Frequency Transient of Three-Dimensional Perturbations in Shear Flows. Similarity Properties and Wave Packets Linear Formation

Abstract

The present thesis deals with the non-modal linear analysis of 3D perturbations in wall flows. In the first part,a solution to the Orr-Sommerfeld and Squire IVP, in the form of orthogonal functions expansion, is researched. The Galerkin method is successfully implemented to numerically compute approximate solutions for bounded flows. The Chandrasekhar functions revealed to ensure a fifth order of accuracy. The focus of the subsequent analysis is on the transient behavior of the perturbation frequency and phase velocity. The results confirm recent observations about a jump in the temporal evolution of the frequency of the wall-normal velocity signal, considered as the end of an Early Transient. After this jump, the wave frequency for Plane Couette flow experiences a periodic modulation about the asymptotic value, which is motivated and investigated in detail. A new result is the presence of a second frequency jump for the wall-normal vorticity. This fact, together with the possibility for different values of the signals asymptotic frequency, shows the existence of an Intermediate Transient. Moreover, a connection between the frequency jumps and the establishing of a self-similarity condition in time for both the velocity and vorticity profiles is found and investigated for both Plane Couette flow and Plane Poiseuille flow. Eventually, through superposition of waves with limited wavenumber range, a wave packet is reconstructed for Plane Couette flow and Blasius boundary-layer flow . The linear spot evolution revealed to have many common features with the early stages of a turbulent spot, particularly the streaky structure and the spot shape.