A High Order Continuous Galerkin Spectrally Stabilized Level-Set Approach for Incompressible Two-Phase Flows

Algebraic momentum-preserving scheme for incompressible two-phase flows at large density and viscosity ratios: Application to droplet impact problems

Improved phase field model for two-phase incompressible flows: Sharp interface limit, universal mobility and surface tension calculation

In this work, we propose an improved discretization, in terms of stability and accuracy, for the incompressible two-phase Darcy flows in a heterogeneous porous medium with discontinuous capillary forces. For this purpose, the total velocity formulation of the model is used. The coupled system is composed of a degenerate parabolic equation for the non-wetting phase and a pressure equation for the total velocity. We combine a positive Vertex Approximation Gradient (VAG) type scheme for the gradient fluxes with a hybrid upwinding of the mobilities. This approach entails a maximum principle on the saturations, which remain in their physical ranges. Energy estimates are obtained by selecting key approximations of the fluxes. These stability results allow to prove the existence of discrete solutions. Numerical experiments on complex test-cases show the robustness of the new approach in terms of the accuracy as well as the nonlinear convergence. Comparison to the usual phase potential upwinding approach and to a previous hybrid upwinding scheme are also provided.

Multi-phase fluid flows can be observed almost everywhere in nature and have a wide of range applications in engineering and natural processes such as atomization of jets and sprays, breaking waves, emulsions, boiling phenomenon, ship hydrodynamics, waterfalls and bubbly motion in cooling towers of nuclear power plants etc. In this paper, a physically possible mass conservative level set method of COMSOL multi-physics software has been illustrated to numerically investigate multi-phase fluid flow problem with and without considering surface tension concentrated on the interface. Here, the interface is represented by 0.5 iso- contour of the level set function ???? where the value of ???? is zero for the fluid inside the interface and 1 for the fluid outside the interface. In order to preserve the mass of the individual fluid phase present in actual physical problem, re- initialization procedure is made integral to the level set advection equation which is also known as governing equation of the dynamically evolving moving interface to keep the thickness (i.e. ????) of the interface constant across which the level set function ???? varies smoothly from 0 to 1.The re-initialization process which is also called intermediate step consists of an artificial compressive flux try to compress the interface when its width is enhanced by the diffusion term, thus they are acting in opposite sense. When these two terms are in equilibrium then only finite thickness of the moving interface (i.e. ????) will be obtained. “P1+P1” discretization scheme is employed to discretize the incompressible Navier stokes equation whereas “linear” discretization technique is implemented to discretize the governing equation for the dynamically evolving interface. Single bubble rising problem in a matrix involving large density ratio (i.e. 1000) with and without considering surface tension have been numerically investigated using the proposed physically possible mass conservative level set method. However, transient evolution of a single bubble in a horizontal developing flow field without considering surface tension has also been numerically simulated to illustrate the robustness of this proposed numerical method. In all these case studies excellent mass conservation of the secondary fluid i.e. the bubble has been reported by using the proposed mass conservative level set method. Few benchmark incompressible two-phase flow problems in a rectangular geometry which includes merging of two-bubble having same density in a matrix with and without considering surface tension and rising of a single bubble involving different density ratios, viscosity ratios and also different magnitude of surface tension have been numerically computed for the purpose of proposed model validation.

Regularity propagation of global weak solutions to a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with chemotaxis and active transport

In this paper, we construct a unified framework of first- and second-order schemes in time for thermodynamically consistent modeling of two-phase incompressible and immiscible flow in porous media with rock compressibility. We rigorously prove that the proposed schemes can preserve the energy dissipation law, conserve the mass of each phase as well as pore volumes and are bounds-preserving for both phases without any restrictions on the time step size. Moreover, the proposed schemes can achieve local mass conservation for both phases and preserve the pore volumes when the saturation of the whole region is within the bounds, and we only need to solve one linear system and several linear algebraic equations, whereas for points where saturation is outside the bounds, we just need to solve an additional nonlinear algebraic equation and hence the proposed schemes can achieve global mass conservation. The developed schemes exhibit high efficiency owing to a substantial reduction in the scale of nonlinear computations and their straightforward implementation. The key point is that we propose a new Lagrange multiplier method that is based on the judgment of saturation bound. This addresses the limitation of the classical Lagrange multiplier method, which is only capable of ensuring global mass conservation. Finally, a variety of illustrative numerical examples including several benchmark problems are provided to validate the accuracy and efficiency of the proposed schemes.

We developed a numerical method for simulating a stationary droplet with a high density ratio on a solid surface using an improved lattice Boltzmann method (LBM) for incompressible two-phase flows. This method uses the phase-field model and wettability of a solid surface is determined from the wetting boundary condition based on a solid surface free energy density. We first reviewed existing solid surface free energy densities from both physical and numerical viewpoints. We then proposed a new one as a cosine power form, which was derived by improving drawbacks in the existing forms. We next implemented a new wetting boundary condition in the LBM based on the proposed density function form. Using the developed method, we conducted three-dimensional simulations of a stationary droplet on a flat solid surface and investigated the range of static contact angles θ that can be computed with the numerical stability. As a result, we successfully demonstrated stable simulations over a wide range of θ, including the superhydrophobic (θ > 150◦) and superhydrophilic (θ < 30◦) surfaces. Furthermore, the simulated contact angles showed good agreement with the theoretical values predicted by Young’s equation within the range of 30◦ ≲ θ ≲ 150◦, confirming the validity of the proposed approach.

High-fidelity phase-field model (PFM) for molten pool dynamics in metal additive manufacturing (MAM) is highly required, which fully takes into account the coupled multi-physics problems of molten pool flows. Herein, we present a thermodynamically consistent non-isothermal PFM for incompressible multiphase flows with thermocapillary effects and phase change. The model is derived from a thermodynamic framework of incompressible two-phase flows, invoking the microforce theory and Coleman–Noll procedure. Features of molten pool flows, such as the large liquid–gas density ratio and temperature gradient, are fully taken into account. The model is further extended to the complicated thermo-solid–liquid–gas flows with laser-induced phase change (melting, solidification, and evaporation). Within the phase-field framework, we derive the smooth function for describing surface tension and recoil pressure acting on the molten pool and establish the thermodynamics of mass and energy losses due to evaporation. Several numerical benchmarks, such as flat gas–liquid interface, single bubble and droplet, and molten plate and particle, are introduced to validate the developed non-isothermal phase-field model. Simulation results show that the area of solidified single track is approximatively linear with the linear energy density. The non-isothermal PFM shows its ability to investigate the molten pool dynamics and pore defect formation during MAM process.

An unconditionally stable hybrid discontinuous Galerkin scheme for a Cahn–Hilliard phase-field model of two-phase incompressible flow with variable densities

Convergence analysis of a fully discrete numerical scheme for the two-phase incompressible ferrofluid flows

High-order bound-preserving finite difference methods for incompressible two-phase flow in porous media

Droplet generation in microfluidic T-junctions is a key process in various chemical and biomedical applications requiring precise size and frequency control. This study presents a numerical investigation of pulsation-controlled droplet formation using a two-phase incompressible laminar flow model with constant surface tension and defined wettability. Simulations were conducted in COMSOL Multiphysics employing the Level Set method, and the model was validated against the benchmark data of Bashir et al., accurately reproducing droplet pinch-off time and morphology under steady co-flow conditions. Pulsatile inlet velocity was then introduced to analyze its influence on droplet dynamics. Results show that at frequencies between 35 and 60 Hz, droplet generation becomes synchronized with the pulsation cycle, producing one droplet per period. Beyond 60 Hz, synchronization is lost, leading to irregular breakup and loss of droplet size control. The droplet length exhibited an approximately linear dependence on pulsation frequency, indicating predictable and tunable droplet formation. These findings demonstrate that simple modulation of the dispersed-phase velocity enables droplet-on-demand operation and robust control of droplet size and generation rate in standard microfluidic T-junctions.

Abstract The main purpose of this paper is to study the existence of local and global strong solutions for a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant in 3D whole space. We first establish the local existence of strong solutions by using the mollifier technique. Then, applying pure energy method and the standard continuity argument, one proves the global existence of strong solutions provided that initial data is sufficiently small.

This study develops an incompressible two-phase flow solver based on the open-source OpenFOAM platform, employing the volume-of-fluid (VOF) method to track the gas–liquid interface and utilizing the MULES algorithm to suppress numerical diffusion. This study provides a comprehensive investigation of the spreading dynamics of droplet pairs near walls, along with the presentation of a corresponding mathematical model. The numerical model is validated through a two-dimensional axisymmetric computational domain, demonstrating grid independence and confirming its reliability by comparing simulation results with experimental data in predicting drConfirmedoplet collision, spreading, and deformation dynamics. The study particularly investigates the influence of surface wettability on droplet impact dynamics, revealing that increased contact angle enhances droplet retraction height, leading to complete rebound on superhydrophobic surfaces. Finally, a mathematical model is presented to describe the relationship between spreading length, contact angle, and Weber number, and the study proves its accuracy. Analysis under logarithmic coordinates reveals that the contact angle exerts a significant influence on spreading length, while a constant contact angle condition yields a slight monotonic increase in spreading length with the Weber number. These findings provide an effective numerical and mathematical tool for analyzing the spreading dynamics of droplet pairs.

In this work, we present a general second-order phase-field model for the transport of insoluble surfactant in incompressible two-phase flows. In this model, a second-order local Allen-Cahn equation is applied for interface capturing, a general form of the simple scalar transport equation [S.S. Jain, J. Comput. Phys. 515, 113277 (2024)0021-999110.1016/j.jcp.2024.113277] is adopted for interface-confined surfactant, and the consistent and conservative Navier-Stokes equations with the surface tension and Marangoni force are used for fluid flows. To solve this model, we further developed a mesoscopic lattice Boltzmann (LB) method, in which the LB model for a surfactant transport equation is proposed under the general LB framework for the convection-diffusion type equation and it can correctly recover the governing equation for surfactant transport. The accuracy of the present LB method is tested by several benchmark problems and the numerical results show that it has a good performance for the transport of the insoluble surfactant in two-phase flows.

We prove existence of weak solutions of a two-incompressible immiscible ‘aqueous and oil’ phase flow model in porous media with three components (water, polymer and oil). This model is obtained by writing down the mass conservation for wetting and non-wetting phase and the mass conservation for polymer component in the wetting phase (water). We get a nonlinear parabolic degenerate system of equations in term of water and oil saturations, and polymer concentration.

Wave deformation and sediment transport nearest the shoreside are among the main reasons for sand erosion and beach profile changes. In particular, identifying the areas of incident-wave breaking and longshore current generation parallel to the shoreline is important for understanding the morphological changes of coastal beaches. In this study, a two-phase incompressible flow model along with a sandy sloping topography was employed to investigate the wave deformation and longshore current generation areas in a circular wave basin model. The finite volume method (FVM) was implemented to discretize the governing equations in cylindrical coordinates, the volume-of-fluid method (VOF) was adopted to differentiate the air–water interfaces in the control cells, and the zonal embedded grid technique was employed for grid generation in the cylindrical computational domain. The water surface elevations and velocity profiles were measured in different wave conditions, and the measurements showed that the maximum water levels per wave were high and varied between cases, as well as between cross-sections in a single case. Additionally, the mean water levels were lower in the adjacent positions of the approximated wave-breaking zones. The wave-breaking positions varied between cross-sections in a single case, with the incident-wave height, mean water level, and wave-breaking position measurements indicating the influence of downstream flow variation in each cross-section on the sloping topography. The cross-shore velocity profiles became relatively stable over time, while the longshore velocity profiles predominantly moved in the alongshore direction, with smaller fluctuations, particularly during the same time period and in measurement positions near the wave-breaking zone. The computed velocity profiles also varied between cross-sections, and for the velocity profiles along the cross-shore and longshore directions nearest the wave-breaking areas where the downstream flow had minimal influence, it was presumed that there was longshore-current generation in the sloping topography nearest the shoreside. The computed results were compared with the experimental results and we observed similar characteristics for wave profiles in the same wave period case in both models. In the future, further investigations can be conducted using the presented circular wave basin model to investigate the oblique wave deformation and longshore current generation in different sloping and wave conditions.

International audience

Summary Despite continuous advancements in numerical simulations of multiphase flow, analytical and semi-analytical solutions remain of significant interest. Earlier efforts related to analytical solutions in heterogeneous porous media primarily focused on linear flow, with limited attention given to radial flow despite its significance in describing fluid movement from wellbores into porous media or vice versa. In this study, we propose a two-step workflow to evaluate the storage capacity and sweep efficiency of incompressible two-phase radial flow in noncommunicating layered media. The first step involves a rigorous mathematical derivation of a general analytical solution for the frontal advance flow problem under viscous-dominated flow conditions. This solution accounts for variability in petrophysical and geometrical properties across different layers. It comprises distinct formulas that describe saturation distribution both before and after the breakthrough of each layer, with the flow rate coupling between layers represented through various combinations of these formulas. In the second step, viscous flow saturation is mapped to its equivalent viscous-gravity flow saturation using a simplified approach outlined in the literature. The workflow is applied to analyze the expected migration of carbon dioxide (CO2) multiplumes during the injection period in shaly-sand bodies within the Captain sandstone units of the Goldeneye field, North Sea, United Kingdom (UK). The analysis reveals that gravitational forces significantly reduce storage efficiency, from more than 30% in viscous flow to approximately 3% in viscous-gravity flow. These findings are verified through industry standard numerical simulation, confirming the accuracy of the proposed workflow. Unlike linear flow, the distribution of injected CO2 among flowing layers in radial flow is characterized by a quasistatic trend that stabilizes shortly after the start of injection. The extension of the proposed approach to model the post-injection period is discussed, in addition to the effects of dissolution, vaporization, and salt precipitation.