D. Zeidan

Validation of hyperbolic model for two-phase flow in conservative form

A mathematical formulation is proposed for the solution of equations governing isentropic gas–liquid flow. The model considered here is a two-fluid model type where the relative velocity between the two phases is implemented by a kinetic constitutive equation. Starting from the conservation of mass and momentum laws, a system of three differential equations is derived in a conservative form for the three principal variables, which are mixture density, mixture velocity and the relative velocity. The governing equations for the mixture offer the novel hyperbolic conservation laws for the description of two-phase flows without any conventional source terms in the momentum or relative velocity equations. The discretisation of the governing equations is based on splitting approach, which is specially designed to allow a straightforward extension to various numerical methods such as Godunov methods of centred-type. To verify the validity of the model, numerical results are presented and discussed. It is demonstrated that the proposed numerical methods have superior overall numerical accuracy among existing methods and models in the literature. The model correctly describes the formation of shocks and rarefactions for the solution of discontinuities in two-phase fluid flow problems, thus verifying the proposed mathematical and numerical investigations.

Numerical resolution for a compressible two-phase flow model based on the theory of thermodynamically compatible systems

This paper is concerned with the numerical solution of the equations governing two-phase gas-solid mixture in the framework of thermodynamically compatible systems theory. The equations constitute a non-homogeneous system of nonlinear hyperbolic conservation laws. A total variation diminishing (TVD) slope limiter centre (SLIC) numerical scheme, based on the splitting approach, is presented and applied for the solution of the initial-boundary value problem for the equations. The model equations and the numerical methods are systematically assessed through a series of numerical test cases. Strong numerical evidence shows that the model and the methods are accurate, robust and conservative. The model correctly describes the formations of shocks and rarefactions in two-phase gas-solid flow.

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

This article is to continue the present authors 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 Lie groups to compressible model of two-phase flows

This paper presents some exact solutions for the drift-flux model of two-phase flows using Lie group analysis. The analysis involves an isentropic no-slip conservation of mass for each phase and the conservation of momentum for the mixture. The present analysis employs a complete Lie algebra of infinitesimal symmetries. Subsequent to these theoretical analysis a symmetry group is established. The symmetry generators are used for constructing similarity variables which reduce the model equations to a system of ordinary differential equations (ODEs). In particular, a general framework is discussed for solving the model equations analytically. As a consequence of this, new classes of exact group-invariant solutions are developed. This provides new insights into the fundamental properties of weak discontinuities and helps one to understand better on existence of solutions.

NUMERICAL SOLUTION FOR HYPERBOLIC CONSERVATIVE TWO-PHASE FLOW EQUATIONS

We outline an approximate solution for the numerical simulation of two-phase fluid flows with a relative velocity between the two phases. A unified two-phase flow model is proposed for the description of the gas–liquid processes which leads to a system of hyperbolic differential equations in a conservative form. A numerical algorithm based on a splitting approach for the numerical solution of the model is proposed. The associated Riemann problem is solved numerically using Godunov methods of centered-type. Results show the importance of the Riemann problem and of centered schemes in the solution of the two-phase flow problems. In particular, it is demonstrated that the Slope Limiter Centered (SLIC) scheme gives a low numerical dissipation at the contact discontinuities, which makes it suitable for simulations of practical two-phase flow processes.

A Robust and accurate Riemann solver for a compressible two-phase flow model

In this paper we analyze the Riemann problem for the widely used drift-flux two-phase flow model. This analysis introduces the complete information that is attained in the representation of solutions to the Riemann problem. It turns out that the Riemann waves have rarefactions, a contact discontinuity and shocks. Within this respect, an exact Riemann solver is developed to accurately resolve and represent the complete wave structure of the gas-liquid two-phase flows. To verify the solver, a series of test problems selected from the literature are presented including validation against independent numerical simulations where the solution of the Riemann problem is fully numerical. In this framework the governing equations are discretized by finite volume techniques facilitating the application Godunov methods of centred-type. It is shown that both analytical and numerical results demonstrate the broad applicability and robustness of the new exact Riemann solver.

Numerical simulation of unsteady cavitation in liquid hydrogen flows

An unsteady cavitation model in liquid hydrogen flow is studied in the context of compressible, two-phase, one-fluid inviscid solver. This is accomplished by applying three conservation laws for mixture mass, mixture momentum and total energy along with gas volume fraction transport equation, with thermodynamic effects. Various mass transfers between phases are utilised to study the process under consideration. A numerical procedure is presented for the simulation of cavitation due to rarefaction and shock waves. Attention is focused on cavitation in which the simulated fluid is liquid hydrogen in cryogenic conditions. Numerical results are in close agreement with theoretical solutions for several test cases. The current numerical results show that liquid hydrogen flow can be accurately modelled using an accurate inviscid approach to describe the features of thermodynamic effects on cavitation.

On the Riemann Problem Simulation for the Drift-Flux Equations of Two-Phase Flows

This work presents computational simulations and analytical techniques for solving the drift-flux two-phase flow model. The model equations are formulated to describe the exact solution of the Riemann problem. The solution is constructed by solving the conservation of mass for each phase and the mixture conservation momentum equation of the two phases under isothermal conditions. Particular attention is given to address the expressions for jump relationships and the Riemann invariants. The performance of the developed Riemann solver is assessed with respect to different test cases selected from the literature. Comparisons with Godunov methods of centred-type are provided to demonstrate the use of the proposed exact and computational framework. Excellent agreement is observed between analytical results and numerical predictions.

Central finite volume schemes on nonuniform grids and applications

We propose a new one-dimensional unstaggered central scheme on nonuniform grids for the numerical solution of homogeneous hyperbolic systems of conservation laws with applications in two-phase flows and in hydrodynamics with and without gravitational effect. The numerical base scheme is a generalization of the original Lax-Friedrichs scheme and an extension of the Nessyahu and Tadmor central scheme to the case of nonuniform irregular grids. The main feature that characterizes the proposed scheme is its simplicity and versatility. In fact, the developed scheme evolves a piecewise linear numerical solution defined at the cell centers of a nonuniform grid, and avoids the resolution of the Riemann problems arising at the cell interfaces, thanks to a layer of staggered cells used intermediately. Spurious oscillations are avoided using a slopes limiting procedure. The developed scheme is then validated and used to solve classical problems arising in gas-solid two phase flow problems. The proposed scheme is then extended to the case of non-homogenous hyperbolic systems with a source term, in particular to the case of Euler equations with a gravitational source term. The obtained numerical results are in perfect agreement with corresponding ones appearing in the recent literature, thus confirming the efficiency and potential of the proposed method to handle both homogeneous and non-homogeneous hyperbolic systems.

Some issues in the simulation of two-phase flows: The relative velocity

In this paper we compare numerical approximations for solving the Riemann problem for a hyperbolic two-phase flow model in two-dimensional space. The model is based on mixture parameters of state where the relative velocity between the two-phase systems is taken into account. This relative velocity appears as a main discontinuous flow variable through the complete wave structure and cannot be recovered correctly by some numerical techniques when simulating the associated Riemann problem. Simulations are validated by comparing the results of the numerical calculation qualitatively with OpenFOAM software. Simulations also indicate that OpenFOAM is unable to resolve the relative velocity associated with the Riemann problem.