SURFACE DISTURBANCES GENERATED BY FLUID FLOW PASTAN OBSTACLE OR OVER TOPOGRAPHY AS PREDICTED BYTHE KORTEWEG-DE VRIES AND THE EULER EQUATIONS

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  • March 22, 2019
Creator
  • Robinson, Quentin
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • The linearized Euler equations and the forced Korteweg-de Vries equation are investigated analytically and numerically as models for the behavior of the surface of a fluid flowing over topography and past an obstacle. Dispersionless and linearized variations of the fKdV equation are compared with the full fKdV equation in various parameter regimes. Ways in which information gained from various approximations to the forced Korteweg-de Vries (fKdV) equations predict the behavior of the solution of the full equation are explored. A critical Froude number parameter value above, which stationary solutions exist, is determined and the stability of the stationary solutions is investigated.The behavior of the dispersionless fKdV equation, which is equivalent to a forced, inviscidBurgers equation, is investigated extensively using the method of characteristics. Exact, analytical solution to the dispersionless, nonlinear approximation to fKdV are derived as well as the amplitude and propagation speed of the shocks obtained from the same approximation.The behaviors of the fKdV equation and its variants are investigated and compared for forcing constant in time and forcing with oscillating amplitude and position. A Wentzel, Kramers, Brillouin approximation is given for dispersionless KdV with low frequency amplitude oscillation in the forcing function. An averaging approximation is given for dispersionless KdV with high frequency amplitude oscillation in the forcing function.The Inverse Scattering Transform is investigated as a diagnostic tool for the behavior of the fKdV equation. The numerical results indicate the emergence of negative eigenvalues of the Schr ̈odinger operator correspond with the emergence of solitons in the solution of the fKdV equation. WKB analysis is used as an application of inverse scattering theory to determine a relationship between the amplitude of the shock in the dispersionless approximation to fKdV and the amplitude of the upstream propagating solitary waves generated by the full equation. All of this information together provides a means of predicting which combinations of parameter values will result in the generation of upstream propagating solitons as well as a novel means of predicting the frequency of soliton generation. Multiple numerical methods and their implementations for solving these equations are discussed. Experiments are carried out in a water recirculating flume and a wave tank. Phenomena predicted by the equations are observed in the experiments and results are compared quantitatively.
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Advisor
  • Camassa, Roberto
  • Adalsteinsson, David
  • Marzuola, Jeremy
  • McLaughlin, Richard
  • Forest, M. Gregory
Degree
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill
Graduation year
  • 2018
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