The Anisotropy of Low Prandtl Number Turbulent Convection

Abstract

A model for homogeneous anisotropic incompressible turbulence is proposed. The model generalizes the GISS model of homogeneous isotropic turbulence; the generalization involves the solution of the GISS equations along a set of integration paths in wavenumber (k-) space. In order to make the problem tractable, these integration paths ("cascade lines") must be chosen in such a way that the behaviour of the energy spectral function along different cascade lines should be reasonably similar. In practice this is realized by defining the cascade lines as the streamlines of a cascade flow; in the simplest case the source of this flow may be identified with the source function of the turbulence. Owing to the different approximations involved, the resulting energy spectral function is not exact but is expected to give good approximative values for the bulk quantities characterising the turbulent medium, and for the measure of the anisotropy itself in particular. The model is then applied to the case of low Prandtl number thermal convection. The energy spectral function and the bulk quantities characterizing the flow are derived for different values of the parameter $S=Ra σ$. The most important new finding is that unlike the anisotropy of the most unstable mode in linear stability analysis the anisotropy of the turbulence does not grow indefinitely with increasing S but it rather saturates to a relatively moderate finite asymptotic value.