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Title: Receptivity of a boundary-layer subject to vertical vibrations and the secondary instability of spanwise modulated shear flows
Author: Pryce, David
ISNI:       0000 0004 5349 5027
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2015
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The viscous drag of a passenger aircraft is influenced significantly by the laminar-turbulent transition in the boundary-layer. Classically the transition process occurs when the turbulence develops as a result of the amplification of instability modes. In the flow past a unswept wing one such mode of instability is observed, Tollmien-Schlichting waves. We deal with the Tollmien-Schlichting waves or, more specifically, with the receptivity of the boundary layer with respect to them. If the surface becomes curved a different instability mode is created: Gortler vortices, which we will investigate as an inviscid secondary instability. Many disturbances can aid the generation of these instabilities but we concentrate on the vibrations of the wing caused by engine noise and the elasticity of the wing itself. This thesis is separated into these two problems of boundary-layer flow over the vibrating wing surface. Firstly we focus on the generation of Tollmien-Schlichting waves due to wing surface vibrations and surface roughness. Piston theory is used to describe the response of the flow outside the boundary layer to wing surface vibrations. Then the perturbations in the Stokes' layer are analysed. The Stokes' layer itself cannot produce a Tollmien-Schlichting wave. Therefore, we will assume that there is a wall roughness. The analysis of the interaction of the Stokes' layer with a wall roughness can be analysed with the help of the Triple deck theory. This allows us to consider the downstream effects of our disturbances and under our flow regime, dependent on the size of wavelength of vibrations, we can Fourier transform and solve our problem for the disturbance pressure. Once we have inverted our solution back into real space with the use of Residue theory we are then able to calculate receptivity coefficients which can be compared to those of previous studies. In the second problem we concentrate on the curved part of the wing where the flow is assumed to be slowly varying in the y direction. This generates Gortler vortices and we expect a sheared base flow with periodicity in the spanwise direction. Using a WKBJ approximation we can derive a multi-scale system of equations, which at leading order can be solved numerically to give eigenvalues and eigenfunctions representing pressure within the boundary-layer. We can only do so with the aid of a two-dimensional problem from which we fix an effective streamwise wavenumber, and an effective maximum growth rate. We use this to create an initial value for our WKBJ eigenvalue. This restricts the values of the streamwise wavenumber that we can calculate solutions for. This also means that when the streamwise wavenumber approaches the effective streamwise wavenumber we get a turning point and hence a breakdown of our WKBJ solutions. We derive and calculate solutions for streamwise wavenumbers away from this limit and discuss the breakdown of these solutions.
Supervisor: Ruban, Anatoly; Hall, Philip Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral