Thermally driven three-dimensional flows and their stability

This thesis focuses on three-dimensional thermally-driven boundary layers with three distinct types of (steady) thermal forcing (i) a streamwise-aligned heat source that is spanwise localised, (ii) a spanwise and streamwise localised heat source and (iii) a non-localised globally driven natural convection problem. What ties these problems together is that the induced three-dimensionality is on a spanwise lengthscale commensurate with the boundary-layer thickness, and so their leading-order flow is governed by the 'boundary-region' equations, attributed to Kemp (1951). A lot of work has been undertaken within this boundary-region framework in an isothermal context, for example forced by injection or roughness, but presently there has been limited work in which temperature variation has been applied. In our first problem (i), we embed a streamwise-aligned spanwise-localised heated strip, which has an O(Re^(-1/2)) spanwise scale, in a horizontal semi-infinite flat plate. At large spanwise distances from the forcing region, the flow takes the form of the two-dimensional Falkner--Skan boundary layer (1930). The three-dimensional flow response is determined numerically and a robust streak-roll structure forms downstream. We undertake an asymptotic analysis at large streamwise coordinates, and obtain the downstream transverse growth of the nonlinear response. We also determine the flow over a streamwise-aligned injection slot, and make qualitative comparisons with the thermally forced problem. A localised heat source with an O(1) streamwise scale and an O(Re^(-1/2)) spanwise scale is then considered (ii), and the corresponding nonlinear problem is solved. For weak thermal forcing, the isolated heated region results in a downstream response which has an algebraically developing velocity field. By seeking a linear bi-global eigenvalue problem in terms of the downstream algebraic spatial growth, we show that there is a new class of linear spanwise-localised algebraically developing disturbances in the Falkner--Skan boundary layer when thermal effects are included. The velocity field of the dominant mode grows (algebraically) downstream, and is driven by a decaying (weak) temperature field. Above a critical pressure gradient the dominant mode decays, and we determine the critical Prandtl number and pressure gradient parameter for downstream algebraic growth. The linear algebraically growing response ultimately gives rise to a nonlinear spanwise-localised streak-roll response downstream. We assess the stability of the streak response to time-harmonic (viscous) disturbances and inviscid Rayleigh modes. Our third problem (iii) is the natural convection flow in a heated vertical corner. This is a globally driven, non-localised problem, however in a high Grashof number regime the flow in a region near to the corner is governed by the boundary-region equations. Two self-similar solutions are determined: (A) a solution first discussed by Riley and Poots (1972), and (B) a new solution which, at large spanwise coordinates, has a coupled linearly-developing crossflow component. We show that the streamwise velocity and temperature fields determined by Riley and Poots (1972) for solution A were surprisingly accurate, given that their computational resources were limited. However, they concluded that there is an in-plane flow away from the corner which is not present in the properly resolved solution. The non-parallel stability of the self-similar profiles is tackled with the parabolised stability equations (PSE).