Data-driven algebraic models of the turbulent Prandtl number for buoyancy-affected flow near a vertical surface

TL;DR

This paper develops machine learning-based algebraic models for the turbulent Prandtl number in buoyancy-affected vertical flows, addressing the infinity anomaly and demonstrating robustness across a range of natural and mixed convection conditions.

Contribution

It introduces novel algebraic models for the turbulent Prandtl number using symbolic regression, overcoming the infinity issue and ensuring coordinate invariance for buoyant vertical flows.

Findings

Models successfully handle the infinity anomaly in $Pr_t$.

Models are robust across a wide range of Rayleigh and Richardson numbers.

Proposed models improve prediction of heat transfer in buoyant flows.

Abstract

The behaviour of the turbulent Prandtl number ($Pr_t$) for buoyancy-affected flows near a vertical surface is investigated as an extension study of {Gibson \& Leslie, \emph{Int. Comm. Heat Mass Transfer}, Vol. 11, pp. 73-84 (1984)}. By analysing the location of mean velocity maxima in a differentially heated vertical planar channel, we {identify an} {infinity anomaly} for the eddy viscosity $\nu_t$ and the turbulent Prandtl number $Pr_t$, as both terms are divided by the mean velocity gradient according to the standard definition, in vertical buoyant flow. To predict the quantities of interest, e.g. the Nusselt number, a machine learning framework via symbolic regression is used with various cost functions, e.g. the mean velocity gradient, with the aid of the latest direct numerical simulation (DNS) dataset for vertical natural and mixed convection. The study has yielded two key outcomes: $(i)$ the new machine learnt algebraic models, as the reciprocal of $Pr_t$, successfully handle the infinity issue for both vertical natural and mixed convection; and $(ii)$ the proposed models with embedded coordinate frame invariance can be conveniently implemented in the Reynolds-averaged scalar equation and are proven to be robust and accurate in the current parameter space, where the Rayleigh number spans from $10^5$ to $10^9 $ for vertical natural convection and the bulk Richardson number $Ri_b $ is in the range of $ 0$ and $ 0.1$ for vertical mixed convection.