Lawrence K. Forbes

Free-surface flow over a semicircular obstruction

The two-dimensional steady flow of a fluid over a semicircular obstacle on the bottom of a stream is discussed. A linearized theory is presented, along with a numerical method for the solution of the fully nonlinear problem. The nonlinear free-surface profile is obtained after solution of an integrodifferential equation coupled with the dynamic free-surface condition. The wave resistance of the semicircle is calculated from knowledge of the solution at the free surface.

Critical free-surface flow over a semi-circular obstruction

Numerical solutions are presented for the problem of two-dimensional “critical” flow of an ideal fluid over a semi-circular obstacle attached to the bottom of a running stream. The upstream Froude number and downstream flow speed are known in advance, and are therefore computed as part of the solution. The dependence of flow behaviour on obstacle size is discussed.

Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution

Two-dimensional periodic waves beneath an elastic sheet resting on the surface of an infinitely deep fluid are investigated using a high-order series-expansion technique. The solution is found to have certain features in common with capillary-gravity waves; specifically, there is a countably infinite set of values of the flexural rigidity of the sheet at which the series solution fails, and these values are conjectured to be bifurcation points of the solution. Limiting waves of maximum height are found at each value of the flexural rigidity investigated. These are characterized by a cusp singularity in the elastic bending moment at the wave crest, and infinite fluid pressure there. For at least one value of the flexural rigidity, the series solution shows that the wave of maximum height also travels with infinite speed.

Rapid computation of static fields produced by thick circular solenoids

A straightforward method is proposed for computing the magnetic field produced by a circular coil that contains a large number of turns wound onto a solenoid of rectangular cross section. The coil is thus approximated by a circular ring containing a continuous constant current density, which is very close to the real situation when wire of rectangular cross section is used. All that is required is to evaluate two functions, which are defined as integrals of periodic quantities; this is done accurately and efficiently using trapezoidal-rule quadrature. The solution can be obtained so rapidly that this procedure is ideally suited for use in stochastic optimization. An example is given, in which this approach is combined with a simulated annealing routine to optimize shielded profile coils for NMR.

Surface waves of large amplitude beneath an elastic sheet. Part 2. Galerkin solution

This study continues the work of Forbes (1986) on periodic waves beneath an elastic sheet floating on the surface of an infinitely deep fluid. The solution is sought as a Fourier series with coefficients that are computed numerically. Waves of extremely large amplitude are found to exist, and results are presented for waves belonging to several different nonlinear solution branches, characterized by different numbers of inflexion points in the wave profiles. The existence of multiple solutions, conjectured in the previous paper (Forbes 1986), is confirmed here by direct numerical computation.

FLOW CAUSED BY A POINT SINK IN A FLUID HAVING A FREE SURFACE

The flow caused by a point sink immersed in an otherwise stationary fluid is investigated. Low Froude number solutions are sought, in which the flow is radially symmetric and possesses a stagnation point at the surface, directly above the sink. A small-Froude-number expansion is derived and compared with the results of a numerical solution to the fully nonlinear problem. It is found that solutions of this type exist for all Froude numbers less than some maximum value, at which a secondary circular stagnation line is formed at the surface. The nonlinear solutions are reasonably well predicted by the small-Froude-number expansion, except for Froude numbers close to this maximum.

Calculating current densities and fields produced by shielded magnetic resonance imaging probes

A method is presented for computing the fields produced by radio frequency probes of the type used in magnetic resonance imaging. The effects of surrounding the probe with a shielding coil, intende...

A note on the flow induced by a line sink beneath a free surface

The problem of withdrawing water through a line sink from a region containing an homogeneous fluid beneath a free surface is considered. Assuming steady, irrotational flow of an ideal fluid, solutions with low Froude number containing a stagnation point on the free surface above the sink are sought using a series substitution method. The solutions are shown to exist for a value of the Froude number up to a critical value of about 1.4. No solutions of this type are found for Froude numbers greater than this value.

Novel target-field method for designing shielded biplanar shim and gradient coils

A new method is presented here for the systematic design of biplanar shielded shim and gradient coils, for use in magnetic resonance imaging (MRI) and other applications. The desired target field interior to the coil is specified in advance, and a winding pattern is then designed to produce a field that matches the target as closely as possible. Both gradient and shim coils can be designed by this approach, and the target region can be located asymmetrically within the coil. The interior target field may be matched at two or more interior locations, to improve accuracy. When shields are present, the winding patterns are designed so that the fields exterior to the biplanar coil are made as small as possible. The method is illustrated here by the design of some transverse gradient and shim coils.

On the effects of non-linearity in free-surface flow about a submerged point vortex

SummaryTwo-dimensional free-surface flow about a point vortex in a stream of infinite depth is investigated. The non-linear problem is formulated in terms of an integrodifferential equation on the exact, unknown location of the free surface, and this equation is then solved numerically. The non-linear results are compared with the predictions of linearized theory and, for positive circulation, it is found that the latter may under-estimate the drag force significantly. For negative circulation, the linearized theory grossly over-predicts the value of the wave resistance, which apparently even becomes zero in a limiting configuration.