Variable Transformation for Explicit Wall-Shear Stress Formula

TL;DR

This paper introduces a variable transformation for wall functions in fluid dynamics that provides an explicit wall-shear stress expression, simplifying implementation and avoiding iterative calculations in CFD simulations.

Contribution

A new variable transformation for wall functions that yields explicit wall-shear stress formulas, enhancing computational efficiency and ease of implementation.

Findings

Transformation enables explicit wall-shear stress expressions.

Law of the wall remains consistent in new variables.

Numerical simulations validate the approach.

Abstract

We formulate a general variable transformation for existing wall functions that allows for an explicit wall-shear stress term. The proposed transformation aims to enable an explicit expression of wall-shear stress and simplify the implementation of existing wall functions in simulation codes through a simple transformation of variables. The transformation is defined by introducing a new velocity scale allowing the definition of the new wall unit variable $(r^+)$ and the corresponding normalized viscosity $(\eta_t^+)$. We also demonstrate that the law of the wall in new variables is equivalent to the one expressed in wall-normal variables $(y^+_{\tau})$ and $(\nu_t^+)$. The new form of the law is particularly suitable for implementations in computational fluid dynamics codes as it does not require an iterative procedure to evaluate the wall shear stress. We show how to transform several known and often used expressions of wall functions with and without pressure gradients to allow for an explicit wall-shear stress expression. Finally, we illustrate the new form of wall functions by performing illustrative numerical simulations.