This thesis focuses on threedimensional thermallydriven boundary layers with three distinct types of (steady) thermal forcing (i) a streamwisealigned heat source that is spanwise localised, (ii) a spanwise and streamwise localised heat source and (iii) a nonlocalised globally driven natural convection problem. What ties these problems together is that the induced threedimensionality is on a spanwise lengthscale commensurate with the boundarylayer thickness, and so their leadingorder flow is governed by the 'boundaryregion' equations, attributed to Kemp (1951). A lot of work has been undertaken within this boundaryregion 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 streamwisealigned spanwiselocalised heated strip, which has an O(Re^(1/2)) spanwise scale, in a horizontal semiinfinite flat plate. At large spanwise distances from the forcing region, the flow takes the form of the twodimensional FalknerSkan boundary layer (1930). The threedimensional flow response is determined numerically and a robust streakroll 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 streamwisealigned 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 biglobal eigenvalue problem in terms of the downstream algebraic spatial growth, we show that there is a new class of linear spanwiselocalised algebraically developing disturbances in the FalknerSkan 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 spanwiselocalised streakroll response downstream. We assess the stability of the streak response to timeharmonic (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, nonlocalised problem, however in a high Grashof number regime the flow in a region near to the corner is governed by the boundaryregion equations. Two selfsimilar 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 linearlydeveloping 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 inplane flow away from the corner which is not present in the properly resolved solution. The nonparallel stability of the selfsimilar profiles is tackled with the parabolised stability equations (PSE).
Date of Award  31 Dec 2022 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Peter Duck (Supervisor) & Richard Hewitt (Supervisor) 

 Asymptotic analysis
 Scientific computing
 Objectoriented programming
 Convection
 Boussinesq approximation
 Flatplate
 Stability
 Buoyancy
 Finitedifference methods
 Corner boundary layers
 Boundaryregion equations
 Parabolised stability equations
 Threedimensional boundary layers
 Applied Mathematics
 CrankNicolson method
Thermally driven threedimensional flows and their stability
Edwards, S. (Author). 31 Dec 2022
Student thesis: Phd