Wall models

Several options are available for the top and bottom boundary conditions:

The stress free condition sets the stresses and vertical derivatives of the velocity equal to zero \[ \tau_{xz} = \tau_{yz} = 0 \qquad \qquad \frac{\partial u}{\partial z} = \frac{\partial v}{\partial z} = 0. \]
The viscous wall condition uses molecular viscosity and calculates the wall stress using the values at the first grid point away from the wall.
The equilibrium wall model applies the local law-of-the-wall expression for the wall stress \[ \tau_w = -\left[ \frac{\kappa}{\ln\left(z/z_0\right)}\right]^2 \left( \tilde{u}^2 + \tilde{v}^2\right)\] where \( \kappa \) is the von Kármán constant, \(z_0\) is the surface roughness height, and velocities have been test filtered at scale \( 2\Delta \).
The integral wall model uses a vertical profile similar to the classical integral method of von Kármán and Pohlhausen.

References

Yang X, Sadique J, Mittal R, Meneveau C. “Integral wall model for large eddy simulations of wall-bounded turbulent flows.” Physics of Fluids 27 (2015). 025112.