In this dissertation, the extended Melnikov's method has been applied to several nonlinear ship dynamics models, which are related to the new generation of stability criteria in the International Maritime Organization (IMO). The advantage of this extended Melnikov's method is it overcomes the limitation of small damping that is intrinsic to the implementation of the standard Melnikov's method. The extended Melnikv's method is first applied to two published roll motion models. One is a simple roll model with nonlinear damping and cubic restoring moment. The other is a model with a biased restoring moment. Numerical simulations are investigated for both models. The effectiveness and accuracy of the extended Melnikov's method is demonstrated. Then this method is used to predict more accurately the threshold of global surf-riding for a ship operating in steep following seas. A reference ITTC ship is used here by way of example and the result is compared to that obtained from previously published standard analysis as well as numerical simulations. Because the primary drawback of the extended Melnikov's method is the inability to arrive at a closed form equation, a 'best fit'approximation is given for the extended Melnikov numerically predicted result. The extended Melnikov's method for slowly varying system is applied to a roll-heave-sway coupled ship model. The Melnikov's functions are calculated based on a fishing boat model. And the results are compared with those from standard Melnikov's method. This work is a preliminary research on the application of Melnikov's method to multi-degree-of-freedom ship dynamics. In the last part of the dissertation, the method of manufactured solution is applied to systems with chaotic behavior. The purpose is to identify points with potential numerical discrepancies, and to improve computational efficiency. The numerical discrepancies may be due to the selection of error tolerances, precisions, etc. Two classical chaotic models and two ship capsize models are examined. The current approach overlaps entrainment in chaotic control theory. Here entrainment means two dynamical systems have the same period, phase and amplitude. The convergent region from control theory is used to give a rough guideline on identifying numerical discrepancies for the classical chaotic models. The effectiveness of this method in improving computational efficiency is demonstrated for the ship capsize models. / Ph. D.

Turbulent boundary layers approximating those found on the NASA Orion Multi-Purpose Crew Vehicle (MPCV) thermal protection system during atmospheric reentry from the International Space Station have been studied by direct numerical simulation, with the ultimate goal of reducing aerothermodynamic heating prediction uncertainty. Simulations were performed using a new, well-verified, openly available Fourier/B-spline pseudospectral code called Suzerain equipped with a ``slow growth'' spatiotemporal homogenization approximation recently developed by Topalian et al. A first study aimed to reduce turbulence-driven heating prediction uncertainty by providing high-quality data suitable for calibrating Reynolds-averaged Navier--Stokes turbulence models to address the atypical boundary layer characteristics found in such reentry problems. The two data sets generated were Ma[approximate symbol] 0.9 and 1.15 homogenized boundary layers possessing Re[subscript theta, approximate symbol] 382 and 531, respectively. Edge-to-wall temperature ratios, T[subscript e]/T[subscript w], were close to 4.15 and wall blowing velocities, v[subscript w, superscript plus symbol]= v[subscript w]/u[subscript tau], were about 8 x 10-3 . The favorable pressure gradients had Pohlhausen parameters between 25 and 42. Skin frictions coefficients around 6 x10-3 and Nusselt numbers under 22 were observed. Near-wall vorticity fluctuations show qualitatively different profiles than observed by Spalart (J. Fluid Mech. 187 (1988)) or Guarini et al. (J. Fluid Mech. 414 (2000)). Small or negative displacement effects are evident. Uncertainty estimates and Favre-averaged equation budgets are provided. A second study aimed to reduce transition-driven uncertainty by determining where on the thermal protection system surface the boundary layer could sustain turbulence. Local boundary layer conditions were extracted from a laminar flow solution over the MPCV which included the bow shock, aerothermochemistry, heat shield surface curvature, and ablation. That information, as a function of leeward distance from the stagnation point, was approximated by Re[subscript theta], Ma[subscript e], [mathematical equation], v[subscript w, superscript plus sign], and T[subscript e]/T[subscript w] along with perfect gas assumptions. Homogenized turbulent boundary layers were initialized at those local conditions and evolved until either stationarity, implying the conditions could sustain turbulence, or relaminarization, implying the conditions could not. Fully turbulent fields relaminarized subject to conditions 4.134 m and 3.199 m leeward of the stagnation point. However, different initial conditions produced long-lived fluctuations at leeward position 2.299 m. Locations more than 1.389 m leeward of the stagnation point are predicted to sustain turbulence in this scenario. / text