Instability and receptivity of boundary layers on concave surfaces and swept wings
Creators: name: Zhao, Difei
Publisher: name: Imperial College London
Year: 2011

description: This thesis studies the instability and receptivity of boundary layers over a concave wall and a swept Joukowski airfoil. The main interest is in excitation of relevant instability waves by free-stream vortical disturbances and in their subsequent linear development. We first consider excitation of Gortler vortices in a Blasius boundary layer over a concave wall. Attention is focused on disturbances with long streamwise wavelengths, to which the boundary layer is most receptive. The appropriate initial-boundary-value problem describing both the receptivity process and the subsequent development of the induced perturbation is formulated for the generic case where the Gortler number GΛ (based on the spanwise wavelength Λ of the disturbance) is of order one. The impact of free-stream disturbances on the boundary layer is accounted for by the far-field boundary condition and the initial condition near the leading edge, both of which turn out to be the same as those given by Leib, Wundrow and Goldstein (J. Fluid Mech. vol. 380, 1999, p.169) for the flatplate boundary layer. Numerical solutions of the initial-value problem show that for a sufficiently small GΛ, the induced perturbation exhibits essentially the same characteristics as streaks occuring in the flat plate case: the streamwise velocity undergoes considerable amplification and then decays. However, when GΛ exceeds a critical value, the induced perturbation exhibits (quasi-)exponential growth. Comparison with local parallel and non-parallel instability theories reveal that the perturbation acquires the modal shape of Gortler vortices rather quickly, but its growth rate differs appreciably from that predicted by local instability theories before the convergence at large downstream distances. Nevertheless, the overall agreement is close enough to indicate that Gortler vortices have been excited by free-stream disturbances. The amplitude of excited Gortler vortices is found to decrease with the frequency. Steady vortices, generated by steady components of free-stream disturbances, tend to be dominant. Detailed quantitative comparisons with experiments were performed. It is found that the eigenvalue approach predicts the modal shape adequately, but only the initial-value approach can accurately predict the evolution of the amplitude as well as the modal shape. An asymptotic analysis is performed on the assumption of GΛ &gt;&gt;1 to map out distinct regimes through which a disturbance of a fixed spanwise wavelength evolves. The centrifugal force enters the play to influence the generation of the pressure when x* ~ΛRΛG−2/3 Λ , where RΛ denotes the Reynolds number based on Λ. The induced pressure leads to full coupling of the momentum equations when x* ~ ΛRΛG−2/5 Λ . This is the crucial regime linking the pre-modal and modal phases of the perturbation because the governing equations admit a countable set of growing asymptotic eigensolutions, which develop into fully fledged Gortler vortices of inviscid nature when x* ~ ΛRΛ. From this position onwards, local eigenvalue formulations are mathematically justified. The generated Gortler vortices continue to amplify and enter the so-called most unstable regime when x* ~ ΛRΛGΛ, and ultimately approach the right-branch regime when x* ~ ΛRΛG2 Λ. We then extend our study to the receptivity of a three-dimensional boundary layer over a swept wing to free-stream vortical disturbances. The base flow is taken to be the boundary layer over a swept Joukowski airfoil. In contrast to the two-dimensional boundary layer, external disturbances with comparable streamwise and spanwise wavelengths are relevant to receptivity. The appropriate initial-boundary-value problem consists of linearised boundary-layer equations supplemented by the initial condition at the leading edge and the boundary condition in the far field, which are derived by applying the rapid distortion theory, and matching the resultant inviscid solution with the boundary-layer solution. It is found that the linearised boundary-layer equations support spatially growing eigenmodes despite the absence of a pressure gradient. The modes may be first excited by free-stream disturbances, and eventually evolve into fully fledged crossflow vortices.