Some Remarks on the Dynamical and Thermal Structure of a Heated Turbulent Fluid

Abstract

Spectral functions for the velocity and temperature fluctuations in heated turbulent flow are obtained in the forms:
F(k)∼ε^2⁄3k^−5⁄3, G(k)∼(K⁄K_H)²Q²ε^−2⁄3k^−7⁄3
for the intermediate scale of turbulence element (turbulon) and
F(k)∼(ε⁄ν)k⁻³, G(k)∼Q²x²k⁻⁵
for the smallest scale. Here k is the turbulon wave number, ε the rate of energy dissipation, Q that of heat transfer, ν the kinematic viscosity, x the heat diffusivity, K the turbulent viscosity, and K_H the turbulent heat diffusivity. Then the concept of smallest turbulon is defined as the intersection of two forms of the spectral function. Thus it turns out that the scale of smallest dynamical turbulon coincides with that of smallest thermal turbulon provided the Prandtl number ν/x is equal to K/K_H.