We develop the immersed interface method (IIM) to simulate a two-fluid flow of two immiscible fluids with different density and viscosity. Due to the surface tension and the discontinuous fluid properties, the two-fluid flow has nonsmooth velocity and discontinuous pressure across the moving sharp interface separating the two fluids. The IIM computes the flow on a fixed Cartesian grid by incorporating into numerical schemes the necessary jump conditions induced by the interface. We present how to compute these necessary jump conditions from the analytical principal jump conditions derived in [Xu, DCDS, Supplement 2009, pp. 838-845]. We test our method on some canonical two-fluid flows. The results demonstrate that the method can handle large density and viscosity ratios, is second-order accurate in the infinity norm, and conserves mass inside a closed interface.

An accurate turbulent mixing model for gravitationally induced instabilities with arbitrarily variable accelerations has been developed to capture the following physical aspects: (1) directed transport, (2) correct buoyancy forces, (3) turbulence diffusion, and (4) geometrical aspects. We present the two-structure two-fluid two-turbulence concept (2SFK), which consistently answers these requirements by identifying the large-scale transport structures in a statistical approach. An example of a 2SFK-based model is given and applied to the Rayleigh–Taylor case.