Non-normal Dynamics and Hydrodynamic Stability

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This thesis explores the interaction of non-normality and nonlinearity incontinuous dynamical systems. A solution beginning near a linearly stable fixed point may grow large by a linear mechanism, if the linearization is non-normal, until it is swept away by nonlinearities resulting in a much smaller basin of attraction than could possibly be predicted by the spectrum of the linearization. Exactly this situation occurs in certain linearly stable shear flows, where the linearization about the laminar flow may be highly non-normal leading to the transient growth of certain small disturbances by factors which scale with the Reynolds number. These issues are brought into focus in Chapter 1 through the study of atwo-dimensional model system of ordinary differential equations proposed by Trefethen, et al. [Science, 261, 1993]. In Chapter 2, two theorems are proved which show that the basin of attraction of a stable fixed point, in systems of differential equations combining a non-normal linear term with quadratic nonlinearities, can decrease rapidly as the degree of non-normality is increased, often faster than inverse linearly. Several different low-dimensional models of transition to turbulence are examined in Chapter 3. These models were proposed by more than a dozen authors for a wide variety of reasons, but they all incorporate non-normal linear terms and quadratic nonlinearities. Surprisingly, in most cases, the basin of attraction of the "laminar flow" shrinks much faster than the inverse Reynolds number. Transition to turbulence from optimally growing linear disturbances, streamwise vortices, is investigated in plane Poiseuille and plane Couette flows in Chapter4. An explanation is given for why smaller streamwise vortices can lead to turbulence in plane Poiseuille flow. In plane Poiseuille flow, the transient linear growth of streamwise streaks caused by non-normality leads directly to a secondary instability. Certain unbounded operators are so non-normal that the evolution of infinitesimal perturbations to the fixed point is entirely unrelated to the spectrum, even as i to infinity. Two examples of this phenomenonare presented in Chapter 5.
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