Linear Stability Analysis of Oblique Couette-Poiseuille flows

Abstract: We perform a detailed numerical study of modal and non-modal stability in oblique Couette-Poiseuille profiles, which are among the simplest examples of three-dimensional boundary layers. Through a comparison with the Orr-Sommerfeld operator for the aligned case, we show how an effective wall speed succinctly characterizes modal stability. Large-scale parameter sweeps reveal that the misalignment between the pressure gradient and wall motion is, in general, destabilizing. For flows that are sufficiently oblique, the instability is found to depend exclusively on the direction of wall motion and not on its speed, a conclusion supported, in part, by the perturbation energy budget and the evolution of the critical layers. Closed forms for the critical parameters in this regime are derived using a simple analysis. Finally, a modified long-wavelength approximation is developed, and the resulting asymptotic eigenvalue problem is used to show that there is no cutoff wall speed for unconditional stability whenever the angle of wall motion is non-zero, in stark contrast to the aligned case. From a non-modal perspective, pseudo-resonance is examined through the resolvent and the $\epsilon$-pseudospectra. An analysis of the unforced initial value problem shows that the maximum energy gain is highly dependent on both the magnitude and direction of the wall velocity. However, the strongest amplification is always achieved for configurations that are only weakly skewed. Finally, the optimal perturbations appear to develop via a lift-up effect induced by an Orr-like mechanism.