Analysis of the Anisotropic Term in the Implicit Algebraic Heat Flux Model in the Case of Low Prandtl Flow in Planar Impinging Jet Case

Numerical simulations of complex or industrial flows are predominantly simulated with Reynolds averaged Navier-Stokes equations. In heat transfer problems, the additional term — the turbulent heat flux — needs to be modelled to be able to solve the average temperature equation. The simple gradient diffusion hypothesis can be used to estimate the turbulent heat flux from the average temperature gradient. However, in recent years, an implicit algebraic heat flux model (AHFM) has seen an increased use in the literature in connection with low Prandtl fluids. The AHFM can account for anisotropy of the turbulence properties, however, the inclusion of the anisotropy is usually avoided due to the instability it introduces. In this paper, we analyze the instability on a planar impinging jet case, combined with a more theoretical approach. We show that in certain regions, the anisotropic term can cause the turbulent heat flux to point in the opposite direction as the molecular heat flux, which inhibits the heat transfer. With certain values of parameters, the anisotropy term can even cause the turbulent heat flux to overpower the molecular heat flux and leads to conditions in which heat would be “conducted” (combination of molecular and turbulent heat flux) from cool regions to hot regions. We present inequalities that limit the model parameters for which the model with the anisotropic term is stable in a generic case. We also apply these limits to the forced convection case of the planar impinging jet and compare the limits to the identified numerical constraint.