Amplitude–Dependent Three–Dimensional Neutral Modes in Plane Poiseuille–Couette Flow at Large Reynolds Number. Issue 1 (31st December 2018) - Kumar, R; Walton, A G
Summary: The non-linear stability of plane Poiseuille–Couette flow to three-dimensional disturbances is investigated asymptotically at large values of the Reynolds numberR 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 sizeΔ=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 ofO(1) . Neutral solutions are found to exist in the range0<V<2 withc_0 = V to leading order, wherec_0 andV 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 limitV → 2 when the disturbance streamwise wavelength decreases toO(R^-1/3 ) and the maximum of the basic flow becomes located at the upper wall.