Odd M. Faltinsen

Water entry of two-dimensional bodies

A numerical method for studying water entry of a two-dimensional body of arbitrary cross-section is presented. It is a nonlinear boundary element method with a jet flow approximation. The method has been verified by comparisons with new similarity solution results for wedges with deadrise angles varying from 4° to 81°. A simple asymptotic solution for small deadrise angles α based on Wagner (1932) agrees with the similarity solution for small α.

Nonlinear wave loads on a slender vertical cylinder

The diffraction of water waves by a vertical circular cylinder is considered in the regime where the wave amplitude A and cylinder radius a are of the same order, and both are small compared to the wavelength. The wave slope is small, and a conventional linear analysis applies in the outer domain far from the cylinder. Significant nonlinear effects exist in the complementary inner domain close to the cylinder, associated with the free-surface boundary condition. Using inner coordinates scaled with respect to a , it is shown that the leading-order nonlinear contribution to the velocity potential includes terms proportional to both A 2 a and A 3 . The wave load which acts on the cylinder near the free surface includes second- and third-harmonic components which are proportional respectively to A 2 a 2 and A 3 a . In a conventional perturbation analysis, where A [Lt ] a , these components would be ordered in magnitude corresponding to the different powers of A , but here they are of the same order. The second- and third-order components of the total force are of comparable magnitude for practical values of the wave slope.

Resonant three-dimensional nonlinear sloshing in a square-base basin

An asymptotic modal system is derived for modelling nonlinear sloshing in a rectangular tank with similar width and breadth. The system couples nonlinearly nine modal functions describing the time evolution of the natural modes. Two primary modes are assumed to be dominant. The system is equivalent to the model by Faltinsen et al. (2000) for the two-dimensional case. It is validated for resonant sloshing in a square-base basin. Emphasis is on finite fluid depth but the behaviour with decreasing depth to intermediate depths is also discussed. The tank is forced in surge/sway/roll/pitch with frequency close to the lowest degenerate natural frequency. The theoretical part concentrates on periodic solutions of the modal system (steady-state wave motions) for longitudinal (along the walls) and diagonal (in the vertical diagonal plane) excitations. Three types of solutions are established for each case: (i) ‘planar’/‘diagonal’ resonant standing waves for longitudinal/diagonal forcing, (ii) ‘swirling’ waves moving along tank walls clockwise or counterclockwise and (iii) ‘square’-like resonant standing wave coupling in-phase oscillations of both the lowest modes. The frequency domains for stable and unstable waves (i)–(iii), the contribution of higher modes and the influence of decreasing fluid depth are studied in detail. The zones where either unstable steady regimes exist or there are two or more stable periodic solutions with similar amplitudes are found. New experimental results are presented and show generally good agreement with theoretical data on effective domains of steady-state sloshing. Three-dimensional sloshing regimes demonstrate a significant contribution of higher modes in steady-state and transient flows.

A local directional ghost cell approach for incompressible viscous flow problems with irregular boundaries

An immersed boundary method for the incompressible Navier-Stokes equations in irregular domains is developed using a local ghost cell approach. This method extends the solution smoothly across the boundary in the same direction as the discretization it will be used for. The ghost cell value is determined locally for each irregular grid cell, making it possible to treat both sharp corners and thin plates accurately. The time stepping is done explicitly using a second order Runge-Kutta method. The spatial derivatives are approximated by finite difference methods on a staggered, Cartesian grid with local grid refinements near the immersed boundary. The WENO scheme is used to treat the convective terms, while all other terms are discretized with central schemes. It is demonstrated that the spatial accuracy of the present numerical method is second order. Further, the method is tested and validated for a number of problems including uniform flow past a circular cylinder, impulsively started flow past a circular cylinder and a flat plate, and planar oscillatory flow past a circular cylinder and objects with sharp corners, such as a facing square and a chamfered plate.

Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth

The modal system describing nonlinear sloshing with inviscid flows in a rectangular rigid tank is revised to match both shallow fluid and secondary (internal) resonance asymptotics. The main goal is to examine nonlinear resonant waves for intermediate depth/breadth ratio 0.1 [lsim ] h / l [lsim ] 0.24 forced by surge/pitch excitation with frequency in the vicinity of the lowest natural frequency. The revised modal equations take full account of nonlinearities up to fourth-order polynomial terms in generalized coordinates and h / l and may be treated as a modal Boussinesq-type theory. The system is truncated with a high number of modes and shows good agreement with experimental data by Rognebakke (1998) for transient motions, where previous finite depth modal theories failed. However, difficulties may occur when experiments show significant energy dissipation associated with run-up at the walls and wave breaking. After reviewing published results on damping rates for lower and higher modes, the linear damping terms due to the linear laminar boundary layer near the tanks surface and viscosity in the fluid bulk are incorporated. This improves the simulation of transient motions. The steady-state response agrees well with experiments by Chester & Bones (1968) for shallow water, and Abramson et al. (1974), Olsen & Johnsen (1975) for intermediate fluid depths. When h / l [lsim ] 0.05, convergence problems associated with increasing the dimension of the modal system are reported.

Wave impact loads: The role of the flip-through

The impact of waves upon a vertical, rigid wall during sloshing is analyzed with specific focus on the modes that lead to the generation of a flip-through [M. J. Cooker and D. H. Peregrine, “A model for breaking wave impact pressures,” in Proceedings of the 22nd International Conference on Coastal Engineering (ASCE, Delft, 1990), Vol. 2, pp. 1473–1486]. Experimental data, based on a time-resolved particle image velocimetry technique and on a novel free-surface tracking method [M. Miozzi, “Particle image velocimetry using feature tracking and Delaunay tessellation,” in Proceedings of the 12th International Symposium on Applications of Laser Techniques to Fluid Mechanics (2004)], are used to characterize the details of the flip-through dynamics while wave loads are computed by integrating the experimental pressure distributions. Three different flip-through modes are observed and studied in dependence on the amount and modes of air trapping. No air entrapment characterizes a “mode (a) flip-through,” engulfm...

Two-dimensional resonant piston-like sloshing in a moonpool

This paper presents combined theoretical and experimental studies of the two-dimensional piston-like steady-state motions of a fluid in a moonpool formed by two rectangular hulls (e.g. a dual pontoon or catamaran). Vertical harmonic excitation of the partly submerged structure in calm water is assumed. A high-precision analytically oriented linear-potential-flow method, which captures the singular behaviour of the velocity potential at the corner points of the rectangular structure, is developed. The linear steady-state results are compared with new experimental data and show generally satisfactory agreement. The influence of vortex shedding has been evaluated by using the local discrete-vortex method of Graham (1980). It was shown to be small. Thus, the discrepancy between the theory and experiment may be related to the free-surface nonlinearity.

NUMERICAL PREDICTIONS OF SHIP MOTIONS AT HIGH FORWARD SPEED

A theory for ship motions at high forward speed is presented. The theory includes interaction between the steady and unsteady flow field. Numerical results for the steady flow and added mass and damping are compared with experimental results.

The effect of hydroelasticity on ship slamming

Wetdeck slamming is studied theoretically by a hydroelastic beam model. The problem is simplified by introducing an initial structural inertia phase and a subsequent free vibration phase. Forward speed effects are included. The theoretical model is validated by comparing with drop tests of elastic plates on waves. The stresses in the plates have a linear dependence on the impact speed and are neither sensitive to the radius of curvature of the waves nor where the waves initially hit. Hydroelasticity is important. The maximum impact pressures can be extremely high and have a stochastic nature even under deterministic environmental conditions, but they are not important for maximum bending stresses.

SLOW DRIFT OSCILLATIONS OF A SHIP IN IRREGULAR WAVES

A procedure to calculate horizontal slow drift excitation forces on an infinitely long horizontal cylinder in irregular beam sea waves is presented. The hydrodynamic boundary-value problem is solved correctly to second order in wave amplitude. Results in the form of second order transfer functions are presented for different two-dimensional shapes. It is concluded that Newmans approximative method is a practical way to calculate slow drift excitation forces on a ship in beam sea and suggested that it may be used in a more general case. Applications of the results for moored ships are discussed.