On the motion of a weakly bouyant heat source near an interface

Summary An asymptotic solution to the problem of a slightly bouyant flow, induced by the motion of a submerged point heat source which is governed by the Oseen-Boussinesq approximation, is presented. Two cases are discussed in particular, one which involves a motion of the heat source beneath a free surface and the other near a rigid boundary. The thermal boundary conditions on these two interfaces are assumed to be that of the mixed Cauchy type. Closed-form expressions are obtained for the temperature field as well as for the velocity and the pressure distributions induced in the fluid. The general solution thus obtained is illustrated by calculating both the thermal and kinematic signatures on the free surface for some particular cases.     

Axisymmetric waves in electrohydrodynamic flows

The formation of nonlinear axisymmetric waves on inviscid irrotational liquid jets in the presence of radial electric fields is considered. Gravity is neglected but surface tension is considered. Electrohydrodynamic waves of arbitrary amplitude and wavelength are computed using finite-difference methods. Particular attention is paid to nonlinear traveling waves. In the first class of problems, an electric field generated by placing the liquid jet inside a hollow cylindrical electrode held at constant voltage, its axis coinciding with that of the jet, is studied. The jet is assumed to be a perfect conductor whose free surface is stressed by the electric field acting in the hydrodynamically passive annulus. In the second class of problems, the annular gas is a perfect conductor that transmits a constant voltage onto the liquid/gas surface. The liquid axisymmetrically wets a constant-radius cylindrical rod electrode placed coaxially with respect to the hollow outer electrode, and held at a different constant voltage. The fluid dynamics and electrostatics need to be addressed simultaneously in the inner region. Axisymmetric interfacial waves influenced by surface tension and electrical stresses are computed in both cases. The computations are capable of following highly nonlinear solutions and predict, for certain parameter values, the onset of interface pinching accompanied with the formation of toroidal bubbles. For given wave amplitudes, the results suggest that, for the former case, the electric field delays bubble formation and reduces wave steepness, while for the latter case the electric field promotes bubble formation, all other parameters being equal.     

Low Reynolds number flow due to impulsively started circular cylinder

Summary The authors obtain solutions for the title problem when the cylinder is imparted a constant velocity or a constant acceleration. It is assumed that initially both the solid obstacle and the infinite expanse of liquid surrounding it are at rest. The velocity and acceleration imparted to the cylinder are of finite magnitude, rectilinear and in a direction perpendicular to the cylinder axis. Both solutions are expressed by means of three distinct matched expansions. These analyses are valid as long as the Reynolds number, Re, is small.Early in the processes under discussion and throughout the exterior of the cylinder, both flows are unsteady Stokesian to within a small error. Later they are represented by inner and outer expansions. The solution structure and the nature of the expansions suggest that close to the cylinder the effects of viscosity and transiency dominate the flow field. It is only in the outer fields, and some time after motion has commenced that the effect of vorticity-convection plays a significant role. For the case in which a steady rectilinear velocity is imparted to the cylinder, both the inner and outer expansions derived here approach those which were obtained previously by Proudman and Pearson for a steady flow pattern. From the leading terms in the two expansions representing the flows close to the cylinder, approximate expressions for the time-dependent drag are obtained.     

On the wake in the low-Reynolds-number flow behind an impulsively started circular cylinder

Summary This analysis completes the authors' singular perturbations type of solution of the title problem, when the cylinder acquires instantaneously a uniform velocity. As reported the solution consists of three expansions which represent the flow in three different space-time subdomains. It is shown here that there exists a fourth subdomain. An appropriate additional expansion is developed and matched with the other three. This latter expansion represents the flow late in the process in a wake region. This wake extends all the way downstream to infinity. Its width is comparable to the diameter of the obstacle.     

Irrotational axisymmetric flow about a prolate spheroid in cylindrical duct

Summary A solution to the problem of potential flow about a prolate spheroid placed axially symmetric in a circular duct has been derived. The solution is in the form of a distribution of vortex rings over the surface of the spheroid. The vortex strength is expressed in terms of an infinite series of Legendre polynomials and the analysis yields an infinite set of equations for determining the coefficients of this series. An expression for the velocity distribution on the surface of the spheroid as well as the longitudinal added mass coefficients of the spheroid are derived in terms of the coefficients of the Neumann series expansion of the vortex sheet strength. Numerical results are presented for various spheroids and different blockages. Also given is a comparison between the present method and few available approximate methods.     

Detection of the electromagnetic field induced by the wake of a ship moving in a moderate sea state of finite depth

The wake of a ship and ambient sea waves induce disturbances in the Earth??s geomagnetic field. Numerical simulations are used to examine the feasibility of detecting these disturbances for moderate sea states. It is assumed that the electromagnetic disturbances are sampled by an air-borne magnetometer moving steadily along a rectilinear path. Spectral analysis of the samples is performed. Numerical simulations indicate that the magnetic field induced by the noisy sea environment and that by the wake of a moving ship have different spectral characteristics. Typically, the peak of the body-induced magnetic-field spectrum is located in the range of frequencies where the corresponding value of the wind wave??s spectrum is less significant. It is illustrated that the feasibility of electromagnetic detection of a ship??s wake depends strongly on the ship??s depth-based Froude number.     

Hydrodynamics of deformable contiguous spherical shapes in an incompressible inviscid fluid

Summary An exact solution to the two-body interaction problem is presented for the case of spherical shapes moving in an incompressible and inviscid fluid. The spheres are assumed to translate in an arbitrary manner and to undergo radial deformation (or pulsation). The problem is formulated in terms of spherical harmonics and the force experienced by the spheres is obtained by employing the Lagally theorem. The expressions for the force are given as an infinite sum of coefficients which are found by solving an infinite set of linear equations. Three main geometries are considered, namely, two spheres exterior to each other, one sphere in the interior of the other and sphere in a rectangular channel. Numerical values for the added-mass coefficients as well as for the hydrodynamic forces are found for the case of rigid sphere moving toward or parallel to a rigid wall or a free surface, and a pulsating sphere in the proximity of these boundaries. Also given are numerical values for the transverse and the longitudinal addedmass coefficients for a sphere moving in a rectangular channel for different channel-blockage ratios.     

On the travelling wave instability caused by moving planar heat sources: the rigid problem

This analysis deals with the convective travelling wave instability appearing in a fluid medium at rest and contained between two horizontal rigid plates, subjected to the same sinusoidal temperature distribution, moving at a uniform speed and in the same direction. The temperature distribution is caused by travelling planar heat sources with a time harmonic output. A three-dimensional coordinate system is used and the small parameter in this problem represents the ratio between the buoyancy and inertial forces. For a finite, yet small , asymptotic expansions are assumed for the velocity, pressure, temperature and the Reynolds number. The mean motion generated by the Reynolds stresses is calculated separately. By keeping the Prandtl number fixed and by using long length and time scales, successive linearized perturbation equations are considered. Two successive amplitude equations are analyzed and their solution yields the mathematical form of these travelling waves, their group velocity and the elevation above the critical Reynolds number.     

Fundamental problems in hydrodynamics of ellipsoidal forms

In this study, we tackle with one single effort, three fundamental problems in ship hydrodynamics, namely the attraction force exerted on a single ship moving rectilinearly close and parallel to a rigid wall(berthing), the determination of the associated added-mass coefficients(maneuvering) and finally evaluating the forces exerted on a ship in a two-ship crossing scenarios. The novelty of the present approach relies on using Weinblum's proposition to assimilate the ship's hull by an "equivalent" tri-axial ellipsoid, as well as the employment of ellipsoidal harmonics expansion techniques.     

Elimination of corners in the mapping of a closed curve

Summary A transformation, which maps the exterior or the interior of a simple closed curve with corners into the exterior or interior respectively of a simple smooth (corner-free) closed curve, is introduced. Symmetry properties are shown to be preserved by the transformation and a numerical procedure for applying the proposed transformation to an arbitrary curve is presented.Smooth curves, resulting from the application of the corner-eliminating transformation to a square and to a sixcornered double ship section are also given.