In this paper, a third-order exponential time differencing scheme, named ETDRK3, was investigated for large time stepping in the computation of hypersonic non-equilibrium flow. The second-order Harten-TVD scheme was used for the spatial discretization. The efficient implementation of the scheme with diagonalization of Jacobin matrix was established and carried out for the semi-cylindrical around flow. Current observations showed that the numerical results were in good agreement with those obtained by the classical explicit three-stage Runge-Kutta scheme (RK3) and implicit LU scheme. Efficiency assessments promised the effectiveness of the ETDRK3 scheme. The rationality of the application of this scheme was proved by its preferable accuracy and efficiency.
With the development of hypersonic vehicle, the numerical simulation of hypersonic flow field has been the frontier of aerodynamics research, of which hypersonic non equilibrium flow is one of the typical representatives. In recent years, significant development of high accuracy spatial discretization schemes has been made for computational fluid dynamics (CFD). Although these spatial schemes perform well in the simulation of hypersonic flow field and can be implemented efficiently by parallel techniques, their computational expense on time marching direction is still high. For time marching computation, the explicit schemes are widely used because of their ease of application. However, in the presence of highly stretched grids, the Courant-Friedrichs-Lewy (CFL) condition leads to the usage of a tiny time-step, consuming a large CPU time. The implicit scheme is theoretically proved insensitive to the CFL condition. Nevertheless, the usage of implicit scheme needs to solve a large linear system at each iteration step, making the computation expensive. Recently, mathematicians have been developing an explicit method with large time step known as exponential time differencing [
The governing equation is the two-dimensional N-S equation with chemical reaction source term in general coordinates
∂ Q ∂ t + ∂ F ∂ ξ + ∂ G ∂ η = ∂ F v ∂ ξ + ∂ G v ∂ η + J ⋅ S , (1)
where,
Q = ( ρ 1 , ⋯ , ρ n s , ρ , ρ u , ρ v , e ) T F = ( ρ u 1 , ⋯ , ρ u n s , ρ u , ρ u 2 + p , ρ u v , ( e + p ) u ) T G = ( ρ v 1 , ⋯ , ρ v n s , ρ v , ρ v u , ρ v 2 + p , ( e + p ) v ) T F v = ( ρ D 1 ∂ Y 1 ∂ x , ⋯ , ρ D s ∂ Y s ∂ x , 0 , τ x x , τ x y , u τ x x + v τ x y + q x ) T G v = ( ρ D 1 ∂ Y 1 ∂ y , ⋯ , ρ D s ∂ Y s ∂ y , 0 , τ y x , τ y y , u τ y x + v τ y y + q y ) T S = ( ω ˙ 1 , ⋯ , ω ˙ n s , 0 , 0 , 0 , 0 , 0 ) T (2)
in the proceeding expressions, ρ k , k = 1 , 2 , ... , n s is the density of the species.
ρ = ∑ k = 1 n s ρ k is the total density, u , v are the speed in the general direction, e is
the energy, P is the pressure, D s denotes the diffusion coefficient and Y s is the mass fraction of specie s .
The quantity J is the coordinate transformation matrix.
The chemical source terms w s represent the production of species from finite rate chemical reactions [
The start point of our derivation is the spatial discretization. In the study, we used the second-order Harten-Yee TVD scheme [
d Q d t = R ( Q ) , (3)
with Q = Q ( x , t ) is the exact solution, R is the right hand term obtained by the spatial discretization above.
Splitting the right hand term of Equation (3) into
R = ( α A n + β B n ) Q + N ( Q , t ) , (4)
where, α = β = 1 are adjustable parameters and
A n = ∂ ξ ∂ x A | Q i j n + ∂ ξ ∂ y B | Q i j n B n = ∂ η ∂ x A | Q i j n + ∂ η ∂ y B | Q i j n . (5)
In Equation (5), the terms A = ∂ F ∂ Q , B = ∂ G ∂ Q denote the Jacobin matrix of the
non-viscous flux F and G , the term N ( Q , t ) = R − ( α A n + β B n ) Q is a non-linear remainder. For notation simplification, we note K = α A n + β B n and Q n = Q ( x , t n ) . Then the Equation (3) can be written as
d Q d t = R n + K ( Q ( t ) − Q n ) + N ( Q ( t ) ) (6)
Multiplying the both sides with e − K t and then integrating over a single time step [ t n , t n + h ] , we can obtain the basic expression of exponential time differencing method [
Q n + 1 = e K h Q n + e K h ∫ 0 h e − K τ N ( Q ( t n + τ ) , t n + τ ) d τ . (7)
Define a function φ ( z ) = ∫ 0 1 e z ( 1 − s ) d s = e z − 1 z , and then the Equation (7) can be
expressed as
Q n + 1 = Q n + h K φ ( h K ) Q n + h ∫ 0 1 e h K ( 1 − s ) N ( Q ) d s . (8)
Various ETD schemes come from the approximation of the integral in (8) [
k 1 = ϕ ( 1 2 h K ) N ( U n ) k 2 = ϕ ( 1 2 h K ) N ( U n + 4 3 h k 1 ) U n + 1 = U n + h 16 ( 13 k 1 + 3 k 2 ) . (9)
In the practical implementation, diagonalize K in
K = R Λ R − 1 , (10)
In the expression (10), R and R − 1 are the right and left eigenvector matrix, Λ is the characteristic matrix with the diagonal elements
λ 1 = θ − k c , λ 2 = λ 3 = θ , λ 4 = θ + k c , (11)
where
k 1 = α ∂ ξ ∂ x + β ∂ η ∂ x , k 2 = α ∂ ξ ∂ y + β ∂ η ∂ y , k = k 1 2 + k 2 2 , k ˜ 1 = k 1 k , k ˜ 2 = k 2 k , θ = k 1 u + k 2 v , θ ˜ = θ k = k ˜ 1 u + k ˜ 2 v . (12)
Founding that for a related exponential function φ we have
φ ( R Λ R − 1 ) = R φ ( Λ ) R − 1 = R d i a g [ φ ( λ 1 ) , φ ( λ 2 ) , φ ( λ 3 ) , φ ( λ 4 ) ] R − 1 . (13)
The evaluation of the exponential related function of Jacobin φ can be converted to the evaluation of the related exponential function of the diagonal elements of Λ , the efficiency of implementation will be much improved.
The first comparison parameter used in this paper is the CFL number defined as
C F L = | λ max | Δ t Δ x , (14)
In expression (13) λ max is the maximum of the eigenvalues. Greater CFL number indicates larger time step and better efficiency. Classical explicit schemes such as the various Runge-Kutta schemes are usually inefficient to solve complex flow problems because of the restriction of CFL number. However, the exponential time differencing scheme do not suffer from this limitation and can run a larger CFL number. To assess the efficiency of the ETDRK3 scheme developed, the baseline solutions computed by the third-order TVD Runge-Kutta scheme noted as RK3 scheme and implicit LU scheme [
In order to assess the accuracy of the scheme, a first norm residual defined as the maximum of the pressure difference between current time step and previous time step on the mesh was used and expressed as
R L 1 = max | P n + 1 − P n | , (15)
A residual of quantity 10−3 can be regarded as the convergence condition.
The test case is a hypersonic two-dimensional cylindrical round flow problem at Mach number 20 which represents a typical external flow application. The free stream conditions are given as follow:
ρ ∞ = 1.0269 × 10 − 3 kg/m 3 ,
T ∞ = 270.65 K ,
Mass fraction N 2 = 0.7655
Mass fraction O 2 = 0.2345
The boundary conditions used in the calculations were as follows: Along the inflow plane, free stream values are maintained. Along the outflow plane, values are obtained by extrapolation. A constant temperature of 1000 K was maintained on the body surface that was assumed to be non-catalytic. Nonslip and zero pressure gradient conditions were enforced. The 180 × 200 grid given by the algebraic generation method is shown in
Figures 3-6 represent separately the pressure distribution, the temperature distribution, the mass fraction of N2 and the mass fraction of O2 computed by ETDRK3 (CFL0.7), RK3 (CFL0.3) and LU (CFL0.7) at 45,000 steps. We can see a clear shock wave in the three figures. After the shock wave the pressure and temperature augmented, chemical reaction happened significantly which leads
great change of species. The results above coincide with the theory [
After the accuracy assessment above, the efficiency of the three schemes were compared by counting the Wall time at 45,000 iterations and the CPU time for a i7-2600 CPU 3.40 GHz.
In this paper, we used the ETDRK3 scheme in the computation of hypersonic non-equilibrium flow. This scheme was compared with explicit RK3 scheme and implicit LU scheme. The numerical results were parallel with that of theory. The convergence comparison revealed that the ETDRK3 scheme could achieve the
| Scheme name | CFL number, wall time and CPU time for 45,000 steps | ||
|---|---|---|---|
| CFL | Wall time/s | CPU time/min | |
| RK3 | 0.3 | 5.08 | 922.5 |
| LU | 0.7 | 11.3 | 577.8 |
| ETDRK3 | 0.7 | 12.3 | 319.9 |
a. wall time and CPU time of the three schemes for 45000 steps.
same accuracy as LU scheme. Efficiency assessment showed that ETDRK3 scheme cost much less CPU time than the two other schemes while outperforms in term of wall time. We can conclude from the assessments that the ETDRK3 scheme is reasonable an alternative time marching scheme in hypersonic chemical non-equilibrium flow. Further study will focus on larger CFL number for ETD schemes. The application of ETD schemes on three-dimensional problems will also be investigated in the future work.
This work was supported in part by National Nature Science Foundation of China (NSFC 91530325).
Dai, S.R. and Wu, S.P. (2018) Explicit Large Time Stepping with Third-Order Exponential Time Differencing Scheme for Hypersonic Chemical Non-Equilibrium Flow. Journal of Applied Mathematics and Physics, 6, 174-182. https://doi.org/10.4236/jamp.2018.61017