Conservative formulation for compressible multiphase flows

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.

International Journal for Numerical Methods in Fluids, 2008

This paper outlines the development of a two-phase flow model based on the theory of thermodynamically compatible systems of hyperbolic conservation laws. The conservative hyperbolic governing equations are numerically implemented in conjunction with the second-order MUSCL method and the GFORCE flux, while for the reduced isentropic model the first-order Godunov method is also derived. Results are presented for the water-air shock tube and water-faucet test problems.

ESAIM: Proceedings, 2013

We give in this paper a short review of some recent achievements within the framework of multiphase flow modeling. We focus first on a class of compressible two-phase flow models, detailing closure laws and their main properties. Next we briefly summarize some attempts to model two-phase flows in a porous region, and also a class of compressible three-phase flow models. Some of the main difficulties arising in the numerical simulation of solutions of these complex and highly non-linear systems of PDEs are then discussed, and we eventually show some numerical results when tackling twophase flows with mass transfer.

Shock Waves, 2018

This paper presents an efficient and robust numerical framework to deal with multiphase real-fluid flows and their broad spectrum of engineering applications. A homogeneous mixture model incorporated with a real-fluid equation of state and a phase change model is considered to calculate complex multiphase problems. As robust and accurate numerical methods to handle multiphase shocks and phase interfaces over a wide range of flow speeds, the AUSMPW+_N and RoeM_N schemes with a system preconditioning method are presented. These methods are assessed by extensive validation problems with various types of equation of state and phase change models. Representative realistic multiphase phenomena, including the flow inside a thermal vapor compressor, pressurization in a cryogenic tank, and unsteady cavitating flow around a wedge, are then investigated as application problems. With appropriate physical modeling followed by robust and accurate numerical treatments, compressible multiphase flow physics such as phase changes, shock discontinuities, and their interactions are well captured, confirming the suitability of the proposed numerical framework to wide engineering applications. Communicated by D. Zeidan and H. D. Ng.

Quarterly of Applied Mathematics, 2007

Governing equations for two-phase compressible flow with different phase pressures and temperatures are presented, the derivation of which is based on the formalism of thermodynamically compatible hyperbolic systems and extended irreversible thermodynamics principles. These equations form a hyperbolic system in conservationlaw form. A two-phase isentropic flow model proposed earlier and the hyperbolic model for heat transfer underlie the developed theory of this paper. A set of interfacial exchange processes such as pressure relaxation, interfacial friction, temperature relaxation and phase transition is taken into account by source terms in the balance equations. It is shown that the heat flux relaxation limit of the governing equations can be written in the Baer-Nunziato form, in which the Fourier thermal conductivity diffusion terms for each phase are included.

Journal of Computational Physics, 2003

We have recently proposed, in , a compressible two-phase unconditionally hyperbolic model able to deal with a wide range of applications : interfaces between compressible materials, shock waves in condensed multiphase mixtures, homogeneous two-phase flows (bubbly and droplet flows) and cavitation in liquids. One of the difficulties of the model, as always in this type of physical problems, was the occurrence of non conservative products. In [21], we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces, not in the case of shock waves. This model was extended to several space dimensions in . In this paper, thanks to a deeper analysis of the model, we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities. This analysis provides a more accurate estimate of closure terms, but also an accurate resolution method for the conservative fluxes as well as non-conservative terms even for situations involving discontinuous solutions. The accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems.

2012

An efficient Eulerian numerical method is considered for simulating multiphase flows governed by general equation of state (EOS). The method allows interfaces between phases to diffuse in a transitional region over a small number of computational cells. The seven-equation model of Saurel and Abgrall [Saurel, R. and Abgrall, R., A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys. 150 (1999), 425 467] is employed to describe the compressible multiphase flows. For one dimensional flow the model which is strictly hyperbolic consists of seven equations. These equations are the volume fraction evolution equation and the conservation equations (mass, momentum and energy) for each phase. The solution of the hyperbolic equations is obtained using HLL Riemann solver. In the present work various equations of state (EOSs) have been discussed. Error analysis, number of time steps and CPU time comparisons between EOSs have been presented. Well known test...

2008

Here t > 0 and x ∈ R; moreover v > 0 is the specific volume, u the velocity, λ the mass density fraction of vapor in the fluid. Then λ ∈ [0, 1], with λ = 0 characterizes the liquid and λ = 1 the vapor phase; intermediate values of λ model mixtures of the two pure phases. The pressure is p = p(v, λ); under natural assumptions the system is strictly hyperbolic. We refer to [4, 3] for more information on the model. System (1) has close connections to a system considered by Peng [6]. A comparison of the two models is done in [1].

Proceedings of ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis, 2012

The simulation of multiphase compressible flows through high pressure nozzles is presented. The study uses the developed numerical approach. There are many important engineering applications which are concerned with multiphase flows and convergent-divergent nozzles. This work presents the developed extension of the model and numerical algorithm based on the so called parent model earlier introduced by Saurel and Abgrall [Saurel, R. and Abgrall, R., A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows, J. Comput. Phys. 150 (1999), 425 - 467]. This model which consists of conservation laws for each phase complemented with the volume fraction evolution equation is modified by adding a source term to simulate area variation. The model is strictly hyperbolic and non-conservative due to the existence of non-conservative terms. The model is able to deal with compressible and incompressible flows. Moreover, it can deal with mixtures and pure fluids, where each fluid has its own pressure and velocity. The presence of velocity and pressure relaxation terms in the governing equations has made the velocity and pressure relaxation processes essential to tackle the boundary conditions at the interface. The interface separating phases is considered as a numerical diffusion zone in this method. The model is solved using an efficient Eulerian numerical method. A second order Godunov-type scheme with approximate Riemann solver is used to enable capturing of a physical interface by the resolution of the Riemann problem. The solution is obtained by splitting the hyperbolic part and source terms parts in the numerical algorithm. The source terms, including relaxation parts of the model, are tackled in succession using Strang splitting technique. The governing equations are solved at each computational cell using the same numerical algorithm for the whole domain including the interface. The main aim of this work has been to study different flow regimes with respect to pressure boundary conditions through the numerical solutions of single and multiphase flows. The performance of the programme has been verified via well established benchmark test problems for multiphase flows.

Journal of Computational Physics, 2007

In this paper, we propose a new approach to compute compressible multifluid equations. Firstly, a single-pressure compressible multifluid model based on the stratified flow model is proposed. The stratified flow model, which defines different fluids in separated regions, is shown to be amenable to the finite volume method. We can apply the conservation law to each subregion and obtain a set of balance equations 1. Secondly, the AUSM + scheme, which is originally designed for the compressible gas flow, is extended to solve compressible liquid flows. By introducing additional dissipation terms into the numerical flux, the new scheme, called AUSM +-up, can be applied to both liquid and gas flows. Thirdly, the contribution to the numerical flux due to interactions between different phases is taken into account and solved by the exact Riemann solver. We will show that the proposed approach yields an accurate and robust method for computing compressible multiphase flows involving discontinuities, such as shock waves and fluid interfaces. Several one-dimensional test problems are used to demonstrate the capability of our method, including the Ransom's water faucet problem and the air-water shock tube problem. Finally, several two dimensional problems will show the capability to capture enormous details and complicated wave patterns in flows having large disparities in the fluid density and velocities, such as interactions between water shock wave and air bubble, between air shock wave and water column(s), and underwater explosion.