On new unstable three-dimensional oblique modes examplified by plane Couette flow

Squire proved that temporally unstable 2D modes are the most unstable ones. In the present work, we expand the Squire theorem by extending it to spatial instabilities allowing for growth in both the streamwise and in the spanwise direction, i.e. we imply complex wave numbers $\alpha$ and $\beta$, i.e. growth rates are then tightly coupled in both directions. The key result is that we can thus generate critical Reynolds numbers that are significantly lower than those of 2D modes, and, if no unstable modes exist at a finite Reynolds number as for plane Couette flow, unstable modes can be calculated at a finite Reynolds number. The complex $\alpha$ and $\beta$ represent an oblique mode structure, which brakes spanwise reflection symmetry. The new modes are generic, but we apply them to the plane Couette flow. For this flow, oblique-type modes at finite amplitude are documented in the literature in experiments and simulations. The present theory shows that no explicit limit can be given for the critical Reynolds number, because it depends on the fact which modes can be realised in streamwise and spanwise directions and these in turn depend on the size of the domain under consideration. By means of highly accurate simulations, we can verify the oblique modes for the Couette flow.