## A Novel Rough Wall Boundary Condition for LES of high Reynolds Number Flows

##### Abstract

The interactions between rough surfaces and fluid flows play an important role in turbulence simulation. The understanding of roughness elements at the wall (i.e., buildings and terrain features) to aerodynamics flow is crucial in wind energy from farm identification and assessment to turbine blade design. In this work, we propose a novel rough-wall boundary condition for LES to simulate flows over rough surfaces at high Reynolds numbers. The proposed rough-wall boundary condition consists of two parts: (1) smooth-wall modeling for high Reynolds number flow; (2) wall-modeling for roughness surface. To reduce the computational costs for high Reynolds number flow, a wall-modeling mesh is applied at the bottom of the boundary layer following (Kawai and Larsson 2012). In this procedure, the wall-modeling mesh will obtain velocity from LES mesh, solve for the shear stress according an equilibrium equation of boundary layer, and provides the calculated wall shear stress back to LES mesh. To verify the smooth wall-modeling LES part, the simulation of high Reynolds number flow in a channel is performed. The Reynolds number of the verification case is Re__8=u_c 8/u,,3.01x 10^5 and the thickness of the wall model is h_wm=0.18. The comparison of normalized streamwise velocity between the experiment, wall-modeling LES and pure LES are shown in the figure below. It is noted that the LES mesh of the modeling LES and pure LES are the same, but the wall-modeling LES will update the shear stress at the wall via wall modeling. Therefore, the wall-model LES results are the combination of the results of the wall-modeling part below h_wm and the LES part above h_wm. From the figure, it is can be seen that the wall modeling improves the results of LES when using relatively coarse grid at the boundary. Another part of the present model is the simulation of the influence of roughness elements. In the presented rough wall boundary condition, the flow around the roughness element, at the inner region of turbulent boundary layer, is not fully resolved. Instead, a one-layer roughness mesh is used to resolve the geometry of roughness elements. On the roughness mesh, the roughness geometry is adequately represented via the surface elevation. By projecting the instantaneous pressure onto the roughness surface, the instantaneous roughness shear stress is obtained. Since the smooth-wall and roughness shear stress are obtained, the total wall shear stress is obtained by adding the two parts. Then, the so obtained total wall shear stress is used to correct the flow at the near wall region. The LES mesh size, lix^+, liy^+ and liz^+ (in streamwise, wall-normal, and spanwise directions, respectively) in the present simulations can be as large as 50 to 4000, which is favorable for high-Reynolds number flow simulations in applications of wind turbines. Moreover, the presented wall model can solve roughness elements having size of K^+ ranging from 100 to several hundred wall units, which can be used to estimate the influence of roughness elements at different sizes. According to the results from the simulations, the presented rough wall-modeling boundary condition can perform high fidelity simulation for turbulent flow at higher Reynolds number by using a relatively low computational cost. The velocity profiles and Reynolds stress agree favorably with experimental data and numerical results in the literature. Therefore, the merits of the proposed rough-wall model are demonstrated.