BIFURCATIONS IN NONLINEAR PROBLEMS OF HYDRODYNAMIC INSTABILITY OF PLANE PARALLEL SHEAR FLOWS

The main goal of this thesis is to contribute towards the understanding of the transition to turbulence. The nonlinear evolution of instabilities occurring in plane parallel shear flows is investigated by following the first few birfurcations. The basic velocity profile described by a cubic polynomial is chosen, since it is an exact solution of the Navier-Stokes equation for a Boussinesq fluid confined between two inclined parallel plates heated from above. The plates are allowed to move in the opposite directions. The instability mechanism becomes purely hydrodynamic by assuming the limit of a small Prandtl number.; Two- and three-dimensional nonlinear steady solutions are obtained numerically by the Galerkin method and the Newton-Raphson iterative method after the solenoidal velocity field is expanded by a truncated series of orthogonal functions. Stability analysis is carried out for the steady solutions by superimposing infinitesimal disturbances with arbitrary three-dimensional spatial dependence. The resulting equations for the time growth rate as the eigenvalue are solved numerically.; In the case of stationary boundaries, two-dimensional steady solutions bifurcate supercritically, taking the form of transverse vortices. At a higher value of the Grashof number, these transverse vortices become unstable. Three mechanisms of instability are found: one oscillatory and two monotone instabilities. The oscillatory instability is described by the wavy pattern traveling in the spanwise direction, causing the sinuous bending of the vortex. One of the monotone instabilities is distinguished by a subharmonic mode in the streamwise direction with a wavelength in that direction twice that of the undisturbed vortices and a nonzero spanwise wavenumber, showing a varicose bending of the vortex tubes and a vortex-pairing tendency. This mode is found to be most unstable. The other monotone instability, known as the Eckhaus instability, is characterized by disturbances independent of the spanwise coordinate.; The helical vortex-pairing process is clearly exhibited in the finite amplitude three-dimensional steady solutions which develop from the subharmonic monotone instability of two-dimensional vortices. Finally, the question of the stability of the tertiary flow is addressed. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author.) UMI...