Studies on Nonlinear Dispersive Water Waves

Citation

Qu, Wendong (2000) Studies on Nonlinear Dispersive Water Waves. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/19B4-2N21. https://resolver.caltech.edu/CaltechETD:etd-08152006-140314

Abstract

This study investigates the phenomena of evolution of two-dimensional, fully nonlinear, fully dispersive, incompressible and irrotational waves in water of uniform depth in single and in double layers. The study is based on an exact fully nonlinear and fully dispersive (FNFD) wave model developed by Wu (1997, 1999a). This FNFD wave model is first based on two exact equations involving three variables all pertaining to their values at the water surface. Closure of the system of model equations is accomplished either in differential form, by attaining a series expansion of the velocity potential, or in integral form by adopting a boundary integral equation for the velocity field. A reductive perturbation method for deriving asymptotic theory for higher-order solitary waves is developed using the differential closure equation of the FNFD wave theory. Using this method, we have found the leading 15th-order solitary wave solutions. The solution is found to be an asymptotic solution which starts to diverge from the 12th-order so that the 11th-order solution appears to provide the best approximation to the fully nonlinear solitary waves, with a great accuracy for waves of small to moderately large amplitudes. Two numerical methods for calculating unsteady fully nonlinear waves, namely, the FNFD method and the Point-vortex method, are developed and applied to compute evolutions of fully nonlinear solitary waves. The FNFD method, which is based on the integral closure equation of Wu's theory, can provide good performance on computation of solitary waves of very large amplitude. The Point-vortex method using the Lagrange markers is very efficient for computation of waves of small to moderate amplitudes, but has intrinsic difficulties in computing waves of large amplitudes. These two numerical methods are applied to carry out a comparative study of interactions between solitary waves. Capillary-gravity solitary waves are investigated both theoretically and numerically. The theoretical study based on the reductive perturbation method provides asymptotic theories for higher-order capillary-gravity solitary waves. A stable numerical method (FNFD) for computing exact solutions for unsteady capillary-gravity solitary waves is developed based on the FNFD wave theory. The results of the higher-order asymptotic theories compare extremely well with those given by the FNFD method for waves of small to moderate amplitudes. A numerical method for computing unsteady fully nonlinear interfacial waves in two-layer fluid systems is developed based on the FNFD model. The subcritical and supercritical cases can be clearly distinguished by this method, especially for waves of amplitudes approaching the maximum attainable for the fully nonlinear theory.

Thesis Files