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.     

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     

Wall boundary conditions for inviscid compressible flows on unstructured meshes

In this paper we revisit the problem of implementing wall boundary conditions for the Euler equations of gas dynamics in the context of unstructured meshes. Both (a) strong formulation, where the zero normal velocity on the wall is enforced explicitly and (b) weak formulation, where the zero normal velocity on the wall is enforced through the flux function are discussed. Taking advantage of both approaches, mixed procedures are defined. The new wall boundary treatments are accurate and can be applied to any approximate Riemann solver. Numerical comparisons for various flow regimes, from subsonic to supersonic, and for various approximate Riemann solvers point out that the mixed boundary procedures drastically improve the accuracy. © 1998 John Wiley & Sons, Ltd.     

A Free-Lagrange method for unsteady compressible flow: simulation of a confined cylindrical blast wave

A Free-Lagrange numerical procedure for the simulation of two-dimensional inviscid compressible flow is described in detail. The unsteady Euler equations are solved on an unstructured Lagrangian grid based on a density-weighted Voronoi mesh. The flow solver is of the Godunov type, utilising either the HLLE (2 wave) approximate Riemann solver or the more recent HLLC (3 wave) variant, each adapted to the Lagrangian frame. Within each mesh cell, conserved properties are treated as piece-wise linear, and a slope limiter of the MUSCL type is used to give non-oscillatory behaviour with nominal second order accuracy in space. The solver is first order accurate in time. Modifications to the slope limiter to minimise grid and coordinate dependent effects are described. The performances of the HLLE and HLLC solvers are compared for two test problems; a one-dimensional shock tube and a two-dimensional blast wave confined within a rigid cylinder. The blast wave is initiated by impulsive heating of a gas column whose centreline is parallel to, and one half of the cylinder radius from, the axis of the cylinder. For the shock tube problem, both solvers predict shock and expansion waves in good agreement with theory. For the HLLE solver, contact resolution is poor, especially in the blast wave problem. The HLLC solver achieves near-exact contact capture in both problems. Received May 25, 1995 / Accepted September 11, 1995     

An arbitrary Lagrangian–Eulerian method based on the adaptive Riemann solvers for general equations of state

Approximate or exact Riemann solvers play a key role in Godunov‐type methods. In this paper, three approximate Riemann solvers, the MFCAV, DKWZ and weak wave approximation method schemes, are investigated through numerical experiments, and their numerical features, such as the resolution for shock and contact waves, are analyzed and compared. Based on the analysis, two new adaptive Riemann solvers for general equations of state are proposed, which can resolve both shock and contact waves well. As a result, an ALE method based on the adaptive Riemann solvers is formulated. A number of numerical experiments show good performance of the adaptive solvers in resolving both shock waves and contact discontinuities. Copyright © 2008 John Wiley & Sons, Ltd.     

A generalized Rusanov method for the Baer‐Nunziato equations with application to DDT processes in condensed porous explosives

The paper addresses a numerical approach for solving the Baer‐Nunziato equations describing compressible 2‐phase flows. We are developing a finite‐volume method where the numerical flux is approximated with the Godunov scheme based on the Riemann problem solution. The analytical solution to this problem is discussed, and approximate solvers are considered. The obtained theoretical results are applied to develop the discrete model that can be treated as an extension of the Rusanov numerical scheme to the Baer‐Nunziato equations. Numerical results are presented that concern the method verification and also application to the deflagration‐to‐detonation transition (DDT) in porous reactive materials.     

MFCAV近似Riemann解在新型拉氏方法中的应用

Maire等提出了一种新型的有限体积中心型拉氏方法, 该方法大大地改善了一直困扰着一般中心型拉氏方法的虚假网格变形. 然而在计算数值流和移动网格时,该方法只应用了数值黏性较大的弱波近似(weak wave approximatedmethod, WWAM) Riemann解, 而且方法的设计表明其他类型的近似Riemann解不能简单直接地应用上去. 将体平均多流管(multifluid channel on averaged volume, MFCAV)近似Riemann解视为对WWAM的修正,成功将其应用于新型方法中, 数值实验表明应用了MFCAV 的新方法是有效的. 研究为将其他更为有效的近似Riemann解应用于该新型方法中开辟了一条道路.      

MUSTA schemes for multi‐dimensional hyperbolic systems: analysis and improvements

We develop and analyse an improved version of the multi‐stage (MUSTA) approach to the construction of upwind Godunov‐type fluxes whereby the solution of the Riemann problem, approximate or exact, is not required. The new MUSTA schemes improve upon the original schemes in terms of monotonicity properties, accuracy and stability in multiple space dimensions. We incorporate the MUSTA technology into the framework of finite‐volume weighted essentially nonoscillatory schemes as applied to the Euler equations of compressible gas dynamics. The results demonstrate that our new schemes are good alternatives to current centred methods and to conventional upwind methods as applied to complicated hyperbolic systems for which the solution of the Riemann problem is costly or unknown. Copyright © 2005 John Wiley & Sons, Ltd.     

Flux difference splitting for the Euler equations in one spatial co-ordinate with area variation

An approximate (linearized) Riemann solver is presented for the solution of the Euler equations of gas dynamics in one spatial co-ordinate. This includes cylindrically and spherically symmetric geometries and also applies to narrow ducts with area variation. The method is Roe's flux difference splitting with a technique for dealing with source terms. The results of two problems, a spherically divergent infinite shock and a converging cylindrical shock, are presented. The numerical results compare favourably with those of Noh's recent survey and also with those of Ben-Artzi and Falcovitz using a more complicated Riemann solver.     

Improvements to compressible Euler methods for low‐Mach number flows

In the present study improvements to numerical algorithms for the solution of the compressible Euler equations at low Mach numbers are investigated. To solve flow problems for a wide range of Mach numbers, from the incompressible limit to supersonic speeds, preconditioning techniques are frequently employed. On the other hand, one can achieve the same aim by using a suitably modified acoustic damping method. The solution algorithm presently under consideration is based on Roe's approximate Riemann solver [Roe PL. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics 1981; 43 : 357–372] for non‐structured meshes. The numerical flux functions are modified by using Turkel's preconditioning technique proposed by Viozat [Implicit upwind schemes for low Mach number compressible flows. INRIA, Rapport de Recherche No. 3084, January 1997] for compressible Euler equations and by using a modified acoustic damping of the stabilization term proposed in the present study. These methods allow the compressible Euler equations at low‐Mach number flows to be solved, and they are consistent in time. The efficiency and accuracy of the proposed modifications have been assessed by comparison with experimental data and other numerical results in the literature. Copyright © 2000 John Wiley & Sons, Ltd.     

A fast riemann solver with constant covolume applied to the random choice method

The Riemann problem for the unsteady one-dimensional Euler equations together with the constant-covolume equation of state is solved exactly. The solution is then applied to the random choice method to solve the general initial-boundary value problem for the Euler equations. The iterative procedure to find p*, the pressure between the acoustic waves, involves a single algebraic (non-linear) equation, all other quantities follow directly throughout the x–t plane, except within rarefaction fans where an extra iterative procedure is required. The solution is validated against existing exact results both directly and in conjunction with the random choice method.     

Equilibrium real gas computations using Marquina's scheme

Marquina's approximate Riemann solver for the compressible Euler equations for gas dynamics is generalized to an arbitrary equilibrium equation of state. Applications of this solver to some test problems in one and two space dimensions show the desired accuracy and robustness. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

A mass‐fraction‐based interface‐capturing method for multi‐component flow

An interface‐capturing method based on mass fraction is developed to solve the Riemann problem in multi‐component compressible flow. Equations of mass fraction with modified form, which is derived from conservative equations of mass, are employed here to capture the interface. By introducing mass fraction into Euler equations system, as well as other conservative coefficients, a quasi‐conservative numerical model is created. Numerical examples show that the mass fraction model performs well not only in multi‐component fluids modeled by simple stiffened gas equation of state (EOS) but also in that modeled by complex Mie–Grüneisen EOS. Moreover, the mass fraction model is applied to Riemann problem with piecewise EOS; the expression of which depends on density. It is found that the mass fraction model can well adapt to the analytic change in piecewise EOS and produce accuracy solutions with fewer unknown quantities, and the model can be easily extended to m‐component fluid mixture by using only m + 4 equations with no additional conditions. Copyright © 2013 John Wiley & Sons, Ltd.     

Difference scheme for two-phase flow

A numerical method for two-phase flow with hydrodynamics behavior was considered. The nonconservative hyperbolic governing equations proposed by Saurel and Gallout were adopted. Dissipative effects were neglected but they could be included in the model without major difficulties. Based on the opinion proposed by Abgrall that “a two phase system, uniform in velocity and pressure at t = 0 will be uniform on the same variable during its temporal evolution“, a simple accurate and fully Eulerian numerical method was presented for the simulation of multiphase compressible flows in hydrodynamic regime. The numerical method relies on Godunov-typescheme, with HLLC and Lax-Friedrichs type approximate Riemann solvers for the resolution of conservation equations, and nonconservative equation. Speed relaxation and pressure relaxation processes were introduced to account for the interaction between the phases. Test problem was presented in one space dimension which illustrated that our scheme is accurate, stable and oscillation free.     

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.     

Application of the method of lines to unsteady compressible Euler equations

In the sense of method of lines, numerical solution of the unsteady compressible Euler equations in 1D, 2D and 3D is split into three steps: First, space discretization is performed by the first‐order finite volume method using several approximate Riemann solvers. Second, smoothness and Lipschitz continuity of RHS of the arising system of ordinary dimensional equations (ODEs) is analysed and its solvability is discussed. Finally, the system of ODEs is integrated in time by means of implicit and explicit higher‐order adaptive schemes offered by ODE packages ODEPACK and DDASPK, by a backward Euler scheme based on the linearization of the RHS and by higher‐order explicit Runge–Kutta methods. Time integrators are compared from several points of view, their applicability to various types of problems is discussed, and 1D, 2D and 3D numerical examples are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

Riemann solvers with evolved initial conditions

The scope of this paper is three fold. We first formulate upwind and symmetric schemes for hyperbolic equations with non‐conservative terms. Then we propose upwind numerical schemes for conservative and non‐conservative systems, based on a Riemann solver, the initial conditions of which are evolved non‐linearly in time, prior to a simple linearization that leads to closed‐form solutions. The Riemann solver is easily applied to complicated hyperbolic systems. Finally, as an example, we formulate conservative schemes for the three‐dimensional Euler equations for general compressible materials and give numerical results for a variety of test problems for ideal gases in one and two space dimensions. Copyright © 2006 John Wiley & Sons, Ltd.