Turbulent barodiffusion, turbulent thermal diffusion, and large-scale instability in gases

Two effects, turbulent barodiffusion and turbulent thermal diffusion in gases, are discussed. These phenomena are related to the dynamics of a gaseous admixture in compressible turbulent fluid flow with low Mach numbers. Turbulent barodiffusion causes an additional mass flux of the gaseous admixture directed to the maximum of the mean fluid pressure, while turbulent thermal diffusion results in an accumulation of the gaseous admixture in the vicinity of the minimum of the mean temperature of the surrounding fluid. At large Péclet and Reynolds numbers these additional turbulent fluxes are considerably higher than those caused by molecular barodiffusion and molecular thermal diffusion. It is shown that turbulent barodiffusion and turbulent thermal diffusion may contribute to the formation of large-scale inhomogeneous structures in a gaseous admixture advected by a low-Mach-number compressible turbulent velocity field. The large-scale dynamics are studied by considering the stability of the equilibrium solution of the derived evolution equation for the mean number density of the gaseous admixture in the limit of large Péclet numbers. The resulting equation is reduced to an eigenvalue problem for a Schrödinger equation with a variable mass, and a modified Rayleigh-Ritz variational method is used to estimate the lowest eigenvalue (corresponding to the growth rate of the instability). This estimate is in good agreement with obtained numerical solution of the Schrödinger equation.