Effects of uniform injection at the wall on the stability of Couette-like flows

Abstract

A linear stability analysis of a family of nearly parallel wall-bounded flows with injection at the lower wall and suction at the upper one is presented. The mean pressure gradient is such that the streamwise velocity profile remains linear in spite of injection. An asymptotic analysis shows that, for weak injection, the expansion rate of linear perturbations is a linear function of the injection Reynolds number (based on the width of the domain and the injection velocity). This point is confirmed by a numerical solver that also shows that, due to the injection process, the eigenmodes are drastically reorganized in the complex plane. In particular, the eigenvalue distribution is no longer symmetric. Moreover, the imaginary part of the phase velocity tends to increase so that some modes may become linearly unstable. However, the base flow remains stable as long as the injection Reynolds number is lower than a critical value close to 48. It is also found that higher injection rates (injection Reynolds numbers greater than 80) stabilize the flow, as already observed for channel or rotating flows.