Keele Research Repository

The physical mechanisms of why an orderly laminar flow suddenly becomes turbulent are still poorly understood, despite more than a century of incessant efforts. The thesis puts forward a radically new approach to understanding of the fundamental challenge of bypass laminar-turbulent transition. In contrast to the overwhelming majority of other transition studies concerned primarily with linear instabilities, here the focus is on finite amplitude three dimensional (3d) longwave perturbations of boundary layer which are weakly decaying in the linear approximation.
The principal novelty is in simultaneous account of viscous decay, three-dimensionality and nonlinearity. To describe analytically the dynamics of such perturbations Witham type pseudo-differential nonlinear evolution equations have been derived and examined. The equations are derived asymptotically in the distinguished limit: nonlinearity, dispersion and viscous decay (shown to be described by the Rayleigh friction type term) are assumed to be in balance. The two models describe the 3d perturbations in generic semi-infinite uniform boundary layers for unidirectional flows in homogeneous and weakly stratified fluids (the equation for the homogeneous case is the essentially two-dimensional Benjamin-Ono equation modified by the account of the Rayleigh friction term), while the third one examines the effect of confined boundary layer in homogeneous fluids.
The key feature of the derived models is that they support collapse, i.e. a specific blow up with formation of a point singularity in finite time. Self-similar solutions describing behaviour of the solution in the neighbourhood of singularity have been derived for all three models. The phase space of these models is simply organised: there are two attractors corresponding to the unperturbed linearly stable boundary layer and the singularity. The boundary in the space of the initial conditions separating the regimes of collapse and decay has been examined analytically and numerically. For the situations where the Rayleigh friction is negligible, the systems are Hamiltonian and an analytical criterion of vanishing of the Hamiltonian specifies the boundary between the regimes. For other situations the boundary is found by direct numerical simulations of the evolution equations.
The overall conclusion is that neither the Rayleigh friction, nor stratification or effect of a second boundary prevent collapse from happening, nor they change the perturbation amplitude time dependence in the vicinity of the singularity, however, these factors do affect the evolution and can strongly increase the amplitude threshold of collapse, often beyond the range of validity of weakly nonlinear models. Thus, within the framework of the derived weakly nonlinear models a broad class of initial conditions tends to form a singularity, in the process of evolution the emerging patterns strongly resemble the 3d coherent structures observed in the wind tunnel boundary layers.