Graduate Thesis Or Dissertation
 

Development and Analysis of Higher-order Nonlinear Wavemaker Theory for Intermediate to Deep-water Waves Based on Inverse Scattering Transform

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/9g54xq48h

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  • The modeling and analysis of laboratory-generated nonlinear intermediate- to deep-water wave fields, using existing wavemaker theories and analysis tools, is one of the most challenging tasks in ocean science and engineering. On one hand, harmonics function (sine and cosine) -based wavemaker theories result in an inherent (linear) instability of the deep-water waves, which brings about an additional level of complexity to the nonlinear hydrodynamic analysis. On the other hand, implementation of harmonics function-based analytical and data-analysis tools based on linear wave theory and Fourier transform may not be the optimal tools to characterize the (nonlinear) wave fields. One of the most successful (higher-order) nonlinear equations in modeling and analyzing of intermediate to deep-water wave behavior is the (focusing) nonlinear Schrödinger (NLS) equation. This research focuses on the application of the NLS equation on nonlinear generation, modeling, and analysis of the wave field, especially for experimental model basin test applications. The dissertation mainly consists of three manuscripts described in the following: Manuscript 1: Generation of nonlinear intermediate to deep-water waves in any experimental facilities using the (currently) existing wavemaker theories play an important role in shaping the characteristics of the resulting wave field. There are two wavemaker theories known to date - the traditional linear theory and the second-order corrected theory. There have been some validations and experimental results showing the effects of using a second-order corrected wavemaker theory and the improvements on the resulting wave field over linear wavemaker theory. Most of such validations have in mind only the profile of the wave at a near-field distance. In such relatively “close” proximities, the nonlinear behavior, hence the unstable modes in the target deep-water wave train, have not yet developed and the effects of the wavemaker theories on such important aspects of deep-water waves is still unknown. In this article, an attempt has been made to first clearly identify and categorize the mechanisms and procedures involving the formation of unstable wave packets and second, to investigate the effects of different experimental procedures, including the implemented wavemaker theories, on the resulting observed instabilities in the target deep-water wave field. Manuscript 2: A nonlinear wavemaker theory, based on the analytical solution of nonlinear Schrödinger equation using inverse scattering transformation, is proposed. The proposed method is based on the analytical nonlinear solution of the NLS equation and the solution is free from perturbation limitations that surpasses the accuracy of the existing second-order wavemaker theory. The analytical solution of NLS equation is defined as the propagating solution to the original boundary value problem of water waves, Euler equations and associated boundary conditions, and hence, the restrictions and irregularities arise in numerical approximations do not need to be considered. Application of any time marching scheme resulted in a simple set of equations in contrast to the existing second-order wavemaker theory. The resulting intermediate to deep-water wave fields is shown to achieve the cubic nonlinear behavior much faster than the waves generated using existing (first- or second-order) wavemaker theories. This is a major advantage as it minimizes the influence of dissipation effect caused by wave propagation over long distance and important when considering the sensitivity of the nonlinear behavior to dissipation effects in the wave tank. The proposed NLS-based wavemaker theory generates complex intermediate to deep-water waves which mimic breathers and rogue waves from the appropriate model of the underlying physics without having to use the linear phase-focusing scheme based on existing linear wavemaker theory. Manuscript 3: The survivability, safe operation, and design of marine vehicles and wave energy convertors are highly dependent on the accurate characterization and estimation of the energy content of the ocean wave field. In this study, analytical solution of the nonlinear Schrödinger equation (NLS) using periodic inverse scattering transformation (IST) and its associated Riemann spectrum are employed to obtain the nonlinear wave components (eigen functions). These nonlinear wave components are used in two approaches to develop a more accurate definition of the energy content. First, as an ad hoc approach, the amplitudes of the nonlinear wave components are used with linear energy calculation resulting in a semi-linear energy estimate. Next, a novel, mathematically exact definition of the energy content taking into account the nonlinear effects up to fifth order is introduced in combination with the nonlinear wave components, the exact energy content of the wave field is computed. Experimental results and numerical simulations were used to compute and analyze the linear, ad hoc, and exact energy contents of the wave field, using both nonlinear and linear spectra. The ratio of the ad hoc and exact energy estimates to the linear energy content were computed to examine the effect of nonlinearity on the energy content. In general, an increasing energy ratio was observed for increasing nonlinearity of the wave field, with larger contributions from higher-order terms. It was confirmed that the significant increase in nonlinear energy content with respect to its linear counterpart is due to the increase in the nonlinear phase-locked components It is expected that the findings from this research can serve as a useful reference for nonlinear analysis theory and practical experimental laboratory generation of intermediate to deep-water wave fields and to provide scientists and engineers a smooth transition from the traditional linear wave-based approach to a mathematically more rigorous and accurate nonlinear based one.
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  • Ongoing Research
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  • 2019-09-24 to 2021-10-25

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