Self-sustaining dual critical layer states in plane Poiseuille-Couette flow at large Reynolds number

The nonlinear stability of plane Poiseuille-Couette flow subjected to three-dimensional disturbances is studied asymptotically at large Reynolds number R. By analysing the nature of the instability for increasing disturbance size Delta, the scaling Delta = O(R-1/3) is identified at which a strongly nonlinear neutral wave structure emerges, involving the interaction of two inviscid critical layers. The striking feature of this structure is that the travelling wave disturbances have both streamwise and spanwise wavelengths comparable to the channel width, with an associated phase speed of O(1). An alternative method to the classical balancing of phase shifts is proposed, involving vorticity jumps, that uses a global property of the flow-field and enables the amplitude-dependence of the neutral modes to be determined in terms of the wavenumbers and the properties of the basic flow. Numerical computation of the Rayleigh equation which governs the flow outside of the critical layers shows that neutral solutions exist for non-dimensional wall sliding speeds in the range 0 <= V < 2. It transpires that the critical layers merge and the asymptotic structure referred to above breaks down both in the large-amplitude limit and the limit V -> 2 when the maximum of the basic flow becomes located at the upper wall.