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 (Reichardt, 1951; Spalding, 1961; Musker, 1979) or to include some non-equilibrium effects (Yang et al, 2015).
Compressible flow
The log-law (Eqn. \ref{eqn:log-law}) and the models by Reichardt, Spalding and Musker do not apply to compressible flow, so (at least at higher Mach numbers) one must take additional steps. One approach is solve a compressible ODE-based wall-model; another is to use an incompressible model profile (log-law, Reichardt, Spalding, Musker, …) and then transform this to the right compressible state. If the Van Driest transformation is used, the resulting wall-model is algebraic, requiring only a root-finding process for the friction velocity \(u_\tau\) (just like the incompressible models do). One method to do this is as follows:
- Guess \(u_\tau\).
- Compute \(h_{\rm wm}^+\) and the transformed velocity \(U_{\rm VD}^+(h_{\rm wm}^+)\) using the incompressible model profile.
- From \(u_\tau\), also compute the friction Mach number \(M_\tau=u_\tau/a_w\) and then solving for the inner layer heat transfer parameter \(B_q\) from the inner layer energy equation (Huang and Coleman, 1994)
\begin{equation}
\frac{T(h_{\rm wm})}{T_w} = 1 + Pr_e B_q \frac{U(h_{\rm wm})}{u_\tau} – Pr_e \frac{\gamma-1}{2} M_\tau^2 \left(\frac{U(h_{\rm wm})}{u_\tau}\right)^2,
\end{equation} with \(U(h_{\rm wm})\) and \(T(h_{\rm wm})\) coming from the LES at the matching location. Note that \(Pr_e\) is an effective Prandtl number accounting for both viscous and turbulent effects. - Solve the Van Driest transformation
\begin{equation}
U_{\rm VD}^+(h_{\rm wm}^+) = \frac{1}{A}\left[ \sin^{-1}\left( \frac{AU(h_{\rm wm})/u_\tau – B}{C} \right) – \sin^{-1}\left( \frac{B}{C} \right) \right]
\end{equation} with
\begin{equation}
A = M_\tau \sqrt{Pr_e \frac{\gamma-1}{2}}
, \
B = \frac{B_q}{M_\tau} \sqrt{\frac{Pr_e}{2(\gamma-1)}}
, \
C = \sqrt{1+B^2}
,
\end{equation} for a new \(u_\tau\). If it is not converged, repeat from step 2. - After convergence, compute \(\tau_w=\rho_w u_\tau^2\) and \(q_w=\rho_w u_\tau c_p T_w B_q\).
The Van Driest transformation is known to become inaccurate for highly cold walls. For such situations it is possible to use a different transformation, but most other transformations will require numerical quadrature (in addition to root-finding). In principle one could also change to a different temperature-velocity relation, although most such relations require knowledge of the external flow state (e.g., knowledge of the total or recovery temperature), which makes them less suited to wall-models.