Use of direct numerical simulation to study the effect of Prandtl number on temperature fields

Abstract

Abstract The influence of Prandtl number on statistical parameters characterizing turbulent transport and the spatial variation of the mean-square of the temperature fluctuations, θ 2 , is described. The system considered is fully developed flow in a channel for which the bottom wall is heated and the top wall is cooled. Results from direct numerical solutions of the Euler balance equations for Pr=0.3, 1, 3, 10 and Lagrangian studies, in a DNS, of the dispersion of heat markers from wall sources for Pr=0.1–2400 are used. The Eulerian results for Pr=10 are new and of particular interest. A time scale τ θ can be defined from the dissipation of k θ = θ 2 /2 , as τ θ = k θ / ϵ θ . This is analogous to the time scale defined from the dissipation of turbulent kinetic energy, τ = k / ϵ . Prandtl number is found to strongly affect τ θ and the correlation, u i θ /( u i 2 ) 1/2 ( θ 2 ) 1/2 . These results can be understood by recognizing that the spectral density function for temperature fluctuations extends over an increasingly larger range of wavenumbers as Pr increases. The observed effect of Pr on τ θ suggests fundamental problems in developing relations for the turbulent diffusivity by a k θ ττ θ approach analogous to the kτ approach used to describe momentum transport. The use of a gradient transport model to represent the turbulent transport of k θ also has fundamental problems.