Potential-Flow Predictions of a Semi-Displacement Vessel Including Applications to Calm-Water Broaching
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- Institutt for marin teknikk 
As a marine vehicle’s operational speed increases, hydrodynamic pressure plays an increasingly significant role in carrying the vessel’s weight. The shift of importance from hydrostatic to hydrodynamic pressure may cause a vessel, which is stable at rest or low speeds, to become dynamically unstable at high speeds. The nature of the dynamic instability depends mainly on the vessel’s type and speed. Among high speed vessels, semi-displacement mono-hulls are particularly susceptible to a nonoscillatory dynamic instability in sway, roll and yaw known as calm water broaching. This type of dynamic instability is the main reason why semi-displacement monohulls must not operate at Froude numbers higher than 1.2 (Lavis, 1980). The application of linear theory in predicting this type of instability motivated the present study. A linear dynamic model of the vessel, including its hydrodynamic coefficients, is needed for the dynamic stability analysis. Prediction of these coefficients is a challenging problem, requiring the solution of the flow around an advancing, oscillating vessel. A three-dimensional boundary element solver was constructed for this purpose. The free-surface and body boundary conditions were linearized using Neumann-Kelvin linearization. Since the focus here is on high speed vessels, this type of linearization is chosen instead of the double-body linearization. Boundary surfaces were discretized using numerical grid generation methods. Elements ranging from constant to cubic were used to represent the surfaces. Rankine sources and dipoles were distributed on the boundaries. The solver was programmed in a way to allow for the implementation of different boundary integral formulations using elements with different distribution orders in a convenient and compact form. The derivatives of the velocity potential on the body surface were calculated using shape functions. On the free surface, the direction of differentiation is known to be important, especially in the flow around high-speed vessels. Therefore, derivatives on the free surface were calculated using upstream finite difference operators to satisfy the radiation condition and avoid numerical instabilities. Semi-discrete Fourier analysis was used to investigate numerical dispersion and damping of a wave traveling on a discrete free surface with and without current. Different singularity distribution and differentiation methods were considered. Based on these studies, a set of practical guidelines were established to choose suitable differentiation methods and an appropriate number of elements for each problem, and assess the numerical accuracy of the results. Damping zones were introduced around the free surface boundaries in order to absorb the waves and ensure that the radiation condition was satisfied in the time-domain analysis. Three types of flow separation were accounted for indirectly in the present potential-flow solution: trailing edge flow separation by a vortex sheet method, transom stern flow separation by a hollow body model, and cross flow separation by a 2D+t drag model. A series of problems for non-separated potential flows with and without forward speed were solved both in the time-domain and steady-state. The results were validated against experimental and analytical data. The flow around an advancing, surface-piercing flat plate with steady drift was investigated using steady-state and time-domain solvers as an example of the tailseparated flows. A 2D+t cross-flow drag model was adopted in order to consider the cross-flow separation effects, which turned out to be important. Then, the problem was extended by adding oscillatory motions to the surface-piercing plate. The hydrodynamic coefficients in sway, roll and yaw were calculated for a series of Froude numbers and oscillation frequencies. The results were validated against existing experimental and numerical data. Next, the flow around monohull semi-displacement vessels was studied using linear theory. The dry transom stern effects were captured by introducing a hollow body model. The results were validated against experimental and numerical data in terms of free-surface elevation and steady vertical forces. The hollow body model was extended to solve the flow around an advancing semi-displacement vessel with constant drift angle. A simplified 2D+t cross-flow drag model explained the differences between numerical and experimental data. The hydrodynamic coefficients in heave were calculated using the extended hollow body model. This method captured the anticipated sharp drop in the values of added mass and damping close to the transom stern. Finally, dynamic stability in sway-yaw and sway-roll-yaw was investigated using linear stability analysis. A semi-displacement vessel with documented instability issues was chosen for validation. The hydrodynamic properties of the vessel were simplified to those of a flat plate. A sensitivity study was carried out to assess the importance of different parameters in the vessel’s dynamic stability. This simplified analysis predicted the presence of an instability around the reported unstable Froude number. The nature of the instability was, however, different than what has been reported in the literature. Further investigations are needed on this subject.