Velocity potential

About: Velocity potential is a research topic. Over the lifetime, 2427 publications have been published within this topic receiving 40493 citations.

The Deformation of Steep Surface Waves on Water. I. A Numerical Method of Computation

Abstract: Plunging breakers are beyond the reach of all known analytical approximations Previous numerical computations have succeeded only in integrating the equations of motion up to the instant when the surface becomes vertical In this paper we present a new method for following the time-history of space-periodic irrotational surface waves The only independent variables are the coordinates and velocity potential of marked particles at the free surface At each time-step an integral equation is solved for the new normal component of velocity The method is faster and more accurate than previous methods based on a two dimensional grid It has also the advantage that the marked particles become concentrated near regions of sharp curvature Viscosity and surface tension are both neglected The method is tested on a free, steady wave of finite amplitude, and is found to give excellent agreement with independent calculations based on Stokes’s series It is then applied to unsteady waves, produced by initially applying an asymmetric distribution of pressure to a symmetric, progres­sive wave The freely running wave then steepens and overturns It is demonstrated that the surface remains rounded till well after the over­turning takes place

On the Radiation of Sound from an Unflanged Circular Pipe

Abstract: A rigorous and explicit solution is obtained for the problem of sound radiation from an unflanged circular pipe, assuming axially symmetric excitation. The solution is valid throughout the wave-length range of dominant mode (plane wave) propagation in the pipe. The reflection coefficient for the velocity potential within the pipe and the power-gain function, embodying the characteristics of the radiation pattern, are evaluated numerically. The absorption cross section of the pipe for a plane wave incident from external space, and the gain function for this direction, are found to satisfy a reciprocity relation. In particular, the absorption cross section for normal incidence is just the area of the mouth. At low frequencies of vibration, the velocity potential within the pipe is the same as if the pipe were lengthened by a certain fraction of the radius and the open end behaved as a loop. The exact value of the end correction turns out to be 0.6133.

An exact theory of nonlinear waves on a Lagrangian-mean flow

Abstract: An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

The theory of ground-water motion

Abstract: The existing analytical treatments of ground-water flow have mostly been founded upon the erroneous conception, borrowed from the theory of the flow of the ideal frictionless fluids of classical hydrodynamics, that ground-water motion is derivable from a velocity potential. This conception is in conformity with the principle of the conservation of matter but not with that of the conservation of energy. In the present paper it is shown that a more exceptionless analytical theory results if a potential whose value at a given point is defined to be equal to the work required to transform a unit mass of fluid from an arbitrary standard state to the state at the point in question is employed. Denoting this function by $$\Phi$$, it is shown that the differential equation of fluid flow in an isotropic medium is given by $$q = - \sigma$$ grad $$\Phi$$, where q is the flow vector whose magnitude is equal to the volume of fluid crossing a unit of area normal to the flow direction in unit time, and $$\sigma$$ a spec...