Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows

doi:10.1016/j.jcp.2016.05.018

Journal of Computational Physics

Volume 319, 15 August 2016, Pages 129-144

https://doi.org/10.1016/j.jcp.2016.05.018 Get rights and content

Highlights

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It is the first time to extend the sphere function-based GKS for simulation of 3D viscous flows.
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D3Q8 model is firstly presented.
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The complicated sphere function-based GKS is replaced by a simple solution process.
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3D incompressible and compressible viscous flows can be accurately simulated by present scheme.

Abstract

The sphere function-based gas kinetic scheme (GKS), which was presented by Shu and his coworkers [23] for simulation of inviscid compressible flows, is extended to simulate 3D viscous incompressible and compressible flows in this work. Firstly, we use certain discrete points to represent the spherical surface in the phase velocity space. Then, integrals along the spherical surface for conservation forms of moments, which are needed to recover 3D Navier–Stokes equations, are approximated by integral quadrature. The basic requirement is that these conservation forms of moments can be exactly satisfied by weighted summation of distribution functions at discrete points. It was found that the integral quadrature by eight discrete points on the spherical surface, which forms the D3Q8 discrete velocity model, can exactly match the integral. In this way, the conservative variables and numerical fluxes can be computed by weighted summation of distribution functions at eight discrete points. That is, the application of complicated formulations resultant from integrals can be replaced by a simple solution process. Several numerical examples including laminar flat plate boundary layer, 3D lid-driven cavity flow, steady flow through a 90° bending square duct, transonic flow around DPW-W1 wing and supersonic flow around NACA0012 airfoil are chosen to validate the proposed scheme. Numerical results demonstrate that the present scheme can provide reasonable numerical results for 3D viscous flows.

Introduction

In the computational fluid dynamics (CFD) community, the gas kinetic scheme (GKS) [1], [2], [3], [4], [5], [6], [7], [8] has been developed into a powerful tool for simulation of fluid flows. Like the popular Riemann solvers [9], [10], [11], [12], [13], [14], the GKS is intended to evaluate the numerical fluxes at the cell interface. As a result, the GKS can be easily embedded in the existing Riemann solver-based CFD codes. On the other hand, different from the conventional Riemann solvers, the GKS treats the inviscid and viscous fluxes as a single entity instead of using a Riemann solver to evaluate the inviscid fluxes and a smooth function approximation to compute the viscous fluxes. In addition, the GKS has a clear physical process (streaming and collision) for evaluation of conservative variables and numerical fluxes, and the scheme can be well applied to both incompressible and compressible flows without unphysical phenomenon such as “Carbuncle phenomenon”. This makes the GKS be more appealing than conventional Riemann solvers.

In the GKS, the local solution of continuous Boltzmann equation with the Maxwellian distribution function is usually used to calculate the numerical fluxes at the cell interface [15], [16], [17], [18], [19], [20], [21], [22]. Due to complexity of the Maxwellian distribution function, these GKSs are usually more complicated and less efficient than the conventional Riemann solvers [9], [10], [11], [12], [13], [14]. In particular, for the 3D viscous flows, the derivation and implementation of the GKS can be very tedious [20], [21]. To improve the computational efficiency of the GKS and simplify its derivation, Shu and his coworkers [23] proposed the sphere function-based GKS for simulation of 3D inviscid compressible flows. In the method, the Maxwellian distribution function is first simplified into the sphere function with assumption that all the particles are concentrated on a spherical surface. Thus, the integrals in the infinity domain of phase velocity space for the Maxwellian function can be reduced to the surface integrals along the spherical surface for the sphere function. As shown in [23], since the expressions of the sphere function-based GKS are relatively simple, it is easier to be implemented than the corresponding Maxwellian distribution function-based GKS. However, the presented sphere function-based GKS can only be used to simulate inviscid compressible flows in the previous work. In theory, we can follow the idea of developing the circular function-based GKS for viscous flows [8] to extend the sphere function-based GKS to viscous flow regime. However, the resultant formulations could be very complicated, which may not be easy for the application by new users. To develop a simple and efficient sphere function-based GKS for simulation of 3D viscous flows motivates the present work.

For the sphere function-based GKS [23], the integrals for conservation forms of moments are along the spherical surface. In addition, the Maxwellian distribution function is reduced to a simple form, which only depends on macroscopic flow variables on the spherical surface. These simplifications enable us to represent the spherical surface in the phase velocity space by using certain discrete points. That is, the integrals along the spherical surface for conservation forms of moments can be approximated by integral quadrature. In this process, the basic requirement is that the conservation forms of moments for the sphere function-based GKS, which are needed to recover 3D Navier–Stokes equations, can be exactly satisfied by weighted summation of distribution functions at discrete points. It was found in this work that, the model with 8 discrete points on the spherical surface, which forms the D3Q8 discrete velocity model, can exactly match the integrals. In this way, the macroscopic flow variables and numerical fluxes can be computed by weighted summations of distribution functions at 8 discrete points, and the application of complicated formulations resultant from integrals is avoided.

The rest of the paper is organized as follows. The development of D3Q8 model from sphere function-based GKS is presented in Section 2. In Section 3, the developed D3Q8 model is utilized to solve the 3D Navier–Stokes equations. Numerical examples and discussions are arranged in Section 4. In the last section (Section 5), some concluding remarks are given.

Section snippets

Sphere function-based GKS and 7 conservation forms of moments

The existing sphere function-based GKS [23] is developed to simulate the 3D inviscid compressible flows. Thus, only 5 conservation forms of moments for recovering the Euler equations are used. To develop discrete velocity model for simulation of viscous flows, the 7 conservation forms of moments for recovering the Navier–Stokes equations are needed. So, at beginning, the sphere function-based GKS and its 7 conservation forms of moments for recovering the Navier–Stokes equations are introduced.

3D Navier–Stokes equations discretized by finite volume method

For the convenience of derivation, we first introduce the local coordinate system. In the local coordinate system, direction 1 is taken as the normal direction pointing always outwards of the cell interface, directions 2 and 3 are chosen as two tangential directions of the cell interface, which are mutually orthogonal. The discrete form of Navier–Stokes equations given by finite volume method (FVM) can be written as [23] $\frac{d W_{I}}{d t} = - \frac{1}{Ω_{I}} \sum_{i = 1}^{N_{f}} F_{i} S_{i}$ where I is the index of a control volume, $Ω_{I}$ and $N_{f}$

Numerical examples

In this section, the developed discrete GKS is validated by simulating some 3D incompressible and compressible viscous flows. For temporal discretization of Eq. (10), the three-stages Runge–Kutta method [13] is used. In order to get a stable numerical solution for compressible flows, some numerical dissipation is needed to be incorporated into the solver. A simple way to incorporate numerical dissipation into the present scheme is to modify the dimensionless collision time given by Eq. (16) as $τ_{0}$

Conclusions

In this work, the sphere function-based GKS proposed in our previous work for simulation of inviscid compressible flows is extended to solve the viscous incompressible and compressible flows. Different from the existing sphere function-based GKS [23], which directly uses the continuous sphere function to evaluate numerical fluxes at the cell interface, the present scheme is based on the discrete velocity model developed by the sphere function. By using certain discrete points to represent the

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Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows

Highlights

Abstract

Introduction

Section snippets

Sphere function-based GKS and 7 conservation forms of moments

3D Navier–Stokes equations discretized by finite volume method

Numerical examples

Conclusions

Comput. Math. Appl.

J. Comput. Phys.

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