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

Direct numerical simulations and a self-similar analysis of the single-fluid Boussinesq 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 \,≤\, 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;Cₒ,h₀)\,=\, Cₒ Agt²+ √AgCₒh1/2₀ t+h₀ ; the result for h(t;Cₒ,h₀) 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 ~ Cₒ Agt² 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, , the Taylor microscale, , and the Kolmogorov dissipation scale, η . The DNS and the self-similar analysis are in accord showing \,~\, Agt² , \,~\, t1/2 and η \,~\, ((A²g²/ν³)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 t² and the smallest scales decrease as t-1/4 , emerges. As a consequence the bandwidth of the turbulence spectrum grows as t9/4 and is consistent with the Rₜ3/4 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;Cₒ,h₀) ; (ii) , as demonstrated mathematically for self-similar Rayleigh–Taylor turbulence and numerically by the DNS, is not a universal constant; (iii) there is statistically significant correlation between decreasing α and lower low-wavenumber loading of the initial spectrum.