Researchers have understood the significance of boundary layers in the process of separation since \citet{Prandtl_1904} seminal paper. Nowadays, society brings up new challenges for both researchers and engineers, and understanding the underlying issues of flow separation is essential, as this area covers a large spectrum of applications. This thesis focuses on understanding unsteady boundary-layer separation with large Reynolds number $\Rey$, using theoretical and numerical tools. In particular, employing perturbation methods allows us to deduce many mathematical and physical information about the problems studied that experiments would fail to highlight.

The first part of this thesis deals with the unsteady flow generated by the rotation of a plane disk of finite radius. A free jet is generated from the rim of the disk, and it propagates away as time increases with a finite speed. Solving numerically the boundary-layer equations reveals that at the front of the jet, termed as a ``pseudo-shock'', the axial velocity is singular, while the radial and circumferential velocities experience a jump. On the pseudo-shock's scale, we show that as the jet penetrates the stagnant fluid, it ejects the fluid from the boundary layer into the region at the outer edge of the boundary layer, which is governed by the Euler equations. Their solution show that the ejected fluid is redirected upstream and reinjected back into the boundary layer through the ``entrainment effect'' process. An unsteady wall jet is discussed in Part~\ref{part:WJ} where similar conclusions are drawn overall.

The third part concerns the quasi-steady, supersonic flow past a flat plate where a shock wave/expansion fan impinges upon the boundary layer and moves in the downstream direction $\widehat{x}$ parallel to the plate, with the dimensionless velocity $\overline{V}_\textrm{sh}$. This situation may occur when a shock wave, generated by an aircraft, hits another one during formation flying. Assuming $\Rey^{-1/8}\ll\overline{V}_\textrm{sh}\ll1$, the interaction region may be described in the framework of the \textit{triple-deck theory}, and the lower deck of the three-tiered structure is divided into two regions: an inviscid one, and a near-wall viscous sublayer $1b$. Setting the strength of the expansion fan such that the velocity at the outer edge of $1b$ is equal to zero at $\widehat{x}=0$, the streamline slope in $1b$ develops a singularity. A new shorter region is added to smooth it out, and where the pressure grows in the upstream direction, such that it finishes at a finite position where it becomes singular.