 1/2 r(u h ) κ\γ + hκ 1/2 ρ(u h ) κ\γ + hκ 1/2 r Γ (u h ) κ Γ + hκ 1/2 ρ Γ (u h ) κ Γ 7 / 35

hp-refinement with anisotropic element subdivision hp-refinement: After having selected an element for refinement, e.g. by residual-based or adjoint-based refinement indicators, decide whether to split the element in subelements, i.e. use h-refinement, when the solution (or the adjoint solution) is smooth/regular increase the polynomial degree, i.e. use p-refinement, when the solution is non-smooth (shocks, sharp trailing edges,... ) The decision is based on the decay of the Legendre series coefficients. Anisotropic element subdivision: After having selected an element for h-refinement decide upon the specific refinement case based on anisotropic error estimation or on an anisotropic jump indicator: the jump of the discrete solution over element faces is associated with the approximation quality orthogonal to the face 8 / 35

The L1T2 high lift configuration full mesh zoom of mesh Coarse mesh of 4740 elements. Grid lines are given by polynomials of degree 4. 9 / 35

Turbulent flow around the L1T2 high lift configuration Freestream conditions: M = 0.197, α = 20.18 and Re = 3.52 10 6 Mach number and streamlines turbulent intensity 10 / 35

Turbulent flow around the L1T2 high lift configuration Freestream conditions: M = 0.197, α = 20.18 and Re = 3.52 10 6 11 / 35

Turbulent flow around the L1T2 high lift configuration Freestream conditions: M = 0.197, α = 20.18 and Re = 3.52 10 6 12 / 35

Turbulent flow around the L1T2 high lift configuration Freestream conditions: M = 0.197, α = 20.18 and Re = 3.52 10 6 13 / 35

Turbulent flow around the L1T2 high lift configuration 1.00 global DG(1) DG(1) hp(σ 0 = 1.0) CL C L ref 0.10 0.01 1e+05 1e+06 degrees of freedom lift convergence hp-adaptive mesh Tobias Leicht 14 / 35

Turbulent flow around the L1T2 high lift configuration 1.00 global DG(1) DG(1) aniso DG(1) hp(σ 0 = 1.0) aniso hp(σ 0 = 1.0) CL C L ref 0.10 0.01 1e+05 1e+06 degrees of freedom lift convergence hp-adaptive mesh Tobias Leicht 14 / 35

Turbulent flow around the L1T2 high lift configuration 1.00 DG(1) aniso DG(1) hp(σ 0 = 1.0) aniso hp(σ 0 = 1.0) CL C L ref 0.10 0.01 1 10 CPU time lift convergence hp-adaptive mesh Tobias Leicht 14 / 35

Turbulent flow around the L1T2 high lift configuration 1 isotropic 1e-3 1e-6 1 1e-3 1e-6 1 1e-3 hp isotropic hp anisotropic A suitable solution-adaptive mesh can improve the solver behavior. 1e-6 0 100 200 300 400 500 600 convergence: residual vs. nonlinear iterations Tobias Leicht 15 / 35

The DLR-F6 wing-body configuration without fairing The original mesh of 3.24 10 6 elements has been agglomerated twice. The elements of the coarse mesh of 50618 elements are curved based on additional points taken from the original mesh geometry curved mesh with lines given by polynomials of degree 4 16 / 35

Subsonic turbulent flow around the DLR-F6 wing-body Modification of the DPW III test case: M = 0.5 (instead of M = 0.75) α = 0.141 (instead of target lift C l = 0.5) Re = 5 10 6 coarse mesh 2 nd order solution DG solutions on coarse mesh of 50618 curved elements. 3 rd order solution 4 th order solution 17 / 35

Subsonic turbulent flow around the DLR-F6 wing-body TAU on original grid 3 rd order solution after one refinement of coarse mesh 18 / 35

Subsonic turbulent flow around the DLR-F6 wing-body TAU on original grid 3 rd order solution after one refinement of coarse mesh 19 / 35

Subsonic turbulent flow around the DLR-F6 wing-body Adjoint-based refinement for C d : Mesh after 2 adjoint-based refinement steps Density adjoint 20 / 35

Subsonic turbulent flow around the DLR-F6 wing-body Adjoint-based refinement for C d : Convergence of C d (global mesh refinement): 0.05 0.045 p=1, global p=2, global p=3, global TAU 0.04 Mesh after 2 adjoint-based refinement steps 0.035 0.03 0.025 Density adjoint 0.02 1e+06 1e+07 degrees of freedom 20 / 35

Subsonic turbulent flow around the DLR-F6 wing-body Adjoint-based refinement for C d : Mesh after 2 adjoint-based refinement steps Convergence of C d (global & anisotropic h-refinement): 0.05 0.045 0.04 0.035 p=1, global p=2, global p=3, global residual-based adjoint-based(cd) adj(cd) + est. TAU 0.03 0.025 Density adjoint 0.02 1e+06 1e+07 degrees of freedom 20 / 35

Subsonic turbulent flow around the DLR-F6 wing-body Adjoint-based refinement for C d : Mesh after 2 adjoint-based refinement steps Convergence of C d (global, anisotropic h- & hp-refinement): 0.05 0.045 0.04 0.035 p=1, global p=1, adjoint-based(cd) p=1, adj(cd) + est. hp, adjoint-based(cd) hp, adj(cd) + est. TAU 0.03 0.025 Density adjoint 0.02 1e+06 1e+07 degrees of freedom 20 / 35

The VFE-2 delta wing with medium rounded leading edge The original mesh of 884 224 elements has been agglomerated twice. The elements of the coarse mesh of 13 816 elements are curved based on additional points taken from the original mesh geometry original mesh curved coarse mesh with lines with straight lines given by polynomials of degree 4 21 / 35

Fully turbulent flow around the VFE-2 delta wing configuration Underlying flow case U.1 in the EU-project IDIHOM The VFE-2 delta wing with medium rounded leading edge at two different flow conditions: U.1b: RANS-kω, subsonic flow at M = 0.4, α = 13.3 and Re = 3 10 6 U.1c: RANS-kω, transonic flow at M = 0.8, α = 20.5 and Re = 2 10 6 22 / 35

Subsonic flow around the VFE-2 delta wing U.1b: Fully turbulent flow at M = 0.4, α = 13.3 and Re = 3 10 6 coarse mesh with 13 816 curved elements c p distribution 5 th order solution vs. experiment (PSP) 23 / 35

Subsonic flow around the VFE-2 delta wing U.1b: Fully turbulent flow at M = 0.4, α = 13.3 and Re = 3 10 6 residual-based refined mesh with 84 348 curved elements c p distribution 4 th order solution vs. experiment (PSP) 24 / 35

Subsonic flow around the VFE-2 delta wing U.1b: Fully turbulent flow at M = 0.4, α = 13.3 and Re = 3 10 6 4 th -order solution on residual-based refined mesh with 84 348 curved elements 25 / 35

Subsonic flow around the VFE-2 delta wing U.1b: Fully turbulent flow at M = 0.4, α = 13.3 and Re = 3 10 6 4 th -order solution on residual-based refined mesh with 84 348 curved elements 2 nd -order solution on residual-based refined mesh with 562 892 curved elements 25 / 35

Shock-capturing based on artificial viscosity (1) N sc (u h, v) ε(u h ) u h : vdx ε klm (u h ) xl uh m xk v m dx, κ κ κ κ For the compressible Navier-Stokes equations (2nd order DG discretization), 1 ε klm (u h ) = C ε δ kl h 2 β k R m (u h ), k, l = 1,..., d, m = 1,..., n, R m (u h ) = n R q (u h ), m = 1,..., n, q=1 where R(u h ) = (R q (u h ), q = 1,..., n) is the residual of the PDE given by R(u h ) = (F c (u h ) F v (u h, u h )). 1 R. Hartmann. Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 51(9 10):1131 1156, 2006. 27 / 35

Shock-capturing based on artificial viscosity (2) N sc (u h, v) ε(u h ) u h : vdx ε klm (u h ) xl uh m xk v m dx, κ κ κ κ For the RANS-kω equations (2nd and higher order discretization), 2 ε klm (u h ) = C ε b k b l hκ 2 f p(u h ) Rp(u h) + s p(u + h, u h ) p, b = p p + ε R p(u h ) = Xd+2 m=1 p Xd+2 R m(u h ), s p(u + h, u h ) = u m m=1 p u m s m(u + h, u h ), with R(u h ) = F c (u h ), Z Z s m(u + h, u + h ) v h dx = `H(u h, u h, n ) F c (u + h ) n + m v h ds κ κ 2 F. Bassi et. al. Very high-order accurate Discontinuous Galerkin Computation of transonic turbulent flows on Aeronautical configurations, ADIGMA, NNFMMD 113, 2010. 28 / 35

Shock-capturing based on artificial viscosity (combines 1 and 2) N sc (u h, v) κ κ ε(u h ) u h : vdx κ κ ε klm (u h ) xl u m h xk v m dx, For the RANS-kω equations (2nd and higher order discretization) ε klm (u h ) = C ε δ kl h2 k f p(u h ) Rp(u h), k, l = 1,..., d, m = 1,..., d + 2, p R p(u h ) = Xd+2 m=1 p u m R m(u h ), R(u h ) = S(u h, u h ) F c (u h ) + F v (u h, u h ), h i = h i /(degree + 1), where h i is the dimension of the element κ in the x i -coordinate direction 29 / 35

Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = 2 10 6 refined mesh with 201 259 curved elements c p distribution 4 th order solution vs. experiment (PSP) 30 / 35

Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = 2 10 6 4 th -order solution on residual-based refined mesh with 201 259 curved elements 31 / 35

Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = 2 10 6 c p 32 / 35

Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = 2 10 6 c p and M = 1 isosurface 33 / 35

Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = 2 10 6 c p and M = 1 isosurface zoom of M = 1 isosurface 34 / 35

Summary Higher-order discontinuous Galerkin methods Error estimation and adaptive mesh refinement for force coefficients Residual-based mesh refinement Numerical results for aerodynamic flows around the 3-element L1T2 high-lift configuration the DLR-F6 wing-body configuration the VFE-2 delta wing configuration (subsonic and transonic) Computations have been performed with the DLR-PADGE code Thank you 35 / 35