Self-similar spectra in Rayleigh-Taylor turbulence

The study considers a perfect fluid. The author begins by recalling how one can evaluate the characteristic transition times to small-scale turbulence in isotropic turbulence, having determined the time of divergence of enstrophy. A semi-local model derived from the EDQNM theory and corresponding to a particular case of the BHRZ model is proposed. For these models, the analysis leading to self-similar solutions for the kinetic energy spectrum is recalled, as well as the influence of the reverse transfer in k⁴ (backscatter). The study is then considering the turbulence generated by instability of Rayleigh-Taylor (RT). Phenomenological arguments involving in particular the backscatter lead to propose a Karman-Howarth type spectrum of energy which evolves in a self-similar way. This spectrum is proportional to (A_tg)⁷t¹²k⁴ at small k, and to (A_tg)^4/3t^2/3k^-5/3 at large k, where A_t is the number of Atwood. Certain constants of the model are evaluated by comparison with the direct numerical simulations of Youngs, then the possibility of such a spectrum is studied in the framework of a simple spectral model, where a source term of RT type is considered in the isotropic spectral equation. Finally, the self-similar spectrum is of the E(k, t) = (A_tg)³t⁴f(A_tgt^2k) form. The influence of instability in the relaxation time of triple correlations is considered. Finally, some sub-mesh models are presented that could validate these phenomenological predictions in large-scale simulations. These are the models of Kraichnan's turbulent viscosity, of the structure function, and of the filtered structure function