Wall Model

LES simulations usually present low resolution near wall boundaries, resulting in a poor representation of the turbulent boundary layer. The use of turbulent models to describe the flow near wall can provide more accurate values of wall stresses at boundary positions, hence wall models are adopted in turbulent flow Large-eddy simulations with a high \(\Delta y^{+}\).

For a point on a surface, as illustrated below, the wall model bridges the extreme points towards normal direction, and considers a 1D flow respective to normal and tangential velocity coordinates.

Being \(x^{+}_{n}=x_{n}u^{*}/\nu\) the normal distance from a surface position, the application range of the wall model is parametrized from \(x^{+}_{n,\mathrm{P_{0}}}=0\) to \(x^{+}_{n,\mathrm{P_{N}}}=h_{\mathrm{wm}}^{+}\).

The friction velocity at the wall \(u^{*}\) is defined as:

(1)\[u^{*}=\sqrt{\frac{\sigma_{\mathrm{w}}}{\rho}}\]

Calculated using the wall stress from previous time step or directly when the log law approximation is adopted.

Note

An iterative procedure can also be applied to achieve a higher precision for the stress estimation, though the small time step from LBM should prevent significant changes of stress magnitude for one time step, making suitable to use the stress from previous time step.

Application

The particularities that must be adopted for the use of wall models with the LBM boundary conditions and the immersed boundary method are described in separate sections.

Note

Despite the differences in the implementation, the modeling is still the same for both types of solid boundary representation.