Entropy stable schemes for compressible flows on unstructured meshes

Entropy stable schemes for compressible flows on unstructured meshes Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore deep@math.tifrbng.res.in http://math.tifrbng.res.in/ deep SCPDE-2014, Hong Kong 9th November 2014

Work done with: Praveen Chandrashekar, TIFR-CAM, Bangalore Siddhartha Mishra, Seminar for Applied Mathematics, ETH Zurich Ulrik S. Fjordholm, NTNU, Trondheim Funded by: AIRBUS Group Corporate Foundation Chair in Mathematics of Complex Systems, established in TIFR/ICTS, Bangalore

Conservation laws Consider the (hyperbolic) system Examples U t + f 1 x + f 2 y = 0 (x, t) R2 R + U(x, 0) = U 0 (x) x R 2 Shallow water equations (Geophysics) Euler equations (Aerodynamics) MHD equations (Plasma physics) Non linearities = disc. soln. = weak (distributional) soln.

Entropy framework Entropy-entropy flux pair (η(u), q(u) = (q 1 (U), q 2 (U)) t η(u) + div (q(u)) 0 where V = η (U) entropy variables. Single out a physically relevant solution d dt η(u)dx 0 = U(., t) L 2 C No global existence, uniqueness results for generic multidimensional systems

Entropy stable schemes Cartesian meshes Tadmor (1987): Entropy conservative flux. Lefloch et al. (2002): Higher order entropy conservative schemes. Explicit entropy stable schemes by Roe and Ismail (2009), PC (2013). Fjordholm et al. (2011): Entropy stable ENO schemes (TeCNO). Madrane et al. (2012): First order entropy stable schemes on unstructured meshes.

Outline 2D conservation laws: Discretisation (unstructured) Entropy conservative/stable schemes High-order schemes (sign property) Numerical results 2D Navier-Stokes equations: Global entropy relations Discrete analogue Boundary conditions Numerical results

Discretization Discretize domain into triangles T, with nodes i, j, k,etc Primary cell: k n T i T l n T e i T e n T e j j e n e i

Discretization Discretize domain into triangles T, with nodes i, j, k,etc Primary cell: Dual control volumes C i Median dual Voronoi dual

Discretization Discretize domain into triangles T, with nodes i, j, k,etc Primary cell: Dual control volumes C i Median dual Voronoi dual Suitable for complex geometries!!

Semi-discrete scheme du i dt + 1 C i F ij = 0 j i U i cell average over C i F ij = F(U i, U j, n ij ) is the numerical flux satisfying 1 Consistency: 2 Conservation: F(U, U, n) = F(U, n) := f 1 (U)n 1 + f 2 (U)n 2 F(U 1, U 2, n) = F(U 2, U 1, n) U 1, U 2, n

First order entropy stable flux 1 Entropy conservative flux F ij = F ij s.t. Sufficient condition (Tadmor) where dη(u i ) dt + 1 C i qij = 0 j i V ijf ij = ψ(u j, n ij ) ψ(u i, n ij ) ψ(u, n) := V(U) F(U, n) q(u, n) }{{} q 1n 1+q 2n 2 (entropy potential) Entropy variable based dissipation F ij = F ij 1 2 D ij V ij, D ij = D ij 0 = dη(u i) dt + 1 C i q ij 0 1 A. Madrane, UF, SM, and E. Tadmor. Entropy conservative and entropy stable finite volume schemes for multi-dimensional conservation laws on unstructured meshes. (in review) j i

First order entropy stable flux Kinetic energy and entropy conservative flux (PC) for Euler equations. Dissipation operator D ij = R ij Λ ij R ij R ij scaled eigenvectors of F U (Barth) Λ ij = diag[ λ 1,..., λ n ] Roe type KEPES Kinetic energy and entropy conservative flux + Roe dissipation

High-order diffusion V ij O( x ij ) = first order flux Idea: For each x ij, reconstruct V in C i, C j with polynomials V i (x), V j (x) respectively. V ij = V i (x ij ), V ji = V j (x ij ), V ij = V ji V ij Sign property (Fjordholm et al.) For each x ij, define the scaled entropy variables Z = Rij V. Then numerical flux F ij = F ij 1 2 R ijλ ij Z ij = F ij 1 2 D ij V ij, V ij = (R T ij) 1 Z ij is entropy stable if the sign property holds for Z componentwise. sign( Z ij ) = sign( Z ij )

Second order (limited) reconstruction i 1 i j + 1 j Define Z i = RijV i, Z j = RijV j The componentwise reconstructed scaled variables are Z ij =Z i + 1 ) ( 2 minmod f ij, } b ij Z ji =Z j 1 ) need h Z i = Rij ( h V i 2 minmod f ji, b ji

Transonic flow past NACA-0012 airfoil angle of attack = 2 degrees, freestream M = 0.85 KEPES KEPES-TeCNO Mach number, 20 equally spaced contours between 0.5 and 1.5 ROE (MUSCL)

Subsonic flow past a cylinder freestream M = 0.3, symmetric, isentropic flow KEPES KEPES-TeCNO KEPES2 Scheme Minimum Maximum Percent deviation from s KEPES 2.07147 2.08695 +0.747 % -0.000 % KEPES-TeCNO 2.07147 2.07208 +0.029 % -0.000 % KEPES2 2.07139 2.07153 +0.003 % -0.004 % Table : Physical entropy bounds, with freestream s = 2.07147

2D Navier-Stokes Equations Initial boundary valued problem U t + f 1 x + f 2 y = g 1 x + g 2 x = (x, y) Ω R 2, t R + y U(x, 0) = U 0 (x) x Ω B(U(x, t)) = h(x, t) x Ω

2D Navier-Stokes Equations Initial boundary valued problem U t + f 1 x + f 2 y = g 1 x + g 2 x = (x, y) Ω R 2, t R + y U(x, 0) = U 0 (x) x Ω B(U(x, t)) = h(x, t) x Ω Specific entropy pair for symmetrization η(u) = ρs γ 1, q 1(U) = ρus γ 1, q 2(U) = ρvs γ 1

2D Navier-Stokes Equations Initial boundary valued problem U t + f 1 x + f 2 y = g 1 x + g 2 x = (x, y) Ω R 2, t R + y U(x, 0) = U 0 (x) x Ω B(U(x, t)) = h(x, t) x Ω Specific entropy pair for symmetrization η(u) = ρs γ 1, q 1(U) = ρus γ 1, q 2(U) = ρvs γ 1 Viscous fluxes in terms of entropy variables g 1 = K 11 (V) V x + K 12(V) V y, where [ ] K11 K K = 12 0 K 21 K 22 g 2 = K 21 (V) V x + K 22(V) V y

Global entropy relation Integrating NSE against V gives us d η = q(u, n) K V, V + V G(U, n) dt Ω Ω Ω Ω q(u, n) + V G(U, n) where and ( ) x V V = y V Ω Ω G(U, n) = g 1 (U)n 1 + g 2 (U)n 2 R 8 obtained from V = ( x V, y V ) R 4 2

Semi-discrete scheme Notations j i = { all vertices j neighbouring vertex i } i T = { all vertices i belonging to triangle T } T i = { all triangles T having vertex i } Γ = { all boundary edges of the primary mesh } Γ i = { all boundary edges of the primary mesh having vertex i } C i du i dt = j i F ij + T i G T nt i 2 F ie + ne G e 2 e Γ i e Γ i

Choosing flux terms Interior inviscid flux F ij is an entropy stable flux

Choosing flux terms Interior inviscid flux F ij is an entropy stable flux Interior viscous flux G T = (G T 1, G T 2 ) is chosen as G T α = K T α1 h xv T + K T α2 h y V T, α = 1, 2

Choosing flux terms Interior inviscid flux F ij is an entropy stable flux Interior viscous flux G T = (G T 1, G T 2 ) is chosen as G T α = K T α1 h xv T + K T α2 h y V T, α = 1, 2 Boundary inviscid flux F ie is chosen as F ie = F E ie = 1 F ie = F ie ( U, U b, n e ) F ρ ie 2 (pb i n e) + u i F ρ ie F E ie [ ( ρ u 2 2 + γp γ 1, F ρ ie (ρ = ub n e 2 ) u b ne 2 ] BC implemented weakly i ) [ + (p b p) u n e 2 i ] i

Choosing flux terms Interior inviscid flux F ij is an entropy stable flux Interior viscous flux G T = (G T 1, G T 2 ) is chosen as G T α = K T α1 h xv T + K T α2 h y V T, α = 1, 2 Boundary inviscid flux F ie is chosen as F ie = F ie ( U, U b, n e ) Boundary viscous flux G e = (G e 1, G e 2) is chosen as G e n e = G Te n e + C e

Discrete global entropy relation Pre-multiplying the scheme by V i and summing over all nodes. [ d ( ) ( ) ] ρs u b n η i C i e ρs u b n e dt γ 1 2 γ 1 2 i e Γ i j [ ] (V b i + V b j) + G Te (Vi + Vj) n e + C e 2 2 e Γ which is consistent with d η dt Ω Ω q(u, n) + Ω V G(U, n)

Homogeneous boundary conditions Slip BC (Euler) Continuous setup d η 0 dt Ω Discrete setup d dt η i C i 0 i

Homogeneous boundary conditions Slip BC (Euler) Continuous setup d η 0 dt Ω Discrete setup d dt η i C i 0 i No-slip BC with zero heat flux Continuous setup G e n e = 0 τ n e 0 d dt Ω η 0 ( C e = 0 0 0 Discrete setup G e 0 n e = τ h n e, 0 ( ( κ h V Te,(4) ) )) n R(V e,(4) ) 2 e d dt η i C i 0 i

Numerical results: Lid driven cavity Top lid moving with velocity (u = 1, v = 0). No-slip BC on other 3 faces. Comparing with data of Ghia et al. 1 0.8 0.3 0.2 0.1 0.6 0 y 0.4 0.2 Referrence Roe KEPES TECNO v 0.1 0.2 0.3 0.4 Referrence Roe KEPES TECNO 0 0.2 0 0.2 0.4 0.6 0.8 u u on vert. line through center 0.5 0 0.2 0.4 0.6 0.8 1 x v on hor. line through center

Flow past a cylinder Inflow: ( ) y(h y) u = 4u m H 2, 0, u m = 1.5, Re = 100 back

Flow past a cylinder Lift Inflow: 0.6 ( ) y(h y) u = 4u m H 2, 0, u m = 1.5, Re = 100 0.4 0.2 0-0.2-0.4-0.6 3 4 5 6 7 8 9 Time

Conclusion Second order (limited) entropy stable finite volume scheme for system of conservation laws, on unstructured meshes. Discretized viscous fluxes for Navier-Stokes equations with appropriate boundary (homogeneous) fluxes to obtain global entropy stability. Next steps: Higher order finite volume schemes? Entropy stable boundary conditions for general BC? Parellelisation and 3D implementation.

Thank you.

2D Euler Equations where Choose ρ ρu ρv U = ρu ρv, f 1(U) = ρu 2 + p ρuv, f 2(U) = ρuv ρv 2 + p E u(e + p) v(e + p) η(u) = ρs γ 1, ( ) 1 p E = ρ 2 u2 + (γ 1)ρ q 1(U) = ρus γ 1, q 2(U) = ρvs γ 1 s = ln (p) γ ln (ρ) (physical entropy) γ s γ 1 β u 2 V = 2βu, β = ρ 2p = 1 RT 2β

Entropy conservative flux for Euler Kinetic energy and entropy conservative flux F,ρ ρu n F = F,m1 F,m2 = pn 1 + uf,ρ pn 2 + vf,ρ F,e F,e where u n = un 1 + vn 2, p = ρ 2β, [ ] F,e 1 = 2(γ 1) β 1 2 u 2 F,ρ + u F,m The crucial property for kinetic energy preservation (Jameson) F m = pn + uf ρ for any consistent approximations of p and F ρ.

Entropy stable KEPES flux For Euler equations, we choose Roe type dissipation where D ij = R ij Λ ij R ij 1 1 0 1 R = u añ 1 u ñ 2 u + añ 1 v añ 2 v ñ 1 v + añ 2 S 1 2 1 H auñ 2 u 2 uñ 2 vñ 1 H + auñ [ ] ρ S = diag 2γ, (γ 1)ρ ρ, p, γ 2γ Λ = diag [u n a, u n, u n, u n + a] ñ unit face normal, uñ = u ñ γp a = speed of sound in air ρ H = a2 γ 1 + u 2 specific enthalpy 2

Second order (limited) reconstruction i 1 i j + 1 j The forward differences f ij = Z j Z i, f ji = Z j+1 Z j = [Z i +2(x j x i ) h Z j ] Z j The backward differences b ij = Z i Z i 1 = Z i [Z j 2(x j x i ) h Z i ], b ji = Z j Z i h Z i = R ij h V i

Entropy stability is important 1D modified shocktube test: (ρ L, u L, p L ) = (1.0, 0.75, 1.0), (ρ R, u R, p R ) = (0.125, 0.0, 0.1) 1.5 1 1 0.8 Exact KEPES ROE 1 0.8 Exact KEPES ROE density 0.6 0.4 x velocity 0.5 Exact KEPES pressure 0.6 0.4 ROE 0.2 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Roe s solver (without fix) gives entropy violating shock. x 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

Gradients 2 3 3 n1,2 ni,2 n T 1 n T 2 i 1 T ni,1 n5,i n5,1 1 2 4 5 h Z i = R ij h V i n T 3 h V T = 1 T h V Te = 1 T (V i + V j ) nk T } 2 {{} trapezoidal rule [ (V b i + V b j ) 2 + (V j + V k ) 2 n e + (V j + V k ) 2 n T i + (V k + V i ) 2 n T j ] n Te i + (V k + V i ) n Te j 2

Gradients 2 3 3 n1,2 ni,2 n T 1 n T 2 i 1 T ni,1 n5,i n5,1 1 2 4 5 h Z i = R ij h V i n T 3 h V i = T h V T T i T T i

Time integration: SSP-RK3: Explicit strong stability preserving RK3 du dt = L(u) v (0) = v n v (1) = v (0) + tl(v (0) v (2) = 3 4 v(0) + 1 [v (1) + tl(v (1)] 4 v (3) = 1 3 v(0) + 2 [v (2) + tl(v (2)] 3 v n+1 = v (3)

Step in wind tunnel Supersonic flow past forward step, M=3. Corner is the center of a rarefaction fan = singular point. Shocks undergo reflections.

2D Navier-Stokes Equations g 1 (U) = U t + f 1 x + f 2 y = g 1 x + g 2 y 0 τ 11 τ 21 uτ 11 + vτ 21 Q 1, g 2(U) = 0 τ 21 τ 22 uτ 21 + vτ 22 Q 2 where τ 11 = 4µ u 3 x 2µ ( v u 3 y, τ 12 = τ 21 = µ y v ), x τ 22 = 4µ 3 v y 2µ 3 u x, Q = (Q 1, Q 2 ) = κ T µ dynamic coeff. of viscosity κ coeff. of heat conductance

Inviscid boundary flux Boundary inviscid flux F ie is chosen as F ρ ie F ie = 1 2 (pb i n e) + u i F ρ ie, F ρ Fie E ie (ρ = ub n e 2 [ ( Fie E = ρ u 2 2 + γp ) ] u b [ ne + (p b p) u n e γ 1 2 2 i ) i ] i

Numerical results: Viscous shock tube problem Balancing effects of physical viscosity, the heat flux and artificial numerical viscosity (ρ L, u L, p L ) = (1.0, 0.0, 1.0) (ρ R, u R, p R ) = (0.125, 0.0, 0.1) P r = 1.0, h 1.0 10 2 κ = µc p P r We consider three flows regimes depending on µ. ND Numerical dissipation present Visc Physical viscosity present Heat Heat flux present

Numerical results: Viscous shock tube problem Regime 1: µ = 2.0 10 4 1 0.9 0.8 Visc+Heat+ND Visc+ND Visc Visc+Heat 1.2 1 Visc+Heat+ND Visc+ND Visc Visc+Heat density 0.7 0.6 0.5 x vleocity 0.8 0.6 0.4 0.4 0.3 0.2 0.2 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.6 0.8 1 x Severe oscillations in the absence of ND.

Numerical results: Viscous shock tube problem Regime 2: µ = 2.0 10 3 1 0.9 0.8 0.7 Visc+Heat+ND Visc+ND Visc Visc+Heat 1 0.9 0.8 0.7 Visc+Heat+ND Visc+ND Visc Visc+Heat density 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 x x vleocity 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 x

Numerical results: Viscous shock tube problem Regime 2: µ = 2.0 10 3 0.75 0.7 Visc+Heat+ND Visc+ND Visc Visc+Heat density 0.65 0.6 0.55 0.5 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 x

Numerical results: Viscous shock tube problem Regime 3: µ = 2.0 10 2 1 0.9 0.8 0.7 Visc+Heat+ND Visc+ND Visc Visc+Heat 1.2 1 0.8 Visc+Heat+ND Visc+ND Visc Visc+Heat density 0.6 0.5 x vleocity 0.6 0.4 0.4 0.3 0.2 0.2 0 0.1 0 0.5 1 1.5 2 x 0.2 0 0.5 1 1.5 2 x

Numerical results: Viscous shock tube problem Regime 2: µ = 2.0 10 2 0.75 0.7 Visc+Heat+ND Visc+ND Visc Visc+Heat density 0.65 0.6 0.55 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 x

Numerical results: Laminar flat-plate boundary layer Re = 10 5, M = 0.1 Slip BC on inlet portion Farfield BC on In-flow Adiabatic no-slip BC on flat plate Freestream pressure BC on top and outlet Inlet portion Flat plate

Numerical results: Laminar flat-plate boundary layer Comparing with Blasius semi-analytical solution on the vertical line through plate center 7 6 Blasius ROE KEPES-TECNO 7 6 Blasius ROE KEPES-TECNO 5 5 4 4 eta eta 3 3 2 2 1 1 0 0 0.2 0.4 0.6 0.8 1 1.2 u/uinf Streamwise velocity 0 0 0.2 0.4 0.6 0.8 1 1.2 v*sqrt(2*rex)/uinf Vertical velocity