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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.
1/--pages
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