An implicit GKS for hypersonic equilibrium air flows Weidong Li,lwd_1982.4.8@163.com Contribution The implicit LU-SGS scheme The gas kinetic BGK scheme has strong and solid gas kinetic foundation and demonstrated outstanding performance on compressible flow simulations. Using the equilibrium gas model to simulate hypersonic and high temperature gas flows is a low cost way to know about the "real gas effects". In this work, We extend the gas kinetic BGK scheme to simulate inviscous high temperature real gas effects. Through an effective γ method,which is obtained through polynomial curve fits of thermodynamic properties of high temperature equilibrium air, the effects of excitation of vibration energy, dissociation and ionization in high temperature flows on the internal degree of freedom of gases can be simulated by the present gas kinetic scheme(GKS). Moreover, to accelerate the convergence of the present scheme, we also developed a matrix-free LU-SGS implicit time marching scheme for our scheme. The present method can be applicable to efficiently simulate hypersonic thermo-chemical equilibrium air flows. Taking moments of the Botlzmann-BGK equation and integrating it on a controlling volume Ωi : The Boltzmann-BGK equation The equilibrium Maxwellian distribution g is defined as e 1 f (x, y, t, u, v, ξ) = τ −λ[(u−U )2 +(v−V )2 +ξ 2 ] 5−3γ γ−1 ∫ t ′ ′ ′ g(x , y , t , u, v)e −(t−t′ )/τ dt′ + e−t/τ f0 (x − ut, y − vt), 0 where x′ = x − u(t − t′ ) and y ′ = y − v(t − t′ ). To obtain an implicit scheme,we need a semi∑N F 1 ∫ △t ∫ ∂Qi discrete finite volume scheme: Ωi ∂t + Ri (Q) = 0, where Ri = j=1 △t [ 0 Ψn un f dΞ]j Sj and △t = min(△t1 , ..., △tncell ). Linearizing Ri (Q) and rearranging the equations,we have: (L + D + ∑ ∑ ∂F(Qj ) ∂F(Qj ) 1 1 n n U) △ Q = −R (Q), with L = 2 j∈L(i) [ ∂Qj − (Λc )ij I]Sj ,U = 2 j∈U (i) [ ∂Qj − (Λc )ij I]Sj ∑ Ωi 1 and D = [ (△t)imp + 2 j∈N (i) (Λc )ij Sj ]I. Finally,the implicit system can be solved in the LU-SGS i procedure: ∑ 1 ∗ −1 n ∗ ∗ △Q = D {−R − [△F − (Λ ) △ Q c ij i j j ]Sj } 2 j∈L(i) −1 ∑ D n ∗ n n [△F − (Λ ) △ Q ]S △Q = Q − c ij j j j i i 2 j∈U (i) ∂F ∂Q △Q, which is matrix-free. Results ∂f ∂f ∂f g−f +u +v = , ∂t ∂x ∂y τ λ g = ρ( ) π The distribution function at the cell interface can be constructed by the analytical integral solution of the Botlzmann-BGK equation as where △F ≈ Gas Kinetic Model K+2 2 NF ∫ ∑ ∂Qi 1 { Ψfj [(u⃗i + v⃗j)·n⃗j ]dΞ}Sj = 0, + ∂t Ωi j=1 Several inviscous hypersonic equilibrium airflow cases are simulated and the simulation condition is fixed as: p∞ = 101325.0P a, T∞ = 1000K, M∞ = 8.0, AoA = 0o , and pref = 101330P a , ρ 2p . where K = + 1. and λ = f and g at any point in space and time satisfy ∫ Ψ(f − g)dΞ = 0, ∫ and (ρ, ρU, ρV, E)T = with Case1: Hypersonic equilibrium air flow around a 10% circular arc airfoil. Ψf dΞ, u2 + v 2 + ξ 2 T Ψ = (1, u, v, ) , 2 and dΞ = dudvdξ1 dξ2 ...dξK . With the Chapman-Enskog expansion { g, Euler equation f= ∂g ∂g ∂g g − τ ( ∂t + u ∂x + v ∂y ), NS equation Case2: Supersonic equilibrium air flow past a double-wedge For the 2D compressible Euler equations ∫ ∂g ∂g ∂g Ψ( +u + v )dΞ = 0, ∂t ∂x ∂y namely, ∂Q ∂F ∂G + + = 0, ∂t ∂x ∂y with an equation of state p = p(ρ, e). Perfect gas: p = (γ − 1)ρe, where γ = 1.4. Equilibrium gas: p = (γ − 1)ρe, where γ is variable and a function of any two independent thermodynamic variables obtained by approximate curve fits developed by Srinivasan et al. [1]. Case3: Supersonic equilibrium air flow over a diamond-shaped airfoil References [1] Srinivasan S, Tannehill JC, Weilmuenster KJ. Simplified Curve Fits for the Thermodynamic Properties of Equilibrium Air. In Tech. Rep.; ISU-ERI-Ames-86041; 1986. [2] Weidong Li, M. Kaneda, K. Suga. An Implicit Gas Kinetic BGK Scheme for High Temperature Equilibrium Gas Flows on Unstructured Meshes In Computers & Fluids,93(2014),100-106.

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