  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.
