An immersed boundary method with y⁺-adaptive wall function for smooth wall shear

Immersed boundary (IB) method with a wall function for high Reynolds number flows is attractive when the boundary is complex and evolving. However, when combined with IB method, existing wall functions cannot produce smooth wall shear stress, which is important for many processes such as erosion. The root cause is the discontinuity between the log-law and laminar layers in most wall functions. A wall function in IB methods is typically enforced through IB cells. However, for complex and evolving boundaries, IB cells can be located in either the log-law layer or the laminar layer, which follow different laws. To remedy this, a new method is introduced with a y⁺-adaptive strategy. The idea is that when an IB cell is too close to an IB, it is replaced by a neighboring fluid cell further away from the boundary. Consequently, all IB cells are in the log-law layer. The resulting wall shear stress is much smoother. This adaptive strategy is a compromise between how accurate the location of an IB wall is represented and the smoothness of simulated wall shear. Example cases in 1D, 2D, and 3D show the y⁺-adaptive wall function produces results compared well with theory and experiments.