A sequel to a rough Godunov scheme: application to real gases

Abstract

We present here an approximate Riemann solver to compute Euler equations using real gas state laws. The scheme is based on an earlier proposition, called VFRoe, introduced by one of the authors. It makes use of non conservative variables in order to preserve numerically Riemann invariants through the contact discontinuity. Detailed investigation of actual rate of convergence of the scheme is reported. The study also includes a comparison with original VFRoe and Roe schemes.

Introduction

We are interested in numerical solutions of initial-value hyperbolic systems of conservation laws:

$$\text{∂}\text{W}\text{∂t}\text{+}\text{∂}\text{F}\text{W}\text{∂x}\text{=0}$$

$$\text{W}\text{x,}\text{0}\text{=}\text{W}{}_0\text{x}$$

where $\text{W}\text{x,}\text{t}\text{∈}\text{R}{}^{\mathrm{p}}$ and the flux $\text{F}\text{W}$ is such that the Jacobian matrix given by $\text{A}\text{W}\text{=}\text{∂}\text{F}\text{∂}\text{W}$ is diagonalizable with real eigenvalues.

Finite volume schemes in conservation form have proven their efficiency to solve such problems. The most popular ones are based on the resolution of local Riemann problems at each interface. The first one, proposed by Godunov [1], is based on the exact solution of the one-dimensional Riemann problem associated with Eq. (1). It enjoys very good properties in the one-dimensional case (in particular, it fulfills the positivity of density and pressure for the Euler system) but its main drawbacks are its cost and its lack of robustness when the integration of Riemann invariants cannot be done analytically. Schemes based on Approximate Riemann solvers were introduced to avoid these difficulties. The commonly used Roe scheme [2] requires the knowledge of analytical eigenvalues and eigenvectors of the Jacobian matrix and, above all, the definition of a matrix of linearization $\text{A}\text{W}{}_{\mathrm{L}}\text{,}\text{W}{}_{\mathrm{R}}\text{∈M}{}_{\mathrm{p}}\text{R}$ such that:

$$\text{A}\text{W}{}_{\mathrm{L}}\text{,}\text{W}{}_{\mathrm{R}}\text{is}\text{diagonalizable}\text{with}\text{real}\text{eigenvalues}$$

$$\text{F}\text{W}{}_{\mathrm{R}}\text{−}\text{F}\text{W}{}_{\mathrm{L}}\text{=}\text{A}\text{W}{}_{\mathrm{L}}\text{,}\text{W}{}_{\mathrm{R}}\text{W}{}_{\mathrm{R}}\text{−}\text{W}{}_{\mathrm{L}}\text{.}$$

$$\text{A}\text{W}\text{,}\text{W}\text{=}\text{∂}\text{F}\text{W}\text{∂}\text{W}\text{.}$$

Remark that condition (5) may be replaced by: A is a continuous mapping of $\text{R}{}^{\mathrm{p}}\text{×}\text{R}{}^{\mathrm{p}}$ in $\text{M}{}_{\mathrm{p}}\text{R}$. The main losses in comparison with Godunov’s scheme are the violation of the entropy law (a sonic entropy correction must be added), and, for instance, when dealing with the Euler system, the occurence of nonpositive density or pressure values. Moreover, finding a matrix satisfying Roe’s condition (4) may sometimes be difficult or even impossible for some systems: Euler with real gas, some two-phase flow models and some complex turbulence models for example. This fact has motivated the development of an alternative to the Roe scheme. This simple Finite Volume scheme, called VFRoe, was introduced in Refs. [3], [4], [5] to approximate solutions of a two-phase flow model. Following the conservative approach of the Godunov scheme, the VFRoe flux function identifies with the physical one. As with the Roe scheme, the Riemann problems at each interface are linearized. Roe’s condition is not necessary here, since one does not need to fulfill a consistency relation with the integral form of the conservation laws. Let us note two drawbacks of the VFRoe scheme, already finded with the Roe scheme: it admits entropy-violating stationary discontinuities, and it can produce nonpositive density or pressure values when it is applicated to the Euler system for example. Nonetheless, its behavior in many configurations is quite good and its application field is wider than Roe’s one. Let us mention another approach recently considered in Ref. [6] for the resolution of a two phase flow in a pipe.

We propose here an extension of the VFRoe scheme: VFRoe using non conservative variables, noted here VFRoencv. Although it differs from VFRoe by the resolution of a linearized system written in non conservative variables, it may be used for smooth and non smooth flow. In this paper, the VFRoencv scheme is presented and tested on the Euler set of equations for equilibrium real gas with various equations of state (EOS). Many classical numerical schemes have been extended to include real-gas effects. Extensions of exact Riemann solver have been obtained by Collela and Glaz [7] (with a model based on an approximation of γ across the waves), by Saurel et al. [8] and by Letellier and Forestier [9]. For equilibrium real gases, different approaches have been considered for generalized Roe average. Grossman and Walters [10] have introduced an equivalent γ to relate c² and h as in perfect gases. The more classical approach is to find $\text{χ}\text{̄}$ and $\text{κ}\text{̄}$ satisfying $\text{Δ}\text{p=}\text{χ}\text{̄}\text{Δ}\text{ρ+}\text{κ}\text{̄}\text{Δ}\text{ϵ}$ (or an equivalent form) so that relation (4) holds. Here, the greatest difficulty comes from the approximation of (generally pressure-) derivatives. Different formulas have been obtained by Glaister [11], Liou et al. [12], Vinokur and Montagné [13], Abgrall [14] in the case of chemically equilibrium flow, Toumi [15], Buffat and Page [16] who choose the temperature, instead of the pressure, as the relevant thermodynamic parameter. The ‘simplest’ extensions of the Roe scheme cannot be applied in practice if the EOS is a non-convex function for example, whereas others need the evaluation of integrals. A new technique using relaxation of energy allows the extension of classical approximate Riemann solvers used in perfect gas to treat real gases; it has been introduced by Coquel and Perthame [17], and tested by In [18]. The approach using a kinetic scheme has also been extended (see El Amine [19]) to non-isentropic real fluids. The VFRoencv scheme seems to be an interesting alternative according to numerical test cases presented here (see also Refs. [20], [21]). It is a simple scheme, sometimes cheaper than previous ones.

In the next section, we briefly recall the Roe and Godunov schemes ([22]). Section 3 introduces the VFRoencv-type scheme on a general hyperbolic system (1). In Section 4, we propose a VFRoencv scheme when focusing on Euler equations. We also prove some relevant properties, especially in the case of ideal gas. We will discuss in Section 5 the results of several test cases, together with comparisons with other schemes when possible, for several equations of state. Extensions to second order, with the M.U.S.C.L. method and to multidimensional in space, are also presented in Section 6. An appendix is devoted to boundary conditions. In particular, a study in the case of rigid wall boundary condition with ‘mirror state’ technique is given (comparison between Godunov, Roe and VFRoencv approaches).