Fluid Velocity Potential Research Articles

Green's function solution for axisymmetric vibration of flexible liquid-filled storage tanks

The dynamic response of a flexible, liquid-filled cylindrical storage tank subjected to high frequency vertical ground motions is investigated theoretically. The tank motion is idealized as that of a uniform, elastic, thin-walled cylindrical shell undergoing radial displacements; all axial deformations are neglected. The fluid is assumed to be inviscid and linearly compressible and to undergo small-amplitude, irrotational motion. Both the structural and fluid motions are expressed in terms of appropriate Green's functions. The axisymmetry of the response then leads to a pair of coupled line integral equations for the fluid velocity potential and its normal derivative on the walls of the tank. These equations are then solved numerically. Results are presented which illustrate the influence of the frequency of ground excitation and the various geometric and material parameters on the hydrodynamic pressure distribution and associated dynamic response of several example structures.

Hydrodynamic interactions in floating cylinder arrays—II. Wave radiation

An approximate method is utilized to investigate the hydrodynamic interactions between the members of an array of floating circular cylinders which occur when one member undergoes prescribed forced oscillations. The method of solution for the fluid velocity potential involves replacing the radiated/scattered waves emanating from one cylinder in the vicinity of another cylinder by equivalent plane waves together with non-planar correction terms and is essentially a large-spacing approximation. It is possible through this approach to determine the hydrodynamic interactions between the various array members given only the radiation and scattering characteristics of an isolated cylinder. Numerical results are presented for example array configurations consisting of 2–6 cylinders, calculations of the added-mass and added (radiation) damping coefficients have been performed for a range of wave and structural parameters. It is found that for certain parameter combinations the influence of neighboring bodies on the total wave field leads to values of the above quantities on individual array members which differ significantly from those for an isolated cylinder. The present analysis is relevant to the design of multibody wave energy extraction systems and floating offshore platforms.

Hydrodynamic interactions in floating cylinder arrays—I. Wave scattering

The hydrodynamic interactions due to wave scattering between the numbers of an array of stationary, truncated circular cylinders simulating the columns of an idealized tension-leg platform (TLP) are investigated. The method of solution for the fluid velocity potential involves replacing scattered waves by equivalent plane waves together with non-planar correction terms. This technique is, therefore, essentially a large spacing approximation. Use of this approach makes it possible to determine the hydrodynamic interactions between the array members utilizing only the diffraction characteristics of an isolated cylinder. Numerical results are presented for six array configurations consisting of 2–6 cylinders representing the legs of idealized TLPs. Calculations of the wave loads on these cylinders have been performed for a range of wave and structural parameters. It is found that, for certain parameter combinations, the influence of neighbouring bodies on the total wave field leads to hydrodynamic loading on individual columns which is significantly greater than the loading they would experience in isolation. The presented results demonstrate the significance of hydrodynamic interactions between TLP columns and clearly indicate that these effects should be considered by the designers and researchers associated with TLPs.

Three-dimensional wave scattering by elliptical breakwaters

The diffraction of small-amplitude surface waves by floating and submerged stationary elliptical breakwaters in water of arbitrary uniform depth is investigated analytically. The theoretical formulation leads to solutions for the fluid velocity potential in terms of series of Mathieu and modified Mathieu functions of real argument. Numerical results are presented for the wave-induced forces and moments on the structures, the total and differential scattering cross-sections and the variation of water surface elevation around the breakwater perimeter for a range of wave and structural parameters.

Hydrodynamic Interactions Between Submerged Cylinders

An approximate method is presented to investigate the hydrodynamic interactions between a pair of free‐standing, bottom‐mounted, flexible circular cylinders subjected to high‐frequency, horizontal ground motion. The cylinders are aligned parallel to the direction of ground excitation and the response of each is assumed to be one‐dimensional and governed by a beam equation. The solution technique for the fluid velocity potential uses the modified plane‐wave method and involves replacing divergent radiated and scat‐ tered waves by equivalent plane waves together with nonplanar correction terms. Numerical results are presented, which illustrate the influence of the various geometrical and material properties of the cylinders on the hydrodynamic loading and associated dynamic response for several pairs of example structures. It is found that the inclusion of fluid compressibility in the formulation leads to predictions of quite significant interactions between squatty cylinders at high frequencies, even at a ...

Interference effects between flexible cylinders in waves

An approximate method is presented to estimate the hydrodynamic loading and associated structural response of each of a pair of free-standing, bottom-nounted, flexible circular cylinders subjected to a regular train of linear surface waves. The cylinders are aligned parallel to the incident wave direction and the response of each is assumed to be one-dimensional and governed by a beam equation. The solution technique for the fluid velocity potential involves replacing scattered waves by equivalent plane waves together with non-planar, first-correction terms, and can be shown to be a large-spacing approximation. Numerical results are presented which show the influence of the various wave and structural parameters on the hydrodynamic loading and dynamic response of the individual cylinders.

Influence of structural flexibility and wave interference on dynamic behavior of idealized offshore platform

An approximate method is presented to estimate the hydrodynamic loading and structural response of an idealized offshore platform subjected to a regular train of linear surface waves. The platform is taken to consist of four bottom-mounted, flexible, circular cylinders supporting a rigid deck and is assumed to be aligned parallel to the incident wave direction. The response of each column is assumed to be one-dimensional and to be governed by linear beam theory. The solution technique for the fluid velocity potential involves replacing scattered waves by equivalent plane waves together with non-planar, first-correction terms, and can be shown to be a large spacing approximation. Numerical results are presented which show the effect of hydrodynamic interference and structural flexibility on the platform response.

Wave Forces on an Elliptic Cylinder

Two approximate methods are presented for the calculation of the wave induced forces and moments on a vertical, surface‐piercing cylinder of elliptic cross section. Both methods provide a substantial reduction in computational effort when Compared with the exact solution which involves the numerical evaluation of Mathieu functions. One method involves the expansion of the exact expressions for the forces and moments for small values of the elliptic eccentricity parameter. The second method is based on Green's theorem and gives rise to an integral equation for the fluid velocity potential on the cylinder surface. Numerical results are presented for a range of relevant parameters and show excellent agreement with the computed values of the exact solution.

Analysis of fluid-structure interactions. a direct symmetric coupled formulation based on the fluid velocity potential

We present a symmetric finite element method for solving fluid-structure interaction problems. The formulation uses velocity potentials and a hydrostatic pressure as unknowns in each fluid region, and displacements as unknowns in the solid. The hydrostatic pressure is an unknown variable at only one node per fluid region. A C matrix (multiplied by time derivatives of the nodal variables, but not a damping matrix) enforces the coupling between the variables. The resulting matrix equations are banded and symmetric, making them easy to incorporate in standard displacement-based finite element codes. Several test cases indicate that this approach works well for static, transient, and frequency analyses.

Wave diffraction by elliptical breakwaters in shallow water

The scattering of waves by both floating and submerged stationary elliptical breakwaters is investigated by means of linearised shallow water wave theory. This formulation leads to solutions for the fluid velocity potential in terms of Mathieu functions of real argument. Expressions are derived for the wave-induced forces and moments on the structures and their total and differential scattering cross-sections. Numerical results are presented for a range of wave and structural parameters. The present analysis serves as a prelude to a more comprehensive study of the problem without the shallow water restriction.

Inhomogeneous mixmaster universes: Some exact solutions

Algorithms for generating new exact solutions of the Einstein-Klein-Gordon field equations, which describe inhomogeneous universes with S 3 topology of spatial sections, are developed. The known exact vacuum and stiff-fluid solutions with S 3 topology are used as an input. The methods developed are further applied to derive inhomogenous generalizations of Bianchi type IX solutions and inhomogeneous S 3 Gowdy models with gravitational and scalar waves. It is shown that the new solutions, which are generalizations of the Bianchi type IX models, permit identification of the scalar field with the velocity potential of the stiff irrotational fluid. The latter result is further used to study the growth rate of density perturbations of the isotropic and anisotropic Bianchi type IX universes in a fully nonlinear relativistic regime. The role of anisotropy of the rate of growth of density perturbations is studied in detail.

Dynamic fluid effects in liquid‐filled flexible cylindrical tanks

AbstractThe velocity potential of a compressible fluid is found by Galerkin's method. Free surface displacements and a flexible tank wall are assumed. Explicit expressions for the impulsive mass, the impulsive moment and the overturning moment are derived for wave number m = 1. In the case of m ≥ 1 the dynamic effect of the fluid is represented by a fictitious apparent mass in explicit form.

Notes on the Linearised Equation for the Velocity Potential of the Steady Supersonic flow of a Compressible Fluid

This paper deals with solutions of the “linearised” differential equation of the velocity potential of a compressible fluid in steady supersonic motion. The linearised equation is identical in form with that of two-dimensional subsonic wave motion. Known solutions of the two-dimensional wave equation can be interpreted in terms of supersonic flow. Some of these, which have been used by various investigators, are brought together and are shown to be special cases of the general solution. Two new types of solution, which are of special interest and are appropriate to special systems of coordinates, are developed.

The Refractive Index in Electron Optics

The application of methods of geometrical optics for the investigation of electron trajectories requires the knowledge of the refractive index $\ensuremath{\mu}$ of the equivalent optical problem. A general formula for $\ensuremath{\mu}$ in terms of the electrostatic potential $V$ and of the magnetic vector potential A is known. If $\mathbf{A}\ensuremath{\ne}0$ this formula contains a unit vector s parallel to the electron velocity, which is not known in general. It is therefore important that s be eliminated. This can be done only if a suitable integral of the equations of motion of the electron is known. If $V$ and A are symmetric about an axis, s may be eliminated from $\ensuremath{\mu}$ by means of a momentum integral (Glaser). This elimination is generalized in the present paper to include cylindrical fields (not necessarily symmetrical) as well as some more general fields with a transversal field component. The result is reached by interpreting the magnetic scalar potential $U$ as a velocity potential of an ideal fluid and by making use of the corresponding stream function $w$ to obtain an expression for $\ensuremath{\mu}$.