M. J. Philipp Hack

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 o...

Low-complexity stochastic modeling of spatially-evolving flows

Low-complexity approximations of the Navier-Stokes (NS) equations are commonly used for analysis and control of turbulent flows. In particular, stochastically-forced linearized models have been successfully employed to capture structural and statistical features observed in experiments and high-fidelity simulations. In this work, we utilize stochastically-forced linearized NS equations and the parabolized stability equations to study the dynamics of flow fluctuations in transitional boundary layers. The parabolized model can be used to efficiently propagate statistics of stochastic disturbances into statistics of velocity fluctuations. Our study provides insight into the interaction of the slowly-varying base flow with streamwise streaks and Tollmien-Schlichting waves. It also offers a systematic, computationally efficient framework for quantifying the influence of stochastic excitation sources (e.g., free-stream turbulence and surface roughness) on velocity fluctuations in weakly non-parallel flows.