Vortex sheet motion in incompressible Richtmyer–Meshkov and Rayleigh–Taylor instabilities with surface tension

Abstract

Motion of a planar interface in incompressible Richtmyer–Meshkov (RM) and Rayleigh–Taylor (RT) instabilities with surface tension is investigated numerically by using the boundary integral method. It is shown that when the Atwood number is small, an interface rolls up without regularization of the interfacial velocity. A phenomenon known as “pinching” in the physics of drops is observed in the final stage of calculations at various Atwood numbers and surface tension coefficients, and it is shown that this phenomenon is caused by a vortex dipole induced on the interface. It is also shown that when the surface tension coefficient is large, finite amplitude standing wave solutions exist for the RM instability. This standing wave solution is investigated in detail by nonlinear stability analysis. When gravity is taken into account (RT instability), linearly stable but nonlinearly unstable motion can occur under a critical condition that the frequency of the linear dispersion relation in the system is equal to zero. Further, it is shown that the growth rate of bubbles and spikes under this critical motion is neither of the exponential type nor of the power law type at both the linear stage and the asymptotic stage.