The influence of unsteady streaks on the stability of flat plate boundary layers

Abstract

The natural mechanism for transition to turbulence in flat-plate boundary layers is the growth and breakdown of Tollmien-Schlichting (TS) waves. In the presence of significant free-stream turbulence (FST) however, streamwise velocity perturbations, known as Klebanoff modes or streaks, amplify inside the boundary layer. These distortions alter the stability characteristics of the boundary layer, and the natural mechanism is bypassed, leading to earlier transition. Herein, a model is employed to describe the Klebanoff distortions: one Fourier component of the FST is used along with its signature inside the shear region to force the boundary layer and stimulate streaks. Varying the parameters of the forcing mode causes streaks with different frequencies and amplitudes. A base flow which is periodic in two dimensions is formed, and its linear stability is investigated using Floquet theory. Two modes emerge as the most unstable, and their eigenvalues are tracked whilst varying streak frequency and amplitude. The ‘inner’ mode, is related to the TS wave, but its growth rate is enhanced by unsteady streaks. The ‘outer’ mode is a high-frequency instability of the streaks at the edge of the boundary layer. It has no counterpart in the undisturbed boundary-layer. The critical streak amplitude for the outer mode is calculated for different streak frequencies and it agrees more closely with experiments than previous analyses which assumed the streaks to be steady. The current analysis indicates that increasing the frequency of the streaks can enhance their instability. In fact an optimum frequency exists for free-stream disturbances to penetrate the shear and stimulate unstable streaks. Direct numerical simulations with streaks and secondary-instability eigenmodes are conducted. The simulations show that both the inner and outer mode can grow to nonlinear amplitudes and cause boundary-layer transition to turbulence.