The method of Lie group transformations is used to derive all group-invariant similarity solutions of the unsteady two-dimensional laminar boundary-layer equations. A new method of nonlinear superposition is then used to generate further similarity solutions from a group-invariant solution. Our results are shown to include all the existing solutions as special cases. A detailed analysis is given to several classes of solutions which are also solutions to the full Navier–Stokes equations and which exhibit flow separation.

The structure of homogeneous turbulence subject to high shear rate has been investigated by using three-dimensional, time-dependent numerical simulations of the Navier–Stokes equations. The instantaneous velocity fields reveal that a high shear rate produces structures in homogeneous turbulence similar to the ‘streaks’ that are present in the sublayer of wall-bounded turbulent shear flows. Statistical quantities such as the Reynolds stresses are compared with those in the sublayer of a turbulent channel flow at a comparable shear rate made dimensionless by turbulent kinetic energy and its dissipation rate. This study indicates that high shear rate alone is sufficient for generation of the streaky structures, and that the presence of a solid boundary is not necessary.

Evolution of the statistical correlations is examined to determine the effect of high shear rate on the development of anisotropy in turbulence. It is shown that the streamwise fluctuating motions are enhanced so profoundly that a highly anisotropic turbulence state with a ‘one-component’ velocity field and ‘two-component’ vorticity field develops asymptotically as total shear increases. Because of high shear rate, rapid distortion theory predicts remarkably well the anisotropic behaviour of the structural quantities.

The stability of a family of tanh mixing layers is studied at large Mach numbers using perturbation methods. It is found that the eigenfunction develops a multilayered structure, and the eigenvalue is obtained by solving a simplified version of the Rayleigh equation (with homogeneous boundary conditions) in one of these layers which lies in either of the external streams. Our analysis leads to a simple hypersonic similarity law which explains how spatial and temporal phase speeds and growth rates scale with Mach number and temperature ratio. Comparisons are made with numerical results, and it is found that this similarity law provides a good qualitative guide for the behaviour of the instability at high Mach numbers.

In addition to this asymptotic theory, some fully numerical results are also presented (with no limitation on the Mach number) in order to explain the origin of the hypersonic modes (through mode splitting) and to discuss the role of oblique modes over a very wide range of Mach number and temperature ratio.

An analytical study on the Eckhaus instability of moderately nonlinear thermal Rossby waves is developed. A solvability condition of the lowest order is derived. The condition not only produces results that agree reasonably well with the earlier Galerkin formulation, but also leads to some new findings that are otherwise difficult to discover by the previous method. Over a wide range of parameters, this paper reports the existence of a branch of the stability limit that corresponds to a pair of disturbances with a finite, rather than an infinitesimal wavenumber modulation. As the Prandtl number tends to a small value, the asymmetry between the two branches of the stability limit becomes very pronounced, which is manifested as a severely distorted stability region.

An analysis of the flow over a leading edge with distributed roughness is presented. The analysis is focused on a small neighbourhood of the stagnation line. The roughness is assumed to have a small amplitude and to be symmetric with respect to the stagnation line. Results show that roughness acts as a source of streamwise vorticity. The existence of a universal form of the flow field for long-wavelength roughness is demonstrated. It is shown that surface stresses tend to eliminate roughness if erosion or wall flexibility are admitted. The heat flow tends to concentrate at the tips of the roughness and this may lead to the generation of large thermal stresses along the surface of the leading edge.