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

Abstract

The behaviour of the turbulent Prandtl number (Prt) for buoyancy-affected flows near a vertical surface is investigated as an extension study of Gibson & Leslie, 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 νt and the turbulent Prandtl number Prt, 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 Prt, 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 105 to 109 for vertical natural convection and the bulk Richardson number Rib is in the range of 0 and 0.1 for vertical mixed convection.