The decay of kinetic energy and temperature variance in three-dimensional isotropic turbulence

Abstract

With the aid of the eddy‐damped quasinormal Markovian statistical theory (EDQNM) developed by Orszag [J. Fluid Mech. 4 1, 363 (1970)], the dynamics of a passive scalar (such as the temperature in a slightly heated flow) in three‐dimensional isotropic turbulence is studied. Starting initially with kinetic energy and temperature spectra exponentially decreasing above a wavenumber k_i, it is shown that in the limit of zero viscosity (ν→0) and conductivity (κ→0) the temperature gradient variance diverges at a finite critical time t_c, together with the enstrophy. After t_c, the kinetic energy and temperature variance are dissipated at finite rates that are independent of ν and κ if both are small. Afterward, the decay laws of the temperature variance and the wavenumber k_θ characteristic of the temperature spectrum maximum are determined analytically when the temperature is initially injected at k_θ≫k_i. First, a Richardson law for the temperature integral scale is demonstrated without any assumption on the low k behavior of the temperature spectrum. Second, it is shown that k_θ(t) rapidly catches up with k_i(t), which explains some ‘‘anomalous’’ temperature decay behavior, as observed experimentally by Warhaft and Lumley [J. Fluid Mech. 8 8, 659 (1978)]. The analytical analysis of the latter phenomenon generalizes, for an arbitrary infrared temperature spectrum, an earlier study of Nelkin and Kerr [Phys. Fluids 2 4, 1754 (1981)].