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

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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 ki, 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 tc, together with the enstrophy. After tc, 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 kθ characteristic of the temperature spectrum maximum are determined analytically when the temperature is initially injected at kθ≫ki. 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 kθ(t) rapidly catches up with ki(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)].