Incompressible Two-phase Flow Research Articles

A conservative sharp interface method for two-dimensional incompressible two-phase flows with phase change

A conservative sharp interface method is proposed in this work to simulate two-dimensional/axisymmetric incompressible two-phase flows with phase change. In this method, we use the cut cell method to generate unstructured meshes near the interface, of which the cell edges overlap with the interface at each time step. On such mesh, the mass and heat transfer during phase change and all the jump conditions can be incorporated into the calculation of fluxes at the cell edges, to ensure that they are strictly satisfied at the interface in a sharp manner. The governing equations, including the incompressible Navier–Stokes equations, heat equation, and vapor mass fraction equation, are discretized by a second-order finite volume method in the arbitrary Lagrangian–Eulerian framework. To well couple the mass, heat, momentum, and interface evolution, the solution procedure is carefully designed and performed with several techniques. In such a way, the sharp discontinuity of the velocity, stress, temperature gradient, and vapor fraction, caused by the mass/heat transfer during phase change, can be simulated accurately and robustly. The performance of this method is systematically examined by cases of phase change at or below the saturated temperature, including vapor bubble in superheated liquid, film boiling, droplet evaporation at different relative humidity conditions, droplet evaporation under gravity, and droplet evaporation under forced convection. The applicability of the present method for incompressible two-phase flows with phase change is well demonstrated by comparing the numerical results with the benchmark, theoretical or experimental ones.

Quantitative comparison of dam break and bubble flows based on CF-VOF and IR-VOF schemes

In this study, an incompressible and immiscible two-phase flow solver was employed to assess the accuracy of phase-interface capturing methods in simulating two-phase flows. Three benchmark investigations were conducted involving two dam break flows with deformable free surfaces and the upward motion of a gas bubble through a liquid column. Four phase-interface capturing schemes, namely, MULES, HRIC and CICSAM from the Color Function Volume of Fluid (CF-VoF), and Interface Reconstruction Volume of Fluid (IR-VoF), were utilized. Comparative analysis of the phase interfaces indicated that CICSAM and IsoAdvector consistently provided sharp interfaces across all cases when compared to MULES and HRIC. The mass conservation performance of all schemes excelled in the dam-break case with relatively simple free-surface development and the rising bubble case. An examination of the local pressure distribution over time revealed that methods yielding diffusive interfaces produced inaccurate results. Thus, the importance of employing denser grids or schemes that yield sharper interfaces, such as CICSAM and IsoAdvector, to enhance simulation accuracy was underscored. Overall, the results of this analysis confirm that the IsoAdvector scheme, which provides sharp interfaces, is a reliable option for simulating incompressible immiscible two-phase flows.

TwoPhaseFlow: A Framework for Developing Two Phase Flow Solvers in OpenFOAM

We present a new OpenFOAM based open-source framework, TwoPhaseFlow, enabling fast implementation and testing of new phase change and surface tension force models for two-phase flows including interfacial heat and mass transfer. Capitalizing on the runtime-selection mechanism in OpenFOAM, the new models can easily be selected and benchmarked against analytical solutions and existing models. The framework currently includes the following three interface curvature calculation methods for surface tension: 1) the height function method, 2) the parabolic fit method and 3) the reconstructed distance function method. As for phase change, two models are available: 1) Interface heat resistance and 2) direct heat flux. These can be combined in three solvers: 1) InterFlow for isothermal, incompressible two-phase flow, 2) compressibleInterFlow for compressible, non-isothermal two-phase flow and 3) multiRegionPhaseChangeFlow for compressible, non-isothermal two-phase flow with conjugated heat transfer. By design, addition of new models and solvers is straightforward and users are encouraged to contribute their specific models, solvers, and validation cases to the library.

Some remarks on simplified double porosity model of immiscible incompressible two-phase flow

The paper is devoted to the derivation, by linearization, of simplified homogenized models of an immiscible incompressible two-phase flow in double porosity media in the case of thin fissures. In a simplified double porosity model derived previously by the authors the matrix-fracture source term is approximated by a convolution type source term. This approach enables to exclude the cell problem, in form of the imbibition equation, from the global double porosity model. In this paper we propose a new linear version of the imbibition equation which leads to a new simplified double porosity model. We also present numerical simulations which show that the matrix-fracture exchange term based on this new linearization procedure gives a better approximation of the exact one than the corresponding exchange term obtained earlier by the authors.

Investigations on the shallow water wave attenuation over continuous porous oyster reef-like structure

Observations over the decades have found that oyster reefs in shallow estuarine environments have the potential to protect shorelines or offshore structures by dissipating wave energy. Laboratory experiments were conducted to observe the shallow water wave attenuation in different terrains of the continuous porous oyster reef-like structures. To better reproduce this process, a numerical wave flume that is within the OpenFOAM scheme was developed in this study. The volume-averaged Reynolds-Averaged Navier–Stokes (VARANS) equations were solved for two-phase incompressible flow around the porous structures and the VOF method was used for tracking the free surface. The model was first validated by the present laboratory experiments. Subsequently, dimensional analysis was conducted to address the factors influencing wave attenuation over continuous porous reef-like structures. The model was then applied to evaluate the impacts of wave factors and morphological factors of porous oyster reef-like structures on the wave attenuation. By the end, an empirical formula was proposed to predict the wave transmission coefficient based on the numerical results and the experimental data.

Inverse asymptotic treatment: Capturing discontinuities in fluid flows via equation modification

A major challenge in developing accurate and robust numerical solutions to multi-physics problems is to correctly model evolving discontinuities in field quantities. These manifest themselves as interfaces between different phases in multi-phase flows, or as shock and contact discontinuities in (either single- or multi-phase) compressible flows. A plethora of bespoke discretization schemes have been developed to capture both types of discontinuities in physics-based simulations. However, when a quick response is required to rapidly emerging challenges such as the need to design novel hypersonic vehicles, the complexity of such implementations impedes a swift transition from problem formulation to computation. This is exacerbated by the need to compose multiple interacting physics, which may cause specialized schemes to break unexpectedly as governing equations need to be changed and/or added. We introduce “inverse asymptotic treatment” (IAT) as a unified framework for capturing discontinuities in fluid flows that enables building directly computable models based on standard, off-the-shelf numerics. By capturing discontinuities through modifications at the level of the governing equations, rather than via reliance on specialized discretization schemes, IAT can seemlessly handle additional physics and thus makes it easier for novice end users to quickly obtain numerical results for a variety of multi-physics scenarios. This strategy also facilitates the use of a “multi-physics” compiler that automates the conversion of the modified PDEs to numerical source code readable by popular computational frameworks like OpenFOAM. We outline IAT in the context of phase-field modeling of two-phase incompressible flows, and then demonstrate its generality by showing how localized artificial diffusivity (LAD) methods for single-phase compressible flows can be viewed as instances of IAT. Through the real-world example of a laminar hypersonic compression corner, we illustrate IAT’s ability to, within a span of just a few months, generate a directly computable model whose wall metrics predictions for sufficiently small corner angles come close to that of NASA’s state-of-the-art VULCAN-CFD solver. Finally, we propose a novel LAD approach via “reverse-engineered” PDE modifications, inspired by total variation diminishing (TVD) flux limiters, to eliminate the problem-dependent parameter tuning that plagues traditional LAD. Through canonical numerical tests, we demonstrate that this “limiter-inspired” LAD approach, when combined with second-order central differencing, can robustly and accurately model compressible flows.

On a Navier–Stokes–Cahn–Hilliard system for viscous incompressible two-phase flows with chemotaxis, active transport and reaction

On a Navier–Stokes–Cahn–Hilliard system for viscous incompressible two-phase flows with chemotaxis, active transport and reaction

Fully discrete, decoupled and energy-stable Fourier-Spectral numerical scheme for the nonlocal Cahn–Hilliard equation coupled with Navier–Stokes/Darcy flow regime of two-phase incompressible flows

In this paper, we introduce fully discrete Fourier-Spectral numerical scheme to solve the nonlocal Cahn–Hilliard equation coupled with Navier–Stokes/Darcy equations, which represent a phase-field model for two-phase incompressible flow in either the free flow regime or a Hele-Shaw cell. The proposed scheme achieves full decoupling while maintaining linearity and energy stability through a combination of the Scalar Auxiliary Variable (SAV) method, which discretizes the nonlinear potential, and the “Zero-Energy-Contribution” (ZEC) method, which handles the coupled nonlinear advective/surface tension terms. The efficiency of this scheme is attributed to its linear decoupling structure and the fact that it requires only a few elliptic equations with constant coefficients to be solved at each time step. We rigorously establish the scheme’s unconditional energy stability. Further, some numerical simulations are provided in both 2D and 3D to show its effectiveness, including its accuracy, stability, and efficiency.

Flow and mixing dynamics in face-to-face and rear-end collisions of pairs of equal-sized droplets

In this work, the behaviors of pairs of equal-sized droplets in rear-end and face-to-face collisions were simulated using the improved lattice Boltzmann method for incompressible two-phase flows. First, the time evolution of the droplet shape was investigated by tracing colored particles, and this was compared between the rear-end and face-to-face collisions. For collinear collisions, the droplet shapes in the rear-end collisions were found to be similar to those in the face-to-face collisions. However, the behaviors of the tracer particles were different: the droplets in the rear-end collisions mixed more easily than those in the face-to-face collisions. For offset collisions, it was found that the rolling motion of the coalesced droplet accelerates the mixing inside it in both face-to-face and rear-end collisions. A new index—the total mixing intensity—was introduced, and the droplet mixing can be quantitatively evaluated by calculating its value. The results indicate that the droplet mixing process of a collinear collision can be characterized by the velocity ratio, which is defined as the ratio of the center-of-mass velocity to the relative impact velocity.

A conservative finite volume method for incompressible two-phase flows on unstructured meshes

A robust finite volume framework for solving incompressible two-phase flow problems based on an explicit dual-time stepping based artificial compressibility formulation has been developed by Parameswaran and Mandal (2019) for structured meshes. This artificial compressibility framework is extended to unstructured meshes in the present paper. In order to further enhance the applicability to real life engineering problems, the surface tension effect is also modeled here. To deal with arbitrary unstructured mesh topology, least square and Green-Gauss integral based methods are used. The efficacy of the proposed method is demonstrated using a set of scalar advection based test problems and a set of standard incompressible two-phase flow test problems involving gravitational, viscous and surface tension forces. The numerical results show improvement in accuracy compared to the results of the structured mesh formulation.

Fully-coupled parallel solver for the simulation of two-phase incompressible flows

In the framework of the in-house code Fugu, a fully-coupled solver is developed for massively parallel simulations of three-dimensional incompressible multiphase flows. The linearized momentum and continuity equations arising from the implicit solution of the fluid velocities and pressure are solved simultaneously. The method uses a BiCGStab(2) (Dongarra et al., 1998) iterative solver with an original preconditioner for the velocity block and an approximation of the inverse of the Schur complement. This is achieved by using PFMG or SMG from HYPRE and an efficient sparse matrix–vector multiplication using the CSR storage format. The construction and the tracking of the interface separating the different involved phases is based on a conservative VOF method. Test cases, such as a spherical bubble rising in quiescent liquid and the free fall of a dense sphere, are performed to validate the models, especially in the presence of strong density and viscosity ratios between fluids. Other cases, such as the phase inversion, demonstrate the ability of the new fully-coupled solver to solve two-phase problems with more than 1 billion degrees of freedom with excellent scalability.

A mass-preserving level set method for simulating 2D/3D fluid flows with deformed interface

This research paper focuses on the development of a novel interface-capturing solver for predicting two-phase flow in 2D/3D Cartesian meshes within the framework of Eulerian approaches. The primary objective is to ensure mass conservation and accurate capture of interface topology. To achieve this, a mass-preserving level set advection equation formulated as a scalar sign-distance function is proposed. The key innovation of the Eulerian solver lies in the incorporation of a scalar speed function for robust reconstruction of the classical level set equation. The effectiveness and reliability of the proposed flow solver in solving incompressible two-phase viscous flow equations are demonstrated through several benchmark problems.

Discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

In this paper, we present a discontinuous Galerkin (DG) method for the simulation of incompressible and immiscible two-phase flow in fractured porous media with capillary pressure. The hybrid-dimensional fracture models are considered in which fractures are treated as (d−1)-dimensional interfaces in the d-dimensional domain and fluid exchange between fractures and surrounding matrix is taken into account. The approximation schemes combining the DG method with the fully implicit backward Euler time discretization are formulated for models with a single fracture and intersecting fractures network, and the stability and unique solvability are established. Optimal error estimates in the discrete H1-norm for saturation and pressure are derived. Numerical experiments in two-dimensional geometry are conducted to verify the accuracy of our theoretical analysis. In addition to the backward Euler formulation, we also implement the algorithm with a kind of conventional IMplicit Pressure Explicit Saturation (IMPES) method.

Discrete exterior calculus discretization of two-phase incompressible Navier-Stokes equations with a conservative phase field method

We present a discrete exterior calculus (DEC) based discretization scheme for incompressible two-phase flows. Our physically-compatible exterior calculus discretization of single phase flow is extended to simulate immiscible two-phase flows with discontinuous changes in fluid properties such as density and viscosity across the interface. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The discrete counterpart of these smooth equations is obtained by substituting with discrete differential forms and discrete exterior calculus operators. We prove the boundedness of the method for the forward Euler and predictor-corrector time integration schemes in the DEC framework. With a proper choice of two free parameters, the scheme remains phase field bounded without the requirement of any ad hoc mass redistribution. We verify the scheme against several standard test cases (for interface capturing) comprising not only the flat domains but also the curved domains, leveraging the advantage that DEC operators are independent of the coordinate system. The results show excellent properties of boundedness, mass conservation and convergence. Moreover, we demonstrate the ability of the scheme to handle large density and viscosity ratios as well as surface tension in the simulation of various two phase flow physical phenomena on flat or curved surfaces.

A two-grid mixed finite element method of a phase field model for two-phase incompressible flows

A two-grid mixed finite element method of a phase field model for two-phase incompressible flows

A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure

Abstract In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.

Improved hybrid Allen-Cahn phase-field-based lattice Boltzmann method for incompressible two-phase flows.

In this work we develop an improved phase-field based lattice Boltzmann (LB) method where a hybrid Allen-Cahn equation(ACE) with a flexible weight instead of a global weight is used to suppress the numerical dispersion and eliminate the coarsening phenomenon. Then two LB models are adopted to solve the hybrid ACE and the Navier-Stokes equations, respectively. Through the Chapman-Enskog analysis, the present LB model can correctly recover the hybrid ACE, and the macroscopic order parameter used to label different phases can be calculated explicitly. Finally, the present LB method is validated by five tests, including the diagonal translation of a circular interface, two stationary bubbles with different radii, a bubble rising under the gravity, the Rayleigh-Taylor instability in two-dimensional and three-dimensional cases, and the three-dimensional Plateau-Rayleigh instability. The numerical results show that the present LB method has a superior performance in reducing the numerical dispersion and the coarsening phenomenon.

Well-posedness of a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with surfactant

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.

Central-moment discrete unified gas-kinetic scheme for incompressible two-phase flows with large density ratio

In this paper, we proposed a central moment discrete unified gas-kinetic scheme (DUGKS) for multiphase flows with large density ratio and high Reynolds number. Two sets of kinetic equations with central-moment-based multiple relaxation time collision operator are employed to approximate the incompressible Navier-Stokes equations and a conservative phase field equation for interface-capturing, respectively. In the framework of DUGKS, the first moment of the distribution function for the hydrodynamic equations is defined as velocity instead of momentum. Meanwhile, the zeroth moments of the distribution function and external force are carefully defined such that a artificial pressure evolution equation can be recovered. Moreover, the Strang splitting technique for time integration is employed to avoid the calculation of spatial derivatives in the force term at cell faces. For the interface-capturing equation, two equivalent DUGKS methods that deal with the diffusion term differently using a source term as well as a modified moments of the equilibrium distribution function are presented. Several benchmark tests that cover a wide a range of density ratios (up to 1000) and Reynolds numbers (up to 105) are subsequently carried out to demonstrate the capabilities of the proposed scheme. Numerical results are in good agreement with the theoretical and experimental data.

Classical solutions to a model of two-phase incompressible flows with variable density

We study the global well-posedness of three-dimensions Allen-Cahn-Navier-Stokes equations, a diffuse-interface model for two-phase incompressible flows with different densities. We first prove the local-in-time existence of classical solutions with finite initial energy. The key point is to carefully design an energy norm. The derivative f′(ϕ) of the physical relevant energy density f(ϕ) produces a damping effect near equilibrium ϕ=±1. We thereby establish the unique global classical solution near equilibrium under small size of initial energy.