Ph.D Student | Toledo Yaron |
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Subject | Refraction and Diffraction of linear and Nonlinear Waves |

Department | Department of Civil and Environmental Engineering |

Supervisor | Professor Yehuda Agnon |

Full Thesis text |

The irrotational flow of an incompressible homogeneous inviscid fluid is generally a three-dimensional problem. In this work, we deal with two types of approximated equations that reduce the water wave problem to a two-dimensional one. The first is the high-order Boussinesq (HOB) type equations, and the second is the mild-slope type equations. This model is fully nonlinear and is solved in the time domain. The mild-slope (MS) type equations, on the other hand, are posed in the frequency domain. They are essentially linear, but can be coupled with nonlinear terms as a set of equations in order to take into account nonlinear interactions.

In the first part of the work, an accurate finite-difference (AFD) numerical method is developed specifically for HOB equations. The method solves the water-wave flow with much higher accuracy compared to the standard finite-difference method (SFD) for the same computer resources. It is first developed for linear water waves and then for the nonlinear problem. It is presented for a horizontal bottom, but can be used for variable depth as well. The method can be developed for other equations as long as they use Padé approximation. Finally, the results of the AFD and the SFD are compared with the accurate solution for nonlinear progressive waves and nonlinear standing ones. The agreement of the AFD to the accurate solution is found to be excellent compared to the SFD one.

The complementary mild-slope equation (CMSE) is a depth-integrated equation, which models a linear time-harmonic water wave flow. For 2D problems, it was shown to give better agreements to exact linear theory comparing to other MS-type equations. Unlike these other MS-type models, the CMSE is derived in terms of a stream-function vector rather than in terms of a velocity potential or the wave height. For the 3D case, this complicates the governing equation and creates difficulties in formulating an adequate number of boundary conditions. In the second main part of the work, the CMSE is rederived from Hamilton's principle using the Irrotational Green-Naghdi equations with a correction for the 3D case. The additional boundary conditions are constructed using the irrotationality condition .A parabolic approximation is applied, and the model is solved in comparison with a wave tank experiment. The results have excellent agreement. Then, the CMSE model is extended up to second order to enable nonlinear coupling between frequency components. The nonlinear model is compared to laboratory experiments and accurate numerical ones, and gives good agreements as well.