The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a discretization of the Navier-Stokes equation that advects mass and momentum in a consistent manner. Incompressible flow with cap
illary forces is modeled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul dInterface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressibility is satisfied. To improve the stability of these methods momentum fluxes are advected in a manner consistent with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a 3mm-diameter falling raindrop and for an atomizing flow in air-water conditions.
The two-phase mixing layer formed between parallel gas and liquid streams is an important fundamental problem in turbulent multiphase flows. The problem is relevant to many industrial applications and natural phenomena, such as air-blast atomizers in
fuel injection systems and breaking waves in the ocean. The velocity difference between the gas and liquid streams triggers an interfacial instability which can be convective or absolute depending on the stream properties and injection parameters. In the present study, a direct numerical simulation of a two-phase gas-liquid mixing layer that lie in the absolute instability regime is conducted. A dominant frequency is observed in the simulation and the numerical result agrees well with the prediction from viscous stability theory. As the interfacial wave plays a critical role in turbulence transition and development, the temporal evolution of turbulent fluctuations (such as the enstrophy) also exhibits a similar frequency. In order to investigate the statistical response of the multiphase turbulence flow, the simulation has been run for a long physical time so that time-averaging can be performed to yield the statistically converged results for Reynolds stresses and the turbulent kinetic energy (TKE) budget. An extensive mesh refinement study using from 8 million to about 4 billions cells has been carried out. The turbulent dissipation is shown to be highly demanding on mesh resolution compared to other terms in TKE budget. The results obtained with the finest mesh are shown to be not far from converged results of turbulent dissipation which allow us to obtain estimations of the Kolmogorov and Hinze scales. The computed Hinze scale is significantly larger than the size of droplets observed and does not seem to be a relevant length scale to describe the smallest size of droplets formed in atomization.
