A volume-of-fluid method for incompressible free surface flows

This paper proposes a hybrid volume-of-fluid (VOF) level-set method for simulating incompressible two-phase flows. Motion of the free surface is represented by a VOF algorithm that uses high resolution differencing schemes to algebraically preserve both the sharpness of interface and the boundedness of volume fraction. The VOF method is specifically based on a simple order high resolution scheme lower than that of a comparable method, but still leading to a nearly equivalent order of accuracy. Retaining the mass conservation property, the hybrid algorithm couples the proposed VOF method with a level-set distancing algorithm in an implicit manner when the normal and the curvature of the interface need to be accurate for consideration of surface tension. For practical purposes, it is developed to be efficiently and easily extensible to three-dimensional applications with a minor implementation complexity. The accuracy and convergence properties of the method are verified through a wide range of tests: advection of rigid interfaces of different shapes, a three-dimensional air bubble's rising in viscous liquids, a two-dimensional dam-break, and a three-dimensional dam-break over an obstacle mounted on the bottom of a tank. The standard advection tests show that the volume advection algorithm is comparable in accuracy with geometric interface reconstruction algorithms of higher accuracy than other interface capturing-based methods found in the literature. The numerical results for the remainder of tests show a good agreement with other numerical solutions or available experimental data. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical investigations of the XFEM for solving two-phase incompressible flows

This paper discusses the application of the extended finite element method (XFEM) to solve two-phase incompressible flows. The Navier–Stokes equations are discretised using the Taylor–Hood finite element. To capture the different discontinuities across the interface, kink or jump enrichments are used for the velocity and/or pressure fields. However, these enrichments may lead to an inappropriate combination of interpolations. Different polynomial enrichment orders and different enrichment functions are investigated; only the stable combination will be used afterward.

In cases with a surface tension force, the accuracy mainly relies on the precise computation of the normal and curvature. A novel method for computing normal vectors to the interface is proposed. This method employs successive mesh refinements inside the cut elements. Comparisons with analytical and numerical solutions demonstrate that the method is effective. Moreover, the mesh refinement improves the sub-integration in the XFEM and allows for a precise re-initialisation procedure.     

Numerical simulation of 1-D unsteady two-phase flows with shocks

In the present paper, random-choice method (RCM) and second-order GRP difference method, which are high resolution methods used for pure gas flows with shocks, are extended and employed to study the problem of one-dimensional unsteady two-phase flows. The two-phase shock wave and the flow field behind it in a dusty gas shock tube are calculated and the time-dependent change of the fiow parameters for the gas antiparticle phase are obtained. The numerical results indicate that both the two methods can give the relaxation structure of the two-phase shocks with a sharp discontinuous front and that the GRP method has the advantages of less time-consuming and higher accuracy over the RCM method.     

Lie symmetry solutions for two-phase mass flows

We apply Lie symmetry method to a set of non-linear partial differential equations, which describes a two-phase rapid gravity mass flow as a mixture of solid particles and viscous fluid down a slope (Pudasaini, J. Geophys. Res. 117 (2012) F03010, 28 pp [1]). In order to systematically explore the mathematical structure and underlying physics of the two-phase mixture flow, we generate several similarity forms in general form and construct self-similar solutions. Our analysis generalizes the results, obtained by applying the Lie symmetry method to relatively simple single-phase pressure-driven gravity mass flows, to the two-phase mass flows that include several dominant driving forces and strong phase-interactions. Analytical and numerical solutions are presented for the symmetry-reduced homogeneous and non-homogeneous systems of equations. Analytical and numerical results show that the new models presented here can adequately describe the dynamics of two-phase debris flows, and produce observable phenomena that are consistent with the physics of the flow. The solutions are strongly dependent on the choice of the symmetry-reduced model, as characterized by the group parameters, and the physical parameters of the flows. These solutions reveal strong non-linear and distinct dynamic evolutions, and phase-interactions between the solid and fluid phases, namely the phase-heights and phase-velocities.     

An interface-capturing method for incompressible two-phase flows. Validation and application to bubble dynamics

We report on the development and applications of an interface-capturing method aimed at computing three-dimensional incompressible two-phase flows involving high density and viscosity ratios, together with capillary effects. The numerical approach borrows some features to the Volume of Fluid method (since it is essentially based on the transport of the local volume fraction of the liquid) as well as to the Level Set technique (as no explicit reconstruction of the interface is carried out). The transport of the volume fraction is achieved by using a flux-limiting Zalesak scheme and the fronts are prevented from spreading in time by a specific strategy in which the velocity at nodes crossed by the interface is modified to keep the thickness of the transition region constant. As shown on several test cases, this algorithm allows the interface to deform properly while maintaining the numerical thickness of the transition region within three computational cells whatever the structure of the local flow field. The full set of governing equations is then used to investigate some fundamental aspects of bubble dynamics. More precisely we focus on the evolution of shape and rise velocity of a single bubble over a wide range of physical parameters and on head-on and side-by-side interactions between two rising bubbles.     

The use of drift-flux techniques for the analysis of horizontal two-phase flows

New data is presented for horizontal air/water two-phase flow having various flow regimes. It is shown that drift-flux models are able to correlate these data and that the drift velocity, V_gj, is normally finite.     

Unified macro-to-microscale method to predict two-phase frictional pressure drops of annular flows

The study considers the prediction of pressure gradients in adiabatic gas–liquid annular two-phase flow in the macro-to-microscale range. Twenty-four empirical correlations have been tested against an experimental data bank drawn together in this study containing 3908 points for eight different gas–liquid combinations and 22 different tube diameters, covering microscale and macroscale channels from 0.517 to 31.7 mm in diameter. The correlations of Lombardi, Friedel and Baroczy-Chisholm were found to be the best existing methods when considering macroscale data only, while the microscale database was best predicted by the correlations of Lombardi, Müller-Steinhagen and Heck and the homogeneous model with the two-phase viscosity defined according to Cicchitti. A new correlating approach based on the vapor core Weber number, capable of providing physical insight into the flow, was proposed and worked better than any of the existing methods for the macroscale database. This new macroscale method was then extended to cover microscale conditions, resulting in one unified method for predicting annular flows from the macroscale to the microscale covering both laminar and turbulent liquid films. The macroscale method optimized for microchannels worked better than any of the other methods considered.     

Time marching integral equation method for unsteady transonic flows

In this paper, we have proposed a time marching intregral equation method which does not have the limitation of the time linearized integral equation method in that the latter method can not satisfactorily simulate the shock-wave motions. Firstly, a model problem—one dimensional initial and boundary value wave problem is treated to clarify the basic idea of the new method. Then the method is implemented for 2-D and 3-D unsteady transonic flow problems. The introduction of the concept of a quasi-velocity-potential simplifies the time marching integral equations and the treatment of trailing vortex sheet condition. The numerical calculations show that the method is reasonable and reliable.     

A SOLVER USING NEWTON‘S METHOD FOR UNSTEADY VISCOUS FLOWS

A solver is developed for time-accurate computations of viscous flows based on the conception of Newton‘s method. A set of pseudo-time derivatives are added into governing equations and the discretized system is solved using GMRES algorithm. Due to some special properties of GMRES algorithm, the solution procedure for unsteady flows could be regarded as a kind of Newton iteration. The physical-time derivatives of governing equations are discretized using two different approaches, I.e., 3-point Euler backward, and Crank-Nicolson formulas, both with 2nd-order accuracy in time but with different truncation errors. The turbulent eddy viscosity is calculated by using a version of Spalart~Allmaras one-equation model modified by authors for turbulent flows. Two cases of unsteady viscous flow are investigated to validate and assess the solver, I.e., low Reynolds number flow around a row of cylinders and transonic bi-circular-arc airfoil flow featuring the vortex shedding and shock buffeting problems, respectively. Meanwhile, comparisons between the two schemes of timederivative discretizations are carefully made. It is illustrated that the developed unsteady flow solver shows a considerable efficiency and the Crank-Nicolson scheme gives better results compared with Euler method.     

The setting up of the equations for two-phase flows by the kinetic theory method

The fundamental equations for two-phase flows are deduced from the Boltzmann's equation. The collision terms are treated with a method similar to what is used in the classical kinetic theory for handling the transport properties of dense gases. It is shown that collision pressure and collision thermal flux exist in gas-particle flows in addition to the general partial pressure and partial thermal flux. Their physical natures are quite different from those of the general partial pressure and partial thermal flux. The applicability of the binary collision assumption and the molecular chaos assumption to gas-particle flows is also discussed. Finally, the equations for two-phase flows obtained by the method of the kinetic theory are compared with those obtained by average continuum models and by the model of particle clouds. The results from the kinetic theory show clearly the physical significance of various parameters and clarify some confusing concepts. Institute of Mechanics, Academia Sinica     

THE SOLUTION FOR TRANSIENT TWO-PHASE FLOW BY SPLIT FLUX VECTOR METHOD

In this paper the transient two-phase flow equations and their eigenvalues are first intreduced. The flux vector is then split into subvectors which just contain a specially signed eigenvalue. Using one-sided spatial difJerence operators finite difference equations and their solutions are obtained. Finally comparison with experiment shows the predicted results produce good agreement with experimental data.     

Averaged topological equations for dispersed two-phase flows

A general relationship between the volume fraction and the specific interfacial area for averaged dispersed two-phase flows is proposed. This relationship, expressed as a basic set of two scalar evolution equations and two vectorial non-uniformity state equations, is an analytical result obtained by a systematic approach using the derivatives of some generalized functions and a local volume-averaging technique. The proposed set of equations was expressed for measurable macroscopic parameters of the system and has the same generality as the averaged transport equations of two-phase flows. By combination of the basic set of equations, called the averaged topological equations (ATEs), second-order ATEs for the volume fraction were found. The second-order ATEs were expressed both by a Lagrangian formulation and by a Eulerian formulation. The importance and physical meaning of the ATEs developed in this study were clarified within the framework of the theory of kinematic waves.     

modeling of multicomponent multiphase mixture flows on the basis of the density-functional method

Examples of numerical calculations of isothermal flows of two-phase two-component mixtures based on the density-functional method are presented. Using this method, the following problems are calculated in the two-dimensional formulation: drop impact on a liquid layer, drop rupture in a Couette flowfield, wetting-angle formation for a drop on a solid surface, development of Rayleigh-Taylor and Kelvin-Helmholtz instability on a gas-liquid interface.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 101–114.Original Russian Text Copyright © 2004 by Demyanov and Dinariev.     

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.     

Numerical analysis of density flows in adiabatic two-phase fluids through characteristic finite element method

The purpose of this study is to analyze the density flow in adiabatic two-phase fluids through the characteristic finite element method. The fluids are assumed to be liquids. The equations of conservations of mass and momentum for the adiabatic flows and the Birch–Murnaghan equation of state are employed as the governing equations. The employed finite element method is a combination of the characteristic method and the implicit method. The governing equations are divided into two parts: the advection part and the non-advection part. The characteristic method is applied to the advection part. The Hermite interpolation function, which is based on the complete third-order polynomial interpolation using triangular finite element is employed for the interpolation of both velocity and density. Using the discontinuity conditions, an interface translocation method can be derived. The interface of the two flow densities are interpolated through the third-order spline function, using which the curvature of the interface can be directly computed. For the numerical study, the development of density flow over the Tokyo bay is presented. It is detected out that high density area is abruptly diffused over the whole area. According to the differences in the two densities, various flow patterns are computed.     

Improved subgrid scale model for dense turbulent solid-liquid two-phase flows

The dense solid-phase governing equations for two-phase flows are obtained by using the kinetic theory of gas molecules. Assuming that the solid-phase velocity distributions obey the Maxwell equations, the collision term for particles under dense two-phase flow conditions is also derived. In comparison with the governing equations of a dilute two-phase flow, the solid-particle‘s governing equations are developed for a dense turbulent solid-liquid flow by adopting some relevant terms from the dilute two-phase governing equations. Based on Cauchy-Helmholtz theorem and Smagorinsky model, a second-order dynamic sub-grid-scale (SGS) model, in which the sub-grid-scale stress is a function of both the strain-rate tensor and the rotation-rate tensor, is proposed to model the two-phase governing equations by applying dimension analyses. Applying the SIMPLEC algorithm and staggering grid system to the two-phase discretized governing equations and employing the slip boundary conditions on the walls, the velocity and pressure fields, and the volumetric concentration are calculated. The simulation results are in a fairly good agreement with experimental data in two operating cases in a conduit with a rectangular cross-section and these comparisons imply that these models are practical.     

Improved THINC/SW scheme for computing incompressible two-phase flows

In this paper, we propose an improved tangent of hyperbola for interface capturing with slope weighting (THINC/SW) scheme for computing incompressible two-phase flows with surface tension on fixed Eulerian grids. The new scheme possesses the following major new properties in comparison with the original THINC/SW scheme: (i) providing a simple and accurate approach for generating a smooth level set (LS) field from the discontinuous volume-of-fluid field, (ii) determining the interface slope from the cogenerated LS field, and (iii) evaluating the surface tension force by the cogenerated LS field. We verified the proposed scheme with the widely used advection benchmark tests and multifluid simulations. Numerical results reveal that the new scheme can not only improve the solution quality of interface capturing but also increase the accuracy of curvature estimates and suppress the parasitic currents.     

Numerical simulation and homogenization of two-phase flow in heterogeneous porous media

A mathematically rigorous method of homogenization is presented and used to analyze the equivalent behavior of transient flow of two incompressible fluids through heterogeneous media. Asymptotic expansions and H-convergence lead to the definition of a global or effective model of an equivalent homogeneous reservoir. Numerical computations to obtain the homogenized coefficients of the entire reservoir have been carried out via a finite element method. Numerical experiments involving the simulation of incompressible two-phase flow have been performed for each heterogeneous medium and for the homogenized medium as well as for other averaging methods. The results of the simulations are compared in terms of the transient saturation contours, production curves, and pressure distributions. Results obtained from the simulations with the homogenization method presented show good agreement with the heterogeneous simulations.     

低浓度固液两相流的颗粒相动理学模型

用广义Fokker-Planck扩散模型描述液相湍动对颗粒的挟带作用，用修正的BGK模型描述粒间碰撞效应，建立了封闭的颗粒相PDF输运方程．运用Chapman-Enskog迭代法求得方程的二阶近似解，获得颗粒相脉动速度二阶矩和三阶矩闭合关系．模型与颗粒流模型相容，与液相湍流闭合模型是否相容依赖于扩散模型的具体形式，并据此比较了不同的涡一颗粒作用模型．模型与二维明渠流轻质沙和天然沙试验资料符合很好．表明细小粒径颗粒能够充分跟随水流运动；大粒径颗粒的相间平均速度差和壁面滑移速度明显，近壁区内的颗粒沿流向和垂向脉动强度都可能大于水流，并存在一定程度的颗粒碰撞效应．     

Origin and quantification of coupling between relative permeabilities for two-phase flows in porous media

An extended formulation of Darcy's two-phase law is developed on the basis of Stokes' equations. It leads, through results borrowed from the thermodynamics of irreversible processes, to a matrix of relative permeabilities. Nondiagonal coefficients of this matrix are due to the viscous coupling exerted between fluid phases, while diagonal coefficients represent the contribution of both fluid phases to the total flow, as if they were alone. The coefficients of this matrix, contrary to standard relative permeabilities, do not depend on the boundary conditions imposed on two-phase flow in porous media, such as the flow rate. This formalism is validated by comparison with experimental results from tests of two-phase flow in a square cross-section capillary tube and in porous media. Coupling terms of the matrix are found to be nonnegligible compared to diagonal terms. Relationships between standard relative permeabilities and matrix coefficients are studied and lead to an experimental way to determine the new terms for two-phase flow in porous media.