Towards front-tracking based on conservation in two space dimensions III, tracking interfaces

This is the third paper in the series of our conservative front-tracking method. In this paper, we describe how our method tracks fluid interfaces in multi-fluid flows. Two important ingredients in our conservative front-tracking method in tracking fluid interfaces are: (1) the velocities and pressures of the left and right cell-averages in a discontinuity cell are respectively maintained to be equal to each other, and in doing so the physics that the normal velocity and pressure are continuous cross the interface is simulated but that the tangential velocity may be discontinuous is ignored, and (2) a so-called numerical surface dissipation is designed on the tracked interface to eliminate possible numerical instability there, and we believe that this numerical surface dissipation is a good substitute for the missing physical dissipation acting on the interface. We then present numerical simulation of Haas–Sturtevant’s two shock–bubble interaction experiments [J.F. Haas, B. Sturtevant, Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities, J. Fluid Mech. 181 (1987) 41–76] using this method. Our numerical results are in good agreement with the experimental and other numerical results in the early times of the flow. Moreover, our numerical results are also in good agreement with the experimental results in the later times of the flow and give clear pictures of the bubble deformation then, which show that the right boundaries of the bubble behave just as Rychtmyer–Meshkov instabilities with the shock coming from either heavy or light gases.