Amplitude–Dependent Three–Dimensional Neutral Modes in Plane Poiseuille–Couette Flow at Large Reynolds Number

The non-linear stability of plane Poiseuille–Couette flow to three-dimensional disturbances is investigated asymptotically at large values of the Reynolds number $R$ based on channel half-width and the maximum velocity of the Poiseuille component. The asymptotic theory, aimed at a detailed understanding of the physical mechanisms governing the amplitude-dependent stability properties of the flow, shows that the phase shifts induced across the critical layer and a near-wall shear layer are comparable when the disturbance size $\Delta=O(R^{-4/9} )$. In addition, it emerges that at this crucial size both streamwise and spanwise wavelengths of the travelling wave disturbance are comparable with the channel width, with an associated phasespeed of $O(1)$. Neutral solutions are found to exist in the range $0<V<2$ with $c_0 = V$ to leading order, where $c_0$ and $V$ are non-dimensional quantities representing the dominant phasespeed of the non-linear travelling waves and the wall sliding speed respectively. Moreover, these instability modes exist at sliding speeds well in excess of the linear instability cut-off. The amplitude equation governing these modes is derived analytically and we further find that this asymptotic structure breaks down in the limit $V \rightarrow 2$ when the disturbance streamwise wavelength decreases to $O(R^{-1/3} )$ and the maximum of the basic flow becomes located at the upper wall.