Exact Eddy-Viscosity Equation for Turbulent Wall Flows—Implications for Computational Fluid Dynamics Models

Emmanuel Plaut (1) , Stefan Heinz (2)

1 LEMTA - Laboratoire d'Energétique et Mécanique Théorique et Appliquée
2 UW - University of Wyoming

For turbulent channel flow, pipe flow and zero-pressure gradient boundary layer, Heinz (2018, 2019) yields analytical formulas for the eddy-viscosity as a product of a function of + (the wall-normal distance scaled in inner units) and a function of / (the same scaled in outer units). By calculating the eddy-viscosity turbulent diffusion term, we construct for those flows an exact high-Reynolds number equation with one production and two dissipation terms. One dissipation term is universal, peaks near the wall, and scales mainly with +. The second, smaller one, is flow-dependent, peaks in the wake, and scales mainly with /. The production term is flow-dependent, peaks in between, and scales similarly. The universal dissipation term implies a length scale analogous to the von Karman length scale used in the SAS models of Menter. This length scale also appears in the production term. This confirms the relevance of these length scales. An asymptotic analysis of all terms in the budget in the limit of infinite Reynolds numbers is provided. This yields a test bench of existing RANS models with a similar equation. We show that some models, e.g. the one of Spalart & Allmaras, do not respect the flow physics: they display a production peak in the near-wall region. We modify the most promising model, a SAS model. As a step forward towards a solution to the wall damping problem, the equation of our model behaves much more correctly in the near-wall region.

Domaines: Milieux fluides et réactifs

Emmanuel Plaut, Stefan Heinz. Exact Eddy-Viscosity Equation for Turbulent Wall Flows—Implications for Computational Fluid Dynamics Models. AIAA Journal, 2022, 60 (3), pp.1347-1364. ⟨10.2514/1.J060761⟩. ⟨hal-03708277⟩