Incompressible Two-phase Flow Research Articles

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

In this work, we present a conservative Allen–Cahn (CAC) equation and investigate its unconditionally maximum principle-preserving linear numerical scheme. The operator splitting strategy is adopted to split the CAC model into a conventional AC equation and a mass correction equation. The standard finite difference method is used to discretize the equations in space. In the first step, the temporal discretization of the AC equation is performed by using the energy factorization technique. The discrete version of the maximum principle-preserving property for the AC equation is unconditionally satisfied. In the second step, we apply mass correction by using an explicit Euler-type approach. Without the constraint of time step, we estimate that the absolute value of the updated solution is bounded by 1. The unique solvability is analytically proved. In each time step, the proposed method is easy to implement because we only need to solve a linear elliptic type equation and then correct the solution in an explicit manner. Various computational experiments in two-dimensional and three-dimensional spaces are performed to confirm the performance of the proposed method. Moreover, the experiments also indicate that the proposed model can be used to simulate two-phase incompressible fluid flows.

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

In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time.

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

We propose a new surfactant transport model on the interface based on the phase-field method. The new model has two features: (a) preventing surfactants on the interface from leaking out of the interface due to numerical diffusion even when the diffusion coefficient is extremely low, and (b) satisfying the conservation of surfactant mass. The concentration profile on the interface is matched to the surface delta function to prevent numerical diffusion, and the finite volume method is applied to the resulting model to satisfy the conservation of surfactant mass. The proposed model is incorporated in a weakly compressible scheme and applied in several numerical tests related to incompressible two-phase flows with surfactants. Numerical results show that the proposed model can be applied to various incompressible two-phase flow simulations with surfactant.

An Efficient Regularized Lattice Boltzmann Method to Solve Consistent and Conservative Phase Field Model for Simulating Incompressible Two-Phase Flows

An Efficient Regularized Lattice Boltzmann Method to Solve Consistent and Conservative Phase Field Model for Simulating Incompressible Two-Phase Flows

High-Order Bound-Preserving Local Discontinuous Galerkin Methods for Incompressible and Immiscible Two-Phase Flows in Porous Media

High-Order Bound-Preserving Local Discontinuous Galerkin Methods for Incompressible and Immiscible Two-Phase Flows in Porous Media

A Compressible Formulation of the One-Fluid Model for Two-Phase Flows

In this paper, we introduce a compressible formulation for dealing with 2D/3D compressible interfacial flows. It integrates a monolithic solver to achieve robust velocity–pressure coupling, ensuring precision and stability across diverse fluid flow conditions, including incompressible and compressible single-phase and two-phase flows. Validation of the model is conducted through various test scenarios, including Sod’s shock tube problem, isothermal viscous two-phase flows without capillary effects, and the impact of drops on viscous liquid films. The results highlight the ability of the scheme to handle compressible flow situations with capillary effects, which are important in computational fluid dynamics (CFD).

Immiscible Two-Phase Flow in Porous Media: Effective Rheology in the Continuum Limit

AbstractWe consider steady-state immiscible and incompressible two-phase flow in porous media. It is becoming increasingly clear that there is a flow regime where the volumetric flow rate depends on the pressure gradient as a power law with an exponent larger than one. This occurs when the capillary forces and viscous forces compete. At higher flow rates, where the viscous forces dominate, the volumetric flow rate depends linearly on the pressure gradient. This means that there is a crossover pressure gradient that separates these two flow regimes. At small enough pressure gradient, the capillary forces dominate. If one or both of the immiscible fluids percolate, the volumetric flow rate will then depend linearly on the pressure gradient as the interfaces will not move. If none of the fluids percolate, there will be a minimum pressure gradient threshold to mobilize the interfaces and thereby get the fluids moving. We now imagine a core sample of a given size. The question we pose is what happens to the crossover pressure gradient that separates the power-law regime from the high-flow rate linear regime and the threshold pressure gradient that blocks the flow at low pressure gradients when the size of the core sample is increased. Based on analytical calculations using the capillary bundle model and on numerical simulations using a dynamical pore-network model, we find that the crossover pressure gradient and the threshold pressure gradient decrease with two distinct power laws in the size. This means that the power-law regime disappears in the continuum limit where the pores are infinitely small compared to the sample size.

Fully higher-order coupling of finite element and level set methods for two-phase flow with a new explicit projection method

This work presents the development of a higher-order finite element method for modeling unsteady, incompressible and inviscid two-phase flows using the level set method. A new explicit projection method is introduced to solve the incompressible Euler equation, where the pressure and the velocity fields are solved separately. To implement high order temporal schemes, the Hodge decomposition in this projection method is not applied to the velocity but to its time derivative. A discontinuous Galerkin discretization is used to solve both the level set and the nonlinear advection terms present in the Euler equations, while a Galerkin discretization is used to solve the pressure Poisson equation. The present method is a fully high order numerical approach, both in space and temporal dimensions. Numerical results provided by several 2D benchmark test cases in naval hydrodynamics (i.e hydrostatic equilibrium, sloshing and dam break) validate the computational code, illustrating the power of the proposed method. Compared to the CFD codes based on low order VOF/finite-volume methods, the present approach demonstrates greater accuracy on coarse meshes, leading to reduced CPU time.

The subdivision-based IGA-EIEQ numerical scheme for the Navier–Stokes equations coupled with Cahn–Hilliard phase-field model of two-phase incompressible flow on complex curved surfaces

We develop an accurate and robust numerical scheme for solving the incompressible hydrodynamically coupled Cahn–Hilliard system of the two-phase fluid flow system on complex surfaces. Our algorithm leverages a number of efficient techniques, including the subdivision-based isogeometric analysis (IGA) method for spatial discretization, the explicit Invariant Energy Quadratization (EIEQ) method for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling, and the projection method for the Navier–Stokes equation to facilitate fully decoupled type implementations. The integration of these methodologies results in a fully discrete scheme with desired properties such as linearity, second-order temporal accuracy, full decoupling, and unconditional energy stability. The implementation of the scheme is straightforward, requiring the solution of a few elliptic equations with constant coefficients at each time step. The rigorous stability proof of unconditional energy stability and the implementation procedure are given in detail. Numerous numerical simulations on complex curved surfaces are carried out to verify the effectiveness of the proposed numerical scheme.

A lattice Boltzmann model for incompressible gas and liquid two-phase flows combined with free-surface method

A novel lattice Boltzmann (LB) model is proposed to study the gas and liquid two-phase flows with large density and viscosity ratios. In the model, both the gas and liquid phases are considered as viscous incompressible fluids, which are governed separately by the two-relaxation-time LB equations. They are coupled by a momentum exchange method at the interface. The interaction between the gas and liquid phases is explicitly described and naturally involved in the model. The interfacial conditions in the model are validated by the benchmarks of the layered Poiseuille flow and the Laplace law. The feasibility of combining this model with the bubble model and the wetting scheme is proven through transient flow problems of single bubble rising and capillary intrusion. The validity of this model is confirmed by more complex flows including solid–liquid–gas coupling and droplet breaking problems by simulating shearing a droplet on a substrate and a droplet falling on a liquid film. The results demonstrate that the present model can be used to describe both the gas and the liquid flows. This work provides a solution to model the simulation of the dynamical behaviors of multi-phase flows.

A conservative sharp interface method for incompressible viscous two-phase flows with topology changes and large density ratio

Topology change of interfaces often leads to unresolved interface structures such as tiny droplets/bubbles, which requires the regularization of the order parameter and consequently causes artificial modification of volume fraction of the fluids. This could give rise to unphysical oscillation of pressure and cause the failure of computation, in particular for two-phase flows involving large density ratio. We develop a robust conservative sharp-interface method for incompressible viscous two-phase flows with topology changes and large density ratio. The method consists of a cut-cell meshing approach that dynamically generates unstructured meshes in the vicinity of the interface, a second-order finite volume scheme in the Arbitrary Lagrangian–Eulerian framework, and a projection method. The jump conditions at the interfaces are incorporated into the calculation of fluxes at the cell faces of the unstructured meshes that coincide with the reconstructed interfaces, to ensure that they are strictly satisfied at the interface. Unlike a pseudo-compressible formulation used in the previous study, the projection method prevents the oscillation of pressure in the presence of topology change. Special treatments are proposed to further improve the robustness of the method when dealing with unresolved interface structures, including removal of tiny droplets, adjustment of spatial discretization and restriction of the maximum curvature calculated. In such a way, two-phase flows involving topology change and large density ratio can be numerically resolved in a sharp-interface, efficient and robust manner. The performance of the proposed method is systematically examined by a series of numerical tests, including spurious currents, oscillation of a suspended droplet, droplet deformation in shear flows, breakup of a liquid thread, rising bubbles and partial coalescence of a droplet with a pool. The numerical results are compared against analytical solution, benchmark results or experimental data, and good agreement has been achieved.

Discretization schemes for the two simplified global double porosity models of immiscible incompressible two-phase flow

We present the discretization schemes for the two simplified homogenized models of immiscible incompressible two-phase flow in double porosity media with thin fractures. The two models were derived previously by the authors by different linearizations of the nonlinear local problem called the imbibition equation which appears in the homogenized model after passage to the limit as ε → 0. The models are fully homogenized with the matrix-fracture source terms expressed as a convolution.

Numerical simulation for the conserved Allen–Cahn phase field model of two-phase incompressible flows by an efficient dimension splitting method

A second-order, stable, and efficient numerical method is proposed for the simulation of high-dimensional two-phase incompressible flow. The model considers the coupled system of incompressible Navier–Stokes equations with variable density and the conserved Allen–Cahn equations. To save storage and computing costs, a dimension splitting scheme is proposed, which combines a second-order pressure correction method with a splitting effect and an alternating direction implicit method to split a two- or three-dimensional problem into a series of one-dimensional sub-problems that can be solved in parallel. By reconstructing the time derivative terms, the variable coefficient velocity equations are transformed into constant coefficient linear equations. In addition, a second-order stabilization term is added to enhance stability. Finally, a post-processing method is proposed which can preserve both the maximum bound principle and the mass conservation. For space discretization, the weighted essentially non-oscillatory scheme and central difference scheme are used on the marker and cell grid, which can effectively avoid numerical oscillation caused by high Reynolds numbers. Numerical experiments verify the convergence, stability, parallel efficiency, and effectiveness of post-processing, and demonstrate some simulations of two-phase incompressible flows.

Lattice Boltzmann method for studying dynamics of single rising bubble in shear-thickening power-law fluids

Bubble motion in non-Newtonian fluids is widely present in various industrial processes such as crude oil extraction, enhancement of boiling heat transfer, CO<sub>2</sub> sequestration and wastewater treatment. System containing non-Newtonian liquid, as opposed to Newtonian liquid, has shear-dependent viscosity, which can change the hydrodynamic characteristics of the bubbles, such as their size, deformation, instability, terminal velocity, and shear rate, and ultimately affect the bubble rising behaviors. In this work, the dynamic behavior of bubble rising in a shear-thickened fluid is studied by using an incompressible lattice Boltzmann non-Newtonian gas-liquid two-phase flow model. The effects of the rheological exponent <i>n</i>, the Eötvös number (<i>Eo</i>), and the Galilei number (<i>Ga</i>) on the bubble deformation, terminal velocity, and the shear rate are investigated. The numerical results show that the degree of bubble deformation increases as <i>Eo</i> grows, and the effect of <i>n</i> on bubble deformation degree relates to <i>Ga</i>. On the other hand, the terminal velocity of the bubbles increases monotonically and nonlinearly with <i>Ga</i> for given <i>Eo</i> and <i>n</i>, and the effect of <i>n</i> on the terminal velocity of the bubbles turns stronger as <i>Ga</i> increases. When <i>Ga</i> is fixed and small, the terminal velocity of the bubble increases and then decreases with the increase of <i>n</i> at small <i>Eo</i>, and increases with the increase of <i>n</i> when <i>Eo</i> is large; but when <i>Ga</i> is fixed and large, the terminal velocity of the bubbles increases with the increase of <i>n</i> in a more uniform manner. In addition, regions with high shear rates can be found near the left end and right end of the bubble. The size of these regions grows with <i>Eo</i> and <i>Ga</i>, exhibiting an initial increase followed by a decrease as <i>n</i> increases. Finally, the orthogonal experimental method is used to obtain the influences of the aforementioned three factors on the shear rate and terminal velocity. The order of influence on shear rate is <i>n</i>, <i>Ga</i> and <i>Eo</i> which are arranged in descending order. For the terminal velocity, <i>Ga</i> has the greatest influence, followed by <i>n</i>, and <i>Eo</i> has the least influence.

Decay properties of the Stokes semigroup for two-phase incompressible viscous flows

Decay properties of the Stokes semigroup for two-phase incompressible viscous flows

The extended Discontinuous Galerkin method for two-phase flows with evaporation

In this work, we present an extended Discontinuous Galerkin method for simulating transient, incompressible two-phase flows, which include heat transfer and thermally driven evaporation at the interface of single-component systems. This expands our previous work to include the consideration of non-material interfaces and a coupling between velocities and temperature gradients at the interface. The phase boundary is represented by the zero-set of a level set function, while effects due to surface tension are treated by the Laplace-Beltrami formulation. This sharp interface model allows for a sub-grid accurate representation of the solution fields. By using compactly supported polynomial solutions, discontinuities at the interface can be sharply represented without employing additional reconstruction schemes. The approach is validated through well-known evaporation test cases. This includes two 1D test cases, known as Stefan and Sucking problem, a 2D film boiling and finally the 3D growth of a vapor bubble, known as Scriven test case.