# The SquAd derivation: A Square Additive approach to the turbulent Prandtl number

## TL;DR

This paper introduces a new formula for the turbulent Prandtl number in liquid metals, improving simulations of turbulent thermal flows.

## Contribution

A novel formula for the turbulent Prandtl number is derived using square additivity of molecular and turbulent parameters.

## Key findings

- A new formula for the turbulent Prandtl number is derived using local variables from two-equations turbulence models.
- The formula avoids degeneracy and can approximate a Kays-like formulation.
- The approach may enhance simulations of turbulent thermal flows in liquid metals.

## Abstract

Liquid metals have been chosen as primary coolant of innovative nuclear systems under current development. They present a very high thermal conductivity and hence a very low molecular Prandtl number. This feature challenges the modeling of turbulent thermal flows applying the Reynolds analogy. This paper addresses this challenge. A new formula for the turbulent Prandtl number is derived in terms of local variables available from two-equations turbulence models. The derivation is a direct consequence of the expected square additivity of the molecular and turbulent parameters defining the effective viscosity and the effective conductivity. The formula does not degenerate and leads to a Kays like formulation if approximated. While constrained by the quality of the turbulent viscosity modeling, it has the potential to improve the numerical simulation of turbulent thermal flows.