Amplitude–Dependent Three–Dimensional Neutral Modes in Plane Poiseuille–Couette Flow at Large Reynolds Number. Issue 1 (31st December 2018)

• Record Type: Journal Article
• Authors: Kumar, R; Walton, A G
• Is Part Of: Quarterly journal of mechanics and applied mathematics. Volume 72:Issue 1(2019:Feb.)
• Page Start: 87
• Page End: 130
• Publication Date: 2018-12-31
• Subjects: Mechanics -- Mathematics -- Periodicals; Applied mathematics -- Periodicals; 530.1
• DOI: 10.1093/qjmam/hby022
• Languages: English
• ISSNs: 0033-5614

Abstract

Summary: 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.