Mean Temperature Profiles in Turbulent Thermal Convection

TL;DR

This paper develops a new model for predicting mean temperature profiles in turbulent thermal convection by relating eddy thermal diffusivity to the stream function, achieving excellent agreement with numerical simulations across a wide range of parameters.

Contribution

It introduces a novel relation based on Prandtl's mixing length model to solve the thermal boundary layer equation for general Prandtl numbers, providing a closed-form temperature profile.

Findings

Predicted temperature profiles match direct numerical simulations well.

Model works across a broad range of Prandtl and Rayleigh numbers.

Provides a practical analytical tool for turbulent thermal convection analysis.

Abstract

To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al., Phys. Rev. Lett. 114 (2015), the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated respectively as a cubic and a linear function of the distance from the plate. In the present work we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from 10^7 to 10^9.