An exact Riemann solver for detonation products

Numerical methods based upon the Riemann Problem are considered for solving the general initial-value problem for the Euler equations applied to real gases. Most of such methods use an approximate solution of the Riemann problem when real gases are involved. These approximate Riemann solvers do not yield always a good resolution of the flow field, especially for contact surfaces and expansion waves. Moreover, approximate Riemann solvers cannot produce exact solutions for the boundary points. In order to overcome these shortcomings, an exact solution of the Riemann problem is developed, valid for real gases. The method is applied to detonation products obeying a 5th order virial equation of state, in the shock-tube test case. Comparisons between our solver, as implemented in Random Choice Method, and finite difference methods, which do not employ a Riemann solver, are given.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

A linearized Riemann solver for the steady supersonic Euler equations

A very simple linearization of the solution to the Riemann problem for the steady supersonic Euler equations is presented. When used locally in conjunction with the Godunov method, computing savings by a factor of about four relative to the use of exact Riemann solvers can be achieved. For severe flow regimes, however, the linearization loses accuracy and robustness. We then propose the use of a Riemann solver adaptation procedure. This retains the accuracy and robustness of the exact Riemann solver and the computational efficiency of the cheap linearized Riemann solver. Numerical results for two- and three-dimensional test problems are presented.     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.     

The Riemann problem for a hyperbolic model of two-phase flow in conservative form

This article is to continue the present author's work (International Journal of Computational Fluid Dynamics (2009) 23 (9), 623–641) on studying the structure of solutions of the Riemann problem for a system of three conservation laws governing two-phase flows. While existing solutions are limited and found quite recently for the Baer and Nunziato equations, this article presents the first instance of an exact solution of the Riemann problem for two-phase flow in gas–liquid mixture. To demonstrate the structure of the solution, we use a hyperbolic conservative model with mechanical equilibrium and without velocity equilibrium. The Riemann problem solution for the model equations comprises a set of elementary waves, rarefaction and discontinuous waves of various types. In particular, such a solution treats both the wave structure and the intermediate states of the two-phase gas–liquid mixture. The resulting exact Riemann solver is fully non-linear, direct and complete. On this basis then, we use locally the exact Riemann solver for the two-phase flow in gas–liquid mixture within the framework of finite volume upwind Godunov methods. In order to demonstrate the effectiveness and accuracy of the proposed solver, we consider a series of test problems selected from the open literature and compare the exact and numerical results by using upwind Godunov methods, showing excellent oscillation-free results in two-phase fluid flow problems.     

HLLC scheme for the preconditioned pseudo-compressibility Navier–Stokes equations for incompressible viscous flows

A new HLLC (Harten-Lax-van leer contact) approximate Riemann solver with the preconditioning technique based on the pseudo-compressibility formulation for numerical simulation of the incompressible viscous flows has been proposed, which follows the HLLC Riemann solver (Harten, Lax and van Leer solver with contact resolution modified by Toro) for the compressible flow system. In the authors' previous work, the preconditioned Roe's Riemann solver is applied to the finite difference discretisation of the inviscid flux for incompressible flows. Although the Roe's Riemann solver is found to be an accurate and robust scheme in various numerical computations, the HLLC Riemann solver is more suitable for the pseudo-compressible Navier--Stokes equations, in which the inviscid flux vector is a non-homogeneous function of degree one of the flow field vector, and however the Roe's solver is restricted to the homogeneous systems. Numerical investigations have been performed in order to demonstrate the efficiency and accuracy of the present procedure in both two- and three-dimensional cases. The present results are found to be in good agreement with the exact solutions, existing numerical results and experimental data.     

Dissipation matrix and artificial heat conduction for Godunov‐type schemes of compressible fluid flows

This paper aims to reassess the Riemann solver for compressible fluid flows in Lagrangian frame from the viewpoint of modified equation approach and provides a theoretical insight into dissipation mechanism. It is observed that numerical dissipation vanishes uniformly for the Godunov‐type schemes in the sense that associated dissipation matrix has zero determinant if an exact or approximate Riemann solver is used to construct numerical fluxes in the Lagrangian frame. This fact connects to some numerical defects such as the wall‐heating phenomenon and start‐up errors. To cure these numerical defects, a traditional numerical viscosity is added, as well as the artificial heat conduction is introduced via a simple passage of the Lax–Friedrichs type discretization of internal energy. Copyright © 2016 John Wiley & Sons, Ltd.     

近似黎曼解对高超声速气动热计算的影响研究

高超声速流场计算一般采用TVD型格式,这些格式中,大多采用了不同形式的近似黎曼解. 通过分析和数值验证,论述了激波捕捉格式中近似黎曼解的耗散性质,说明其对高超声速热流计算的影响. 数值实验证明,采用低耗散格式可大大提高热流计算精度,降低热流计算对网格的依赖程度,从而获得精确的热流数值解.      

An approximate solution of the Riemann problem for a realisable second-moment turbulent closure

Abstract. An approximate solution of the Riemann problem associated with a realisable and objective turbulent second-moment closure, which is valid for compressible flows, is examined. The main features of the continuous model are first recalled. An entropy inequality is exhibited, and the structure of waves associated with the non-conservative hyperbolic convective system is briefly described. Using a linear path to connect states through shocks, approximate jump conditions are derived, and the existence and uniqueness of the one-dimensional Riemann problem solution is then proven. This result enables to construct exact or approximate Riemann-type solvers. An approximate Riemann solver, which is based on Gallou?t's recent proposal is eventually presented. Some computations of shock tube problems are then discussed. Received 2 March 1999 / Accepted 24 August 2000     

Numerical modeling of multiphase first-contact miscible flows. Part 1. Analytical Riemann solver

In this series of two papers, we present a front-tracking method for the numerical simulation of first-contact miscible gas injection processes. The method is developed for constructing very accurate (or even exact) solutions to one-dimensional initial-boundary-value problems in the form of a set of evolving discontinuities. The evolution of the discontinuities is given by analytical solutions to Riemann problems. In this paper, we present the mathematical model of the problem and the complete Riemann solver, that is, the analytical solution to the one-dimensional problem with piecewise constant initial data separated by a single discontinuity, for any left and right states. The Riemann solver presented here is the building block for the front-tracking/streamline method described and applied in the second paper.     

A fast Godunov method for the water‐hammer problem

An efficient Godunov‐type numerical method with second‐order accuracy was developed to simulate the water‐hammer problem in piping. The exact solutions of the Riemann problem were analysed and illustrated on the intriguing solution diagram by properly introducing dimensionless variables within reasonably practical ranges. Based on the solution diagram, an efficient fast Riemann solver was also developed. Moreover, small perturbation analysis was performed to demonstrate the relations between the primitive variables, velocity and pressure, for the Riemann problem. The typical shock‐tube problem and the water‐hammer problem were implemented as sample ones to test the numerical method. It was shown that the present numerical method incorporated with Van Leer's flux limiter is a promising one to simulate fluid transient problem for piping in the present study. Copyright © 2002 John Wiley & Sons, Ltd.