Modelling the effects of convergence on the Richtmyer-Meshkov instability in planar geometry

In spherical or cylindrical geometry, the Richtmyer-Meshkov instability (RMI) and Rayleigh-Taylor instability behave differently to planar geometry due to the Bell-Plesset effect. The Bell-Plesset encompasses the effects on the growth rate of the mixing layer that arise from the geometric convergence of the mixing layer which is a result of the changing radius and wavelength, as well as the change in density of the fluids because of the compression of the mixing layer. To understand the effect of convergence, a potential flow model was derived for the linear regime of RMI in planar geometry with transverse strain. This model was compared to two-dimensional direct-numerical simulations of single-mode RMI. The model and simulation showed agreement within the linear regime and demonstrated that a compressive strain-rate increases the growth rate, whilst an expansive transverse strain-rate decreases the growth rate. The investigation into the effects of transverse strain-rates were extended to a three-dimensional multi-mode narrowband case. The effects of the linear-regime model are not observed to occur for developing multi-mode mixing layer. Instead, the compressive transverse strain caused a decrease in mixing layer growth and vice versa. The ability for buoyancy-drag and RANS models to model this effect of convergence is discussed, as well as potential corrections to the models.