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Access Type
Open Access
Document Type
thesis
Degree Program
Mechanical Engineering
Degree Type
Master of Science in Mechanical Engineering (M.S.M.E.)
Year Degree Awarded
January 2008
Month Degree Awarded
September
Keywords
Hydrodynamic Stability, Operator Theory, Non-normality, Non-hermitianness, Bi-orthogonality, Transient Growth
Abstract
Whether it is designing and controlling super-efficient high speed transport systems or understanding environmental fluid flows, a key question that arises is: what state does the fluid take and why? An answer to this question lies in understanding the hydrodynamic stability properties of the flow as a function of parameters. While much work has been done in this area in the past, there are many open questions that need to be addressed. Here we study the effect of spatial domain size, number of modes, non-hermitianness and non-normality on the finite time and infinite time stability properties of a standing, viscous shock flow problem.
It has been shown that the above problems are not only non-normal but also non-hermitian, when the base flow has shear. The eigenvalue problems corresponding to infinite spatial domain, finite spatial domain, Forward and L2 adjoint problems are solved exactly by converting the linear partial differential equations into nonlinear Riccati equations. In the finite domain case, the full time dependent solutions are obtained analytically using bi-orthogonal basis functions.
In the infinite domain case, the point spectrum of the forward operator is shown to be unbounded and that of the adjoint operator to be empty. In the unbounded case, the spectrum fills the entire area on one side of a parabola in the complex plane and is connected. As the fluid viscosity decreases the width of the parabola increases and in the limit of zero viscosity covers almost entire left half plane(LHP). On the other hand, as the fluid viscosity increases the width of parabola decreases and in the limit of infinite viscosity becomes negative real axis, which is the spectrum of heat equation. The spectrum of adjoint problem is empty for all values of the viscosity and prescribed velocity.
In the finite spatial domain case, the point spectrum lies in the open left half plane for all Reynolds numbers and hence asymptotically stable. The results obtained showed that perturbations grow substantially large for finite time before they decay at large times. It is also found that retainig right number of modes is crucial for observing transient growth phenomena. Finally, the linear results are compared with the nonlinear finite amplitude simulation results.
The relevance of current results to other fluid flows is presented.
DOI
https://doi.org/10.7275/535279
First Advisor
Kumar M Bobba