A hyperbolic model of multiphase flow

arXiv: Analysis of PDEs, 2006

We consider a hyperbolic system of three conservation laws in one space variable. The system is a model for fluid flow allowing phase transitions; in this case the state variables are the specific volume, the velocity and the mass density fraction of the vapor in the fluid. For a class of initial data having large total variation we prove the global existence of solutions to the Cauchy problem.

ESAIM: Proceedings, 2013

We discuss a model for the flow of an inviscid fluid admitting liquid and vapor phases, as well as a mixture of them. The flow is modeled in one spatial dimension; the state variables are the specific volume, the velocity and the mass density fraction λ of vapor in the fluid. The equation governing the time evolution of λ contains a source term, which enables metastable states and drives the fluid towards stable pure phases. We first discuss, for the homogeneous system, the BV stability of Riemann solutions generated by large initial data and check the validity of several sufficient conditions that are known in the literature. Then, we review some recent results about the existence of solutions, which are globally defined in time, for λ close either to 0 or to 1 (corresponding to almost pure phases). These solutions possibly contain large shocks. Finally, in the relaxation limit, solutions are proved to satisfy a reduced system and the related entropy condition. Résumé. On discute un modèle pour l'écoulement d'un fluide non visqueux admettant phases liquides et de vapeur, ainsi qu'un mélange d'entre eux. L'écoulement est modélisé dans une dimension spatiale; les variables d'état sont le volume spécifique, la vitesse et la fraction de densité de masse λ de la vapeur dans le liquide. L'équation régissant l'évolution temporelle de λ contient un terme de source, ce qui permet desétats métastables et conduit le fluide vers de phases stables pures. Nous discutons d'abord, pour le système homogène, la stabilité BV des solutions de Riemann générés par des grandes données initiales et vérifions la validité de plusieurs conditions suffisantes qui sont connues dans la littérature. Ensuite, nous passons en revue quelques résultats récents sur l'existence de solutions, qui sont definies pour tous les temps, pour λ soit près de 0 ou de 1 (correspondantà des phases presque pures). Ces solutions sont susceptibles de contenir des grands chocs. Enfin, dans la limite de la relaxation, les solutions sont prouvèes satisfaire un système réduit et la condition d'entropie.

An Introduction to Reservoir Simulation Using MATLAB/GNU Octave

We will return to models having three phases and more than one component per phase in Chapter 11. For now, however, we assume that our system consists of two immiscible phases.

2013

The conservative hyperbolic formulation for compressible multiphase flow for the case of four phase flow is presented. The properties of governing equations are described and numerical results for some Riemann test problems are shown.

Comptes Rendus Mathematique, 2007

The Two-Fluid Model, an averaged model widely used in the modeling of two-phase compressible flows, generally fails to be hyperbolic in its basic formulation. However, interfacial forces such as the interfacial pressure term and the virtual mass force, bringing new differential terms to the system can change the previous analysis and make the problem hyperbolic. The case where the two phases are incompressible has been studied by Stuhmiller ([1]) in 1977, but till now, no proof of their efficiency in rendering the model hyperbolic exists in the compressible case. The aim of this paper is to detail the effects these forces have on the hyperbolicity of the Two-Fluid Model in the compressible case. We characterise the location and topology of the non hyperbolic regions, and propose a closure for the interfacial pressure that makes the system unconditionnally hyperbolic.

The flow problems considered in previous chapters are concerned with homogeneous fluids, either single phases or suspensions of fine particles whose settling velocities are sufficiently low for the solids to be completely suspended in the fluid. Consideration is now given to the far more complex problem of the flow of multiphase systems in which the composition of the mixture may vary over the cross-section of the pipe or channel; furthermore, the components may be moving at different velocities to give rise to the phenomenon of "slip" between the phases. Multiphase flow is important in many areas of chemical and process engineering and the behaviour of the material will depend on the properties of the components, the flowrates and the geometry of the system. In general, the complexity of the flow is so great that design methods depend very much on an analysis of the behaviour of such systems in practice and, only to a limited extent, on theoretical predictions. Some of the more important systems to be considered are: Mixtures of liquids with gas or vapour. Liquids mixed with solid particles ("hydraulic transport"). Gases carrying solid particles wholly or partly in suspension ("pneumatic transport"). Multiphase systems containing solids, liquids and gases. Mixed materials may be transported horizontally, vertically, or at an inclination to the horizontal in pipes and, in the case of liquid-solid mixtures, in open channels. Although there is some degree of common behaviour between the various systems, the range of physical properties is so great that each different type of system must be considered separately. Liquids may have densities up to three orders of magnitude greater than gases but they do not exhibit any significant compressibility. Liquids themselves can range from simple Newtonian liquids such as water, to non-Newtonian fluids with very high apparent viscosities. These very large variations in density and viscosity are responsible for the large differences in behaviour of solid-gas and solid-liquid mixtures which must, in practice, be considered separately. For, all multiphase flow systems, however, it is important to understand the nature of the interactions between the phases and how these influence the flow patterns — the ways in which the phases are distributed over the cross-section of the pipe or duct. In design it is necessary to be able to predict pressure drop which, usually, depends not only on the flow pattern, but also on the relative velocity of the phases; this slip velocity will influence the holdup , the fraction of the pipe volume which is occupied by a particular phase. It is important to note that, in the flow of a

Quarterly of Applied Mathematics, 2015

Derivation of governing equations for multiphase flow on the base of thermodynamically compatible systems theory is presented. The mixture is considered as a continuum in which the multiphase character of the flow is taken into account. The resulting governing equations of the formulated model belong to the class of hyperbolic systems of conservation laws. In order to examine the reliability of the model, the one-dimensional Riemann problem for the four phase flow is studied numerically with the use of the MUSCL-Hancock method in conjunction with the GFORCE flux.

Computers & Mathematics with Applications, 2014

It is well known that classic two-phase flow equation systems have complex characteristic roots and, therefore, constitute an ill-posed initial-value problem. Here we suggest that illposedness is due to working with two different material derivatives for the phases, which have varying velocities, but employing the same position vector for both operators. There follows an analysis of the conditions required for a global treatment of both phases, but using only one material derivative for both phases, now coherent with only one position vector. Consequently, new global mass-and momentum-conservation equations for a twophase flow without mass exchange are derived by strictly following the classic Reynolds' transport theorem. The new global mass-conservation equation proposed would only be valid if the 'zero-net-mass-flux' condition, another independent equation, was fulfilled. We also found that the new equation system is well-posed, i.e. its two characteristic roots are real if a new relation between velocities and densities is satisfied. Finally, we have highlighted the strong connections of new conservation laws with classic treatments, and also shown that minor modifications of the current equation system would turn it into a hyperbolic one, thus easing the computational solution of this complex problem.

VTT PUBLICATIONS, 1996

Numerical flow simulation utilising a full multiphase model is impractical for a suspension possessing wide distributions in the particle size or density. Various approximations are usually made to simplify the computational task. In the simplest approach, the suspension is represented by a homogeneous single-phase system and the influence of the particles is taken into account in the values of the physical properties. The multiphase nature of the flow cannot, however, be avoided when the concentration gradients are large and the dispersed phases alter the hydrodynamic behaviour of the mixture or when the distributions of the particles are studied. In many practical applications of multiphase flow, the mixture model is a sufficiently accurate approximation, with only a moderate increase in the computational effort compared to a single-phase simulation. The interest of applying computational fluid dynamics in industrial multiphase processes has increased during the last few years. Fluidised beds, polymerisation processes, settling tanks, chemical reactors, gas dispersion in liquids and air-lift reactors are typical examples in process industry. Modelling of multiphase flows is, however, very complicated. Full multiphase modelling requires a large computing power, especially if several secondary phases need to be considered. In this study, we investigate the mixture model, which is a simplification of the full models. This approach is a considerable alternative in simulating dilute suspensions of solid particles or small bubbles in liquids.

Communications in Mathematical Sciences, 2014

We study a class of models of compressible two-phase flows. This class, which includes the Baer-Nunziato model, is based on the assumption that each phase is described by its own pressure, velocity and temperature and on the use of void fractions obtained from averaging process. These models are nonconservative and non-strictly hyperbolic. We prove that the mixture entropy is non-strictly convex and that the system admits a symmetric form.