The effect of wall waviness on shear flow instabilities
This thesis is concerned with the effect of wall waviness on shear flow instability, specifically for the incomprehensiblc flow in a channel. We investigate the stability of the flow in a channel with fixed wavy boundaries using two methods. Firstly the disturbance evolution is calculated using the parabolised stability equations (PSE), which apply to the flow stability at finite Reynolds number, and are solved using a finite-difference marching scheme, marching in the downstream direction. Secondly we employ the triple-deck formulation for channel flow which is valid at asymptotically high Reynolds number and the problem is solved using Floquet theory, making use of the periodic coefficients appearing in the disturbance equations. The mean flow for the PSE analysis is obtained by linearising the Navier-Stokes equations using a perturbation method, valid for small amplitude boundary waviness, Δ. We solve the linear PSE using this periodic mean flow, and it is found that increasing Δ stabilises plane Poiseuille flow near the nose of the neutral curve but has a destabilising effect on the lower branch for higher Reynolds numbers. The nonlinear PSE are used to study thc stability of 2-D finite amplitude waves, and are able to demonstrate the existence of suporcritical equilibrium amplitude solutions, as well as threshold amplitudes separating growing and decaying solutions in the subcritical regime. Wall waviness is found to have a stabilising effect on subcritical disturbances, raising the amplitude needed for instability to occur. Using Floquet theory and decomposing the disturbance equations into Fourier modes enables the high Reynolds number problem to be formulated as an eigenvalue problem. The waviness is found to be able to produce a destabilising effect in agreement with the results for the linear PSE near the lower branch. The method of multiple scales is used to study the wavy channel flow stability at high Reynolds number in the limit of small Δ, which gives an O(Δ 2) correction to the flat boundary eigenvalue, λ. When λ = ±i μ, for boundary wavenumber, μ, we find a degeneracy in the intermediate O(Δ) system of equations due to a resonance ktween a neutrally stable flat-boundary T-S wave and the boundary wave of equal wave-length. New asymptotic scalings are derived in this case to obtain a valid solution.