Mean equation based scaling analysis of fully-developed turbulent channel flow with uniform heat generation

Abstract

Multi-scale analysis of the mean equation for passive scalar transport is used to investigate the asymptotic scaling structure of fully developed turbulent channel flow subjected to uniform heat generation. Unlike previous studies of channel flow heat transport with fixed surface temperature or constant inward surface flux, the present flow has a constant outward wall flux that accommodates for the volumetrically uniform heat generation. This configuration has distinct analytical advantages relative to precisely elucidating the underlying self-similar structure admitted by the mean transport equation. The present analyses are advanced using direct numerical simulations (Pirozzoli et al., 2016) that cover friction Reynolds numbers up to δ+=4088 and Prandtl numbers ranging from Pr=0.2–1.0. The leading balances of terms in the mean equation are determined empirically and then analytically described. Consistent with its asymptotic universality, the logarithmic mean temperature profile is shown analytically to arise as a similarity solution to the mean scalar equation, with this solution emerging (as δ+→∞) on an interior domain where molecular diffusion effects are negligible. In addition to clarifying the Reynolds and Prandtl number influences on the von Kármán constant for temperature, kθ, the present theory also provides a couple of self-consistent ways to estimate, kθ. As with previous empirical observations, the present analytical predictions for kθ indicate values that are larger than found for the mean velocity von Kármán constant. The potential origin of this is briefly discussed.