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

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

Many multiphase flows relevant to natural phenomena and technical applications are characterized by large density ratios between the phases or fluids. Numerical simulations of such flows are especially challenging if the phase interface has a complex shape and is subject to large shear. This scenario is typical for atomization of liquids under ambient conditions. In this paper we discuss some of the reasons why one-fluid level-set based methods are prone to become unstable for high-density ratio/high shear atomizing flows and present a consistent rescaled momentum transport method that addresses the identified shortcomings. We present results obtained with the new method for a number of high-density ratio test cases, including the advection of a 1 000 000:1 density ratio impulsively accelerated drop, a 1000:1 density ratio damped surface wave, and the collapse of a water column in air under ambient conditions. The new method shows significantly improved results compared to standard level set based single-fluid methods.

In this paper, an integrated smoothed particle hydrodynamics (SPH) model for complex interfacial flows with large density ratios is developed. The discrete continuity equation and acceleration equation are obtained by considering the time derivative of the volume of particle and Eckart's continuum Lagrangian equation. A continuum surface force model is used to meet the fact that surface force may not be distributed uniformly on each side of the interface. An improved boundary condition is imposed to model wall free‐slip and no‐slip condition for interfacial flows with large density ratios. Particle shifting algorithm (PSA) is added for interfacial flows by imposing the normal correction near the interface, called as Interface‐PSA. Then four representative numerical examples, including droplet deformation, Rayleigh‐Taylor instability, dam breaking, and bubble rising, are presented and compared well with reference data. It is demonstrated that inherent interfacial flow physics can be well captured, including surface tension and the dynamic evolution of the complex interfaces.

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.

Level set methods have been widely used in image processing and computer vision. In conventional level set formulations, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually destroy the stability of the evolution. Therefore, a numerical remedy, called reinitialization, is typically applied to periodically replace the degraded level set function with a signed distance function. However, the practice of reinitialization not only raises serious problems as when and how it should be performed, but also affects numerical accuracy in an undesirable way. This paper proposes a new variational level set formulation in which the regularity of the level set function is intrinsically maintained during the level set evolution. The level set evolution is derived as the gradient flow that minimizes an energy functional with a distance regularization term and an external energy that drives the motion of the zero level set toward desired locations. The distance regularization term is defined with a potential function such that the derived level set evolution has a unique forward-and-backward (FAB) diffusion effect, which is able to maintain a desired shape of the level set function, particularly a signed distance profile near the zero level set. This yields a new type of level set evolution called distance regularized level set evolution (DRLSE). The distance regularization effect eliminates the need for reinitialization and thereby avoids its induced numerical errors. In contrast to complicated implementations of conventional level set formulations, a simpler and more efficient finite difference scheme can be used to implement the DRLSE formulation. DRLSE also allows the use of more general and efficient initialization of the level set function. In its numerical implementation, relatively large time steps can be used in the finite difference scheme to reduce the number of iterations, while ensuring sufficient numerical accuracy. To demonstrate the effectiveness of the DRLSE formulation, we apply it to an edge-based active contour model for image segmentation, and provide a simple narrowband implementation to greatly reduce computational cost.

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.

A consistent and conservative volume distribution algorithm and its applications to multiphase flows using Phase-Field models

Simulation of a single droplet impact onto a thin liquid film using the lattice Boltzmann method

SummaryThe parametric level set approach is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, level set methods can be directly coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Moreover, the parametric level set scheme can not only inherit the primary advantages of the conventional level set methods, such as clear boundary representation and the flexibility in handling topological changes, but also alleviate some undesired features from the conventional level set methods, such as the need for reinitialization. However, in the existing radial basis function–based parametric level set method, it is difficult to identify the range of the design variables. Besides, the parametric level set evolution often struggles with large fluctuations during the optimization process. Those issues cause difficulties both in numerical stability and in material property mapping. In this paper, a cardinal basis function is constructed based on the radial basis function partition of unity collocation method to parameterize the level set function. The benefit of using cardinal basis function is that the range of the design variables can now be clearly specified as the value of the level set function. A distance regularization energy functional is also introduced, aiming to maintain the desired signed distance property during the level set evolution. With this desired feature, the level set evolution is stabilized against large fluctuations. In addition, the material properties mapped from the level set function to the finite element model can be more accurate.

Simulation of incompressible two-phase flows with large density differences employing lattice Boltzmann and level set methods

An improved lattice Boltzmann model for high gas and liquid density ratio in composite grids

In Sussman, Smereka, and Osher [ J. Comp. Phys., 94 (1994), pp. 146--159], a numerical was presented for computing incompressible air--water flows using the level set method. Crucial to the above method was a new iteration method for maintaining the level set function as the signed from the zero level set. In this paper we implement a along with higher order difference schemes in order to make the iteration method more accurate and efficient. Accuracy is measured in terms of the new computed signed function and the original level set function having the same zero level set. We apply our redistancing to incompressible flows with noticeably better resolved results at reduced cost. We validate our results with experiment and theory. We show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface. We perform basic accuracy checks and more stringent tests involving complicated interfacial structures. As with all level set schemes, our method is easy to implement.

High fidelity anisotropic adaptive variational multiscale method for multiphase flows with surface tension

A stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics

The Nonlinear Large-Eddy Simulation (nLES) method [G. C. Burton, “The nonlinear large-eddy simulation method (nLES) applied to Sc≈1 and Sc⪢1 passive-scalar mixing,” Phys. Fluids 20, 035103 (2008)] is employed in the first numerical study of multimode miscible Rayleigh–Taylor instability (RTI) in the ultrahigh Atwood-number regime above A≥0.90. The present work focuses on the dynamics of turbulent mixing at the large density ratios that may be encountered in certain astrophysical contexts and engineering applications. Using the initial condition from the landmark (N=30723) direct numerical study of Cabot and Cook [W. Cabot and A. W. Cook, “Reynolds number effects on Rayleigh-Taylor instability with possible implications for type-Ia supernovae,” Nat. Phys. 2, 562 (2006)], the nLES method is first validated in simulations of A=0.5 RTI mixing and is shown to recover important statistical measures of the mixing process, such as bubble and spike growth rates and mixing efficiency reported in that study, but at the significantly coarser resolutions typical of most large-eddy simulations. The first simulations of RTI at Atwood numbers A>0.90 are then used to explore the effects of varying the density ratio on mixing dynamics at very high Atwood numbers 0.75≤A≤0.96. Spike heights and mixing layer growth rates are shown to be strongly affected by the initial density ratio. An empirical power-law scaling relationship is shown to predict nearly exactly the variation in the ratio of spike to bubble heights as a function of Atwood number. Mixing efficiency is shown to be influenced by the initial density difference but the competition between increased molecular mixing and entrainment largely cancel, producing a relatively modest variation in these flow parameters when compared with the intermediate Atwood-number case. Late-time power spectra show the appearance of an inertial range, indicating that the mixing layer has transitioned to a fully turbulent state. The role of bubble and spike structures in the interscale transfer of kinetic energy is explored for the first time for high Atwood-number RTI flows. Bubble heads (stems) are shown to produce forward (reverse) transfer due to compressive (extensional) straining at intermediate Atwood number. At high Atwood number, however, spike stems are shown to produce forward transfer due to compressive straining generated from the larger spike penetration velocity and the differing spike morphology produced at the higher Atwood number. The study indicates that stable and accurate simulations of ultrahigh Atwood-number mixing may be conducted using the nLES method on grids significantly coarser than used previously to examine RTI flows.