Elsevier

Journal of Computational Physics

Volume 295, 15 August 2015, Pages 322-339
Journal of Computational Physics

A three-dimensional explicit sphere function-based gas-kinetic flux solver for simulation of inviscid compressible flows

https://doi.org/10.1016/j.jcp.2015.03.058Get rights and content

Abstract

In this work, a truly three-dimensional (3D) flux solver is presented for simulation of inviscid compressible flows. Like the conventional multi-dimensional gas-kinetic scheme, in the present work, the local solution of 3D Boltzmann equation at the cell interface is used to evaluate the flux. On the other hand, different from most of the existing gas-kinetic schemes, which are constructed from Maxwellian distribution function, the present flux solver is derived from a simple distribution function defined on the spherical surface in the phase velocity space. As a result, the explicit expression of flux vector at the cell interface can be simply given. Since the simple distribution function is defined on the spherical surface, for simplicity, it is termed as sphere function hereafter. In addition, to simulate fluid flow problems with strong shock waves, the non-equilibrium part of the distribution function is regarded as numerical dissipation and involved in evaluating the inviscid flux at the cell interface. The weight of the non-equilibrium part is controlled by introducing a switch function which ranges from 0 to 1. In the smooth region, the switch function takes a value close to zero, while around the strong shock wave, it tends to one. To validate the proposed flux solver, several transonic, supersonic and hypersonic inviscid flows are simulated. Numerical results showed that the present solver can provide accurate numerical results for three-dimensional inviscid flows with strong shock waves.

Introduction

Over the past half century, the computational fluid dynamics (CFD) has been developed into a very powerful tool for solving the fluid flow problems in industrial applications. The CFD is to apply a numerical method to solve governing equations (Euler/Navier–Stokes equations) on the computer. Currently, there are a number of numerical methods available in CFD [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Among them, the finite volume method (FVM) [4] is widely used, and it will be applied in this work. The key issue in applying FVM is to develop an appropriate numerical scheme, which is also known as flux solver, for evaluation of fluxes at the cell interface.

To obtain an appropriate flux solver for calculation of inviscid flux, most of conventional schemes simplify the fluid flow problems into a series of Riemann problems and then solve one-dimensional (1D) Euler equations to get local analytical solution [12] or approximate solution [13], [14], [15], [16]. For multi-dimensional problems, these solvers usually solve 1D Riemann problem along the normal direction to the cell interface and use approximate methods to consider contributions from tangential velocity. Godunov scheme [12] is the pioneering work in this field, which solves 1D Riemann problem at the cell interface analytically. Due to very large computational effort required for pursuing exact solution of the Riemann problem and the lack of necessary numerical dissipation for capturing strong shock waves, this scheme is not popular in practical applications. Inspired by the analytical Riemann solver, a number of approximate Riemann solvers were presented. Some of the representative approximate Riemann solvers are the HLLC scheme [13], Roe scheme [14], van Leer scheme [15] and AUSM scheme [16]. These flux solvers are also constructed from 1D Riemann problem, but they solve it along the normal direction to the cell interface approximately. Also aiming at the calculation of inviscid flux, Shu and his coworkers [17], [18], [19], [20] developed a lattice Boltzmann flux solver (LBFS). In LBFS, the inviscid flux at the cell interface is evaluated by local reconstruction of solution using one-dimensional compressible lattice Boltzmann model. That is to say, in LBFS, the 1D Riemann problem is constructed by the equilibrium distribution function at two sides of the cell interface, and the flux at the cell interface is calculated by conservation forms of moments. Note that, in LBFS, the contributions from tangential velocity are also evaluated approximately.

An alternative approach for evaluation of fluxes at the cell interface is the gas-kinetic scheme. Some of the representative gas-kinetic schemes include the work of Pullin [21], Mandal and Deshpande [22], Chou and Baganoff [23], Xu and his coworkers [24], [25], Yang et al. [26], [27] and Shu and his coworkers [28], [29]. Different from the conventional flux solver [12], [13], [14], [15], [16] and LBFS [17], [18], [19], [20], the multi-dimensional Boltzmann equation is solved locally at the cell interface in the gas-kinetic scheme. This makes the solution of gas-kinetic scheme be more credible. As a result, this scheme can be viewed as a truly multi-dimensional flux solver. Furthermore, the gas-kinetic scheme 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”. However, due to the use of Maxwellian distribution function [21], [22], [23], [24], [25] or the general distribution function [26], [27], most of the gas-kinetic schemes are usually more complicated and less efficient than the traditional Riemann solvers [13], [14], [15], [16]. In addition, to the best of our knowledge, there is no any explicit formulation of Maxwellian distribution function-based gas-kinetic scheme found in the literature to compute the conservative variables and numerical fluxes for multi-dimensional flow problems. Only the solution process is provided. This makes it difficult for new users to implement the schemes. To improve the computational efficiency of the gas-kinetic scheme, a circular function-based gas-kinetic scheme was presented by Shu and his coworkers [28], [29] for the two-dimensional (2D) case. Firstly, the original Maxwellian distribution function, which is a function of the phase velocity and phase energy, is simplified into the function of phase velocity only. The effect of the phase energy is embodied as particle potential energy. Then, the simplified Maxwellian function is further reduced to the circular function, with assumption that all the particles are concentrated on a circle. By applying the circular function, two circular function-based gas-kinetic schemes were presented for simulation of 2D inviscid flows [28] and viscous flows [29] respectively. Since the explicit expressions of these solvers are directly given, the circular function-based gas-kinetic schemes usually require much less computational time than the corresponding Maxwellian function-based gas-kinetic schemes. Besides the above works, in the lattice Boltzmann framework, Zadehgol and Ashrafizaadeh [30] introduced a new equilibrium distribution function (EDF) to simulate the nearly incompressible flows. In the EDF, the particle velocity magnitude is assumed as a constant and the particles are distributed along a circle. By applying the simplified EDF to discrete phase velocity space, several lattice velocity models were developed. Similar works have also been done by Qu et al. [31], Li and Zhong [32] and Li et al. [37]. It is noted that different from the work of Shu and his coworkers [28], [29], in which the inviscid and viscous fluxes are evaluated simultaneously and the macroscopic governing equations are solved, the discrete velocity model is utilized and the discrete velocity Boltzmann equation (DVBE) is usually solved in these works [30], [31], [32], [37].

Inspired by the circular function-based gas-kinetic scheme for the 2D case, a sphere function-based gas-kinetic scheme for the 3D case is presented in this work. With assumption that all the particles are concentrated on a sphere, the Maxwellian distribution function can be simplified as a sphere function, and the integral in the infinite domain of phase velocity space for the Maxwellian function can be reduced to the surface integral along the spherical surface for the sphere function. By applying the sphere function, a 3D flux solver for simulation of 3D compressible inviscid flows is developed. Like the conventional gas-kinetic schemes, the present scheme can be viewed as a truly 3D flux solver since it is based on local solution of 3D Boltzmann equation and all macroscopic flow velocity components are involved in the flux expression. It inherits the advantages of conventional gas-kinetic scheme [21], [22], [23], [24], [25], [26], [27]. In the meantime, since the integrals for conservation forms of moments are greatly simplified, the sphere function-based gas-kinetic scheme is simpler and more efficient than the conventional Maxwellian function-based gas-kinetic scheme. The present work is the first time to give explicit formulations to evaluate conservative variables and numerical fluxes for the 3D case. On the other hand, like other Riemann solvers, to simulate compressible flows with strong shock waves, introducing numerical dissipation (viscous flux) into the scheme is necessary. According to Chapman–Enskog analysis [33], it is known that the non-equilibrium part of the distribution function contributes to the viscous flux. Thus, in this work, the non-equilibrium part of distribution function, which is controlled by a switch function ranging from 0 to 1, is viewed as the numerical dissipation and involved in evaluation of the inviscid flux at the cell interface. In the smooth region such as in boundary layer, the switch function takes a value close to zero, while around the strong shock wave, it tends to one. To validate the developed scheme, some test cases, such as explosion in a box, forward facing step, flows around NACA0012 airfoil, flows around ONERA M6 wing, hypersonic flow around a hemisphere and two-dimensional Riemann problem, are simulated. Numerical results showed that the present solver can provide accurate results for the transonic flow, supersonic flow and hypersonic flow as compared with the reference data or the results of conventional flux solvers.

Section snippets

From Maxwellian function to sphere function

The motivation of this work is to develop a simple and efficient 3D gas-kinetic scheme to improve the computational efficiency of existing gas-kinetic schemes, and to provide explicit formulations to compute the conservative variables and numerical fluxes so that other researchers can easily apply them. To do this, we first simplify the Maxwellian function to a simple distribution function by assuming that the mass, momentum and energy of all particles in the phase velocity space are

Numerical examples

To validate the proposed sphere function-based gas-kinetic scheme for simulation of inviscid compressible flows, the explosion in a box, forward facing step, flows around NACA0012 airfoil, flows around ONERA M6 wing, hypersonic flow around a hemisphere and two-dimensional Riemann problem are considered. In this work, the conservative variables at two sides of cell interface are interpolated from those at cell centers and Venkatakrishnan's limiter [35] is used. For the temporal discretization to

Conclusions

This paper first simplified the Maxwellian function to a sphere function. In the meantime, the integral in the infinity domain of phase velocity and phase energy for the Maxwellian function is reduced to the surface integral along the spherical space. By applying the sphere function, a 3D gas-kinetic scheme for simulation of compressible inviscid flows is developed, and explicit expressions for computation of conservative variables and flux vector at the cell interface are provided. Since the

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