CNCS Graduate Certificate Recipient

Michael R. Gustafson II

Thesis Title: Analytical and Experimental Control of Horton-Rogers-Lapwood Convection

Ph.D. Final Defense Date:  December 2, 1999

Ph.D. Dissertation Committee:

Laurens E. Howle (Chair)
Robert P. Behringer
Henry S. Greenside
Edward J. Shaughnessy
Gary A. Ybarra

Abstract:

This work presents a theoretical and experimental investigation of applying proportional and differential control to delay and suppress Horton-Rogers-Lapwood convection in a porous medium. The technique used for control exploits information obtained by shadowgraphy. This work represents the first time the effects of proportional and differential control on convection in a porous medium have been analyzed and presented experimentally. The focus is to theoretically determine how far the controller can delay the onset of convection and then to test this experimentally.

The first section of this work develops a mathematical model for Horton-Rogers-Lapwood convection in a medium of infinite horizontal extent. It revisits the historical progression of the problem of convection in a porous medium and discusses various models and assumptions. It then presents a linear stability analysis including the theoretical modeling of the shadowgraphic signal and controller. Finally, theoretical stability results are shown which demonstrate the importance of various physical and thermal properties of the medium and the bounding materials. Specifically, the thickness and conductivity of the boundaries are shown to be important in determining the controller gain needed to obtain maximum stabilization of the no-motion state.

The second section of this work presents the experimental apparatus, the evolution of the apparatus to its current form, and data obtained from various experiments to test the effectiveness of both proportional and differential control. Comparisons between porous convection and bulk fluid convection are made, as well as comparisons between theoretical and experimental results. The method of creating and recording the shadowgraphic signal is explained, as are the mechanism behind the controller, the method of delivering heat to the lower surface, and the specifics of the porous medium. Data are presented which show the controller's ability to delay and suppress convection. Also, the presence of a Hopf bifurcation to time-dependent convection -- not predicted by the theoretical model -- is shown. This bifurcation is thought to be a result of controller delay. The effects of differential control both on delaying and suppressing convection and on the frequency and amplitude of time-dependent convection is presented.

Finally, conclusions are drawn as to the applicability of theory to the experimental apparatus and the usefulness of shadowgraphy when working with a porous medium. Specifically, the success of the experiments performed for this work indicates that simple proportional and differential control can be used to suppress convection in a porous medium heated from below.