This work belongs to the theory of fluid flows at high Reynolds number, characterised by low viscosity and, to a lesser extent, high speeds. It is a mathematically intensive field covering most aspects of applied mathematics, from Partial Differential equations to Numerical Analysis and Complex Analysis. Two of the central areas of research within this particular branch of fluid dynamics are the boundary-layer laminar-turbulent transition and
the boundary-layer separation. Both of these phenomena have a high impact on the aerodynamic
performance of aircraft wings, turbine blades and other fast-moving objects such as rockets. As a consequence, their study is relevant both for industrial applications and to the scientific community as a whole.
The mathematical element pertaining to the analysis of these two phenomena is the consistent use of asymptotic methods, such as matched asymptotic expansions and multiple scales analysis. In particular, this work will be based on the viscous-inviscid interaction theory originally discovered by Lin (1955) and further developed by the work of Neiland (1969) and Stewartson & Williams (1969).
In the first chapter of this thesis, we will study the initial stages of the laminar-turbulent transition of a compressible boundary layer on a swept wing. The instability of interest will be the stationary crossflow vortex, which is known to be the main instability mode in three-dimensional boundary layers on a swept wing. We will focus on two specific aspects
of the transition process, namely the receptivity and linear stability analysis of the flow.
The receptivity mechanism introduced in our work is a roughness of size comparable with that of the boundary layer thickness. This justifies our restriction to the study of the inviscid instability of the flow and the use of the impermeability condition on the roughness.
The remaining two chapters are concerned with the viscous-inviscid interaction of the boundary layer in the vicinity of a surface discontinuity. We will first study the behaviour of the subsonic flow exposed to the singular pressure gradient
dp/dx = κ(x0 − x)−1/3, as x → x0.
It forms when the body contour has a point x = x0 near which
yw = (x0 − x)5/3.
We will show how logarithmic terms need to be included in the solution. We then study a similar problem, this time in the context of an incoming transonic flow near a point of curvature discontinuity. We will show that in both cases the boundary layer experiences an “extreme acceleration”.