A class of structurally complete approximate Riemann solvers for trans- and supercritical flows with large gradients

Highlights

• Quantify the error in Riemann solvers by omitting the expansion wave under trans- and supercritical conditions.

• Highlight the strong coupling between the magnitude and prevalence of entropy violations with respect to the thermodynamic state and gradients of the flow.

• Propose a Structurally complete Approximate Riemann Solver (StARS) that restores the expansion wave for a non-ideal equation of state using an analytical expression.

• Show the linear convergence of the error, using a low-order numerical scheme, by restoring the expansion wave under transonic conditions.

• Quantify the computational cost of the StARS modification to classical Riemann solvers and compare with other entropy fixes.

Abstract

The wave structure of approximate Riemann solvers has a significant impact on the accuracy and computational requirements of finite volume codes. We propose a class of structurally complete approximate Riemann solvers (StARS) and provide an efficient means for analytically restoring the expansion wave to pre-existing three-wave solvers. The method analytically restores the expansion, is valid for arbitrary thermodynamics, and has comparable complexity to the popular Harten-Hyman entropy fix. The StARS modification is applied to a Roe scheme, resulting in Roe-StARS with noticeable improvements in unsteady transcritical and supercritical conditions with large flow gradients. A novel scaling analysis is performed on the flow conditions that cause rarefaction fluxes and the magnitude of errors if the rarefaction is omitted. Four test cases are examined: a transcritical shock tube, a shock tube with periodic bounds resulting in interfering shocks and rarefactions, a two-dimensional Riemann problem, and a “gradient” Riemann problem—a variant on the traditional Riemann problem featuring an initial gradient of varying slope rather than an initial step function. The results highlight the complex causes and effects of entropy violations, and encourage further study of StARS-type solvers for modern flow problems in which high flow speeds, large gradients, and non-ideal thermodynamics are increasingly common.

Introduction

Flux computations are essential to the finite-volume method in Computational Fluid Dynamics (CFD). In Godunov schemes, used for compressible flows, the numerical fluxes at the intercell boundaries are evaluated by solving either an approximate or an exact Riemann problem. Many approximate Riemann solvers have been developed for practical use in CFD [1], and are typically non-iterative to prioritize computational speed. Such solvers also tend to assume a perfect gas to relate the thermodynamic variables. From these flux estimates it is possible to construct efficient numerical schemes for simulating flow problems of interest to scientists and engineers [2]. With the increasing interest in systems operating at thermodynamic conditions that depart from the ideal assumptions—often characterized by highly non-linear thermodynamic coupling and computationally expensive evaluation of fluid properties [3]—the accuracy and efficiency of the Riemann solvers become increasingly relevant.

For nearly four decades, there has been a continual pursuit to improve the representation of the wave structure within Riemann solvers. This is because improvements to the design of the Riemann solver can result in increased accuracy at minimal additional expense as compared to leveraging more involved discretization schemes. In some cases, it is impossible to resolve certain flow phenomena if the Riemann solver omits the necessary waves [4]. For instance, the HLLC solver [5] restored the missing contact discontinuity in the HLL flux [6], leading to improved resolution of material interfaces and sharp physical features in contexts such as supersonic and shallow water flows. Also, the HLLE [7] and HLLEM [8] solvers addressed issues with wavespeeds to ensure positively conservative results, particularly under vacuum conditions. To date, the highest fidelity approximate solvers consist of three-wave models [9], [10], [5], [11], [12], named as such because they model the three spatially distinct waves that are observed in the solution to the one-dimensional Riemann problem.

A major limitation of most approximate solvers is that rarefactions in the solution to the Riemann problem are simplified to discontinuous jumps—the spatially varying nature of rarefactions is lost despite their presence in exact solutions as well as in the underlying physical problem. This occurs due to the consideration of piecewise constant states between the wave fronts, resulting from the linearization of the governing equations. At a fully subsonic or supersonic state (Fig. 1.a), the rarefaction does not enclose the cell interface and the omission of an exact expansion wave is benign; the intercell flux is determined by other regions in the solution. However, if a rarefaction is present in a transonic scenario (Fig. 1.b), and it is approximated as a discontinuous jump, one can prove that the resulting weak solution violates the entropy condition. In the context of this work, we denote a transonic scenario when the head of the rarefaction is subsonic while the tail is supersonic. Because transonic scenarios are more prevalent at higher pressures and temperatures, the issue of entropy violations is especially relevant to the study of trans- and supercritical flows. Harten and Hyman [13], Osher [14], and Quirk [15] were among the first to explore entropy issues and fluxes due to transonic rarefactions in Riemann solvers. They showed that entropy violations are consistent with the mathematical definition of hyperbolic conservation laws but are thermodynamically inconsistent for the purposes of simulating real-world flows. Entropy-violating solutions frequently contain nonphysical phenomena such as expansion waves that suddenly decay into a shock front, also called expansion or rarefaction shocks.

Various entropy fixes have been developed over the years [13], [16], and they are generally modeled after Harten and Hyman's [13] approach of introducing a new intermediate state to approximate the lost rarefaction wave. The new intermediate state is often treated as a constant; alternatively, it can be linearly or polynomially interpolated between known states. This has the effect of introducing additional diffusivity in the flux terms to mitigate any expansion shocks. It has also been shown, in the case of perfect gases, that it is possible to calculate the flux analytically [17], [2], [1]. Even so, a simple and analytically correct means of restoring the expansion wave for arbitrary Riemann solvers—and especially under non-ideal thermodynamics—has not yet been demonstrated.

Recent contributions in the area of entropy stability have instead focused on extending existing entropy concepts to new applications. Few fundamental improvements have been made to the design of the Riemann solver itself. For example, studies have investigated entropy violations in boundary conditions [18], higher dimensions [19], [20], multicomponent flows [21], or hybridized Riemann solvers that switch or average between different flux estimates [22], [23]. The ideal gas assumption is usually made, and any entropy fixes follow the classical implementation or with minor optimizations. Other works have also examined entropy stability in the context of discontinuous Galerkin schemes [24], [25], [26], magnetohydrodynamics [27], Lagrangian gas dynamics [28], relativistic hydrodynamics [29], and nonclassical dense gases where rarefaction shocks are physically admissible [30], [31], [32], [33], [34]. The case of a single-species gas with arbitrary state equation obeying the Euler equations has thus far been overlooked. Studying this particular problem would facilitate the analysis of nonphysicalities attributable only to the Riemann solver.

We propose a class of structurally complete approximate Riemann solvers (StARS) that use recent derivations by Wang and Hickey [35] to analytically restore the expansion wave in pre-existing three-wave solvers. By structurally complete and approximate, it is meant that StARS provides explicit non-iterative means to compute: 1) wave speeds associated with the method of characteristics, i.e. normal shocks, contact discontinuities, rarefaction heads, and rarefaction tails; and 2) primitive and conservative variables as well as fluxes in each region between these waves—in particular, expansion waves are analytically reconstructed and not approximated as a constant or interpolated state. The result is a class of efficient approximate solvers that offer improved accuracy, the benefits of which are especially apparent under transonic flux conditions. Most importantly, this property of structural completeness is valid for both ideal and non-ideal thermodynamics. In this study, the Roe solver [36], whose entropy properties have been widely studied, is compared to a structurally complete version of the Roe solver (dubbed Roe-StARS) across compressible flow test cases where entropy violations arise. We also perform a comprehensive scaling analysis of flow conditions that give rise to transonic rarefactions, yielding a clear conceptual understanding of the thermodynamic and flow conditions in which such rarefactions occur. It is shown that transonic fluxes are particularly prevalent in trans- and supercritical flows with large thermophysical gradients.

The remaining paper is organized in the following manner. Section 2 describes the governing equations, thermodynamics, and numerical methods relevant to general aspects of Riemann solvers as well as the specific choice of solver and numerical scheme adopted in this study. Section 3 describes the general approach in restoring the expansion wave to an arbitrary three-wave solver, so that transonic fluxes are correctly accounted for. Section 4 conducts a scaling analysis of the flow conditions under which transonic fluxes occur and the errors that can arise if they are omitted. Finally, section 5 provides numerical results for a transcritical shock tube, shock tube with periodic bounds resulting in interfering shocks and rarefactions, a so-called gradient Riemann problem, and a two-dimensional Riemann problem. Provided at the end are a brief conclusion and appendices containing supporting derivations.