Rayleigh-Taylor turbulence: self-similar analysis and direct numerical simulations

(D)irect numerical simulations and a self-similar analysis of the single-fluid Boussmesq Rayleigh-Taylor instability and transition to turbulence are used to investigate Rayleigh-Taylor turbulence. The Schmidt, Atwood and bulk Reynolds numbers are Sc = 1. A = 0.01, Re less than or equal to 3000. High-Reynolds-number moment self-similarity, consistent with the the energy cascade interpretation of dissipation, is used to analyse the DNS results. The mixing layer width obeys a differential equation with solution h(t - C-o, h(0)) = (1)/(4)C(o)Agt(2) + rootAgC(o)h(0)(1/2) t + h(0); the result for h(t; C-o, h(0)) is a rigorous consequence of only one ansatz, self-similarity. It indicates an intermediate time regime in which the growth is linear and the importance of a virtual origin. At long time the well-known h similar to (1)/(4)C(o)Agt(2) scaling dominates. The self-similar analysis indicates that the asymptotic growth rate is not universal. The scalings of the second-order moments, their dissipations, and production-dissipation ratios, are obtained and compared to the DNS. The flow is not self-similar in a conventional sense there is no single length scale that scales the flow. The moment similarity method produces three different scalings for the turbulence energy-containing length scale, e, the Taylor microscale, lambda, and the Kolmogorov dissipation scale, eta. The DNS and the self-similar analysis are in accord showing l similar to Agt(2), lambdasimilar tot(1/2) and etasimilar to((A(2)g(2)/v(3))t)(-1/4) achieving self-similar behaviour within three initial eddy turnovers of the inception of the turbulence growth phase at bulk Reynolds numbers in the range of Re = 800-1000 depending on initial conditions. A picture of a turbulence in which the largest scales grow, asymptotically, as t(2) and the smallest scales decrease as t(-1/4), emerges. As a consequence the bandwidth of the turbulence spectrum grows as t(9/4) and is consistent with the R-t(3/4) Kolmogorov scaling law of fully developed stationary turbulent flows. While not all moments are consistent, especially the dissipations and higher-order moments in the edge regions, with the self-similar results it appears possible to conclude that: (i) the turbulence length scales evolve as a power of h(t; C-o, h(0)); (n) alpha, as demonstrated mathematically for self-similar Rayleigh-Taylor turbulence and numerically by the DNS, is not a universal constant; (111) there is statistically significant correlation between decreasing a and lower low-wavenumber loading of the initial spectrum.