Modern Physics Letters B, Vol. 23, No. 3 (2009) 313−316 World Scientific Publishing Company
A LATTICE BOLTZMANN METHOD-BASED FLUX SOLVER AND ITS APPLICATION TO SOLVE SHOCK TUBE PROBLEM
C. Z. JI*, C. SHU†,‡ and N. ZHAO* *
Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Nanjing Jiangsu, 210016, China † Departmen of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Received 1 June 2008 This paper presents an approach, which combines the conventional finite volume method (FVM) with the lattice Boltzmann Method (LBM), to simulate compressible flows. Similar to the Godunov scheme, in the present approach, LBM is used to evaluate the flux at the interface for local Riemann problem when solving Euler/Navier-Stokes (N-S) equations by FVM. Two kinds of popular compressible Lattice Boltzmann models are applied in the new scheme, and some numerical experiments are performed to validate the proposed approach. From the sharper shock profile and higher computational efficiency, numerical results demonstrate that the proposed scheme is superior to the conventional Godunov scheme. It is expected that the proposed scheme has a potential to become an efficient flux solver in solving compressible Euler/N-S equations. Keywords: Finite volume method; Lattice Boltzmann method; Riemann problem; flux evaluation.
1. Introduction The finite volume method (FVM), employed firstly by McDonald1 for the simulation of inviscid flows, is the most popular numerical discretization scheme in the computational fluid dynamics. FVM needs to evaluate fluxes at the cell interface. Among various inviscid flux solvers, the Godunov scheme2 is a pioneer work, which uses analytical solution of the local Riemann problem at the interface. On the other hand, we have to indicate that although the exact solution of Riemann problem exists, it is appeared in the nonlinear form. One has to apply an iterative procedure to get the solution at the interface for the application of Godunov scheme. The process will take a considerable time. To reduce the computational effort for the solution of Riemann problem, many researchers proposed approximate Riemann solvers. Among notable solvers, the solver proposed by Roe3 is the most popular one and is widely used in practice. Lattice Boltzmann method (LBM) 4 has been developed into a new tool for simulating fluid flows and modeling complicated physical phenomena in recent years. Unlike conventional Navier-Stokes solvers, it considers flows to be composed of a collection of ‡
Corresponding author, e-mail:
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pseudo-particles that are represented by a set of density distribution functions. These fluid particles hold collision and propagation over a discrete lattice mesh. Several lattice Boltzmann models4,5 for incompressible flows have been proposed and successfully applied to solve many real flow problems. However, these models are not suitable for compressible flows as they are limited to the flow with small Mach number. Some attempts have been made to develop compressible lattice Boltzmann model. Among the existing lattice Boltzmann models6,7 for compressible flows, the one proposed by Kataoka and Tsutahara7 has a simple and rigorous theoretical background. It takes flexible ratio of specific heat and is superior in computational efficiency. However, it encounters numerical instability which leads to difficulty of its application to high-Machnumber flows. Recently, Qu and Shu8 proposed a lattice Boltzmann model for inviscid compressible flows at high Mach number. It is different from other models in the way that the conventional Maxwellian distribution function is replaced by a circular function. Numerical experiments indicate that supersonic flows with weak and strong shock waves can be simulated successfully by this model. To sum up, we have to indicate that all the current lattice Boltzmann models for compressible flows are quite complicated. Their application may not be as efficient as conventional Euler/N-S solvers. On the other hand, we may be able to use the explicit nature of LBM to find exact solution of Riemann problem so that the flux at the interface can be simply evaluated. This motivates the present work. The flux evaluation solver developed in this paper is similar to Godunov scheme. The present paper is arranged in the following order. In the second section, FV-LBM is described. A numerical example and analysis are shown in the third section. In last section, some concluding remarks are given. 2. Finite Volume-Lattice Boltzmann Method (FV-LBM) for Compressible Flows 2.1. Governing equations In FVM, a domain is discretized into many small control volumes which are usually called cells. The variation of flow states in every cell is expressed as the sum of fluxes across all the interfaces of the cell and it can be expressed as: K ∂Q Ω = − ∑ ( Fi + Di ) si . ∂t i
(1)
where Q is the vector of conserved variables (density ρ, moment ρv and energy ρE), Ω is the volume of control cell, K is the number of interfaces of the control volume, Fi and Di are inviscid flux and viscous flux (only for viscous flows) across the i-th interface, and si is the area of the interface. On the other hand, the flow field can also be simulated by LBM. The Lattice Boltzmann equation with BGK model can be written as,
(
)
fi r + eiδ t , t + δ t − fi ( r , t ) = −
1
eq fi ( r , t ) − fi ( r , t ) ) ( τ
( i = 0,1, ..., M ) .
(2)
A LBM-based Flux Solver and Its Application
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where fi is the density distribution function, which depends on position r in the physical space, the particle discrete velocity ei and time t, fieq is its corresponding equilibrium state, which depends on the local macroscopic variables ρ, p and u; τ is the single relaxation parameter related to the hydrodynamic viscosity, δt is the time step and M is the number of discrete particle velocities. The macroscopic density, velocity and internal energy can be obtained by mass, momentum and energy conservation forms. 2.2. LBM-based flux solver When FVM is applied to solve compressible N-S/Euler equations, the key issue is to evaluate the inviscid flux Fi at the interface:
(
)
F ≈ Fˆ Ql , Qr .
(3)
where Ql and Qr are flow states on the two sides of the interface. To evaluate the flux F by LBM, we apply LBM to the local Riemann problem, as shown in Fig. 1. In this work, we only focus on the Euler equations. This means that the viscous effect is not considered. As we know, the collision term in LBM corresponds to the viscous term in N-S equations. Therefore, for inviscid flows, equation (2) can be reduced to
(
)
fi r + eiδ t , t + δ t = fi ( r , t )
(i = 0,1,..., M )
(4)
To apply equation (4) to the local Riemann problem, we set the initial distribution function fi(r, t) to be equilibrium distribution function fieq(r, t), which can be computed by using macroscopic variables (ρL, pL, uL) and (ρR, pR, uR) with formulations given in [7-8]. Then, we can get the distribution function at the interface, which further gives macroscopic variables ρ, p and u from conservation laws of mass, momentum and energy. Substituting ρ, p, u into the flux expression can directly give the flux at the interface. Obviously, this process is similar to the Godunov scheme. The difference is that ρ, p, u are given by LBM in this work. ( ρL , p L , uL )
( ρ R , p R ,u R )
Interface
Fig. 1. Configuration a Riemann problem.
3. Numerical Results and Discussion To validate the proposed method, the shock tube problem is chosen for simulation. The initial condition is set as, ( ρ L , u L , pL ) = (1.0, 0.0,1.0), − 0.5 ≤ x < 0, ( ρ R , uR , pR ) = (0.125, 0.0, 0.1), 0 < x ≤ 0.5 (5) The spatial mesh spacing is taken as ∆x = 1/100 and time step is chosen as ∆t = 0.001. Two LB models [7-8] are all tested as flux solver in FVM respectively and the obtained
316 C. Z. Ji, C. Shu & N. Zhao
1 Exact solution FV-LBM Godunov-FVM
0.8
Density
Density
numerical results are the same. As shown in Fig. 2, our scheme gives a steeper shock profile as compared to the results obtained from Godunov scheme. Figure 2(a) shows the computed density profile at t = 0.22. Figure 2(b) enlarges the part containing the shock wave for a closing view.
0.4 Exact solution FV-LBM Godunov-FVM
0.35 0.3
0.6
0.25 0.4
0.2 0.15
0.2 -0.4
-0.2
X
0
(a)
0.2
0.4
0.15
0.2
0.25
0.3
0.35
0.4
X
(b)
Fig. 2. Comparison with Godunov scheme and exact solution.
4. Conclusions This paper presents a LBM-based flux evaluation solver for the solution of Euler equations. In this method, LBM is used to solve the local Riemann problem across the interface directly. The proposed method is well validated by its application to solve a shock tube problem. Numerical results showed that shock wave, contact discontinuity and rarefaction wave are all well captured, and the present results are closer to the exact solution than those obtained by the conventional Godunov scheme. As the solution at the interface is explicitly given in the present method, it is easier to implement than the Godunov scheme, and takes less time to evaluate the flux. Acknowledgments This work was supported by the National Natural Science Foundation of China (10728206). References 1. 2. 3. 4. 5. 6. 7. 8.
J. Blazek, Computational Fluid Dynamics: Principles and Applications (Elsevier, 2001). S. Godunov, Math. Sbornik, 47, 357−393 (1959). P. Roe, Journal of Computational Physics, 43, 357−372 (1981). S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond (Oxford, 2001). F. Alexander, S. Chen, J. Stering, Phys. Rev. E, 47, R2249−R2252 (1993). C. Sun, Journal of Computational Physics, 161, 70−84 (2000). T. Kataoka, M. Tsutahara, Phys. Rev. E, 69, 056702 (2004). K. Qu, C. Shu and Y. T. Chew, Phys. Rev. E, 75, 036706 (2007).
