  Nonlinear Richtmyer--Meshkov instability and mixing transition induced by a
Mach 1.45 shock and subsequent re-shock at an interface between two ideal gases
(sulfur hexafluoride and air) with high Atwood number are studied with
second-moment analysis using data from high-resolution compressible
Navier--Stokes simulations. The analysis first addresses the importance of two
second-order moments: turbulent mass flux and density-specific-volume
covariance, together with their transport equations. These quantities play an
essential role in the development of Favre-averaged Reynolds stress and
turbulent kinetic energy in this variable-density flow. Then, grid
sensitivities and the time evolution of the turbulent quantities which include
the second-moments are investigated, followed by a detailed study of the
transport equations for the second-moments including the Reynolds stress and
the turbulent kinetic energy with well-resolved data before re-shock. After
re-shock, budgets of the same but large-scale turbulent quantities are studied
with the effects of the subfilter-scale stress taken into account. The budgets
of these large-scale quantities are shown to have an insignificant influence
from the numerical regularization. Finally, the effects of the subfilter-scale
stress on the budgets of the large-scale turbulent quantities with different
degrees of filtering are also examined.
