Hang Ding

A diffuse-interface immersed-boundary method for two-dimensional simulation of flows with moving contact lines on curved substrates

We propose an approach to simulate flows with moving contact lines (MCLs) on curved substrates on a Cartesian mesh. The approach combines an immersed boundary method with a three-component diffuse-interface model and a characteristic MCL model. The immersed boundary method is able to accurately enforce the no-slip boundary condition at the solid surface, thereby circumventing the penetration of the gas and the liquid into the solid by convection. On the other hand, using the three-component diffuse-interface model can prevent the gas and liquid from infiltrating into the solid substrate through the diffusive fluxes during the interface evolution. A combination of these two methods appears to effectively conserve the mass of the phases in the computation. The characteristic MCL model not only allows the contact lines to move on the curved boundaries, but makes the gas-liquid interface to intersect the solid object at an angle in consistence with the prescribed contact angle, even with the variation of surface tangent at the solid substrate. We examine the performance of the approach through a variety of numerical experiments. The mass conservation and interface shapes at equilibrium were tested through the simulation of drop spreading on a circular cylinder. The dynamic behaviors of moving contact lines were validated by simulating the droplet spreading on a flat substrate, and we compared the numerical results against theoretical predictions and previous experimental observations. The method was also applied to the simulations of flows with curved boundaries and moving contact lines, such as drop impact on a sphere and water entry of a sphere. Finally, we studied the penetration process of a two-dimensional drop into a porous substrate that consists of a cluster of circular cylinders.

On the diffuse interface method using a dual-resolution Cartesian grid

We investigate the applicability and performance of diffuse interface methods on a dual-resolution grid in solving two-phase flows. In the diffuse interface methods, the interface thickness represents a cut-off length scale in resolving the interfacial dynamics, and it was found that an apparent loss of mass occurs when the interface thickness is comparable to the length scale of flows [24]. From the accuracy and mass conservation point of view, it is desirable to have a thin interface in simulations. We propose to use a dual-resolution Cartesian grid, on which a finer resolution is applied to the volume fraction C than that for the velocity and pressure fields. Because the computation of C field is rather inexpensive compared to that required by velocity and pressure fields, dual-resolution grids can significantly increase the resolution of the interface with only a slight increase of computational cost, as compared to the single-resolution grid. The solution couplings between the fine grid for C and the coarse grid (for velocity and pressure) are delicately designed, to make sure that the interpolated velocity is divergence-free at a discrete level and that the mass and surface tension force are conserved. A variety of numerical tests have been performed to validate the method and check its performance. The dual-resolution grid appears to save nearly 70% of the computational time in two-dimensional simulations and 80% in three-dimensional simulations, and produces nearly the same results as the single-resolution grid. Quantitative comparisons are made with previous studies, including Rayleigh Taylor instability, steadily rising bubble, and partial coalescence of a drop into a pool, and good agreement has been achieved. Finally, results are presented for the deformation and breakup of three-dimensional drops in simple shear flows.

Diffuse interface simulation of ternary fluids in contact with solid

In this article we developed a geometrical wetting condition for diffuse-interface simulation of ternary fluid flows with moving contact lines. The wettability of the substrate in the presence of ternary fluid flows is represented by multiple contact angles, corresponding to the different material properties between the respective fluid and the substrate. Displacement of ternary fluid flows on the substrate leads to the occurrence of moving contact point, at which three moving contact lines meet. We proposed a weighted contact angle model, to replace the jump in contact angle at the contact point by a relatively smooth transition of contact angle over a region of diffuse contact point of finite size. Based on this model, we extended the geometrical formulation of wetting condition for two-phase flows with moving contact lines to ternary flows with moving contact lines. Combining this wetting condition, a Navier-Stokes solver and a ternary-fluid model, we simulated two-dimensional spreading of a compound droplet on a substrate, and validated the numerical results of the drop shape at equilibrium by comparing against the analytical solution. We also checked the convergence rate of the simulation by investigating the axisymmetric drop spreading in a capillary tube. Finally, we applied the model to a variety of applications of practical importance, including impact of a circular cylinder into a pool of two layers of different fluids and sliding of a three-dimensional compound droplet in shear flows.

Simulation of compressible two-phase flows with topology change of fluid-fluid interface by a robust cut-cell method

We develop a robust cut-cell method for numerical simulation of compressible two-phase flows with topology change of the fluid-fluid interface. In cut cell methods the flows can be solved in the finite volume framework and the jump conditions at the interface are resolved by solving a local Riemann problem. Therefore, cut cell methods can obtain interface evolution with high resolution, and at the same time satisfactorily maintain the conservation of flow quantities. However, it remains a challenge for the cut cell methods to handle interfaces with topology change or very high curvature, where the mesh is not sufficiently fine to resolve the interface. Inappropriate treatment could give rise to either distorted interface advection or unphysical oscillation of flow variables, especially when the regularization process (e.g. reinitialization in the level set methods) is implemented. A robust cut-cell method is proposed here, with the interface being tracked by a level set function. The local unphysical oscillation of flow variables in the presence of topology change is shown to be greatly suppressed by using a delayed reinitialization. The method can achieve second-order accuracy with respect to the interface position in the absence of topology changes of interface, while locally degrading to first-order at the interface region where topology change occurs. Its performance is examined through a variety of numerical tests, such as Rayleigh collapse, shock-bubble interaction, and shock-induced bubble collapse in water. Numerical results are compared against either benchmark solutions or experimental observations, and good agreement has been achieved qualitatively and/or quantitatively. Finally, we apply the method to investigating the collapse process of two tandem bubbles in water.

Fluid–structure interaction involving dynamic wetting: 2D modeling and simulations

Abstract In this paper, we propose a hybrid model to compute the capillary force acting on moving solid objects, and combine it with the diffuse-interface immersed-boundary method in Liu and Ding (2015) [18] to simulate fluid–structure interaction (FSI) involving dynamic wetting. Dynamic wetting is very important in the dynamic interaction between fluid–fluid interfaces and small moving objects. Numerical simulations of these flow problems require accurate computation of the capillary force acting on the structure, which depends on the instantaneous position of and the effective surface tension at the moving contact line. In order to achieve this, we use the diffuse-interface immersed-boundary method to simulate the dynamic wetting on moving objects, and propose a hybrid model to compute the effective surface tension at the contact line. Specifically, a diffuse interface model is used for the interface profile out of equilibrium, e.g. at the onset of formation or detachment of contact lines, and a sharp interface model is used for the interface profile at equilibrium. The performance of the method is examined by a variety of numerical experiments. We simulate the sinking of a circular cylinder due to gravity, and study the capillarity-dominated impact dynamics of a solid sphere on a water pool. In both cases the numerical results are quantitatively compared against the experimental data, and good agreements have been achieved. The momentum conservation of the system is carefully checked by studying head-on collision between a drop and a solid sphere. Finally, we apply the method to the self-assembly process of multiple floating cylinders on water surface.