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.     

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 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 Riemann solver and upwind methods for a two‐phase flow model in non‐conservative form

We present a theoretical solution for the Riemann problem for the five‐equation two‐phase non‐conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non‐conservative methods to solve the general initial‐boundary value problem for the two‐phase flow model in non‐conservative form. The basic upwind scheme constructed is the non‐conservative analogue of the Godunov first‐order upwind method. Second‐order methods in space and time are then constructed via the MUSCL and ADER approaches. The methods are systematically assessed via a series of test problems with theoretical solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

The reactive Riemann problem for thermally perfect gases at all combustion regimes

In this work we analyze the reactive Riemann problem for thermally perfect gases in the deflagration or detonation regimes. We restrict our attention to the case of one irreversible infinitely fast chemical reaction; we also suppose that, in the initial condition, one state (for instance the left one) is burnt and the other one is unburnt. The indeterminacy of the deflagration regime is removed by imposing a (constant) value for the fundamental flame speed of the reactive shock. An iterative algorithm is proposed for the solution of the reactive Riemann problem. Then the reactive Riemann problem and the proposed algorithm are investigated from a numerical point of view in the case in which the unburnt state consists of a stoichiometric mixture of hydrogen and air at almost atmospheric condition. In particular, we revisit the problem of 1D plane‐symmetric steady flames in a semi‐infinite domain and we verify that the transition from one combustion regime to another occurs continuously with respect to the fundamental flame speed and the so‐called piston velocity. Finally, we use the ‘all shock’ solution of the reactive Riemann problem to design an approximate (‘all shock’) Riemann solver. 1D and 2D flows at different combustion regimes are computed, which shows that the approximate Riemann solver, and thus the algorithm we use for the solution of the reactive Riemann problem, is robust in both the deflagration and detonation regimes. Copyright © 2009 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.     

A general Riemann solver for Euler equations

In this paper, we present a general Riemann solver which is applied successfully to compute the Euler equations in fluid dynamics with many complex equations of state (EOS). The solver is based on a splitting method introduced by the authors. We add a linear advection term to the Euler equations in the first step, to make the numerical flux between cells easy to compute. The added linear advection term is thrown off in the second step. It does not need an iterative technique and characteristic wave decomposition for computation. This new solver is designed to permit the construction of high‐order approximations to obtain high‐order Godunov‐type schemes. A number of numerical results show its robustness. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

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.     

A general approximate‐state Riemann solver for hyperbolic systems of conservation laws with source terms

An approximate‐state Riemann solver for the solution of hyperbolic systems of conservation laws with source terms is proposed. The formulation is developed under the assumption that the solution is made of rarefaction waves. The solution is determined using the Riemann invariants expressed as functions of the components of the flux vector. This allows the flux vector to be computed directly at the interfaces between the computational cells. The contribution of the source term is taken into account in the governing equations for the Riemann invariants. An application to the water hammer equations and the shallow water equations shows that an appropriate expression of the pressure force at the interface allows the balance with the source terms to be preserved, thus ensuring consistency with the equations to be solved as well as a correct computation of steady‐state flow configurations. Owing to the particular structure of the variable and flux vectors, the expressions of the fluxes are shown to coincide partly with those given by the HLL/HLLC solver. Computational examples show that the approximate‐state solver yields more accurate solutions than the HLL solver in the presence of discontinuous solutions and arbitrary geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Solution to the Riemann problem for drift-flux model with modified Chaplygin two-phase flows

In this paper, we concern about the Riemann problem for compressible no-slip drift-flux model which represents a system of quasi-linear partial differential equations derived by averaging the mass and momentum conservation laws with modified Chaplygin two-phase flows. We obtain the exact solution of Riemann problem by elaborately analyzing characteristic fields and discuss the elementary waves namely, shock wave, rarefaction wave and contact discontinuity wave. By employing the equality of pressure and velocity across the middle characteristic field, two nonlinear algebraic equations with two unknowns as gas density ahead and behind the middle wave are formed. The Newton–Raphson method of two variables is applied to find the unknowns with a series of initial data from the literature. Finally, the exact solution for the physical quantities such as gas density, liquid density, velocity, and pressure are illustrated graphically.     

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.     

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     

Riemann solvers and boundary conditions for two‐dimensional shallow water simulations

Most existing algorithms for two‐dimensional shallow water simulations treat multi‐dimensional waves using wave splitting or time splitting. This often results in anisotropy of the computed flow. Both wave splitting and time splitting are based on a local decomposition of the multi‐dimensional problem into one‐dimensional, orthogonal problems. Therefore, these algorithms handle boundary conditions in a very similar way to classical one‐dimensional algorithms. This should be expected to trigger a dependence of the number of boundary conditions on the direction of the flow at the boundaries. However, most computational codes based on alternate directions do not exhibit such sensitivity, which seems to contradict the theory of existence and uniqueness of the solution. The present paper addresses these issues. A Riemann solver is presented that aims to convert two‐dimensional Riemann problems into a one‐dimensional equivalent Riemann problem (ERP) at the interfaces between the computational cells. The ERP is derived by applying the theory of bicharacteristics at each end of the interface and by performing a linear averaging along the interface. The proposed approach is tested against the traditional one‐dimensional approach on the classical circular dambreak problem. The results show that the proposed solver allows the isotropy of the solution to be better preserved. Use of the two‐dimensional solver with a first‐order scheme may give better results than use of a second‐order scheme with a one‐dimensional solver. The theory of bicharacteristics is also used to discuss the issue of boundary conditions. It is shown that, when the flow is subcritical, the number of boundary conditions affects the accuracy of the solution, but not its existence and uniqueness. When only one boundary condition is to be prescribed, it should not be the velocity in the direction parallel to the boundary. When two boundary conditions are to be prescribed, at least one of them should involve the component of the velocity in the direction parallel to the boundary. Copyright © 2003 John Wiley & Sons, Ltd.     

Euler solutions using flux‐based wave decomposition

The Euler equations are solved for non‐hydrostatic atmospheric flow problems in two dimensions using a high‐resolution Godunov‐type scheme. The Riemann problem is solved using a flux‐based wave decomposition suggested by LeVeque. This paper describes in detail, the design and implementation of the Riemann solver used for computing the Godunov fluxes. The methodology is then validated against benchmark cases for non‐hydrostatic atmospheric flows. Comparisons are made with solutions obtained from the National Center for Atmospheric Research's state‐of‐the‐art numerical model. The method shows promise in simulating non‐hydrostatic flows, which are characterized by steep gradients on the meso‐, micro‐ and urban‐scales. Copyright © 2006 John Wiley & Sons, Ltd.     

Shallow flow simulation on dynamically adaptive cut cell quadtree grids

A computationally efficient, high‐resolution numerical model of shallow flow hydrodynamics is described, based on dynamically adaptive quadtree grids. The numerical model solves the two‐dimensional non‐linear shallow water equations by means of an explicit second‐order MUSCL‐Hancock Godunov‐type finite volume scheme. Interface fluxes are evaluated using an HLLC approximate Riemann solver. Cartesian cut cells are used to improve the fit to curved boundaries. A ghost‐cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The numerical model is validated through simulations of reflection of a surge wave at a wall, a low Froude number potential flow past a circular cylinder, and the shock‐like interaction between a bore and a circular cylinder. The computational efficiency is shown to be greatly improved compared with solutions on a uniform structured grid implemented with cut cells. Copyright © 2006 John Wiley & Sons, Ltd.     

Simulation of dam‐ and dyke‐break hydrodynamics on dynamically adaptive quadtree grids

Flooding due to the failure of a dam or dyke has potentially disastrous consequences. This paper presents a Godunov‐type finite volume solver of the shallow water equations based on dynamically adaptive quadtree grids. The Harten, Lax and van Leer approximate Riemann solver with the Contact wave restored (HLLC) scheme is used to evaluate interface fluxes in both wet‐ and dry‐bed applications. The numerical model is validated against results from alternative numerical models for idealized circular and rectangular dam breaks. Close agreement is achieved with experimental measurements from the CADAM dam break test and data from a laboratory dyke break undertaken at Delft University of Technology. Copyright © 2004 John Wiley Sons, Ltd.