Incompressible Two-phase Flow Research Articles

Rational constitutive law for the viscous stress tensor in incompressible two-phase flows: Derivation and tests against a 3D benchmark experiment

Rational constitutive law for the viscous stress tensor in incompressible two-phase flows: Derivation and tests against a 3D benchmark experiment

Geometric structure of parameter space in immiscible two-phase flow in porous media

In a recent paper, a continuum theory of immiscible and incompressible two-phase flow in porous media based on generalized thermodynamic principles was formulated (Transport in Porous Media, 125, 565 (2018)). In this theory, two immiscible and incompressible fluids flowing in a porous medium are treated as a single effective fluid, substituting the two interacting subsystems for a single system with an effective viscosity and pressure gradient. In assuming Euler homogeneity of the total volumetric flow rate and comparing the resulting first-order partial differential equation to the total volumetric flow rate in the porous medium, one can introduce a novel velocity that relates the two pairs of velocities. This velocity, the co-moving velocity, describes the mutual co-carrying of fluids due to immiscibility effects and interactions between the fluid clusters and the porous medium itself. The theory is based upon general principles of classical thermodynamics and allows for many relations and analogies to draw upon in analyzing two-phase flow systems in this framework. The goal of this work is to provide additional connections between geometric concepts and the variables appearing in the thermodynamics-like theory of two-phase flow. In this endeavor, we will encounter two interpretations of the velocities of the fluids: as tangent vectors (derivations) acting on functions or as coordinates on an affine line. The two views are closely related, with the former viewpoint being more useful in relation to the underlying geometrical structure of equilibrium thermodynamics and the latter being more useful in concrete computations and finding examples of constitutive relations. We apply these relatively straightforward geometric contexts to interpret the relations between velocities and, from this, obtain a general form for the co-moving velocity.

A comprehensive study of two-phase flow in deformable porous media

This study explored the initial-boundary value problem of an immiscible, incompressible two-phase flow within a deformable porous medium system, modeled in an N-dimensional (N≥3) Euclidean space. The primary objective was to devise a novel model that effectively captured previously overlooked physical phenomena and established the global existence of a weak solution to the system. After proving the existence of a classical solution to the approximate problem using Schauder's fixed point theorem under a series of reasonable assumptions, we obtained the convergence to a weak solution of the original problem in two different cases by applying the Fréchet–Kolmogorov theorem. Subsequently, we verified the applicability of this model to the commonly used van Genuchten model in the dynamic simulation of oil storage, providing theoretical support for engineering applications.

A class of high-order physics-preserving schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media

A class of high-order physics-preserving schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media

Discrete Exterior Calculus Discretization of Axisymmetric Incompressible Two-Phase Navier-Stokes Equations with a Conservative Phase Field Method

Abstract We present a discrete exterior calculus (DEC) based on the discretization scheme for axisymmetric incompressible two-phase flows, in which the previous work [39] is extended to its axisymmetric version. We first transform the axisymmetric two-phase incompressible Navier-Stokes (NS) equations and the auxiliary conservative phase field (PF) equation into the exterior calculus framework using differential forms and exterior operators. Discretization of these exterior calculus equations is obtained using discrete differential forms and exterior operators. The PF variable, used to capture the interface between the two phases, varies from zero to unity, and preserving these bounds is desirable. Several verification and validation tests are presented to numerically confirm mass conservation, solution boundedness, and convergence properties. Various axisymmetric two-phase flow simulations, including a drop oscillation, a bubble bursting, a rising bubble, and a drop merger, with large density and viscosity ratios and surface tension, demonstrate the excellent performance of DEC dealing with axisymmetric two-phase flows.

An efficient dimension splitting-based multi-threaded simulation approach for the phase-field model of two-phase incompressible flows

This paper presents a study on the fast numerical simulation of the phase-field model for two-phase incompressible flow, which comprises a coupled system of the Cahn–Hilliard and Navier–Stokes equations. To address the practical challenges posed by high storage demands and computational complexity, we aim to introduce a numerical approach that leverages dimension splitting for parallel and multi-threaded implementation. Specifically, we develop a novel splitting method: First, a projection method with a dimension splitting effect is incorporated to solve the phase-variable-coupled Navier–Stokes equation in parallel. Second, the convective Cahn–Hilliard equation is tackled using a space–time operator splitting scheme. It is confirmed that the proposed method can effectively reduce the huge amount of computation and storage in solving two- and three-dimensional problems. At the same time, it also has the advantages of linearity, space–time second-order accuracy, mass conservation, parallel implementation, and easy programming. The mass conservation property, time complexity, and storage requirement are analyzed. The parallel efficiency is shown by numerical verification. A large number of interesting numerical simulations, such as phase separation, two-phase cavity flow, bubble rising, viscous droplet falling, Kelvin–Helmholtz, and Rayleigh–Taylor instabilities, are performed to show the performance of the method and investigate complex two-phase interface problems.

A consistent interphase mass transfer method and level-set formulation for incompressible two-phase flows

A consistent interphase mass transfer method and level-set formulation for incompressible two-phase flows

A semi-Lagrangian meshfree lattice Boltzmann method for incompressible two-phase flows

A semi-Lagrangian meshfree lattice Boltzmann method for incompressible two-phase flows

Numerical simulation of incompressible two-phase flows with phase change process in porous media

Numerical simulation of incompressible two-phase flows with phase change process in porous media

Second-Order, Energy-Stable and Maximum Bound Principle Preserving Schemes for Two-Phase Incompressible Flow

In this paper, we propose several linear fully discrete schemes for the mass-conserved Allen-Cahn-Navier–Stokes equation, based on the generalized stabilized exponential scalar auxiliary variable approach in time and the marker and cell (MAC) scheme in space. It is quite remarkable that our schemes can guarantee second-order accuracy in space provided the maximum bound principle (MBP) is satisfied, whereas most previous work can only possess first-order accuracy in space. We rigorously show that the constructed schemes satisfy the unconditional energy dissipation law and preserve the MBP. Finally, various numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed schemes.

Thermodynamics-like Formalism for Immiscible and Incompressible Two-Phase Flow in Porous Media

It is possible to formulate an immiscible and incompressible two-phase flow in porous media in a mathematical framework resembling thermodynamics based on the Jaynes generalization of statistical mechanics. We review this approach and discuss the meaning of the emergent variables that appear, agiture, flow derivative, and flow pressure, which are conjugate to the configurational entropy, the saturation, and the porosity, respectively. We conjecture that the agiture, the temperature-like variable, is directly related to the pressure gradient. This has as a consequence that the configurational entropy, a measure of how the fluids are distributed within the porous media and the accompanying velocity field, and the differential mobility of the fluids are related. We also develop elements of another version of the thermodynamics-like formalism where fractional flow rather than saturation is the control variable, since this is typically the natural control variable in experiments.

Convergence of a CVFE finite volume scheme for nonisothermal immiscible incompressible two-phase flow in porous media

Convergence of a CVFE finite volume scheme for nonisothermal immiscible incompressible two-phase flow in porous media

A combined mixed finite element method and discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

A combined mixed finite element method and discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

Two-Phase Incompressible Flow with Dynamic Capillary Pressure in a Heterogeneous Porous Media

We prove the existence of weak solutions of a two-incompressible immiscible wetting and non-wetting fluids phase flow model in porous media with dynamic capillary pressure. This model is a coupled system which includes a nonlinear parabolic saturation equation and an elliptic pressure–velocity equation. In the regularized case, the existence and uniqueness of the weak solution are obtained. We let the regularization parameter be η→0 to prove the existence of weak solutions.

Energy-stable smoothed particles hydrodynamics for the incompressible single-phase and two-phase flows

Energy-stable smoothed particles hydrodynamics for the incompressible single-phase and two-phase flows

Unconditionally maximum principle-preserving linear method for a mass-conserved Allen–Cahn model with local Lagrange multiplier

Unconditionally maximum principle-preserving linear method for a mass-conserved Allen–Cahn model with local Lagrange multiplier

Axisymmetric phase-field-based lattice Boltzmann model for incompressible two-phase flow with phase change

In this work, a phase-field-based lattice Boltzmann equation (LBE) model for axisymmetric two-phase flow with phase change is proposed. Two sets of discrete particle distribution functions are employed to match the conserved Allen–Cahn equation and the hydrodynamic equations with phase change effect, respectively. Since phase change occurs at the interface, the divergence-free condition of the velocity field is no longer satisfied due to mass transfer, and the conserved Allen–Cahn equation needs to be equipped with a source term dependent on the phase change model. To deal with these, a novel source term in the hydrodynamic LBE is delicately designed to recover the correct target governing equations. Meanwhile, the LBE for the Allen–Cahn equation is modified with a discrete force term to model mass transfer. In particular, an additional correction term is added into the hydrodynamic LBE to reduce the spurious velocity and improve numerical stability. Several axisymmetric benchmark multiphase problems with phase change, including bubble growing in superheated liquid, D2 law, film boiling, bubble rising in superheated liquid under gravity, and droplet impact on a hot surface, have been conducted to test the performance of the proposed model. Numerical results agree well with analytical solutions and available published data in the literature.

A numerical model for solitary wave breaking based on the phase-field lattice Boltzmann method

This study presents a numerical investigation of a solitary wave breaking over a slope by using the phase-field lattice Boltzmann method. The incompressible two-phase flow equations are solved by using a velocity-based formulation of the two-phase lattice Boltzmann method with a central-moment collision model to accurately simulate wave breaking problems. For interface capture, a phase-field lattice Boltzmann method that ensures mass conservation is employed. The validity of the proposed method is confirmed through solitary wave propagation and transformation problems, and the obtained results are in good agreement with the experimental and calculated results. The proposed method is then employed to analyze wave breaking on a slope, demonstrating strong concordance with experimental data and existing computational findings. By analyzing the instantaneous flow characteristics and the temporal evolution of the variation in kinetic, potential, and total energy from deep to shallow water, the model can reveal the macroscopic characteristics of solitary wave breaking. Because the phase-field model effectively simulates wave breaking and air entrainment, it can depict wave energy dissipation more accurately than the single-phase lattice Boltzmann method with free surface tracking.

The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system

The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system

Conservative transport model for surfactant on the interface based on the phase-field method

Conservative transport model for surfactant on the interface based on the phase-field method