Simulate with complex geometries and complex physics
%ind_tauWv(1)%
x-component of turbulent wall shear stress [N/m^2]
This quantity is computed for each regular boundary point. However it takes effect / makes sense only in the case of BC_v(...) = ( %BND_slip%, ...). If a noslip boundary is concerned, this value shall be negligible.
From the knowledge of \( y_p\), \( k_p\) and \( \mathbf{U}_p\) we compute nondimensionalized values
\begin{align} y^* = \frac{\rho \cdot c_\mu^{1/4} \cdot k_p^{1/2} \cdot y_p }{ \eta }\end{align}
as per definition, and
\begin{align} U^* = \frac{1}{\kappa} \log \left( E y^* \right)\end{align}
as by the logarithmic boundary layer model.
Finally we derive the formulation for the vector-valued turbulent wall stress by
\begin{align} \mathbf{\tau}_W = \frac{\rho \cdot c_\mu^{1/4} \cdot k_p^{1/2} }{ U^*} \cdot \mathbf{U}_p\end{align}
where
\begin{align} \mathbf{U}_p = \mathbf{v}_p-\mathbf{v}_\text{boundary} - C^\text{virt} \cdot \frac{1}{\rho} \nabla p \cdot \tau_p^\text{virt}\end{align}
with \( C^\text{virt} = 1\), currently.
The virtual equilibration time is given by
\begin{align} \tau_p^\text{virt} = \frac{1}{2} \left( \frac{y_v}{\kappa k_p^{1/2}} ln\left(\frac{y_p}{y_v}\right) + \frac{y_p-y_v}{\kappa k_p^{1/2}} + \frac{\rho}{\eta}y_v^2 \right)\end{align}.
Be aware, that
- \( \mathbf{v}_\text{boundary}\) = %ind_v_p(1)% ... %ind_v_p(3)%
- \( \mathbf{v}_p\) = %ind_v(1)% ... %ind_v(3)%