ABSTRACT
The kinetic theory is developed for the mass mixing of two incompressible immiscible fluids due to Rayleigh-Taylor instability (as an example for turbulence in variable-density statistically inhomogeneous incompressible fluids). An expression is derived for the fine grain force in terms of the mass-density and velocity fields. This expression enables the conversion of the Navier-Stokes equation into an exact explicit conservation equation in phase space. The equation is a generalization, to the variable-density case, of the Lundgren equation [T. S. Lundgren, Phys. Fluids 10, 969 (1967)]. The conserved quantity is the fine grain density-velocity distribution (FGDVD). The fine grain mass-density and fluid velocity fields are the two lowest moments of the FGDVD. The joint density-velocity probability density function (DVPDF) is the ensemble average of the FGDVD. Using detailed numerical solutions of the Navier-Stokes equation, it is found that the correlation between the acceleration and the FGDVD is weak. This result identifies a small parameter which enables the derivation, by controlled approximations, of closed equations for the DVPDFs. The lowest order yields the mean-field approximation. It is shown by a numerical solution of the closed kinetic equation in the mean-field approximation that it properly describes the time evolution of the system for periods shorter than the relaxation time. Closure schemes beyond the mean field are discussed.
ABSTRACT
A study--based on simulations and experiments as well as analytical derivations--of the internal structure of the fragmented ("mixed") state induced by the Rayleigh-Taylor instability at the interface between two fluids is presented. The distribution of sizes and the energy spectrum in the fragmented state are derived from the symmetries exhibited by the data and by dimensional analysis.
