In two-phase flow simulations, a difficult issue is usually the treatment of surface tension effects. These cause a pressure jump that is proportional to the curvature of the interface separating the two fluids. Since the evaluation of the curvature incorporates second derivatives, it is prone to numerical instabilities. Within this work, the interface is described by a level-set method based on a discontinuous Galerkin discretization. In order to stabilize the evaluation of the curvature, a patch-recovery operation is employed. There are numerous ways in which this filtering operation can be applied in the whole process of curvature computation. Therefore, an extensive numerical study is performed to identify optimal settings for the patch-recovery operations with respect to computational cost and accuracy.

A front tracking method combined with the real ghost fluid method (RGFM) is proposed for simulations of fluid interfaces in two-dimensional compressible flows. In this paper the Riemann problem is constructed along the normal direction of interface and the corresponding Riemann solutions are used to track fluid interfaces. The interface boundary conditions are defined by the RGFM, and the fluid interfaces are explicitly tracked by several connected marker points. The Riemann solutions are also used directly to update the flow states on both sides of the interface in the RGFM. In order to validate the accuracy and capacity of the new method, extensive numerical tests including the bubble advection, the Sod tube, the shock-bubble interaction, the Richtmyer-Meshkov instability and the gas-water interface, are simulated by using the Euler equations. The computational results are also compared with earlier computational studies and it shows good agreements including the compressible gas-water system with large density differences.

We study the settling of solid particles in a viscous incompressible fluid contained within a two-dimensional channel, where the mass density of the particles is greater than that of the fluid. The fluid-structure interaction problem is simulated numerically using the immersed boundary method, where the added mass is incorporated using a Boussinesq approximation. Simulations are performed with a single circular particle, and also with two particles in various initial configurations. The terminal particle settling velocity and drag coefficient correspond closely with other theoretical, experimental and numerical results, and the particle trajectories reproduce the expected behavior qualitatively. In particular, simulations of a pair of interacting particles similar drafting-kissing-tumbling dynamics to that observed in other experimental and numerical studies.

In this paper, we compute a phase field (diffuse interface) model of Cahn-Hilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids. The generalized Navier boundary condition proposed by Qian et al. [1] is adopted here. We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time. We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable. Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid. We also derive the discrete energy law for the droplet displacement case, which is slightly different due to the boundary conditions. The accuracy and stability of the scheme are validated by examples, results and estimate order.

A thin liquid film subject to a temperature gradient is known to deform under the action of thermocapillary stresses which induce convective cells. The free surface deformation can be thought of as the signature of the imposed temperature gradient, and this study investigates the inverse problem of trying to reconstruct the temperature field from known free surface variations. The present work builds on the analysis of Tan et al. [“Steady thermocapillary flows of thin liquid layers I. Theory”, Phys. Fluids A2 (1990) 313–321, doi:10.1063/1.857781] which provides a long-wave evolution equation for the fluid film thickness variation on nonuniformly heated substrates and proposes a solution strategy for the planar flow version of this inverse problem. The present analysis reveals a particular case for which there exists an explicit, closed-form solution expressing the local substrate temperature in terms of the local film thickness and its spatial derivatives. With some simplifications, this analysis also shows that this solution applies to three-dimensional flows. The temperature reconstruction strategies are successfully tested against “artificial” experimental data (obtained by solving the direct problem for known temperature profiles) and actual experimental data.

An important test of the quality of numerical methods developed to track the interface between two fluids is their ability to reproduce test cases or benchmarks. However, benchmark solutions are scarce and virtually nonexistent for complex geometries. We propose a simple method to generate benchmark solutions in the context of the two-layer flow problem, a classical multiphase flow problem. The solutions are obtained by considering the inverse problem of finding the required channel geometry to obtain a prescribed interface profile. This viewpoint shift transforms the problem from that of having to solve a complex differential equation to the much easier one of finding the roots of a quartic polynomial.

An analysis is developed for the behaviour of a cloud of cavitation bubbles during both the growth and collapse phases. The theory is based on a multipole method exploiting a modified variational principle developed by Miles [“Nonlinear surface waves in closed basins”, J. Fluid Mech.75 (1976) 418–448] for water waves. Calculations record that bubbles grow approximately spherically, but that a staggered collapse ensues, with the outermost bubbles in the cloud collapsing first of all, leading to a cascade of bubble collapses with very high pressures developed near the cloud centroid. A more complex phenomenon occurs for bubbles of variable radius with local zones of collapse, with a complex frequency spectrum associated with each individual bubble, leading to both local and global collective behaviour.