The method of matched asymptotic expansions is used to calculate the lift for a class of thin aerofoils with highincidence flaps or blunt trailing edges. The analysis is based on smallness of the ratio between the flap length (or trailingedge thickness) and the chord of the main foil. Results are compared with exact solutions for bent plates and for triangular and quadrilateral bodies. A procedure for extension to general trailing-edge configurations, including detached flaps, is outlined.

If an aerofoil of chord c has a parabolic nose with radius of curvature r, and is placed at angle-of-attack α to a stream, the laminar boundary layer on its upper surface remains unseparated for α<0.8l8. In the present paper we consider some smooth local modifications to the leading edge. Symmetric modifications of the nature of local sharpening of the nose can improve this result to at least α<0.897. Further improvements are possible for unsymmetrical (e.g. drooped) noses, and an example of a ‘drooped’ nose with α<0.912 is shown.

This is a review of thin-body and slender-body theories, with indications of some new applications. Topics discussed include bodies with near-constant surface pressure, subsonic and supersonic aerodynamics, ship hydrodynamics, slender bodies in Stokes flow, slender footings in elastic media, and slender moonpools. Mathematical features of the thin- and slender-body approximations are also discussed, especially nonlocal convolution terms modelling three-dimensionality in the otherwise two-dimensional near field, end effects, and the role of the logarithm of the slenderness ratio. This review was presented by the first author as the IMA Lighthill Memorial Lecture at the British Applied Mathematics Colloquium (BAMC) 2004.

This note describes a simple numerical method for solution of the lifting surface integral equation of aerodynamics, and provides benchmark computations of up to 7 figure accuracy for flat rectangular wings of arbitrary aspect ratio. The nature of the large aspect ratio limit is also investigated numerically and asymptotically. This enables determination of the limiting behaviour near the wing tips, which is compared to the predictions of lifting line theory. Generalisations to non-rectangular wings are discussed.

In the absence of surface tension, the problem of determining a travelling surface pressure distribution that displaces a portion of the free surface in a prescribed manner has been solved by several authors, and this “planing-surface” problem is reasonably well understood. The effect of inclusion of surface tension is to change, in a dramatic way, the singularity in the integral equation that describes the problem. It is now necessary in general to allow for isolated impulsive pressure, as well as a smooth distribution over the wetted length. Such pressure points generate jump discontinuities in free-surface slope. Numerical results are obtained here for a class of problems in which there is a single impulse located at the leading edge of the planing surface and detachment with continuous slope at the trailing edge. These results do not appear to approach the classical results in the limit as the surface tension approaches zero, a paradox that is resolved in Part II, which follows.

A steady two-dimensional free-surface flow in a channel of finite depth is considered. The channel ends abruptly with a barrier in the form of a vertical wall of finite height. Hence the stream, which is uniform far upstream, is forced to go upward and then falls under the effect of gravity. A configuration is examined where the rising stream splits into two jets, one falling backward and the other forward over the wall, in a fountain-like manner. The backward-going jet is assumed to be removed without disturbing the incident stream. This problem is solved numerically by an integral-equation method. Solutions are obtained for various values of a parameter measuring the fraction of the total incoming flux that goes into the forward jet. The limit where this fraction is one is also examined, the water then all passing over the wall, with a 120° corner stagnation point on the upper free surface.

A correspondence is established between flows of air above stationary water, and flows of water below air at atmospheric pressure. Flows in the latter category are well studied, and all such hydrodynamic flows can be “turned upside-down” to generate flows of air in which the free surface deforms under gravity, due to a balance between aerodynamic and hydrostatic pressures. Examples are given of some exact inverse solutions, and a general semi-inverse approach is outlined for numerical solutions via an integral formulation.

A direct numerical computation is provided for the impedance of a screen consisting of a regular array of slits in a plane wall. The problem is solved within the framework of oscillatory Stokes flow, and results presented as a function of porosity, frequency and viscosity.

A theory is provided for the natural seiching frequencies and radiative decay rates for a shallow-water basin whose connection to open water is restricted by a submerged wall or reef. The transition from an essentially-open basin to a closed basin, as the aperture reduces to zero, is discussed using a matching procedure. Graphs of frequencies and damping factors as functions of aperture size are obtained for idealized two-dimensional shelf configurations, involving a constant-depth shallow basin connected to constant-depth, but not necessarily shallow, open water.

The method of matched asymptotic expanisions is used to solve the problem of flow over a thin airfoil possessing a trailing-edge appendage, which may be of a general character, but is confined to a region of small size compated to the airfoil' chord. A feature of this asymptotic solution is effective de-coupling of the flow problems for the main airfoil and the flap. The special case of an attached flat flap set at an arbitrary angle is solved in detail.

In Part I of this series, surface tension was included in the classical two-dimensional planing-surface problem, and the usual smooth-detachment trailing-edge condition enforced. However, the results exhibited a paradox, in that the classical results were not approached in the limit as the surface tension approached zero. This paradox is resolved here by abandoning the smooth-detachment condition, that is, by allowing a jump discontinuity in slope between the planing surface and the free surface at the trailing edge. A unique solution is obtainable for any input planing surface at fixed wetted length if one allows such jumps at both leading and trailing edges. If, as is the case in practice, the wetted length is allowed to vary, uniqueness may be restored by requiring either, but not both, of these slope discontinuities to vanish. The results of Part I correspond to the seemingly more-natural choice of making the trailing-edge detachment continuous, but it appears that the correct choice is to require the leading-edge attachment to be continuous.

A new numerical method is applied to the problem of inviscid irrotational flow past a semi-infinite stern-like body of general shape. Both smooth-detachment and stagnant-detachment flows are considered, in the context of varying the geometry of the stern to generate very small waves, with the eventual aim of eliminating waves altogether. The results of this work confirm previously published results for the smooth-detachment case, but cast doubt on the existence of waveless solutions for stagnant detachment.

In two-dimensional bow-like flows past a semi-infinite body, one must in general expect a free-surface discontinuity, in the form of a splash or spray jet. However, there is numerical evidence that special body shapes do exist for which this splash is absent. In this study, we first establish conditions on the geometry of the bow in order that it should be splash-free at zero gravity, by solving the mathematical problem exactly. We then obtain solutions for finite non-zero gravity, by solving a non-linear integral equation numerically. A class of splashless body geometries with a downward directed segment at the extreme of the bow, to which the free surface attaches tangentially, is demonstrated in detail.

The acoustic response of a two-dimensional nearly-closed cavity to an excitation through a small opening is examined, using the method of matched asymptotic expansions. The Helmholtz mode of vibration is discussed using a low-frequency expansion of the velocity potential in the cavity interior. The variation in frequency and magnitude of the resonator response is explored, both for the Helmholtz and the natural-frequency modes.

John Henry Michell (1863–1940) published scientific papers only between 1890 and 1902, but included in his 23 papers from that short but productive period are some of the most important contributions ever made by an Australian mathematician. In this article I shall concentrate on the extraordinary 1898 paper “The wave resistance of a ship” Phil. Mag.(5) 45, 106–123. There are many reasons why this paper was an astounding achievement, but perhaps the most remarkable is that the resulting formula has not been improved upon to this day. In the computer age, many efforts have been made to do so, but with little success so far. The formula itself involves a triple integral of an integrand constructed from the offset data for the ship's hull, and even the task of evaluating this triple integral is not a trivial one on today's computers; another reason for admiration of Michell's own heroic hand-calculated numerical work in the 1890's. Lack of a routine algorithm for Michell's integral has inhibited its use by naval architects and ship hydrodynamic laboratories, and there has been a tendency for it to receive a bad press based on unfair comparisons, e.g. comparison of model experiments (themselves often suspect) with inaccurate computations or computations for the wrong hull, etc. The original integral is in fact quite reasonable as an engineering tool, and some new results confirming this are shown. Improvement beyond Michell is however needed in some important speed ranges, and indications are given of recent approaches that may be promising.

This paper demonstrates feasibility of aerodynamically-supported motion of a thin sheet near to a plane wall. Steady equilibrium is possible for uniform sheets only if they are deformable, and a set of possible equilibrium shapes is determined.

In two-dimensional flow past a body close to a free surface, the upwardly diverted portion may separate to form a splash. We model the nose of such a body by a semi-infinite obstacle of finite draft with a smoothly curved front face. This problem leads to a nonlinear integral equation with a side condition, a separation condition and an integral constraint requiring the far-upstream free surface to be asymptotically plane. The integral equation, called Villat's equation, connects a natural parametrisation of the curved front face with the parametrisation by the velocity potential near the body. The side condition fixes the position of the separation point, whereas the separation condition, known as the Brillouin-Villat condition, imposes a continuity relation to be satisfied at separation. For the described flow we derive the Brillouin-Villat condition in integral form and give a numerical solution to the problem using a polygonal approximation to the front face.

Optimal strategies are obtained for two-player games with an alternating staek doubling option. A complete two-parameter analysis is provided for games that must end within two moves, and a recursive procedure then enables a solution for games of any number of moves. Examples are given of relevance to extureme end games in backgammon.

A solution is found for a line source or sink beneath a free surface, at a unique squared Froude number of 12.622.

An inviscid fluid in steady two-dimensional motion in a region bounded by a closed streamline must have constant vorticity. We solve here for some such flows where the boundary is in part free, the fluid velocity magnitude being constant on the free boundary. A trivial example of such a flow is a circular cylinder of fluid rotating about its axis as if rigid, for which the whole circular boundary is free, irrespective of its radius. We now ask what happens to that flow when it comes into contact with solid boundaries. There is no steady flow when the contact is with a finite segment of a single plane wall, but a unique solution exists when the rotating fluid mass is in contact with some concave boundaries. Computed results are obtained for vortices lying inside a parabolic dish, or in a corner between two planes.