Search ETDs:
Numerical study of the KP solitons and higher order Miles theory of the Mach reflection in shallow water
Jia, Yuhan

2014, Doctor of Philosophy, Ohio State University, Mathematics.
In 1970, two Russian physicists Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation to study the stability of the solitary wave solution under the influence of weak perturbations transverse to the propagation direction. The equation is now referred to as the KP equation. The KP equation arises as the leading order approximation of certain physical systems under weak nonlinearity, weak dispersion and quasi-two dimensionality assumptions, and admits several exact solutions, called KP solitons, that are regular, non-decaying and localized along distinct lines in the two-dimensional plane, the $xy$-plane.


The main part of this thesis concerns numerical study of the KP solitons for their application to the Mach reflection problem in shallow water. This problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall. In 1977, Miles proposed a theory to explain the phenomenon, and predicted a fourfold amplification of the incident wave. His theory is based on an asymptotic analysis, and in this thesis we show that his results can be interpreted in terms of the KP solitons.

Since Miles presented his theory, there have been several numerical studies as well as water tank experiments to confirm his theory. However, they all found considerable discrepancies between their results and Miles' predictions, in particular, they could not obtain fourfold amplification of the incident wave. In this thesis, we consider the problem starting from the three-dimensional Euler equation for the irrotational and incompressible fluid. We still use the assumptions, weak nonlinearity, weak dispersion and quasi-two dimensionality, but derive the higher order corrections to the leading order KP equation, which we refer to as the higher order KP equation in this thesis. Employing an asymptotic perturbation theory called the normal form theory, we study the higher order KP equation, and find the higher order corrections to the KP solitons. We then perform numerical simulation of the full Euler equation for the Mach reflection phenomena, and confirm that the solutions to the higher order KP equation well describe the phenomena. Thus, we extend the Miles theory including the higher order corrections, which we call the higher order Miles theory, and show that the results obtained from the extended theory are in good agreement with the numerical and experimental results in the literature.

In this thesis, we construct a higher order KP soliton solutions from the result of the numerical simulation to demonstrate the validity of the higher order KP equation. We also consider the stability of the KP solitons which concerns the robustness of the KP solitons and certain convergence issues of the initial value problem of the KP equation. We find an orbital instability of a solitary wave solution due to the existence of the phase shifts propagating along the transversal direction of the soliton.
Yuji Kodama (Advisor)
Barbara Keyfitz (Committee Member)
Fei-Ran Tian (Committee Member)
140 p.

Recommended Citations

Hide/Show APA Citation

Jia, Y. (2014). Numerical study of the KP solitons and higher order Miles theory of the Mach reflection in shallow water. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

Hide/Show MLA Citation

Jia, Yuhan. "Numerical study of the KP solitons and higher order Miles theory of the Mach reflection in shallow water." Electronic Thesis or Dissertation. Ohio State University, 2014. OhioLINK Electronic Theses and Dissertations Center. 23 Nov 2017.

Hide/Show Chicago Citation

Jia, Yuhan "Numerical study of the KP solitons and higher order Miles theory of the Mach reflection in shallow water." Electronic Thesis or Dissertation. Ohio State University, 2014. https://etd.ohiolink.edu/

Files

Dissertation.pdf (13.56 MB) View|Download