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It is worthy to mention that there are exist-ing semi-empirical methods for roll-damping estimation due to the hull surface and bilge keels (Himeno, 1981, Ikeda, 1977, Tanaka, 1960). Of course, one component of damping is surface-wave generation, which is normally well predicted by potential-flow theory. There have been efforts also to predict viscous damping without relying on empiricism, (Fink and Soh 1974, Brown and Patel 1985, Braathen and Faltinsen 1988, Cozens 1987, Downie et al. 1990). None has yet been able to model the interaction of hull geometry, vorticity generation and free-surface simultaneously.

Recently, Yeung & Vaidhyanathan (1994) have developed a Free-Surface Random-Vortex Method (FSRVM) to model near-surface flows for body geometry that may or may not have sharp edges. Their earlier works involved several important improvements (Yeung et al., 1993) over the Random Vortex Method (RVM) proposed originally by Chorin (1973). An efficient numerical scheme was devised to compute the mutual induction of discrete vortex elements. A boundary-integral formulation was used to handle arbitrary body geometry. These new developments were extended to take into account of free-surface effects. The essence of this method is briefly described in Section 4. FSRVM does not require any prior knowledge of the location of the separation point. Since the method is grid-free, it can represent vortex structures of a rather wide range of scales. Accurate predictions at high Reynolds number is also possible, as illustrated in Section 5 by some direct comparisons of the numerical predictions with experimental results.

Although forces and moments are usually the primary quantities of interest, it is not always appropriate to establish the validity of a mathematical model on forces or moments alone. A more rigorous validation should involve comparing flow patterns also. At the University of California Ship-Model Testing Facility, a Digital Image Particle Velocimetry (DPIV) was developed and implemented to provide quantitative results on the velocity and vorticity fields generated by a body moving in a free surface (Cermelli, 1995). Some of the major findings of such an investigation are reported in Section 3.

Because of the complexity of vortex patterns generated by a realistic hull model, a simplified “canonical” problem was introduced to study the roll motion of a ship section. The canonical problem chosen is that of a surface-piercing plate rolling periodically about the free surface. The flow is primarily two dimensional so that comparison with applicable theoretical results is immediately possible. For the typical frequency range of interest, this rolling plate is designed so that surface-wave generation and and vorticity generation are important. Naturally, besides providing a validation on our theoretical model, this canonical experiment is closely relevant to the understanding of the performance of bilge keels, the prevalent passive device for reducing roll motion. In the last section of the paper, we present predictions of added inertia and “equivalent” linear damping for a rectangular hull section and compare them with some existing data.

2
THE ROLLING-PLATE EXPERIMENT
2.1
Design of experiment

A special model was designed and fabricated to study the vortical structures generated by a plate undergoing rolling motion in a free surface. This experiment was carried out at the Ship-Model Testing Facility of the University of California at Berkeley. The towing tank is 61m(L)×2.44m(W) ×1.52m(D). The plate is oscillated by a hydraulic piston. Flow visualization can be conducted through four observation windows along tank walls.

The plate was built out of one-inch thick acrylic and stiffened longitudinally. It has a natural frequency of 4 Hz, while the range of forced-motion frequency being investigated is less than 0.6 Hz. The clearance between the plate vertical edges and the tank walls was kept to a minimum (0.64cm) to reduce end effects. The plate was hinged at the free surface. Its draft was primarily set to 30.48cm (12 in). Flow visualizations were conducted also with a 15.24cm (6 in) draft.

A plate with a tip that was rectangular in section was found to yield force and moment measurements that were not repeatable, primarily as a result of chaotic vortex shedding from the bottom edges of the plate. The tip of the plate was then modified by the addition of a half circle rod at the bottom of the plate (see Fig.1). This greatly improved the consistency and repeatability of force records.

Plate rotation was achieved by applying horizontal motion to a rod which was hinged to the plate on one side and to (a random motion) hydraulic piston on the other side (Fig.1). A software controller was developed to ensure speedy



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