The continuous spectrum of time-harmonic shear layers

In boundary layers, eigenfunctions which belong to the continuous spectrum of the Orr-Sommerfeld equation have been established as a suitable basis for the expansion of general free-stream vortical disturbances. They are oscillatory in the free stream, and attenuate inside the boundary layer due to shear sheltering. The extent of modal penetration into the shear depends on the disturbance frequency and orientation, with the low-frequency, streamwise elongated modes being the most effective triggers of a high-amplitude streak-like response. The influence of introducing a time-periodic, spanwise mean flow on modal sheltering is investigated. The evaluation of the continuous modes in this case requires a Floquet expansion in the fundamental frequency of the base flow. Appropriate treatment of the free-stream boundedness condition is developed, and quantitative measures of modal sheltering are computed. The time-dependent, spanwise motion is shown to significantly enhance shear sheltering, and to change the optimal orientation of the continuous modes for penetration into the shear. An explanation is provided, in the limit of low frequency of the base flow, using asymptotic analyses. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3687451]