Universality and scaling laws among fingers at Rayleigh–Taylor and Richtmyer–Meshkov unstable interfaces in different dimensions

The study of fingers at gravity-induced or shock-induced unstable interfaces, known as Rayleigh–Taylor instability and Richtmyer–Meshkov instability respectively, is extremely important. It is well known that many factors can affect the growth rates of fingers in the Rayleigh–Taylor and Richtmyer–Meshkov instabilities: density ratio of the fluids in difference phases, finger type (spike or bubble) and the dimension of systems (in two dimensions or in three dimensions). Therefore, conducting systematic investigations for Rayleigh–Taylor and Richtmyer–Meshkov instabilities in different sets of physical parameters is very challenging and time-consuming. This is especially true for fingers in three dimensions. We here present a surprising universality property: once the time and the growth rate are properly scaled, the dominant behavior of all fingers in systems with different density ratios, even in different dimensions, approximately follows a universal relation. This universal scaling relation allows using data for fingers in two dimensions to predict the dominant behavior of fingers in three dimensions. Both numerical results and experimental data in the literature confirm this universality property.