An integrated smoothed particle hydrodynamics model for complex interfacial flows with large density ratios

Consistent mass and momentum transport for simulating incompressible interfacial flows with large density ratios using the level set method

Based on the principle of virtual works, a multiphase smoothed particle hydrodynamics (SPH) model is further developed from the foundation of Hu X Y et al. (2006) and Grenier N et al. (2009). In the present model, the surface tension force implementation suitable for the multiphase flows with a large density ratio is applied, and this allows a good continuity at the multiphase interface. Artificial displacement correction is applied to keep the particles distributing uniformly in the whole flow field, and therefore any artificial viscous term is never needed; this is very important in the numerical simulation of viscous flows since the introduction of artificial viscosity changes the Reynolds number. Background pressure and interface sharpness force are added in the equation of state and the equation of momentum respectively to ensure the multiphase interface stability and smoothness; this is essential in the simulation of multiphase flows with large density difference at the multiphase interface. Two types of viscosity expressions suitable for multiphase flows are introduced and analyzed; the conclusion is that the formula proposed by Morris et al. (1997) and its similarly derived forms can give more accurate results. In the numerical validations, an oscillating droplet test is applied first to confirm the accuracy of the surface tension model and good results are achieved. This demonstrates that the artificial displacement and the interface sharp force will make negligible effects to the surface tension implementation. After that, two classic quantitative benchmarks of rising bubbles are simulated and the results of SPH agree well with the reference data. Moreover, in the two numerical benchmarks, the effect of the artificial displacement, the choice of the viscosity expression, and the type of the kernel function are compared and finally an optimal combination of these numerical aspects is recommended. Based on the above numerical investigations, the splitting process of an initially circular bubble is simulated and the numerical results agree well with the experimental data. In the last numerical case, the process of chasing and merging between two rising bubbles in vertical direction is simulated, based on which the mechanisms of these interesting interactions between two rising bubbles are analyzed. It is demonstrated in the present work that further improved multiphase SPH model may provide a potential method for the research of bubble dynamics.

Smoothed particle hydrodynamics (SPH) is a meshfree, Lagrangian method, highly suitable for numerical simulation of free-surface and interfacial flows. In the SPH literature, several numerical algorithms have been proposed for solving multi-phase fluid flow problems. One of the fundamental distinction between different methods hinges on the density estimation procedure. The present study assesses two such methods viz. the summation and continuity density approach. For these two methods, we perform a comparative analysis of accuracy, convergence, stability and energy conservation. The multi-phase flow simulations involve density ratio r = (), ranging from 1 to 1000. The benchmark problems chosen for the analysis are dam break, bubble rise in a water column, Rayleigh–Taylor instability and non-Boussinesq lock exchange. The numerical solution for these problems is validated with experimental and other numerical benchmark results available in the literature. From detailed multi-phase flow simulations, it was observed that, for low density ratios, the continuity density algorithm provides better accuracy and energy conservation, whereas for high density ratios (), summation density algorithm is preferable.

Heat transfer and phase change in multiphase media involve complex behavior such as multi-physics coupling, unsteady interfaces, and boundary generation/disappearance. The smoothed particle hydrodynamics (SPH) method is used to simulate multicomponent problems (initially solid material surrounded by a fluid) characterized by heat transfer, solid–liquid phase change, and multiphase flow. First, the interface behavior of the melted liquids and ambient fluids are simulated by a multiphase SPH model, where corrected algorithms and techniques are adopted to provide a stable simulation for the interface with a large density ratio. The wetting effect of the solid surface by melted liquids is considered by extending the continuum surface force model to three phases including solid, melted liquid and ambient fluid. Second, a thermal dynamics model is incorporated with the multiphase SPH model. The solid–liquid phase change occurs in a small temperature interval in which the latent heat is incorporated using the equivalent heat capacity. Third, the numerical model is validated using several benchmark examples. Five specially configured cases are simulated to demonstrate the robustness, adaptability, and stability of the model. The intriguing phenomena of evolution of the fluid interface of melted liquids, accumulation of thin-layer liquids, and melted droplet formation are reproduced.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

A simplified method is proposed to implement a wetting boundary condition on curved surfaces within the conservative phase-field lattice-Boltzmann (LB) simulation framework. It combines the idea of Huang et al. [“An alternative method to implement contact angle boundary condition and its application in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces,” Eur. Phys. J. E 41, 17 (2018)] to find the order parameter on the other side of the wall with the conservative Allen–Cahn equation (CACE) for interface evolution solved by the LB equations. It inherits the advantage of the original method using the Cahn–Hilliard equation to avoid complicated interpolations under different geometries. By using the CACE, the boundary condition for the chemical potential is circumvented (making it more simplified), and the overshooting of the order parameter is also greatly suppressed, enabling it to simulate two-phase flows with solid objects of various shapes and wettabilities at large density and viscosity ratios. Several two-dimensional, axisymmetric, and three-dimensional problems, including some previously studied by experiments, were simulated and the capability of the proposed method is demonstrated through its good agreement with theoretical predictions and/or experimental observations.

The cross-sectional shape of the lift pipe of the bubble pump is considered to have a significant impact on the merging of rising bubbles. The effects of cross-sectional shapes of pipes and Reynolds number on the rise and coalescence of near-wall bubbles in lithium bromide solutions are investigated. A hybrid thermal multiphase model incorporating the D3Q15 pseudopotential multiphase lattice Boltzmann model with a large density ratio and the finite difference method is utilized. And the volume translated Peng-Robinson (VTPR) state equation is adopted. It is found that the merging time of rising bubbles near the walls of rectangular, circular, elliptical, and triangular cross-section pipes increases in order at the same Reynolds number. Under the same cross-sectional conditions of the pipes, a suitable increase in the Reynolds number will accelerate the merging of bubbles near the wall. This study may offer valuable insights for informing the design of pipe types for bubble pumps in absorption refrigeration systems.

The standard weakly compressible Smoothed Particle Hydrodynamics (WCSPH) is successfully applied to multi-phase problems involving fluids with similar densities, but when density ratio increases at some order of magnitude, serious instability phenomena occur at the interface. Several remedies have been proposed based on numerical correctives that deviate from standard formulation, increasing the algorithm complexity and, sometimes, the computational cost. In this study, the standard SPH has been adapted to treat free-surface multi-phase flows with a large density ratio through a modified form of the governing equations which is based on the specific volume (i.e. the inverse of particle volume) instead of density: the former is continuous across the fluid interface while the latter is not and generates numerical instability. Interface sharpness is assured without cohesion forces; kernel truncation at the interface is avoided. The model, relatively simple to implement, is tested by simulating two-phase dam breaking for two configurations: kinematic and dynamic features are compared with reference data showing good agreement despite the reduced computational effort.

In this paper, the 3D Navier–Stokes (N–S) equation and Cahn–Hilliard (C–H) equations were solved using a free‐energy‐based lattice Boltzmann (LB) model. In this model, a LB equation with a D3Q19 velocity model is used to recover continuity and N–S equations while another LB equation with D3Q7 velocity model for solving C–H equation (Int. J. Numer. Meth. Fluids, 2008; 56:1653–1671) is applied to solve the 3D C–H equation.To avoid the excessive use of computational resources, a moving reference frame is adopted to allow long‐time simulation of a bubble rising. How to handle the inlet/outlet and moving‐wall boundary conditions are suggested. These boundary conditions are simple and easy for implementation.This model's performance on two‐phase flows was investigated and the mass conservation of this model was evaluated. The model is validated by its application to simulate the 3D air bubble rising in viscous liquid (density ratio is 1000). Good agreement was obtained between the present numerical results and experimental results when Re is small. However, for high‐Re cases, the mass conservation seems not so good as the low‐Re case. Copyright © 2009 John Wiley & Sons, Ltd.

This study proposes a three-dimensional non-orthogonal multiple-relaxation-time (NMRT) phase-field multiphase lattice Boltzmann (PFLB) model within a recently established unified lattice Boltzmann model (ULBM) framework [Luo et al., Phil. Trans. R. Soc. A 379, 20200397, 2021]. The conservative Allen-Cahn equation and the incompressible Navier-Stokes (NS) equations are solved. In addition, a local gradient calculation scheme for the order parameter of the Allen-Cahn equation is constructed with the non-equilibrium part of the distribution function. A series of benchmark cases are conducted to validate the proposed model, including the two-phase Poiseuille flow, Rayleigh-Taylor instability, binary liquid/metal droplet collision, and a bubble rise in water. The present simulation results are in good agreement with existing simulation and experimental data. In the simulation of the co-current two-phase Poiseuille flow, the present model is proven to resolve the discontinuity at the phase interface and provide accurate results at extremely high density ratios (i.e., up to 106). Finally, the proposed model is adopted to simulate two challenging cases: (1) water droplet splashing during its impacting on a thin liquid film and (2) liquid jet breakup. The simulation results demonstrate an excellent agreement with previous experimental results, both qualitatively and quantitatively. In these simulations, the Weber number and Reynolds number reach 105 and 6000, respectively, and the viscosity can be as low as ∼10−4, in the lattice unit.

Comparison of surface tension generation methods in smoothed particle hydrodynamics for dynamic systems

Wave generation and wave–structure impact modelling with WCSPH

A VOSET method combined with IDEAL algorithm for 3D two-phase flows with large density and viscosity ratio

In the subsurface, fluids play a critical role by transporting dissolved minerals, colloids, and contaminants (sometimes over long distances); by mediating dissolution and precipitation processes; and by enabling chemical transformations in solution and at mineral surfaces. Although the complex geometries of fracture apertures, fracture networks, and pore spaces may make it difficult to accurately predict fluid flow in saturated (single‐phase) subsurface systems, well‐developed methods are available. The simulation of multiphase fluid flow in the subsurface is much more challenging because of the large density and/or viscosity ratios found in important applications (water/air in the vadose zone; water/oil, water/gas, gas/oil, and water/oil/gas in hydrocarbon reservoirs; water/air/nonaqueous phase liquids (nonaqueous phase liquids/dense nonaqueous phase liquids) in contaminated vadose zone systems; and gas/molten rock in volcanic systems, for example). In addition, the complex behavior of fluid‐fluid‐solid contact lines and their impact on dynamic contact angles must also be taken into account and coupled with the fluid flow. Here we review the methods that are currently being used to simulate pore‐scale multiphase fluid flow and reactive transport in fractured and porous media. After the introduction, the review begins with an overview of the fundamental physics of multiphase fluids flow followed by a more detailed discussion of the complex dynamic behavior of contact lines and contact angles, an important barrier to accurate pore‐scale modeling and simulation. The main part of the review focuses on five different approaches: pore network models, lattice gas and lattice Boltzmann methods, Monte Carlo methods, particle methods (molecular dynamics, dissipative particle dynamics, and smoothed particle hydrodynamics), and traditional grid‐based computational fluid dynamics coupled with interface tracking and a contact angle model. Finally, the review closes with a discussion of future trends and challenges.

Implementing contact angle boundary conditions for second-order Phase-Field models of wall-bounded multiphase flows