Flows of incompressible inviscid heavy fluids with free or rigid boundary surfaces are considered. For slender streams of fluid, the flow and the free boundaries are represented by a number of different asymptotic expansions in powers of the slenderness ratio. There are three kinds of outer expansions representing respectively jets, which have two free boundaries, wall flows, which have one free and one rigid boundary, and channel flows, which have two rigid boundaries. The flow at the junction of two or more outer flows is represented by an inner expansion. Previously we constructed the three outer expansions and the inner expansion at the junction of a wall flow and a jet (Keller & Geer 1973). Now we construct the inner expansions at the junctions of a channel flow and a jet, a channel flow and a wall flow, and a jet and the two wall flows into which it splits upon hitting a wall. We also match each inner expansion to the adjacent outer expansions. These seven expansions can be combined to solve many problems involving flows of slender streams.

This paper uses asymptotic methods to analyse the flow in a narrow channel of a fluid with temperature-dependent viscosity and internal viscous dissipation. When the Nahme–Griffith number is large we show how the flow evolves from Poiseuille flow with a uniform temperature distribution to a plug flow with hot boundary layers on the walls. An asymptotic solution is obtained for the flow in the region of transition from Poiseuille to plug flow and an explicit equation is derived for the pressure gradient in terms of the local downstream co-ordinate in this transition region.

A stability analysis for high wavenumber perturbations of a Stokes wave of wavenumber k1 and slope ϵ is presented. Except for a correction term the governing equation is shown to be of Hill's type. The analysis predicts instability at wavenumbers k2 = ¼(m + 1)2k1. The two lowest and strongest instabilities are the Benjamin- Feir instability at m = 1, and the quartet resonance at m = 2. Both are incorrectly treated by the present method. For m ≥ 3 the analysis should be asymptotically (ϵ → 0) correct, yielding instability O(ϵm) due to m-fold Bragg-scattering. The non-resonant perturbations behave as predicted by WKBJ theory. The instability is too weak for experimental detection; numerical tests should be possible, but are not available at present.

The effect of viscosity on the inner free surface of a rotating imploding cylindrical liquid shell compressing an ideal gas or magnetic flux load is analysed in the limit of high Reynolds number Re. The condition of vanishing tangential stress on the free surface leads to the formation of a boundary layer of thickness ∼ Re−½. Within this layer the zonal velocity v is reduced by an amount Δv such that Δ v/vRe−½. This results in a requirement of slightly increased rotation in order to satisfy the criterion for suppression of the Rayleigh-Taylor instability on the free surface. Calculations are presented for a model implosion trajectory.

For steady laminar flow with closed streamlines Batchelor (1956) has shown how an integral condition arising from the effect of viscosity can be used with the inviscid flow equations to determine the vorticity distribution when the Reynolds number is large. Here a condition analogous to that used by Batchelor is derived for a class of flows with helical streamlines. An exact integral condition relating the constant axial pressure gradient and the viscous terms is obtained, which when combined with the inviscid flow equations leads to the result that the axial velocity is proportional to the stream function for the motion in the plane normal to the axial velocity.

In the early theoretical study of aquatic animal propulsion either the two-dimensional theory or the large aspect-ratio theory has been generally used. Only recently has the unsteady lifting-surface theory with the continuous loading approach been applied to the study of this problem by Chopra & Kambe (1977). Since it is well known that the continuous loading approach is difficult to extend to general configurations, a new quasi-continuous loading method, applicable to general configurations and yet accurate enough for practical applications, is developed in this paper. The method is an extension of the steady version of Lan (1974) and is particularly suitable for predicting the unsteady lead-edge suction during harmonic motion.

The method is applied to the calculation of the propulsive efficiency and thrust for some swept and rectangular planforms by varying the phase angles between the pitching and heaving motions. It is found that with the pitching axis passing through the trailing edge of the root chord and the reduced frequency k equal to 0·75 the rectangular planform is quite sensitive in performance to the phase angles and may produce drag instead of thrust. These characteristics are not shared by the swept planforms simulating the lunate tails. In addition, when the pitching leads the heaving motion by 90°, the phase angle for nearly maximum efficiency, the planform inclination caused by pitching contributes to the propulsive thrust over a large portion of the swept planform, while, for the rectangular planform, only drag is produced from the planform normal force at k = 0·75. It is also found that the maximum thrust is not produced with maximum efficiency for all planforms considered. The theory is then applied to the study of dragonfly aerodynamics. It is shown that the aerodynamically interacting tandem wings of the dragonfly can produce high thrust with high efficiency if the pitching is in advance of the flapping and the hindwing leads the forewing with some optimum phase angle. The responsible mechanism allows the hindwing to extract wake energy from the forewing.

The three-dimensional evolution of packets of gravity waves is studied using a nonlinear Schrödinger equation (the Davey–Stewartson equation). It is shown that permanent wave groups of the elliptic en and dn functions and their common limiting solitary sech forms exist and propagate along directions making an angle less than ψc = tan−1(1/√2) = 35° with the underlying wave field, whilst, along directions making an angle greater than ψc, there exist permanent wave groups of elliptic sn and negative solitary tanh form. Furthermore, exact general solutions are given showing wave groups travelling along the two characteristic directions at ψc or − ψc. These latter solutions tend to form regions of large wave slope and are used to discuss the waves produced by a ship, in particular the nonlinear evolution of the leading edge of the pattern.

A grid-generated turbulence is subjected to a pure plane strain and the principal axes of the Reynolds stress tensor become those of the strain. This ‘oriented’ homogeneous turbulence is then submitted to a new strain the principal axes of which have a different orientation. We show that in such a situation it is possible to observe a transfer of energy from the fluctuating motion to the mean one. Such transfer is necessarily associated with a forced decay of the anisotropy of the motion. A detailed analysis of the reorientation of the principal axes of the Reynolds stress tensor in the frame of those of the second strain gives an explanation of the evolution of the principal axes of the Reynolds stress tensor in a shear flow.

Strong external disturbances were introduced into a mixing layer in order to test the formation of the quasi two-dimensional coherent eddies and their survival under less than ideal conditions. Velocity and temperature correlation measurements, flow visualization, and the simultaneous use of a large number of sensors suggest that these eddies are very stable in the range of Reynolds numbers considered and they persevere in spite of the external buffeting imposed. Some measurements were carried out in a mixing layer between two parallel streams and some in a mixing layer entraining quiescent surrounding fluid. In both cases the large eddies could be described. as vortex rolls spanning the test section; these rolls may be contorted and sometimes skewed, but they are basically two-dimensional.

A study is made of the flow and the shape of the free surface of a non-Newtonian fluid contained in a flat-bottomed cylindrical vessel performing torsional oscillations of small amplitude. For the case where the fluid depth is small compared with the vessel radius, the solution is shown to have a simple radial dependence except in a boundary-layer region near the side wall. It is shown that under certain circumstances the mean steady second-order components of both free surface curvature and radial–axial flow may be in the reverse direction to those for a Newtonian fluid. It is found that flow reversal may occur at any frequency of oscillation, but that surface curvature reversal cannot occur at low frequencies.

The wavelength of turbulent Taylor vortices at very high Taylor numbers up to 40000Tc, has been measured in long fluid columns with radius ratios η = 0·896 and η = 0·727. Following slow acceleration procedures the wavelength (in units of the gap width) of turbulent axisymmetric vortices was found to be λ = 3·4 ± 0·1 with the small gap and about λ = 2·4 ± 0·1 with the larger gap, and thus in both cases substantially larger than the critical wavelength of laminar Taylor vortices. In the narrow and wide gap the wavelength was, within experimental error, independent of the Taylor number for T > 100Tc. In the experiments with the narrow gap a clear dependence of the value of the wavelength of the turbulent vortices on initial conditions was found. After sudden starts to Taylor numbers > 700Tc the wavelength of steady axisymmetric turbulent vortices was only 2·4 ± 0·05, being then the same as the wavelength of the vortices after sudden starts in the wide gap, and being, within the experimental error, independent of the Taylor number. In the narrow gap all values of the wavelength between λmax = 3·4 and λmin = 2·4 can be realized as steady states through different acceleration procedures. In the wide gap the dependence of the wavelength on initial conditions is just within the then larger experimental uncertainty of the measurements.

Laboratory and numerical experiments have been conducted to study the secular spin-up of both a homogeneous and a thermally stratified rotating fluid in a right cylinder. In these experiments, the angular velocity of the container increases linearly in time from some initial rotation rate at t = 0. A simple quasi-geostrophic model is developed to describe the adjustment of the fluid over the characteristic spin-up time scale to the constant angular acceleration of the basin. Good agreement is found between the observed interior temperature and azimuthal velocity fields and the theory in both the homogeneous and stratified secular experiments. This result is in contrast to the much faster adjustment observed in stratified instantaneous spin-up experiments reported earlier. The main difference between these experimental cases is the inability of secular forcing to excite energetic inertial–gravity-wave transients during the initial phases of secular spin-up. Thus, the asymptotic theory which has filtered out these initial higher-frequency transients is accurate even though the inertial period is not much smaller than the characteristic spin-up time scale.

Steady flow in a cylinder of finite length rotating rapidly at right angles to the earth's gravity and containing a rigid cylindrical float is found under the assumption of small viscosity. The float is modelled by a ‘rigid free surface'. The major new result is a prediction of the rotation rate of the interface, which differs by a factor of O(E1/4) from those given by Greenspan (1976) and Wood (1977) for a rigid interface in an infinitely long cylinder. The difference is accounted for by the appearance of Stewartson layers on the inner boundary in the finite case. The result is supported by an experiment of my own, and is not in conflict with the experimental results given by Greenspan.

Some additional experimental observations are reported: a comparison of free and rigid surface rotation rates; and a brief description of the spin-up from rest.

Hoverming motions, by which an animal (or a helicopter) in stationary fluid generates a downflow to support its weight, entail energy costs that include the induced power (power supplied to that downflow). The simplest classical model for induced power is the actuator-disk model. This paper shows how a relatively insignificant modification can be made to that model to make it aerodynamically self-consistent. The modified simple model of the downflow may be evaluated in fluid that either is unbounded or is bounded below by horizontal ground. Comparison of the calculated induced powers in the two cases (even though made in this paper not for the true axisymmetric flow patterns but for the corresponding two-dimensional flow patterns) appears to give a more satisfactory analysis than was previously available of the observed reduction of induced power associated with proximity to the ground.

The asymptotic properties of Burgers turbulence at extremely large Reynolds numbers and times are investigated by analysing the exact solution of the Burgers equation, which takes the form of a series of triangular shocks in this situation. The initial probability distribution for the velocity u is assumed to decrease exponentially as u → ∞. The probability distribution functions for the strength and the advance velocity of shocks and the distance between two shocks are obtained and the velocity correlation and the energy spectrum function are derived from these distribution functions. It is proved that the asymptotic properties of turbulence change qualitatively according as the value of the integral scale of the velocity correlation function J, which is invariant in time, is zero, finite or infinite. The turbulent energy per unit length is shown to decay in time t as t−1 (with possible logarithmic corrections) or $t^{-\frac{2}{3}}$ according as J = 0 or J ≠ 0.

Further experiments on the detailed study of annular jets are described in which the mean and fluctuating properties in the inner region have been measured. The experiments in the conical jetf have shown, besides the jet† vortices in the outer mixing region, another train of vortices in the inner region. This train of vortices is due to the wake formed by the boundary layer on the surface of the conical bullet.

The experiments in the inner region of the basic annular jet have similar mean velocity profiles to those in the internal recirculating region. Good correlation is found for the location of the vortex centre, the location of reattachment, the minimum and maximum mean static pressure and their locations with the non-dimensional available pressure M°/AiPatm for entrainment behind the interface. A train of wake vortices is generated in the internal recirculating region and a train of jet vortices is found in the inner mixing region. Both types of vortices in the inner region seem to decay fairly rapidly within a distance of one outer diameter D° downstream.

The disturbances associated with the wake vortices in the inner region seem to excite the outer mixing region. This results in another wave or train of vortices observed in the outer shear layer in addition to the jet vortices which are already in existence.

The unsteady flow past a circular disk is studied with hot-wire and microphone probes positioned in planes normal to the axis of symmetry at 3 and 9 disk diameters down-stream. Both the fluctuating velocity and pressure signals are shown to be continuously dominated by large-scale coherent motions enveloping the wake flow as a whole. This suggests narrowband two-point space correlations as an experimental tool for describing spatial coherence and phase characteristics of the basically random signals. The specific symmetry imposed by the axisymmetric boundary conditions of the disk enables a decomposition of the large-scale flow phenomena into relatively simple elementary structures or modes. The resulting azimuthal constituents are quantified in terms of their respective magnitudes and individual power spectra.

The capability of the approach to uncover characteristic features of turbulence as far as its large-scale domain is concerned is demonstrated by a comparison of the present results with certain remarkably different features found in earlier jet flow investigations: the m = 1 and m = 2 modes are found to clearly dominate in wakes whereas the m = 0 and m = 1 modes were dominant in jets in a relevant range of Strouhal numbers. These large-scale coherent structures are more than just an interesting flow phenomenon; they must have a tremendou back-reaction on rigid flow boundaries (particularly if these allow a vibrational response) and may give rise to specific feedback mechanisms.

The analysing technique proposed for studying large-scale flow phenomena injets and wakes removes part of the randomness in the turbulent signals without artificially exciting or forcing them in one way or another. No conditional sampling of the naturally occurring fluctuations is required, either. The method may be applicable to other than strictly axisymmetric flow configurations, too.

The evolution of an eddy in a two-layer shear flow and the radiation field generated by the eddy are investigated by solving the initial value problem. The solution indicates that two wave regimes with different characteristics exist in the x/t plane, where t is the time and x is the distance from the initial disturbance. The near-field (small x/t) behaviour depends critically on the stability of the shear. In a baroclinically stable shear flow, the radiation field of the baroclinic waves is confined to a region the boundary of which expands linearly with time. If the shear flow is baroclinically unstable, the eddy gradually evolves from an isotropic structure into a wave packet that grows exponentially and expands as the square root of time. The far field (large x/t) consists mainly of long barotropic Rossby waves. As x/t increases, the effect of the mean currents becomes weaker and the motion becomes increasingly barotropic in character. If we consider the strong boundary current as a source of baroclinic energy in the ocean, the far-field results show that the mid-ocean eddies can reach large amplitudes independent of the mean currents. The transient motion (small x and t) of the eddy is examined by numerically computing the stream function from the solution. The results show that the combined actions of baroclinic instability and wave dispersion cause an initially isotropic eddy to remain roughly isotropic for a relatively long period of time. This suggests that the initial development of unstable eddies cannot be neglected in studying the meridional scale of the eddy field in the atmosphere. Application of the model to study the dispersion of wind-generated Rossby waves is discussed.