Off-campus UMass Amherst users: To download campus access theses, please use the following link to log into our proxy server with your UMass Amherst user name and password.

Non-UMass Amherst users: Please talk to your librarian about requesting this thesis through interlibrary loan.

Theses that have an embargo placed on them will not be available to anyone until the embargo expires.

Access Type

Open Access

Document Type


Degree Program

Mechanical Engineering

Degree Type

Master of Science in Mechanical Engineering (M.S.M.E.)

Year Degree Awarded

January 2008

Month Degree Awarded



Hydrodynamic Stability, Operator Theory, Non-normality, Non-hermitianness, Bi-orthogonality, Transient Growth


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.


First Advisor

Kumar M Bobba