Temperature dynamics in decaying isotropic turbulence with Joule heat production
Résumé
This paper presents an extension of existing works dealing with the dynamics of a passive scalar in freely decaying isotropic turbulence, by accounting for a production mechanism of the passive scalar itself. The physically relevant case of the temperature dynamics in the presence of Joule heating via the dissipation of the turbulent kinetic energy is selected and analysed by theoretical and numerical means. In particular, the sensitivity of the temperature decay to the non-dimensional parameters Prandtl number (Pr) and Eckert number (Ec), the latter measuring the intensity of the internal energy production mechanism, is investigated. The time behaviour of the global quantities such as the temperature variance θ2¯¯¯(t) and its destruction rate εθ(t) is analysed, and a detailed analysis of the temperature variance spectrum Eθ(k) is provided. In the case of a very strong heating mechanism, some important modifications of the temperature dynamics are observed. The time-decay-law exponents of the global physical quantities assume new values, which are governed only by features of the kinetic energy spectrum, while they depend on the shape of Eθ(k) in the classical free-decay case. The temperature variance spectrum Eθ(k) exhibits two new spectral ranges. One is a convective–production range such that Eθ(k)∝k1/3 is observed for a finite time at all values of Pr. In the case of very diffusive fluids with Pr≪1, a convective–diffusive–production range with Eθ(k)∝k−7/3 is also detected.