J. N. Newman

WAVE EFFECTS ON DEFORMABLE BODIES

Abstract The linearized frequency-domain analysis of wave radiation and diffraction by a three-dimensional body in a fixed mean position is extended to a variety of deformable body motions. These include continuous structural deflections, and also discontinuous motions which can be used to represent multiple interacting bodies. A general methodology is adopted with the body deflection defined by an expansion in arbitrary modal shape functions, and the response in each mode is obtained as a logical extension of the usual analysis for rigid-body modes. Illustrative computations are presented for the bending of a freely-floating barge, and of a vertical column with cantilever support at the bottom. For these structural deflections the use of orthogonal polynomials is emphasized, as an alternative to the more conventional use of natural modes. Also presented are computations for the motions of two rigid barges connected by a hinge joint, and for a finite array of images used to approximate wall effects on a cylinder in a channel. Results from the latter problem are compared with more analytical solutions, and it is shown that practical results can be obtained for the first-order hydrodynamic force coefficients using only a few images.

Algorithms for the free-surface Green function

SummaryNumerical methods are outlined for computing the velocity potential, and its derivatives, for linearized three-dimensional wave motions due to a unit source with harmonic time dependence beneath a free surface. Two distinct cases are considered where the fluid depth is either infinite, or of constant finite depth. Efficient algorithms are developed in both cases, to replace the numerical evaluation of the relevant integrals by multi-dimensional approximations in economized polynomials. This technique is substantially faster than conventional direct methods based on numerical integration.

Wave diffraction by a long array of cylinders

Water wave diffraction by an array of bottom-mounted circular cylinders is analysed under the assumptions of linear theory. The cylinders are identical, and equally spaced along the array. When the number of cylinders is large, but finite, near-resonant modes occur between adjacent cylinders at critical wavenumbers, and cause unusually large loads on each element of the array. These modes are associated with the existence of homogeneous solutions for the diffraction by an array which extends to infinity in both directions. This phenomenon is related to the existence of trapped waves in a channel. A second trapped wave is established, corresponding to Dirichlet boundary conditions on the channel walls, as well as a sequence of higher wavenumbers where ‘nearly trapped’ modes exist.

Distributions of sources and normal dipoles over a quadrilateral panel

SummaryThe potential due to a distribution of sources or normal dipoles on a flat quadrilateral panel is evaluated for the cases where the density of the singularities is constant, linear, bilinear, or of arbitrary polynomial form. The results in the first two cases are consistent with those derived previously, but the present derivation is considered to be simplified. In particular, the constant dipole distribution is derived from a geometric argument which avoids direct integration; this derivation applies more generally on a curvilinear panel bounded by straight edges.Also presented are multipole expansions for the same potentials, suitable for use when the distance to the field point is substantially larger than the panel dimensions. Algorithms are derived to evaluate the coefficients in these expansions to an arbitrary order.

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.

A generalized slender-body theory for fish-like forms

A consistent slender-body approximation is developed for the flow past a fish-like body with arbitrary combinations of body thickness and low-aspect-ratio fin appendages, but with the fins confined to the plane of symmetry of the body. Attention is focused on the interaction of the fin lifting surfaces with the body thickness, and especially on the dynamics of the vortex sheets shed from the fin trailing edges. Explicit results are given for axisymmetric bodies having fins with abrupt trailing edges, and calculations of the total lift force are presented for bodies with symmetric and asymmetric fin configurations, moving with a constant angle of attack. (Modified author abstract)

Absorption of wave energy by elongated bodies

Abstract Slender-body approximations are used to predict the maximum rate of energy absorption by an elongated floating vessel which performs vertical motions of varying amplitude and phase along its length. Simple estimates are derived for the amplitude and phase of particular mode shapes, and for the corresponding power absorption. Specific mode shapes considered include polynomials, trigonometric functions, and piecewise-linear functions intended to represent an articulated raft. An articulated raft with two hinges appears to be optimum from the engineering standpoint.

The second-order wave force on a vertical cylinder

The second-order wave force is analysed for diffraction of monochromatic water waves by a vertical cylinder. The force is evaluated directly from pressure integration over the cylinder, and the second-order potential is derived by Weber transformation of the corresponding forcing function on the free surface. This forcing function is reduced to a form which involves a simple factor inversely proportional to the radial coordinate plus an oscillatory function which decays more rapidly in the far field. This feature alleviates the slow rate of convergence involved in capturing the far-field effect. Benchmark computations are obtained and compared with other works. Asymptotic approximations are derived for long and short wavelengths. The analysis and results are primarily for the case of infinite fluid depth, but the finite-depth case is also considered to facilitate comparison with other computations and to illustrate the importance of finite-depth effects in the long-wavelength asymptotic regime.

The force on a slender fish-like body

The force acting on a fish-like body with combined thickness and lifting effects is analysed on the assumption of inviscid flow. A general expression is developed for the pressure force on the body, which is analogous to the momentum-flux analysis for non-lifting bodies in classical hydrodynamics. For bodies with constant volume, the mean drag (or propulsive) force is expressed in terms of a contour integral around the vortex sheet behind the body. Attention is focused on the case of steady-state motion with constant angle of attack, and the induced drag is analysed for finned axisymmetric bodies using the slender-body approximation developed by Newman & Wu (1973). Unlike earlier results of Lighthill (1970), the lift–drag ratio in this case depends on the body thickness.

Wave-drift damping of floating bodies

Wave-drift damping results from low-frequency, oscillatory- motions of a floating body, in the presence of an incident wave field. Previous works have analysed this effect in a quasi-steady manner, based on the rate of change of the added resistance in waves, with respect to a small steady forward velocity. In this paper the wave-drift damping coefficient is derived more directly, from a perturbation analysis where the low-frequency body oscillations are superposed on the diffraction field. Unlike the case of body oscillations in calm water, where the damping due to wave radiation is asymptotically small for low frequencies, the superposition of oscillatory motions on the diffraction field results in an order-one damping coefficient. All three degrees of freedom are considered in the horizontal plane. The resulting matrix of damping coefficients is derived from pressure integration on the body, and transformed in special cases to a far-field control surface.