Unsteady compressible two-fluid flow model for interface capturing. On the dynamics of a shock-bubble interaction

Multi-fluid flows are found in many applications in engineering and physics. Examples of these flows from engineering are water-air flows in ship hydrodynamics, exhaust-air flows behind rockets, gas-petrolea flows in upstream pipes of oil rigs, air-fuel bubble interaction flows in scramjets and many others. To gain better insight in the behavior of multi-fluid flows, especially two-fluid flows, numerical simulations are needed. We assume that the fluids do not mix or chemically react, but remain separated by a sharp interface. With these assumptions a model is developed for unsteady, compressible two-fluid flow, with pressures and velocities that are equal on both sides of the interface. The model describes the behavior of a numerical mixture of the two fluids (not a physical mixture). This kind of interface modeling is called interface capturing. Numerically, the interface becomes a transition layer between both fluids. The model consists of five equations; mass, momentum and energy equation for the mixture (these are the standard Euler equations), mass equation for one of the two fluids and energy equation for one of the two fluids. This last equation is not conservative, but contains a source term. The source term represents the exchange of energy between the two fluids. The model is discretized by using a finite-volume approximation. The finite-volume method consists of a third-order Runge-Kutta scheme for temporal discretization and a limited second-order spatial discretization. For the flux evaluation Osher's Riemann solver is constructed, which uses a new set of Riemann invariants that was derived for the two-fluid model. The source term is evaluated using the limited state distribution and the wave pattern in the Osher solver. The two-fluid model is validated on several shock tube problems. The results show that the method is pressure-oscillation-free without special precautions, which is not the case for most other two-fluid flow models. The developed method is applied to two shock-bubble interaction problems. The numerical results really show the competence of the two-fluid model.