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

Abstract

For turbulent channel flow, pipe flow, and zero-pressure gradient boundary layer, Heinz yielded recently 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, an exact high-Reynolds-number equation with one production and two dissipation terms is constructed for those flows. 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 scale-adaptive simulation 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 Reynolds-averaged Navier–Stokes models with a similar equation. It is shown that some models, e.g., the one of Spalart and Allmaras, do not respect the flow physics: they display a production peak in the near-wall region. The most promising model, a scale-adaptive simulation model, is modified. As a step forward toward a solution to the wall damping problem, the equation of our model behaves much more correctly in the near-wall region.