Viscosity, Conductivity, and Power Spectra of the Turbulent Convection in Boussinesq Fluids

Abstract

Abstract Turbulent viscosity, turbulent conductivity, and power spectra of the velocity and temperature fluctuations are formulated for Boussinesq fluids on the basis of a new spectral theory of turbulent convection. The temperature fluctuation and the lengthening of the decay time of a turbulent element due to buoyancy are taken into account. The power spectra, and the turbulent viscosity and conductivity are affected, and the combined system is to be solved consistently. As a result, the turbulent viscosity for instance is expressed differently from Heisenberg’s formula and no uncertain numerical factor needs to be introduced. Our theory is applicable whether the convective flux is less or more than the conductive flux. Numerical calculations are performed to determine the power spectra of steady turbulence for the case of the constant temperature gradient (constant conductive flux). The kinetic energy spectrum F(k) is zero at k=k0 (k0 is the minimum wavenumber), increases very rapidly, attains its maximum at k/k0=1.4–1.5, and then tends to be the Kolmogorov spectrum (≈k−5/3) at large wavenumbers k/k0≳3–4. The velocity correlation, the temperature correlation, and the convective flux calculated with the help of the spectrum are substantially smaller than those given by Ledoux et al. (1961).