Kinetic theoretical approach to turbulence in variable-density incompressible, statistically inhomogeneous fluids.
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.
Hazak
G
G
Physics Department, Nuclear Research Center, Negev, Beer Sheva, Israel.
Elbaz
Y
Y
Zalesak
S
S
Wygoda
N
N
Schmitt
A J
AJ