In this paper, we extend using the Runge-Kutta discontinuous Galerkin method together with the front tracking method to simulate the compressible two-medium flow on unstructured meshes. A Riemann problem is constructed in the normal direction in the material interfacial region, with the goal of obtaining a compact, robust and efficient procedure to track the explicit sharp interface precisely. Extensive numerical tests including the gas-gas and gas-liquid flows are provided to show the proposed methodologies possess the capability of enhancing the resolutions nearby the discontinuities inside of the single medium flow and the interfacial vicinities of the two-medium flow in many occasions.

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.

The equilibrium position of a deformable drop in Couette and Poiseuille flows is investigated numerically by solving the full Navier-Stokes equations using a finite difference/front tracking method. The objective of this work is to study the motion of a non-neutrally buoyant drop in Couette and Poiseuille flows with different density ratios at finite Reynolds numbers. Couette flow: The equilibrium position of the lighter drops is higher than the heavier drops at each particle Reynolds number. Also, the equilibrium position height increases with increasing the Reynolds number at a fixed density ratio. At this equilibrium distance from the wall, the migration velocity is zero, while the velocity of the drop in the flow direction and rotational velocity of the drop is finite. It is observed that the equilibrium position is independent of the initial position of the drops and depends on the density ratio and the shear Reynolds number. Poiseuille flow: When the drop is slightly buoyant, it moves to an equilibrium position between the wall and the centerline. The equilibrium position is close to the centerline if the drop lags the fluid but close to the wall if the drop leads the fluid. As the Reynolds number increases, the equilibrium position of lighter drops moves slightly closer to the wall and the equilibrium position of heavier drops moves towards the centerline.

In this paper, a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces. The velocity components along the interface are introduced as two augmented variables and the resulting augmented equation is then solved by the GMRES method. The augmented variables and /or the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines and are then applied to the fluid through the jump conditions. The Stokes equations are discretized on a staggered Cartesian grid via a second order finite difference method and solved by the conjugate gradient Uzawa-type method. The numerical results show that the overall scheme is second order accurate. The major advantages of the present IIM-Stokes solver are the efficiency and flexibility in terms of types of fluid flow and different boundary conditions. The proposed method avoids solution of the pressure Poisson equation, and comparisons are made to show the advantages of time savings by the present method. The generalized two-phase Stokes solver with correction terms has also been applied to incompressible two-phase Navier-Stokes flow.