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

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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

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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

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Conservation laws: Navier-Stokes equation

u = conserved variables f = inviscid flux g = viscous flux ∂u ∂t + ∂f ∂x = ∂g ∂x u =   ρ ρu E   =   ρ m E   , f =   ρu p + ρu2 (E + p)u   =   m p + um (E + p)u   g =   τ uτ − q   , τ = 4 3µ∂u ∂x , q = −κ∂T ∂x µ = coeff. of dynamic viscosity, κ = coeff. of heat conduction p = ρRT, E = p γ − 1 + 1 2ρu2, γ = Cp Cv

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 3 / 39

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Finite volume method

1 n − 1 n j j − 1 j + 1 j − 1/2 j + 1/2 Ωj x = 0 x = 1

uj = Cell average value in j’th cell Ωj = [xj− 1

2, xj+ 1 2]

Semi-discrete FVM ∆x duj dt + fj+ 1

2 − fj− 1 2 = gj+ 1 2 − gj− 1 2

Godunov scheme: exact or approximate Riemann solver fj+ 1

2 = f(uj, uj+1)

Centered approximation for gj+ 1

2

Locally and globally conserves mass, momentum and energy

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 4 / 39

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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

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KEP/Entropy stable schemes 7 Sep 2012 5 / 39

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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

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KEP/Entropy stable schemes 7 Sep 2012 6 / 39

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Kinetic energy

Kinetic energy per unit volume K = 1 2ρu2 satisfies the following equation d dt

Ω

Kdx =

Ω

p∂u ∂x dx−4 3

Ω

µ ∂u ∂x 2 dx ≤

Ω

p∂u ∂x dx 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 fj+ 1

2 = 1

2(fj + fj+1)

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 7 / 39

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KE preserving FVM

∂K ∂t = −1 2u2∂ρ ∂t + u∂(ρu) ∂t = − ∂ ∂x (p + ρu2/2 − 4 3µ∂u ∂x )u+p∂u ∂x −4 3µ ∂u ∂x 2 Centered numerical flux fj+ 1

2 =

  f ρ f m f e  

j+ 1

2

=   f ρ ˜ p + uf ρ f e  

j+ 1

2

, gj+ 1

2 =

  τ ˜ uτ − q  

j+ 1

2

where uj+ 1

2 = 1

2(uj + uj+1), τj+ 1

2 = 4

3µuj+1 − uj ∆x , qj+ 1

2 = −κTj+1 − Tj

∆x

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 8 / 39

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KE preserving FVM

Discrete KE equation

j

∆x dKj dt =

j

 ∆uj+ 1

2

∆x ˜ pj+ 1

2 − 4

3µ

∆uj+ 1

2

∆x 2  ∆x Jameson’s KEP flux fj+ 1

2 =

  ρ u p + uf ρ Hf ρ  

j+ 1

2

But there can be other choices, e.g., f ρ = ρu, f e = ρHu, etc. We are free to choose ˜ p, f ρ, f e in any consistent manner. We determine all flux components (uniquely) from entropy condition.

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 9 / 39

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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 ∂x = 0 = ⇒ U′(u)∂u ∂t +U′(u)f′(u)∂u ∂x = 0 ⇓ ∂U(u) ∂t +∂F(u) ∂x = 0 For discontinuous solutions, only inequality ∂U(u) ∂t + ∂F(u) ∂x ≤ 0

Ω U(u)dx for an isolated system decreases with time

Second law of thermodynamics

Praveen. C (TIFR-CAM)

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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) (vj+1 − vj) · fj+ 1

2 = ψj+1 − ψj

vj ·

∆x duj

dt + fj+ 1

2 − fj− 1 2 = 0

=

⇒ ∆x dUj dt + Fj+ 1

2 − Fj− 1 2 = 0

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 11 / 39

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Entropy conserving FVM

Consistent entropy flux Fj+ 1

2 = vj+ 1 2 · fj+ 1 2 − ψj+ 1 2

In the scalar case (vj+1 − vj) · fj+ 1

2 = ψj+1 − ψj

= ⇒ fj+ 1

2 = ψj+1 − ψj

vj+1 − vj For systems, we have an under-determined problem. Entropy conservative flux of Tadmor (1987) fj+ 1

2 =

1 f(vj+ 1

2(θ))dθ,

vj+ 1

2(θ) = vj + θ(vj+1 − vj)

Cannot be explicitly evaluated, requires numerical quadrature

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 12 / 39

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Entropy condition for Euler equation

Entropy-Entropy flux pair U = − ρs γ − 1, F = − ρus γ − 1, s = ln(p) − ln(ργ) + const. Entropy variables v =  

γ−s γ−1 − βu2

2βu −2β   , β = 1 2RT , ψ = m = ρu Entropy conservative numerical flux for the Euler equations (vj+1 − vj) · fj+ 1

2 = mj+1 − mj

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 13 / 39

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Roe’s entropy conservative flux

Parameter vector and logarithmic average z =   z1 z2 z3   = ρ p   1 u p   , ˆ ϕ(ϕl, ϕr) = ϕr − ϕl ln ϕr − ln ϕl = ∆ϕ ∆ ln ϕ Entropy conserving numerical flux f∗ =   ˜ ρ˜ u ˜ p1 + ˜ uf ρ ˜ Hf ρ   where ˜ ρ = z1ˆ z3, ˜ u = z2 z1 , ˜ p1 = z3 z1 , ˜ p2 = γ + 1 2γ ˆ z3 ˆ z1 + γ − 1 2γ z3 z1 ˜ a = γ ˜ p2 ˜ ρ 1

2

, ˜ H = ˜ a2 γ − 1 + 1 2 ˜ u2

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 14 / 39

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New KEP and entropy conserving flux

Jump in entropy variables in terms of (ρ, u, β) ∆v1 = ∆ρ ˆ ρ +

1

(γ − 1)ˆ β − u2

∆β − 2uβ∆u

∆v2 = 2β∆u + 2u∆β ∆v3 = −2∆β Condition for entropy conservative flux: ∆v · f = ∆(ρu) f ρ∆v1 + f m∆v2 + f e∆v3 = ∆(ρu) = ρ∆u + u∆ρ KEP and Entropy conserving numerical flux f∗ =    ˆ ρu ˜ p + uf ρ

1

2(γ−1)ˆ β − 1 2u2

f ρ + uf m

   , ˜ p = ρ 2β

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 15 / 39

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Uniqueness of KEP/entropy conservative flux

Independent variables: (ρ, u, p) f ρ = ˆ ρu

γ (γ−1) − pˆ ρ (γ−1)ρˆ p

, Depends on γ !!! Independent variables: (β, u, p) f e =

γ

2(γ − 1)ˆ β − 1 2u2

f ρ + uf m,

Flux is not consistent !!!

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

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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.,

j

∆x dKj dt =

j
∆uj+ 1

2

∆x ˜ pj+ 1

2 − 4

3µ ∆uj+ 1

2

∆x 2 ∆x ≤

j

∆uj+ 1

2

∆x ˜ pj+ 1

2

∆x

and

j

∆x dUj dt = −

j
8µβj+ 1

2

3 ∆uj+ 1

2

∆x 2 + κ RTjTj+1 ∆Tj+ 1

2

∆x 2 ∆x ≤ 0

Expect good stability property
No control of density/pressure
Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 17 / 39

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DNS of NS for 1-D Sod test case

(ρ, u, p)L = (1.0, 0.0, 1.0) and (ρ, u, p)R = (0.125, 0.0, 0.1) Re = 2500 based on left sonic speed Density Entropy s

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 x Density KEP ROE−ES KEP−ES KEP−ES(D4) 0.35 0.4 −5 −4 −3 −2 −1 1 2 3 4 5 x 10

−3

x Entropy KEP ROE−ES KEP−ES KEP−ES(D4)

Solution at time t = 0.2 using 500 cells

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 18 / 39

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2-D Taylor-Green vortex: [0, 2π]2, 322 grid

u = − cos(x) sin(y) exp(−2µt) v = sin(x) cos(y) exp(−2µt) p = 500 − 1 4(cos(2x) + cos(2y)) exp(−4µt) Central KEP-EC flux

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 t KE Kinetic Energy vs Time (at different Reynold Numbers [RE]) Exact Solution Num Soln (RE = 1) Num Soln (RE = 10) Num Soln (RE = 100) Num Soln (RE = 1000) Num Soln (RE = infinity) 1 2 3 4 5 6 7 8 9 10 245.344 245.346 245.348 245.35 245.352 245.354 245.356 245.358 t Entropy Entropy vs Time (at different Reynold Numbers [RE]) Exact Solution Num Soln (RE = 1) Num Soln (RE = 10) Num Soln (RE = 100) Num Soln (RE = 1000) Num Soln (RE = infinity)

Total KE Total Entropy

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KEP/Entropy stable schemes 7 Sep 2012 19 / 39

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Scalar dissipation

Numerical flux with scalar dissipation fj+ 1

2 = f∗

j+ 1

2 − 1

2λj+ 1

2Dj+ 1 2,

D = [Dρ, Dm, De]⊤, λ ≥ 0 Choose Dρ = ∆ρ, Dm = ∆(ρu) = u∆ρ + ρ∆u Kinetic energy equation

j

dKj dt ∆x =

j

 ∆uj+ 1

2

∆x ˜ pj+ 1

2 − 4

3µ

∆uj+ 1

2

∆x 2  ∆x −1 2

j

λj+ 1

2ρj+ 1 2(∆uj+ 1 2)2

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 20 / 39

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Scalar dissipation

De = ∆E is not good choice. Choose De =

1

2(γ − 1)ˆ β + 1 2ujuj+1

∆ρ + ρ u∆u +

ρ 2(γ − 1)∆(1/β) Entropy equation

j

dUj dt ∆x =−

j
8µβj+ 1

2

3 ∆uj+ 1

2

∆x 2 + κ RTjTj+1 ∆Tj+ 1

2

∆x 2 ∆x −1 2

j

λj+ 1

2

(∆ρj+ 1

2 )2

ˆ ρj+ 1

2

+ 2ρj+ 1

2 βj+ 1 2 (∆uj+ 1 2 )2

+ ρj+ 1

2

(γ − 1)βjβj+1 (∆βj+ 1

2 )2

Control on all variables: ρ, u, T
Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 21 / 39

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Density-based dissipation

Dρ = ∆ρ, Dm = u∆ρ, De =

1

2(γ − 1)ˆ β + 1 2ujuj+1

∆ρ

Kinetic energy equation, no extra dissipation

j

dKj dt ∆x =

j

 ∆uj+ 1

2

∆x ˜ pj+ 1

2 − 4

3µ

∆uj+ 1

2

∆x 2  ∆x Entropy equation, dissipation based on density

j

dUj dt ∆x =−

j
8µβj+ 1

2

3 ∆uj+ 1

2

∆x 2 + κ RTjTj+1 ∆Tj+ 1

2

∆x 2 ∆x −1 2

j

λj+ 1

2

(∆ρj+ 1

2 )2

ˆ ρj+ 1

2

Control on all variables: ρ, u, T
Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 22 / 39

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Adaptive dissipation (JST)

Blend of second and fourth order dissipation ∆ρj+ 1

2 → ε(2)

j+ 1

2(ρj+1 − ρj) − ε(4)

j+ 1

2(ρj+2 − 3ρj+1 + 3ρj − ρj−1)

with similar expressions for ∆u and ∆T. νj = |pj−1 − 2pj + pj+1| |pj−1 + 2pj + pj+1|, νj+ 1

2 = max(νj, νj+1)

ε(2) and ε(4) are switching functions ε(2)

j+ 1

2 = min(1, κ(2)νj+ 1 2),

ε(4)

j+ 1

2 = max(0, κ(4)−ε(2)

j+ 1

2),

κ(2), κ(4) ≥ 0 Smooth flow: ε(2) = O(∆x2), ε(4) = O(1), D = O(∆x3) Near shocks: ε(2) = O(1), ε(4) = 0 and D = O(∆x).

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 23 / 39

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NS shock structure: KEP-ES + scalar dissipation

M1 = 1.5, γ = 5

3, Pr = 2 3, µ = 5 × 10−4(T/T1)0.8, κ(2) = 1 2, κ(4) = 1 25

−0.15 −0.14 −0.13 −0.12 −0.11 −0.1 0.5 0.6 0.7 0.8 0.9 1 1.1 x Velocity Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 x Pressure Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.045 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 x Shear stress Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 x Heat flux Exact Numerical

N = 50 cells, second and fourth order dissipation

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 24 / 39

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NS shock structure: KEP-ES + scalar dissipation

−0.15 −0.14 −0.13 −0.12 −0.11 −0.1 0.5 0.6 0.7 0.8 0.9 1 1.1 x Velocity Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 x Pressure Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.045 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 x Shear stress Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 x Heat flux Exact Numerical

N = 100 cells, second and fourth order dissipation

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 25 / 39

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NS shock structure: KEP-ES + scalar dissipation

−0.15 −0.14 −0.13 −0.12 −0.11 −0.1 0.5 0.6 0.7 0.8 0.9 1 1.1 x Velocity Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 x Pressure Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.045 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 x Shear stress Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 x Heat flux Exact Numerical

N = 200 cells, second and fourth order dissipation

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 26 / 39

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NS shock structure: KEP-ES + scalar dissipation

−0.15 −0.14 −0.13 −0.12 −0.11 −0.1 0.5 0.6 0.7 0.8 0.9 1 1.1 x Velocity Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 x Pressure Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.045 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 x Shear stress Exact Numerical −0.15 −0.14 −0.13 −0.12 −0.11 −0.1 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 x Heat flux Exact Numerical

N = 200 cells, fourth order dissipation, κ(4) = 1/200

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 27 / 39

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Matrix dissipation

Roe flux fj+ 1

2 = 1

2(fj + fj+1) − 1 2Rj+ 1

2|Λj+ 1 2|R−1

j+ 1

2∆uj+ 1 2

Eigenvectors and eigenvalues

R =   1 1 1 u − a u u + a H − ua

1 2u2

H + ua   , |Λ| = |Λ|Roe = diag [|u − a|, |u|, |u + a|]

Write ∆u in terms of ∆v R−1du = SR⊤dv, S = diag ρ 2γ , (γ − 1)ρ γ , ρ 2γ

Entropy-variable numerical flux (Tadmor)

fj+ 1

2 = f∗

j+ 1

2 − 1

2 Rj+ 1

2|Λj+ 1 2|Sj+ 1 2R⊤

j+ 1

2

Qj+ 1

2

≥0

∆vj+ 1

2

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KEP/Entropy stable schemes 7 Sep 2012 28 / 39

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Matrix dissipation

Entropy equation

∆x dUj dt + Fj+ 1

2 − Fj− 1 2 = −1

4

∆v⊤

j− 1

2 Qj− 1 2 ∆vj− 1 2 + ∆v⊤

j+ 1

2 Qj+ 1 2 ∆vj+ 1 2

≤ 0

with consistent entropy flux Fj+ 1

2 = vj+ 1 2 · f∗

j+ 1

2 − ψj+ 1 2 + 1

2¯ v⊤

j+ 1

2Qj+ 1 2∆vj+ 1 2

Stationary contact waves: Exactly resolved if aj+ 1

2 =

γ

2ˆ βj+ 1

2

, Hj+ 1

2 =

a2

j+ 1

2

γ − 1 + 1 2u2

j+ 1

2

= ⇒ Accurate computation of boundary layers and shear layers

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KEP/Entropy stable schemes 7 Sep 2012 29 / 39

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Matrix dissipation: Kinetic energy stability

Provided |λ1| = |λ3| = λ, λ ≥ 0 the KE equation is

j

dKj dt ∆x =

j

 ∆uj+ 1

2

∆x ˜ pj+ 1

2 − 4

3µ

∆uj+ 1

2

∆x 2  ∆x − 1 2γ

j

a2

j+ 1

2ρj+ 1 2βj+ 1 2λj+ 1 2(∆uj+ 1 2)2

Rusanov form |Λ| = |Λ|Rus = λI, λ = |u| + a KE stable form |Λ| = |Λ|KES = diag [ λ, |u|, λ ] , λ = max(|u−a|, |u+a|) = |u|+a

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Monotone resolution of shocks

Roe (2006) and Ismail and Roe (2009) |Λ| = |Λ|EC1 = diag [|u − a| + β|∆λ1|, |u|, |u + a| + β|∆λ3|] Mach = 1.5

0.2 0.4 0.6 0.8 1 0.8 1 1.2 1.4 1.6 1.8 2 x Density ROE−ES KEP−ES(AC) KEP−ES 0.2 0.4 0.6 0.8 1 0.8 1 1.2 1.4 1.6 1.8 2 x Density ROE−EC1 KEP−EC1(AC) KEP−EC1

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Monotone resolution of shocks

Mach = 4

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x Density ROE−ES KEP−ES(AC) KEP−ES 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x Density ROE−EC1 KEP−EC1(AC) KEP−EC1

Mach = 20

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 x Density ROE−ES KEP−ES(AC) KEP−ES 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 x Density ROE−EC1 KEP−EC1(AC) KEP−EC1

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SLIDE 33

Modified Sod problem

(ρ, u, p)L = (1.0, 0.75, 1.0) and (ρ, u, p)R = (0.125, 0.0, 0.1)

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Density ROE ROE−ES KEP−ES(AC) KEP−ES Exact 0.2 0.25 0.3 0.35 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 x Density ROE ROE−ES KEP−ES(AC) KEP−ES Exact 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Density ROE ROE−EC1 KEP−EC1(AC) KEP−EC1 Exact 0.2 0.25 0.3 0.35 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 x Density ROE ROE−EC1 KEP−EC1(AC) KEP−EC1 Exact

Density at time t = 0.2, N = 100 cells

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 33 / 39

SLIDE 34

Carbuncle and a hybrid scheme

|Λ| = |Λ|Hyb = (1 − φ)|Λ|Roe + φ|Λ|Rus, φ =

∆p

2p

1

2

Primal grid Median dual Voronoi dual

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 34 / 39

SLIDE 35

Carbuncle and a hybrid scheme

Density Pressure Supersonic cylinder, Mach=2: KEP-EC1 flux, Voronoi dual grid Density Pressure Supersonic cylinder, Mach=20: KEP-EC1 flux, Median dual grid

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 35 / 39

SLIDE 36

Carbuncle and a hybrid scheme

KEP-EC1 KEP-ES (KES) KEP-ES (Rus) KEP-ES (Hyb) Supersonic cylinder, Mach=20: Voronoi dual grid, density contours

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 36 / 39

SLIDE 37

Laminar boundary layer: Re = 105, M = 0.1

Inlet portion Flat plate

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7

η u/u∞

Blasius KEP−ES(Hyb) KEP−ES(Rus) 0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7

η v*sqrt(2*Rex)/u∞

Blasius KEP−ES(Hyb) KEP−ES(Rus)

Streamwise velocity Vertical velocity

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 37 / 39

SLIDE 38

Step in wind tunnel: M = 3

(a) (b) Mach number contours, 50 equally spaced contours between 0 and 4.8, (a) KEP-EC1 flux, (b) KEP-ES(Hyb) flux

Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 38 / 39

SLIDE 39

Summary

Novel kinetic energy preserving and entropy conservative flux

Explicit and cheap flux
Semi-discrete FVM: consistent with KE and entropy equation
Scalar and matrix dissipation operators
Hybrid flux for hypersonic flows
Useful for DNS of compressible turbulence ?
Praveen. C (TIFR-CAM)

KEP/Entropy stable schemes 7 Sep 2012 39 / 39