KE and entropy stable schemes for compressible flows Praveen. C - PowerPoint PPT Presentation

KE and entropy stable schemes for compressible flows Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/ praveen Indo-German Conference on Modeling, Simulation and Optimization 5-7 September, 2012 Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 1 / 39

Outline 1 KE and entropy stable flux functions ◮ KEP and entropy conserving flux function ◮ Scalar and matrix dissipation ◮ Numerical examples Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 2 / 39

Conservation laws: Navier-Stokes equation u = conserved variables ∂ u ∂ t + ∂ f ∂ x = ∂ g f = inviscid flux ∂ x g = viscous flux         ρ ρ ρ u m  =  ,  = p + ρ u 2 u = ρ u m f = p + um      E E ( E + p ) u ( E + p ) u   0 τ = 4 3 µ∂ u q = − κ∂ T  , g = τ ∂ x ,  ∂ x u τ − q µ = coeff. of dynamic viscosity , κ = coeff. of heat conduction γ − 1 + 1 p γ = C p 2 ρ u 2 , p = ρ RT , E = C v Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 3 / 39

Finite volume method Ω j x = 0 x = 1 n 0 1 j − 1 j j + 1 n − 1 j − 1 / 2 j + 1 / 2 u j = Cell average value in j ’th cell Ω j = [ x j − 1 2 , x j + 1 2 ] Semi-discrete FVM ∆ x d u j d t + f j + 1 2 − f j − 1 2 = g j + 1 2 − g j − 1 2 Godunov scheme: exact or approximate Riemann solver f j + 1 2 = f ( u j , u j +1 ) Centered approximation for g j + 1 2 Locally and globally conserves mass, momentum and energy Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 4 / 39

• Godunov/upwind schemes dissipate kinetic energy • Conservation alone does not guarantee numerical stability • Consistent evolution of kinetic energy: kinetic energy preserving (KEP) schemes • Entropy condition: second law of thermodynamics • Central schemes: stability via entropy condition and KEP • Kinetic energy preserving: Incompressible flows ◮ Harlow and Welch (1965): Staggered grids Ham (2002): Non-uniform grids Wesseling (1999): General structured grids Morinishi (1998): Fourth order scheme Verstappen et al. (2003): 2/4’th order symmetry preserving Mahesh et al. (2004): Unstructured grids ◮ Sanderse (2012): Energy conserving RK for INS Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 5 / 39

• Kinetic energy preserving: Compressible flows ◮ Jameson (2008): KEP scheme for compressible flow ◮ Subbareddy et al. (2009): Fully discrete implicit KEP scheme ◮ Shoeybi et al. (2010): KEP scheme, unstructured, IMEX-RK ◮ Morinishi (2010): Skew symmetric, staggered grid schemes • Entropy consistent/stable schemes: not fully conservative ◮ Gerritsen et al. (1996): Entropy stable scheme for exponential entropy ◮ Honein et al. (2004): Better entropy consistency using skew-symmetric form, internal energy equation • Entropy consistent/stable schemes: fully conservative ◮ Tadmor (1987): Entropy conservative flux ◮ Lefloch et al. (2002): Higher order entropy conservative schemes ◮ Roe (2006): Explicit entropy conservative flux for Euler equations ◮ Fjordholm et al. (2011): Entropy stable ENO schemes Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 6 / 39

Kinetic energy Kinetic energy per unit volume K = 1 2 ρ u 2 satisfies the following equation � ∂ u � 2 � � � � d p ∂ u ∂ x d x − 4 p ∂ u K d x = µ d x ≤ ∂ x d x d t 3 ∂ x Ω Ω Ω Ω Work done by pressure forces, absent in incompressible flows Irreversible destruction due to molecular diffusion Note: Convection contributes to only flux of KE across ∂ Ω Remark : Central average flux is not stable 2 = 1 f j + 1 2( f j + f j +1 ) Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 7 / 39

KE preserving FVM ∂ K − 1 2 u 2 ∂ρ ∂ t + u ∂ ( ρ u ) = ∂ t ∂ t � ∂ u � 2 − ∂ ∂ x ( p + ρ u 2 / 2 − 4 3 µ∂ u ∂ x ) u + p ∂ u ∂ x − 4 = 3 µ ∂ x Centered numerical flux       f ρ f ρ 0 f m p + uf ρ 2 = = ˜ , g j + 1 2 = τ f j + 1       f e f e u τ − q ˜ j + 1 j + 1 j + 1 2 2 2 where 2 = 1 2 = 4 3 µ u j +1 − u j 2 = − κ T j +1 − T j u j + 1 2( u j + u j +1 ) , τ j + 1 , q j + 1 ∆ x ∆ x Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 8 / 39

KE preserving FVM Discrete KE equation  � 2  �  ∆ u j + 1 ∆ u j + 1 ∆ x d K j 2 − 4 � �  ∆ x d t = 2 p j + 1 ˜ 3 µ 2 ∆ x ∆ x j j Jameson’s KEP flux   ρ u p + uf ρ f j + 1 2 =   Hf ρ j + 1 2 But there can be other choices, e.g., f ρ = ρ u , f e = ρ Hu , etc. p, f ρ , f e in any consistent manner. We We are free to choose ˜ determine all flux components (uniquely) from entropy condition. Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 9 / 39

Entropy condition Entropy-Entropy flux pair: U ( u ) , F ( u ) U ( u ) is strictly convex and U ′ ( u ) f ′ ( u ) = F ′ ( u ) Then, for hyperbolic problem (Euler equation) ∂ u ∂ t + ∂ f U ′ ( u ) ∂ u ∂ t + U ′ ( u ) f ′ ( u ) ∂ u ∂ x = 0 = ⇒ ∂ x = 0 ⇓ ∂ U ( u ) + ∂ F ( u ) = 0 ∂ t ∂ x For discontinuous solutions, only inequality ∂ U ( u ) + ∂ F ( u ) ≤ 0 ∂ t ∂ x � Ω U ( u ) d x for an isolated system decreases with time Second law of thermodynamics Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 10 / 39

Entropy conserving FVM Entropy variables v ( u ) = U ′ ( u ) U ( u ) is strictly convex = ⇒ u = u ( v ) Dual of the entropy flux F ( u ) ψ ( v ) = v · f ( u ( v )) − F ( u ( v )) Entropy conservative flux (Tadmor) ( v j +1 − v j ) · f j + 1 2 = ψ j +1 − ψ j � � ∆ x d u j ∆ x d U j v j · d t + f j + 1 2 − f j − 1 2 = 0 = ⇒ d t + F j + 1 2 − F j − 1 2 = 0 Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 11 / 39

Entropy conserving FVM Consistent entropy flux F j + 1 2 = v j + 1 2 · f j + 1 2 − ψ j + 1 2 In the scalar case 2 = ψ j +1 − ψ j ( v j +1 − v j ) · f j + 1 2 = ψ j +1 − ψ j = ⇒ f j + 1 v j +1 − v j For systems, we have an under-determined problem. Entropy conservative flux of Tadmor (1987) � 1 f j + 1 2 = f ( v j + 1 2 ( θ )) d θ, v j + 1 2 ( θ ) = v j + θ ( v j +1 − v j ) 0 Cannot be explicitly evaluated, requires numerical quadrature Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 12 / 39

Entropy condition for Euler equation Entropy-Entropy flux pair U = − ρ s F = − ρ us s = ln( p ) − ln( ρ γ ) + const. γ − 1 , γ − 1 , Entropy variables   γ − s γ − 1 − β u 2 1  , v = β = 2 RT , ψ = m = ρ u 2 β u  − 2 β Entropy conservative numerical flux for the Euler equations ( v j +1 − v j ) · f j + 1 2 = m j +1 − m j Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 13 / 39

Roe’s entropy conservative flux Parameter vector and logarithmic average     z 1 1 � ρ ϕ r − ϕ l ∆ ϕ  =  , z = z 2 u ϕ ( ϕ l , ϕ r ) = ˆ =   p ln ϕ r − ln ϕ l ∆ ln ϕ z 3 p Entropy conserving numerical flux   ρ ˜ ˜ u f ∗ = uf ρ p 1 + ˜ ˜   ˜ Hf ρ where u = z 2 p 1 = z 3 p 2 = γ + 1 z 3 ˆ + γ − 1 z 3 ρ = z 1 ˆ ˜ z 3 , ˜ , ˜ , ˜ z 1 z 1 2 γ ˆ z 1 2 γ z 1 � 1 � γ ˜ a 2 p 2 γ − 1 + 1 ˜ 2 ˜ u 2 a = ˜ , H = 2 ˜ ρ ˜ Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 14 / 39

New KEP and entropy conserving flux Jump in entropy variables in terms of ( ρ, u , β ) � � ∆ ρ 1 ∆ v 1 = ρ + − u 2 ∆ β − 2 u β ∆ u ( γ − 1)ˆ ˆ β ∆ v 2 = 2 β ∆ u + 2 u ∆ β ∆ v 3 = − 2∆ β Condition for entropy conservative flux: ∆ v · f = ∆( ρ u ) f ρ ∆ v 1 + f m ∆ v 2 + f e ∆ v 3 = ∆( ρ u ) = ρ ∆ u + u ∆ ρ KEP and Entropy conserving numerical flux   ρ u ˆ p = ρ f ∗ = p + uf ρ ˜    , ˜  � � 2 β f ρ + uf m 1 β − 1 2 u 2 2( γ − 1)ˆ Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 15 / 39

Uniqueness of KEP/entropy conservative flux Independent variables: ( ρ, u , p ) ρ u ˆ f ρ = , Depends on γ !!! γ p ˆ ρ ( γ − 1) − ( γ − 1) ρ ˆ p Independent variables: ( β, u , p ) � � γ − 1 f e = f ρ + uf m , 2 u 2 Flux is not consistent !!! 2( γ − 1)ˆ β Conjecture There is a unique two point flux which is kinetic energy preserving, i.e., f m = ˜ p + uf ρ and entropy conservative. Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 16 / 39

FVM for NS equation The semi-discrete finite volume method for NS equations using the centered KEP and entropy conservative flux is stable for the kinetic energy and entropy, i.e., � � 2 � � ∆ u j + 1 � ∆ u j + 1 � ∆ u j + 1 ∆ x d K j 2 − 4 � � � 2 2 2 d t = p j + 1 ˜ 3 µ ∆ x ≤ p j + 1 ˜ ∆ x ∆ x ∆ x ∆ x 2 j j j and � � 2 � � 2 � ∆ u j + 1 � ∆ T j + 1 8 µβ j + 1 ∆ x d U j κ � � d t = − 2 2 + 2 ∆ x ≤ 0 3 ∆ x RT j T j +1 ∆ x j j • Expect good stability property • No control of density/pressure Praveen. C (TIFR-CAM) KEP/Entropy stable schemes 7 Sep 2012 17 / 39