Nonlinear unsteady disturbances generated by the interaction of free-stream vorticity with a laminar boundary layer

As a contribution towards understanding the impact of free-stream perturbations on laminar-to-turbulent boundary layer transition, we calculate the signature of unsteady disturbances engendered by the interaction of free-stream vortical fluctuations with a laminar boundary layer over a flat and a curved plate. We concentrate on low-frequency perturbations which, in the case of a flat plate, induce strong streamwise-elongated components of the boundary-layer signature, known as Klebanoff modes or streaks. In boundary layers over suitably curved concave walls, Klebanoff modes are expected to develop into Gortler vortices. The generation and nonlinear evolution of the induced perturbations, which acquire an O(1) magnitude, are described on a self-consistent and first-principle basis using the mathematical framework of the nonlinear unsteady boundary-region equations (NUBREs), subject to appropriate upstream and far-field boundary conditions. The nonlinear response of a compressible flat-plate boundary layer to free-stream vorticity is investigated first.
The problem is governed by the compressible NUBREs, which are derived herein for the first time. The free-stream flow is studied by including the boundary-layer displacement effect and the solution is matched asymptotically with the boundary-layer flow. The nonlinear interactions inside the boundary layer drive an unsteady two-dimensional flow of acoustic nature in the outer inviscid region through the displacement effect. Analytical solutions are derived by exploiting the well-known analogy with the flow over a thin oscillating airfoil, which is used herein for the first time to study unsteady boundary layers. In the subsonic regime the perturbation is felt from the plate in all directions, while at supersonic speeds the disturbance only propagates within the dihedron defined by the Mach line. Numerical computations are performed for carefully chosen parameters that characterize three practical applications: turbomachinery systems, supersonic flight conditions and wind-tunnel experiments. The results show that nonlinearity plays a marked stabilizing role on the velocity and temperature streaks, and this is found to be the case for low-disturbance environment such as flight conditions. Increasing the free-stream Mach number inhibits the kinematic fluctuations but enhances the thermal streaks, relative to the free-stream velocity and temperature respectively, and the overall effect of nonlinearity becomes weaker. An abrupt deviation of the nonlinear solution from the linear one is observed in the case pertaining to a supersonic wind tunnel. Large-amplitude thermal streaks and the strong abrupt stabilizing effect of nonlinearity are two new features of supersonic flows.
In the second part of the thesis, the generation and nonlinear development of unsteady Gortler vortices in an incompressible boundary layer over a concave plate is studied.
The centrifugal force caused by the concavity of the wall is included in the incompressible NUBREs. The results show that the stabilizing effect on nonlinearity is significantly intensified in the presence of centrifugal forces. Sufficiently downstream the nonlinear vortices generated at different free-stream turbulence levels Tu are stabilized to the same amplitude, suggesting that the initial intensity of the forcing becomes unimportant. At low Tu the perturbation undergoes a quasi-exponential growth with the growth rate being enhanced for lower frequencies and more curved plates. At higher Tu, in the typical range of turbomachinery applications, the Gortler vortices do not exhibit an exponential growth as nonlinearity saturates rapidly, and the wall curvature does not influence the boundary-layer response. Good quantitative agreement with direct numerical simulations and experimental data is obtained.