Dynamics of the passive scalar in compressible turbulent flow: Large-scale patterns and small-scale fluctuations.

Abstract

The work analyzes fluctuations of the passive scalar and large-scale (mean field) effects in a turbulent compressible fluid flow. It is shown that passive scalar transport can be accompanied by slow diffusion of small-scale inhomogeneous fluctuating structures for large P\'eclet numbers, Pe\ensuremath{\gg}1. The origin of the inhibition of the diffusion of small-scale fluctuations of the passive scalar is associated with compressibility (i.e., div u\ensuremath{\propto}\ensuremath{\partial}\ensuremath{\rho}/\ensuremath{\partial}t\ensuremath{\ne}0) of a surrounding fluid flow. The conditions for the slow diffusion of the passive scalar fluctuations in homogeneous and isotropic turbulent flow are found. It is shown that the magnitude of the fluctuations of the passive scalar generated in the presence of external gradient of the mean mass concentration \ensuremath{\nabla}Q in compressible fluid flow can be fairly strong: \ensuremath{\surd}〈${\mathit{q}}^{2}$〉 \ensuremath{\sim}${\mathit{l}}_{0}$ln(Pe)\ensuremath{\Vert}\ensuremath{\nabla}Q\ensuremath{\Vert}, where ${\mathit{l}}_{0}$ is the characteristic scale of the turbulent velocity field. The characteristic spatial scale of a localization of solutions is of the order of ${\mathit{l}}_{0}$/ \ensuremath{\surd}Pe . In addition, compressibility in the stratified turbulent inhomogeneous fluid flow [i.e., div u=-(\ensuremath{\nabla}\ensuremath{\rho}\ensuremath{\cdot}u)/\ensuremath{\rho}\ensuremath{\ne}0] results in formation of large-scale structures for large P\'eclet numbers. The formation of these patterns is caused by the instability of the uniform distribution of the mean passive scalar field whereby an additional nondiffusive component of the flux of passive scalar particles results in a large-scale pattern. The conditions for the excitation of the instability of the mean field are found. Possible environmental applications of these effects are discussed.