Edge waves of frequency ω and longshore wavenumber k in water of depth h(y) = h1H(σy/h1), 0 [les ] y < ∞, are calculated through an asymptotic expansion in σ/kh1 on the assumptions that σ [Lt ] 1 and kh1 = O(1). Approximations to the free-surface displacement in an inner domain that includes the singular point at h = 0 and the turning point near gh ≈ ω2/K2 and to the eigenvalue λ ≡ ω2/σgh are obtained for the complete set of modes on the assumption that h(y) is analytic. A uniformly valid approximation for the free-surface displacement and a variational approximation to Λ are obtained for the dominant mode. The results are compared with the shallow-water approximations of Ball (1967) for a slope that decays exponentially from σ to 0 as h increases from 0 to h1 and of Minzoni (1976) for a uniform slope that joins h = 0 to a flat bottom at h = h1 and with the geometrical-optics approximation of Shen, Meyer & Keller (1968).

A first-order channel model for fluid motion in long homogeneous lakes, as derived in detail by Raggio & Hutter (1982a), is presented. This model describes the motion through spatially one-dimensional boundary-value problems and is deduced by representing each field variable by cross-sectional expansions with a constant and a linear term. Various wave solutions of the governing equations applied to rectangular basins with flat bottom are presented. It is demonstrated that for moderate rotation speeds of the Earth and for elongated basins of a homogeneous fluid the main features of gravitational oscillations are predicted by the model. In particular Kelvin- and Poincaré-type waves are shown to exist. Furthermore, conditions of complete and incomplete reflections of Kelvin waves and free oscillations are discussed. The results corroborate the suitability of the model as far as wave motion in rectangular basins is concerned, but equally elucidate the physics behind them, which is less transparent when attacked with the full theory. The application of the model to basins of different shapes and to a real lake is reserved to a companion paper.

An initial state consisting of sugar solution lying above a denser salt solution in a Hele-Shaw cell is unstable to disturbances that evolve into long, slender fingers. An analysis of the structure of fully evolved (infinitely long) fingers that are independent of the vertical coordinate concludes that fingers with a width of the order of the buoyancy-layer thickness have maximum growth rate. Since effective gravity can be altered by inclining the Hele-Shaw cell toward the horizontal, fingers of different preferred widths can be established. An abrupt change of the angle of inclination changes the preferred width. A stability analysis of the resulting initial-value problem shows that perturbations with a vertical scale of the order of the buoyancy-layer thickness grow, and fluid from each finger penetrates laterally into the two adjacent fingers. The unstable modes resemble those observed experimentally by Taylor & Veronis (1986). It turns out that all vertically uniform fingers, even ones with the preferred width of the basic state, are unstable to a non-oscillatory peturbation that changes straight fingers to ones that have a vertically wavy structure. In all cases the vertical scale of the most unstable disturbance is of the order of the buoyancy-layer thickness. Also included is a discussion of the need for a model describing the transient evolution of fingers and particularly one that contains an analysis of the role of the transition region between the salt-finger zone and the reservoirs above and below.

Dynamically passive transfer of heat across a layer of liquid supporting a progressive, periodic surface wave is approached from a Lagrangian viewpoint. The model layer considered is the region between two constant pressure surfaces of a Gerstner wave and the thermal boundary conditions are that the average temperature of any surface particle remains constant and that there is horizontal homogeneity of the average temperature field.

It is shown that fluctuations in the temperature of any particle are negligibly small for ordinary liquids and a uniformly valid approximation to the average temperature of each particle is presented.

The extent to which the flux of heat through the layer is augmented is computed for typical cases and it is shown to be at most doubled. Indication is given of extensions of the method to other kinds of progressive waves and to situations in which the boundary conditions are unsteady and spatially inhomogeneous.

The paper reports on an investigation into the nonlinear evolution of a wavetrain emanating from a point source in a flat-plate boundary layer. The work was mainly experimental, but calculations using linear and nonlinear stability theory and parabolized stability equations were also performed to support the conclusions. The amplitudes of the disturbances used were very small and, as a consequence, turbulence was not reached within the experimental domain. Nevertheless, interesting nonlinear behaviour was observed. The nonlinear regime was characterized by the appearance of a three-dimensional mean flow distortion in the form of longitudinal streaks. It was possible to distinguish two different stages of the nonlinear regime. In the first stage, the streak pattern displayed relatively low spanwise wavenumbers. The pattern appeared to have grown algebraically from a pair of counter-rotating streamwise vortices. Weakly nonlinear calculations suggested that the vortices arise from the interaction of the spanwise and the wall-normal velocity components. The second stage exhibited more streaks and higher spanwise wavenumbers. This stage was found to be associated with a peak-and-valley structure of the wave amplitude. A remarkable feature was that the streamwise position of the onset of the second stage of the nonlinear regime was not affected by the amplitude of the disturbance. Parabolized stability calculations suggested that the peak-and-valley structure was the outcome of a secondary instability of the fundamental type. An interaction similar to that of the first nonlinear stage involving the waves composing the peak-and-valley structure yielded other streamwise vortices and hence a different streak pattern. The results also suggested that, because the wave amplitudes were very small, the linear stability influenced the nonlinear evolution. Indeed, the onset of the second stage of the nonlinear regime appeared to be associated with the proximity of the second branch of the linear stability diagram.

A radial temperature distribution is applied to the top of a cylinder of rotating stably stratified fluid. Thermal wind shear drives the interior flow. Linearized theory predicts, and laboratory experiments confirm, that when the stratification is large enough it completely suppresses the Ekman pumping into the interior. The interior velocity field, which is primarily azimuthal, responds by satisfying the no-slip boundary conditions without the need of Ekman layers on the horizontal surfaces. Moreover, for large stratification a thermal boundary layer beneath the top surface traps the thermal disturbance applied at the upper surface. The greatest azimuthal velocity occurs at the base of this layer. Below this layer the azimuthal velocity viscously diffuses downward with thermal wind adjusting the temperature. The Rossby radius of deformation based on this layer depth is the cylinder's radius divided by the square root of the Prandtl number. Detailed measurements of the velocity field generated in the cylinder by the heating are compared with the theory in the case where the Ekman layers are eliminated by stratification. The theory and experiments agree qualitatively well over a range of four orders of magnitude of imposed parameters and over a large parameter range the quantitative comparison is also very good.

A global shock solution for the nonlinear shallow water equations (NSWEs) is found by assigning proper seaward boundary data that preserve a constant incoming Riemann invariant during the shock wave evolution. The correct shock relations, entropy conditions and asymptotic behaviour near the shoreline are provided along with an in-depth analysis of the main quantities along and behind the bore. The theoretical analysis is then applied to the specific case in which the water at the front of the shock wave is still. A comparison with the Shen & Meyer (J. Fluid Mech., vol. 16, 1963, p. 113) solution reveals that such a solution can be regarded as a specific case of the more general solution proposed here. The results obtained can be regarded as a useful benchmark for numerical solvers based on the NSWEs.

Dusty-gas flow in laminar boundary layer over a body with a curved surface is considered. In addition to Stokes drag, particles experience a centrifugal force and lift which is due to fluid shear. The body size L is taken to be much greater than the relaxation length of the particle velocity due to the action of Stokes drag, & Lambda;st and is of the same order as or less than the relaxation length due to the action of lift. Λsa. Using an asymptotic approach, momentum equations for the particle phase are reduced to an algebraic equation accounting for the variation of lift coefficient with the shear and the slip velocity. Particle velocity and density are computed for the axisymmetric boundary layer in the neighbourhood of the front stagnation point of a blunt body of size much less than Λsa. It is shown that downstream of some point on the wall (the separation point) particle normal velocity becomes non-zero. As a result particle streamlines turn away from the wall, and a particle-free zone arises. The cause of separation is the lift effect; the centrifugal force cannot make the particle flow separate. This conclusion is extended to the case when L ∼ Λsa. The position of separation for the flow past a sphere is evaluated as a function of the ratio of its radius r′ and relaxation length. Dust flow ceases to separate when this value is greater than a critical value r′c /Λsa ≈ 29.2.

In an extension of an earlier potential-flow theory using conformal mapping and source singularities in the wake region to simulate two-dimensional bluff-body flow, new mapping sequences and additional boundary conditions are presented for the application of the method to lifting airfoils fitted with upper-surface spoilers or lower-surface split flaps of arbitrary size, location, and erection angle. The only empirical input is the base pressure coefficient. Calculations are presented of pressure distribution and lift for several cases of a Joukowsky airfoil fitted with a spoiler or a split flap, and these results are compared with experimental data from wind tunnel tests. Good agreement is found.

In the first part of this paper we introduce a path-integral formalism for the internal-wave field of the ocean. The intent is to show that this type of formalism may be useful in suggesting improvements to current calculations, as it provides a framework for applying a wide variety of approximations that have been and are currently being developed in other areas of physics. We demonstrate the method by deriving equations for a self-consistent field approach (also known as the direct-interaction approximation). The experience in other areas of physics is that the self-consistent field approximation is more reliable than lowest-order perturbation theory. The end result of the DIA is the determination of an effective linear model for the description of internal waves in the deep-ocean environment. In the second part of the paper we obtain Hasselmann's source function by a prescribed limiting process and are able to indicate possible improvements in related calculations by comparing the limiting assumptions with numerically computed values.

An exact solution of the free convective flow of a viscous incompressible fluid from a heated disk, rotating in a vertical plane, is obtained. The non-axisymmetric fluid motion consists of two parts; the primary von Kármán axisymmetric flow and the secondary buoyancy-induced cross-flow. A highly accurate solution of the energy equation is also derived for its subsequent use in the analysis of the cross-flow.

The paper is devoted to the theoretical and experimental investigation of the ring (toroidal) shock wave near the axis of symmetry. The theoretical approach is based upon the Chester–Chisnell–Whitman method. The experimental toroidal shock wave is generated by a novel inducer and visualized by the shadow technique. Attention is paid to the manner of reflection of the shock wave from the axis of symmetry. This reflection appears to be irregular even at small distances from the centre of the ring. This phenomenon is due to the cumulative acceleration of the converging axisymmetric shock front near the axis. The acceleration results in an increase in the incidence angle up to that characteristic of Mach (irregular) reflection.

G. I. Taylor was a happy man who spent a long life doing what he wanted most to do and doing it supremely well. He was a natural scientist whose character and activities were perfectly matched, and that allowed the fullest use of his creative talents. This short article is intended to help the understanding of G. I. Taylor's approach to his research by providing some information about the man and his life, in particular the major events in his early adult years which determined the direction of much of his subsequent research. A list of the now rather numerous articles about the man and his family and his work is appended.

The development of a turbulent boundary layer in a strong adverse pressure gradient can be described by the two-layer model proposed by Stratford (1959), in which the outer part of the flow is assumed to be unmodified by the pressure-rise and the inner part described by two local parameters, the surface stress and the pressure gradient. The description suggests that the modification of the original flow is in some sense self-preserving, and it is shown here that self-preserving development of the modification is consistent with the Reynolds equations of turbulent flow in particular pressure distributions. For these distributions, the predictions of the two-layer model are confirmed without any need to make the sharp and arbitrary distinction between the two parts of the boundary layer.

Viscous, incompressible, axially symmetric flow about an impulsively started rotating sphere is studied in terms of non-steady, partially linearized Navier-Stokes equations. The non-linear centripetal acceleration is included in full, but the other non-linear terms are neglected because of the restriction in interest to the case of large (but subcritical) Reynolds or Taylor numbers, a2Ω/v. Approximate closed-form solutions for u(r,θ,t), v(r,θ,t), w(r,θ,t) are found which satisfy all relevant boundary and initial conditions. The linearization approximation is checked for consistency and a restriction on Ωt is found. The velocity profiles, in the range of validity, are shown to be approximately similar in time, so their shapes may be qualitatively correct for larger values of Ωt. Some comparison with existing steady-state theories is given and the boundary-layer displacement thickness and viscous torque on the sphere are calculated.

This paper addresses the slender laminar flow resulting from the discharge of a low-Mach-number hot gas jet of radius $a$ and moderately large Reynolds number $R_j$ into a cold atmosphere of the same gas. We give the boundary-layer solution for plane and round jets with very small values of the ambient-to-jet temperature ratio $\varepsilon$ accounting for the temperature dependence of the viscosity and conductivity typical of real gases. It is seen that the leading-order description of the jet in the limit $\varepsilon \rightarrow 0$ exhibits a front-like structure, including a precisely defined separating boundary at which heat conduction and viscous shear stresses vanish in the first approximation, so that the temperature and axial velocity remain unperturbed outside the jet. Separate analyses are given for the jet discharging into a stagnant atmosphere, when the jet boundary is a conductive front, and for the jet discharging into a coflowing stream, when the jet boundary appears as a contact surface. We provide in particular the numerical description of the jet development region corresponding to axial distances of order $R_j a$ for buoyant and non-buoyant jets, as well as the self-similar solutions that emerge both in the near field and in the far field. In all cases considered, comparisons with numerical integrations of the boundary-layer problem for moderately small values of $\varepsilon$ indicate that these front descriptions give excellent predictions for the temperature and velocity fields in the near-axis region.

An investigation based on the elongated-body theory of Lighthill (1960, 1970) and Wu (1971) is made of the carangiform swimming motions of fish, a mode of propulsion in which the amplitude of body undulation becomes significant only in the posterior half, or even third, of the length of the fish and the anterior part is relatively inflexible. For typical slender fish performing undulatory swimming motions, three hydro-dynamic features are taken into account: (a) the resistance to the lateral undulatory motions as well as longitudinal (tangential) frictional resistance, (b) the forces of interaction with the water associated with its inertial (virtual-mass) response to the lateral motions of the fish and (c) the reaction forces due to the vortex sheets shed from sharp trailing edges to the rear of the section of maximum span (depth), where a great variety of fins are generally found.

The active tail oscillations give rise to oscillatory side forces, to which the remainder of the body responds passively. These passive yawing motions are studied to find their amplitude, the yawing axis and any associated energy dissipation. The contribution of each of the above three forces is examined and the effects of the oscillation frequency, a slenderness parameter δ of the body, and the shape of the transverse cross-sections are considered.

The present theory is further applied to predict the turning movement when the fish changes direction, the way in which the above forces act during this process being investigated. A small perturbation analysis relative to a uniform rectilinear motion not only reveals whether or not the motion is dynamically stable, but also leads to a full description of the motion that follows a given initial perturbation. For finite perturbations a numerical method is adopted to find the time development of both the direction of motion relative to the direction of the initial uniform motion and the ratio of the kinetic energy of the body to the initial kinetic energy for different magnitudes I0 of the initial impulsive perturbation in the transverse direction. The final angle of turn is obtained in terms of I0 for different values of δ. The loss of kinetic energy during the turn divided by the initial kinetic energy is found to be very small for a small angle of turn.

Detailed analyses are made for rigid fish models in which the distribution of depth (or span) along the body length is quadratic with a maximum at the centre. Agreement of the present analyses with observations is fairly good at least qualitatively and some quantitative estimates are made of the side forces, both oscillatory and impulsive, exerted on the body by the tail. It is found that the vortex sheets shed from trailing edges increase the energy loss due to the yawing oscillations, in addition to the resistive dissipation, but contribute to the directional stability of rectilinear motion.

The linear stability of rapid granular flow down a slope under gravity to the longitudinal perturbations is analysed using hydrodynamic equations. It is demonstrated that the steady flow uniform along the flow direction becomes unstable to long-wavelength perturbations longitudinal to the flow direction for certain parameter ranges to form a density wave, in contrast to the finite-wavelength instability for transverse perturbations (Forterre & Pouliquen 2002). It is shown that the instability can be understood as the long-wave instability of kinematic waves in a quasi-one-dimensional system. The results are compared with our previous molecular dynamics simulations (Mitarai & Nakanishi 2001), where spontaneous density wave formation was found.

Diverse subsonic initial-boundary-value problems (flows in a closed volume initiated by blowing or suction through permeable walls, flows with continuously distributed sources, viscous flows with substantial heat fluxes, etc.) are considered, to show that they cannot be solved by using the classical theory of incompressible fluid motion which involves the equation div u = 0. Application of the most general theory of compressible fluid flow may not be best in such cases, because then we encounter difficulties in accurately resolving the complex acoustic phenomena as well as in assigning the proper boundary conditions. With this in mind a new non-local mathematical model, where div u ≠ 0 in the general case, is proposed for the simulation of unsteady subsonic flows in a bounded domain with continuously distributed sources of mass, momentum and entropy, also taking into account the effects of viscosity and heat conductivity when necessary. The exclusion of sound waves is one of the most important features of this model which represents a fundamental extension of the conventional model of incompressible fluid flow. The model has been built up by modifying both the general system of equations for the motion of compressible fluid (viscous or inviscid as required) and the appropriate set of boundary conditions. Some particular cases of this model are discussed. A series of exact time-dependent solutions, one- and two-dimensional, is presented to illustrate the model.