  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.
