Karl Gustafson

Hopf bifurcation in the driven cavity

Abstract Incompressible two-dimensional calculations are reported for the impulsively started lid driven cavity with aspect ratio two. The algorithm is based on the time dependent stream-function equation, with a Crank-Nicolson differencing scheme for the diffusion terms, and with an Adams-Bashforth scheme for the convection terms. A multigrid method is used to solve the linear implicit equations at each time step. Periodic asymptotic solutions have been found for Re = 10000 and for Re = 5000. The Re = 5000 results are validated by grid refinement calculations. The solutions are shown to be precisely periodic, and care is taken to demonstrate that asymptotic states have been reached. A discussion is included about the indicators that are used to show that an asymptotic state has been reached and to show that the asymptotic state is indeed periodic.

Vortex dynamics of cavity flows

Abstract The unsteady viscous incompressible Navier-Stokes flow in a driven cavity is studied with particular attention to the formation and evolution of vortices and eddies. These are compared in the limit to previous steady flow simulations. Results include new details of the dynamics of secondary eddy separation and subsequent coalescence into sub-primary vortices, and the fine structure of deep cavity flow. Enhanced flow topography is obtained by means of pressure and kinetic energy portraits.

Vortex methods and vortex motion

Vortex phenomena in fluid flows and the experimental, theoretical, and numerical methods used to characterize them are discussed in reviews by leading experts. Chapters are devoted to an overview of vortex methods, the convergence of vortex methods, graphic displays from numerical simulations, physical vortex visualizations, the four principles of vortex motion, the visualization and computation of hovering-mode vortex dynamics, turbulence and vortices in superfluid He, and statistical-mechanics approaches to vortices and turbulence. Extensive photographs and sample computer graphics are provided.

Cavity flow dynamics at higher reynolds number and higher aspect ratio

Abstract A primitive variable unsteady viscous incompressible Navier-Stokes flow simulation in a higher aspect ratio ( A = depth/width = 2) driven cavity at a higher Reynolds number (Re = 10,000) exhibits dynamical features not apparent in previous studies of stationary solutions or at lower Reynolds number or in the more usual unit cavity investigations. The initial dynamics, e.g., between t = 0 sec and t = 70 sec, reveals transient bifurcations between states including two, three, and four interacting vortices in an anterior separation region. Following this, there is an intermediate interval, e.g., between t = 70 sec and t = 140 sec, characterized by a vertical oscillation of the primary vortex related to the general activities of the secondary features of the flow. The long time behavior, e.g., between t = 140 seconds and t = 360 seconds, is one of qualitative smoothing of all secondary features except a persistent oscillation indicating a Hopf bifurcation.

Lectures on computational fluid dynamics, mathematical physics, and linear algebra

This book, an outgrowth of the authors distinguished lecture series in Japan in 1995, identifies and describes current results and issues in certain areas of computational fluid dynamics, mathematical physics, and linear algebra. Notable among these are the authors new notion of numerical rotational release for the understanding of correct solution capture when modelling time-dependent higher Reynolds number incompressible flows, the authors fundamental new perspective of wavelets seen as stochastic processes, and the authors new theory of antieigenvalues which has created an entirely new view of iterative methods in computational linear algebra.

On converse to Koopman's Lemma

Koopmans Lemma states that if a flow Tt is measure preserving for a measure μ on a constant energy surface Ω, then the flow generates a one parameter family of unitary operators Ut on L2 (Ω, μ). We show here a converse, namely that under certain (physically motivated) conditions a unitary operator family Ut can be made to generate a corresponding underlying family Tt of point transformations. This result comes out of questions of independent interest in the study of relationships between reversibility and irreversibility, and has application to the foundations of statistical mechanics. In particular, it establishes the principle often used intuitively in chemistry that a forward moving (e.g., Markov) process that loses information cannot be reversed. In a different setting, it provides the answer to a question in the representation theory of isometries on Lp spaces a Banach-Lamperti theorem). These results also allow an interesting reformulation of Ornsteins isomorphism theorem on Bernoulli systems.

Computation of dragonfly aerodynamics

Abstract Dragonflies are seen to hover and dart, seemingly at will and in remarkably nimble fashion, with great bursts of speed and effectively discontinuous changes of direction. In their short lives, their gossamer flight provides us with glimpses of an aerodynamics of almost extraterrestrial quality. Here we present the first computer simulations of such aerodynamics.

Canonical commutation relations of quantum mechanics and stochastic regularity

In a previous publication (Boletin de la Sociedad Matematica Mexicana 1975) it was established that any weakly stationary linearly regular stochastic process is unitarily equivalent to a quantum mechanical momentum evolution. The object of the present note is, as promised in the previous publication, to amplify some of the details concerning the just mentioned interesting connection, giving in particular a direct proof of the Szego-Kolmolgorov-Krein characterization of regular stationary processes. We also show that although the so-called decaying states without regeneration do not exist for unstable quantum systems, they are natural for regular stationary processes.

Wavelets and stochastic processes

Wavelets are known to have intimate connections to several other parts of mathematics, notably phase-space analysis of signal processing, reproducing kernel Hilbert spaces, coherent states in quantum mechanics, spline approximation theory, windowed Fourier transforms, and filter banks. Here, we establish and survey a new connection, namely to stochastic processes. Key to this link are the Kolmogorov systems of ergodic theory.

Weyl’s Theorems

The purpose of this paper is to briefly describe some recent results and to give some examples that illustrate and make rather sharp the following three theorems concerning perturbation of spectra of linear operators, each often called Weyl’s Theorem in the literature.