Linear and nonlinear stability of plane poiseuille-couette flow
Creators: name: Kumar, Rishi
Publisher: name: Imperial College London
Year: 2020

description: The problem of the response of a viscous fluid in a channel to the dual effects of wall sliding and constant streamwise pressure gradient has received considerable attention from researchers over the years, due to its relevance to applications such as micro-electro-mechanical systems, aerodynamics heating, electrostatic precipitation and also of course the fact that the basic undisturbed profile, plane Poiseuille-Couette flow (PPCF), is an exact Navier-Stokes solution. The purpose of the thesis is to investigate the linear and nonlinear stability of this flow at asymptotically large Reynolds numbers and numerically at finite Reynolds numbers. The advantage of using an asymptotic approach is that it provides useful physical insight, theoretical understanding of underlying physical mechanisms, gives an easier access to the non-linear regime and forms the basis for carrying out numerical work at finite Reynolds numbers for the flow. The linear and nonlinear stability of PPCF to three-dimensional disturbances is investigated asymptotically at large values of the Reynolds numbers R based on channel half-width and the maximum velocity of the Poiseuille component. One of the main achievements of the thesis is demonstrating that three-dimensional nonlinear neutral modes exist in PPCF for disturbances with the magnitude of O(R^-4/9) when R &gt;&gt;1. 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 Δ = 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 phase speed of O(1). Neutral solutions are found to exist in the range 0 &lt; V &lt; 2 with c0 = V to leading order, where c0 and V are non-dimensional quantities representing the dominant phasespeed of the nonlinear 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 V = 0.34. The amplitude equation governing these modes is derived analytically and we further find that this asymptotic structure breaks down in the limit V→ 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. The numerical results from this interaction are found to compare well with full solutions of the Navier-Stokes equations. The second achievement of the thesis is that we demonstrate that three-dimensional nonlinear neutral modes exist in PPCF for disturbances with the magnitude of O(R^-1/3) when R &gt;&gt; 1. By analysing the nature of the instability for increasing disturbance size Δ, the scaling Δ = O(R^-1/3) is identified for 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). A method involving the classical balancing of the phase shifts 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 =&lt; V &lt; 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. The validity of these asymptotic structures is confirmed by comparison with numerical solutions obtained at finite Reynolds numbers.