Avoiding explosive transient growth in laminar planar flows with optimal deceleration profiles*

Perturbations in decelerating laminar flows can exhibit massive growth not observed in either steady or accelerating flows. Some examples of deceleration profiles that exhibit this behavior include linear deceleration,  exponentially decaying deceleration, and the decelerating segment of oscillatory flows. However, the Reynolds number at which this growth becomes substantial varies based on the deceleration profile, indicating that perturbations about some profiles are less prone to transient growth. As mitigating this growth is desirable to avoid turbulent transition, we seek the optimal deceleration profiles for minimizing the transient growth of perturbations in wall-driven channel flow.  We perform this analysis by formulating a constrained optimization problem to minimize transient growth for wall motion that begins with a simple shear profile and decays to no flow. In this problem, we constrain the magnitude of the wall motion. When the wall motion is limited to positive values, we find that the optimal wall motion drastically reduces the growth of perturbations, compared to linear deceleration, by damping growth caused by the Orr mechanism (streamwise perturbations). When the wall motion is allowed to take on negative values, we find the optimal wall motion further reduces the growth of perturbations by damping growth from both the lift-up mechanism (spanwise perturbations) and the Orr mechanism.