UBC Theses and Dissertations
Two dimensional hydrodynamic instabilities in shear flows Guha, Anirban
Hydrodynamic instabilities occurring in two dimensional shear flows have been investigated. First, the process of resonant interaction between two progressive interfacial waves is studied. Such interaction produces exponentially growing instabilities in idealized, homogeneous or density stratified, inviscid shear layers. It is shown that two oppositely propagating interfacial waves, having arbitrary initial amplitudes and phases, eventually phase-lock, provided they satisfy a particular condition. Three types of shear instabilities - Kelvin Helmholtz, Holmboe and Taylor have been studied. The above-mentioned condition provides a range of unstable wavenumbers for each instability type, and this range matches the predictions of the canonical normal-mode based linear stability theory. The non-linear evolution of Kelvin-Helmholtz (KH) instability has been studied. The commonly known manifestation of KH is in the form of spiral billows. However, KH evolving from a piecewise linear shear layer is remarkably different; it is characterized by elliptical vortices of constant vorticity connected via thin braids. Using direct numerical simulation and contour dynamics, it is shown that the interaction between two counter-propagating vorticity waves is solely responsible for this KH formation. The oscillation of the vorticity wave amplitude, the rotation and nutation of the elliptical vortex, and straining of the braids have been investigated. Finally, the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow is studied. The base flow is characterized by the cross flow Reynolds number, Reinj and the dimensionless wall velocity, k. Corresponding to each dimensionless wall velocity, k ∈ [0,1], two ranges of Reinj exist where unconditional stability is observed. In the lower range of Reinj , for modest k we have a stabilization of long wavelengths leading to a cut-off Reinj. As Reinj is increased, we see first destabilization and then stabilization at very large Reinj. Analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien-Schlichting waves.
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