The scattering of water waves by an array of N bottom-mounted vertical circular cylinders is solved exactly (under the assumption of linear water wave theory) using the method proposed by Spring & Monkmeyer in 1974. A major simplification to this theory has been found which makes the evaluation of quantities such as the forces on the cylinders much simpler. New formulae are given for the first and mean second-order forces together with one for the free-surface elevation in the vicinity of a particular cylinder. Comparisons are made between the exact results shown here and those generated using the approximate method of McIver & Evans (1984). The behaviour of the forces on the bodies in the long-wave limit is also examined for the special case of two cylinders with equal radii.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112090002750

The interaction of waves with arrays of vertical circular cylinders

Wave-induced sea floor dynamics

https://msvlab.hre.ntou.edu.tw/grades/Integral-equation(2009)/Y.J.Lin/Near-trapping%20of%20waves%20by%20circular%20arrays%20of.pdf

Near-trapping of waves by circular arrays of vertical cylinders

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112079000975

A high-order cnoidal wave theory

Comparison of methods for computing hydrodynamic characteristics of arrays of wave power devices

This study deals with the interaction of linear, plane water waves with stationary groups of rigid, vertical, circular cylinders under conditions in which the inertial forces on the cylinders dominate over the drag forces. A direct matrix solution as well as multiple scattering as suggested by Twersky (1952) are used to obtain the velocity potential in the vicinity of the cylinders. The groups may consist of a number of cylinders having any geometric arrangement, may have Dirichlet, Neumann, or mixed boundary conditions, and need not have identical diameters. The study represents an extension of the single cylinder case presented by MacCamy and Fuchs in 1954. Basic scattering coefficients for 192 different arrangements of two cylinders are obtained with the aid of a Bessel coordinate transformation and a matrix inversion procedure. The resulting potential function is then applied to calculate force components in the direction of wave advance and orthogonal to it. For the cases considered the former departs as much as 65% from the force on a single cylinder and the mass coefficient is found to range from 1.19 to 3.38 - a not insignificant departure from the often used value of 2.0. Furthermore the orthogonal force may be as large as 67% of the single-cylinder force.

/pdf/interaction-of-plane-waves-with-vertical-cylinders-wf48bgmspo.pdf

Interaction of Plane Waves with Vertical Cylinders

Ocean waves induce dynamic pressure responses in permeable seabeds which result in dynamic loads on buried pipelines. An analytical model is developed for estimating the pore pressure in the soil and the resulting pressure force on buried pipelines. It is assumed that the seabed is rigid, homogeneous, and porous with isotropic permeability, that the pore water is incompressible, that fluid flow in the soil is modeled by Darcy's Law, and that the seabed is infinitely deep. A solution is developed for a circular, rigid pipeline using conformal mapping techniques. The solution is compared with the results of both small and large‐scale tests; reasonable agreement is obtained for the small‐scale tests. Wave‐induced seepage forces are evaluated by integrating the pressure distribution over the pipe surface. The magnitude of the force remains constant but the direction rotates around the cylinder once with the passage of each wave. This force may be of sufficient magnitude to be an important consideration in the...

Wave-induced forces on buried pipelines

The Dupuit-Forchheimer equation describing unconfined ground-water flow is limited by the underlying assumptions inherent in its derivation. A validity criterion is developed from an analysis of the exact equations of motion which shows that, in general, the Dupuit-Forchheimer equation describes a rising water table more accurately than a falling one. In the steady state the criterion reduces to a simple restriction on the free surface slope. For unsteady problems the criterion involves both the free surface slope and the rate of change of the free surface position. For the specific case of flow toward a well it is shown that the commonly used rule of thumb, r/h o > 1.5, may not always be valid and a more general criterion is presented.

Validity of Dupuit-Forchheimer Equation

A higher order theory is presented for symmetrical, non-linear gravity waves As a consequence of the generality employed, the theory includes the full range of possible wave lengths, water depths and wave heights that may be encountered, and brings them into one unified formulation Thus, the theory encompasses both linear and non-linear waves, including Airy waves, Stokes waves, cnoidal waves and the solitary wave Based on the work of Nekrasov, a complex potential in the form of an infinite series is developed to describe the flow field The potential satisfies the bottom (horizontal) condition as well as the kinematic surface condition exactly Furthermore, the dynamic surface condition is satisfied by numerical calculation of the series coefficients which appear in the complex potential The calculation of these coefficients is accomplished by solving a set of non-linear algebraic equations, with the aid of a Newton-Raphson iteration procedure and matrix inversion Coefficients of the complex potential have been obtained for a fifth order analysis and preliminary results are presented in tabular form A brief discussion of the characteristics of the waves, including wave speed, wave shape and the height of the highest possible wave follows.

/pdf/a-higher-order-theory-for-symmetrical-gravity-waves-4jzw9t5fs5.pdf

A higher order theory for symmetrical gravity waves

This study deals with the seepage effects experienced by a large, vertical, circular cylinder resting on a submerged bed of sand when planar water waves interact with it. Potential theory is used to describe the seepage flow field. The sea bottom pressure condition is determined from the water field velocity potential derived by MacCamy and Fuchs (1954) in the case of planar waves diffracted by a large impervious cylinder. Consideration is also given to cylinders with a thin circular base whose diameter exceeds that of the cylinder itself. The problem formulation as well as the initiation of the analysis apply to the general case of a bed of sand with finite depth. For the case of infinite depth of the porous medium, theoretical solutions for the seepage pressure are obtained in the form of infinite integrals. Theoretical solutions for the pressure along the cylinder circular base are then derived, leading by integration to closed form expressions for the wave-induced seepage uplift force and overturning moment exerted on the cylinder. These expressions for the force and moment, which are presented in non-dimensional form are shown to be universal functions of a unique variable. Graphs are provided so that very few computations are required to determine the uplift force and overturning moment exerted on a cylinder. A comparison with various approximate theories reveals the present theory to be the only one which gives reliable results in general. The amplitude and phase angle of the oscillating wave-induced pressure along the cylinder base are determined numerically. Results for the pressure amplitude are presented as non-dimensional ratios to the amplitude of the pressure that would prevail if no cylinder were disturbing the wave field. Expressions for the exit gradient around the cylinder base are also determined. Contours of the ratio of the exit gradient to the one that would prevail in the absence of a cylinder are presented. Laboratory measurements of uplift pressure amplitudes on a circular cylinder show good agreement with theoretical calculations.

https://icce-ojs-tamu.tdl.org/icce/index.php/icce/article/download/3535/3217

Peter L. Monkmeyer

Papers

Interaction of Plane Waves with Vertical Cylinders

Wave-induced forces on buried pipelines

Validity of Dupuit-Forchheimer Equation

A higher order theory for symmetrical gravity waves

Wave-Induced Seepage Effects on a Vertical Cylinder