Optimization of turbulent transition delay effect using quasi-statically transforming wall roughness shape

Boundary-layer transition on swept wings is dominantly caused by the crossflow instability, which is expected to be suppressed by placing artificial roughness elements near the leading edge. It is however difficult to find the optimal roughness shape by using direct numerical simulation (DNS), because a lot of computations are required for assessing a suppression effect due to one roughness shape. In this study, we develop an efficient method to evaluate the suppression effect for a series of roughness shapes by changing a shape parameter quasi-statically and observing the subsequent change of the crossflow mode at a downstream position. Since the mode grows spatially as convective instability, we need to allow for the delay time for the change in the shape to cause the change in the mode. This method is demonstrated for optimizing the height and angle of sinusoidal roughness elements. By applying a volume penalization (VP) method, the height and angle are changed slowly in DNS, where the initial values, rates of change and permeability of the VP method should be chosen appropriately to reproduce the correct results for the fixed shapes. The method developed here shows that the suppression (or laminarizing) effect tends to be improved as the height is increased, but there is a critical height at which flow tripping occurs. Both the laminarization effect and the critical height vary greatly depending on the angle. This result suggests the optimal roughness shape, considering the effectiveness and robustness. For laminar flow control, this method is useful for optimizing the wall roughness shape.