Entropy stable schemes for compressible flows on unstructured meshes - PowerPoint PPT Presentation

Entropy stable schemes for compressible flows on unstructured meshes Deep Ray 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 Deep Ray KEP-ES

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 Deep Ray KEP-ES

Conservation laws Consider the (hyperbolic) system ∂ U ∂t + ∂ f 1 ∂x + ∂ f 2 ( x , t ) ∈ R 2 × R + ∂y = 0 ∀ x ∈ R 2 U ( x , 0) = U 0 ( x ) ∀ Examples • Shallow water equations (Geophysics) • Euler equations (Aerodynamics) • MHD equations (Plasma physics) Non linearities = ⇒ disc. soln. = ⇒ weak (distributional) soln. Deep Ray KEP-ES

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 η ( U ) d x ≤ 0 = ⇒ || U ( ., t ) || L 2 ≤ C d t No global existence, uniqueness results for generic multidimensional systems Deep Ray KEP-ES

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. Deep Ray KEP-ES

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 Deep Ray KEP-ES

Discretization Discretize domain into triangles T , with nodes i, j, k ,etc • Primary cell: n T i l k T n T e j n T e i T e e j i n e Deep Ray KEP-ES

Discretization Discretize domain into triangles T , with nodes i, j, k ,etc • Primary cell: • Dual control volumes C i Median dual Voronoi dual Deep Ray KEP-ES

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!! Deep Ray KEP-ES

Semi-discrete scheme � d U i 1 d t + F ij = 0 | C i | 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: F ( U , U , n ) = F ( U , n ) := f 1 ( U ) n 1 + f 2 ( U ) n 2 2 Conservation: F ( U 1 , U 2 , n ) = − F ( U 2 , U 1 , − n ) ∀ U 1 , U 2 , n Deep Ray KEP-ES

First order entropy stable flux 1 • Entropy conservative flux � d η ( U i ) 1 F ij = F ∗ q ∗ s.t. + ij = 0 ij d t | C i | j ∈ i • Sufficient condition (Tadmor) ∆ V ⊤ ij F ∗ ij = ψ ( U j , n ij ) − ψ ( U i , n ij ) where ψ ( U , n ) := V ( U ) ⊤ F ( U , n ) − q ( U , n ) → (entropy potential) � �� � q 1 n 1 + q 2 n 2 • Entropy variable based dissipation ij − 1 F ij = F ∗ D ij = D ⊤ 2 D ij ∆ V ij , ij ≥ 0 � d η ( U i ) 1 = ⇒ + q ij ≤ 0 d t | C i | j ∈ i 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) Deep Ray KEP-ES

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 Deep Ray KEP-ES

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 = R ⊤ ij V . Then numerical flux ij − 1 ij − 1 F ij = F ∗ 2 R ij Λ ij � Z � ij = F ∗ � V � ij = ( R T ij ) − 1 � Z � ij 2 D ij � V � ij , is entropy stable if the sign property holds for Z componentwise. sign ( � Z � ij ) = sign (∆ Z ij ) Deep Ray KEP-ES

Second order (limited) reconstruction i − 1 j + 1 i j Define Z i = R ⊤ Z j = R ⊤ ij V i , ij V j The componentwise reconstructed scaled variables are � � Z ij = Z i + 1 � ∆ f ij , ∆ b 2 minmod ij need ∇ h Z i = R ⊤ ij ∇ h V i � � Z ji = Z j − 1 ∆ f ji , ∆ b 2 minmod ji Deep Ray KEP-ES

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) Deep Ray KEP-ES

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 Deep Ray KEP-ES

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 Deep Ray KEP-ES

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 q 1 ( U ) = − ρus q 2 ( U ) = − ρvs γ − 1 , γ − 1 , γ − 1 Deep Ray KEP-ES

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 q 1 ( U ) = − ρus q 2 ( U ) = − ρvs γ − 1 , γ − 1 , γ − 1 Viscous fluxes in terms of entropy variables g 1 = K 11 ( V ) ∂ V ∂x + K 12 ( V ) ∂ V g 2 = K 21 ( V ) ∂ V ∂x + K 22 ( V ) ∂ V ∂y , ∂y where � K 11 � K 12 K = ≥ 0 K 21 K 22 Deep Ray KEP-ES

Global entropy relation Integrating NSE against V ⊤ gives us � � � � � � d K · � ∇ V , � V ⊤ G ( U , n ) η = − q ( U , n ) − ∇ V + d t Ω ∂ Ω Ω ∂ Ω � � V ⊤ G ( U , n ) ≤ − q ( U , n ) + ∂ Ω ∂ Ω where G ( U , n ) = g 1 ( U ) n 1 + g 2 ( U ) n 2 and � ∂ x V � � � � ∈ R 8 ∈ R 4 × 2 ∇ V = obtained from ∇ V = ∂ x V , ∂ y V ∂ y V Deep Ray KEP-ES

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 } � � � � G T · n T | C i | d U i G e · n e i = − F ij + 2 − F ie + d t 2 j ∈ i T ∈ i e ∈ Γ i e ∈ Γ i Deep Ray KEP-ES

Choosing flux terms • Interior inviscid flux F ij is an entropy stable flux Deep Ray KEP-ES

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 x V T + K T y V T , G T α = K T α 1 · ∂ h α 2 · ∂ h α = 1 , 2 Deep Ray KEP-ES

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 x V T + K T y V T , G T α = K T α 1 · ∂ h α 2 · ∂ h α = 1 , 2 • Boundary inviscid flux F ie is chosen as � � U , U b , n e F ie = F ie   � � F ρ ρu b ie  i n e ) + u i F ρ  , F ρ n e 2 ( p b 1 F ie = ie = ie 2 F E i ie �� � � u b � � ρ | u | 2 γp ( p b − p ) u n e n e F E ie = + + 2 γ − 1 2 2 i i BC implemented weakly Deep Ray KEP-ES

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 x V T + K T y V T , G T α = K T α 1 · ∂ h α 2 · ∂ h α = 1 , 2 • Boundary inviscid flux F ie is chosen as � � U , U b , n e F ie = F ie • Boundary viscous flux G e = ( G e 1 , G e 2 ) is chosen as G e · n e = G T e · n e + C e Deep Ray KEP-ES

Discrete global entropy relation Pre-multiplying the scheme by V i and summing over all nodes. � � � � � � � � u b u b d ρs ρs η i | C i | ≤ − − n e − n e d t γ − 1 2 γ − 1 2 i j i e ∈ Γ � � � ( V b i + V b j ) ⊤ ⊤ G T e · n e + ( V i + V j ) C e + 2 2 e ∈ Γ which is consistent with � � � d V ⊤ G ( U , n ) ≤ − q ( U , n ) + η d t Ω ∂ Ω ∂ Ω Deep Ray KEP-ES