Wall-stress model: algebraic

The simplest wall-stress models are algebraic, the most famous of which is the log-law where we assume that the mean velocity in the wall-modeled layer is

\begin{equation}\label{eqn:log-law}\tag{1}\frac{U_{\rm wm}(y)}{u_\tau} = \frac{1}{\kappa} \ln \left(\frac{y u_\tau}{\nu} \right) + B\,.\end{equation}

The coupling of this assumed velocity profile to the LES is discussed here, and leads to the matching condition \( U_{\rm wm} (h_{\rm wm}) = \left| \vec{u}_{{\rm wm-top}, \parallel} \right| \,,\) from which one can solve for the friction velocity \(u_\tau\) using either a fixed-point iteration or Newton-Raphson.

Note that the coupling should be done at a height \(h_{\rm wm}\) above the wall that is independent of the LES grid, i.e., the height should be specified without reference to the grid. Once that is specified, the LES grid needs to satisfy resolution criteria that are discussed here.

One could easily extend the assumed velocity profile in Eqn. (\ref{eqn:log-law}) to be more general, e.g. to include the viscous sublayer (Spalding, 1961, or Reichardt, 1951) or to include some non-equilibrium effects (Yang et al, 2015).