Transient growth in accelerating and decelerating laminar planar flows

Although the stability of planar flows with constant pressure gradient or wall motion is well understood, little focus has been put towards understanding the stability properties of unsteady baseflows. Two major challenges when investigating these flows include: 1) determining the laminar flow about which to analyze stability and 2) determining how to incorporate the time-varying linear operator. Here we overcome these challenges by first deriving an analytical solution for laminar profiles of planar flows with arbitrary wall motion and pressure gradient. Then, we study the stability of specific flows by investigating the nonnormal growth of perturbations through the time-varying linearized equations of motion. In particular, we investigate exponentially decaying acceleration and deceleration of wall motion and flow rate. The accelerating cases exhibit growth comparable to stationary flows, while the decelerating cases exhibit massive nonnormal growth -- at a Reynolds number of 800 growth of the decelerating flow is O(10⁵) times larger than growth of the stationary flow. As the rate of deceleration and Reynolds number increase the perturbations become further amplified, and the largest growth rate moves from a spanwise disturbance to a streamwise disturbance upon increasing these values.