The paper develops and discusses some new additions to the available stock of analytical solutions of the nonlinear equations of fluid motion. The motions are steady, two-dimensional and devoid of viscous or other rotational forces (although such forces must have been significant during any starting process). The fluid density is constant.

The solutions are in two groups, referred respectively to Cartesian and polar co-ordinates. In both the stream function is of separable form, i.e. expressible as a product of two functions, each dependent on one co-ordinate. A remarkable variety of motions is revealed. Those that are most significant physically are described as bends (rapid transitions from one rectilinear flow to another) or as loops (closed, non-circular, vortex-type flows). The effects of boundary layers at walls or instability are not explored.

The paper closes with a mention of some preliminary experiments on loop flows in which all streamlines are ellipses and some discussion of the applicability of bend flows. Generalizations to axisymmetric flows and compressible flows are also mentioned briefly.

A theoretical study is presented of the spatial stability of flow in a circular pipe to small but finite axisymmetric disturbances. The disturbance is represented by a Fourier series with respect to time, and the truncated system of equations for the components up to the second-harmonic wave is derived under a rational assumption concerning the magnitudes of the Fourier components. The solution provides a relation between the damping rate and the amplitude of disturbance. Numerical calculations are carried out for Reynolds numbers R between 500 and 4000 and βR [les ] 5000, β being the non-dimensional frequency. The results indicate that the flow is stable to finite disturbances as well as to infinitesimal disturbances for all values of R and βR concerned.

Earlier investigations by the authors have suggested that coherent structures, in the form of two different arrays of vortices, exist in the initial region of coaxial jets. The present investigation was aimed at obtaining further information on the characteristics of the vortices in coaxial jets. Single- and two-point correlation covariance measurements of the fluctuating pressure and the axial and radial velocity fluctuations have been made in the initial region of coaxial jets. Detailed analysis of the correlograms indicates the combined effect of the two vortex trains on the correlograms. Further, the analysis enables the phase relationship between the pressure and axial and radial velocity fluctuations to be obtained. From the correlation results the phase properties obtained within the whole initial region of coaxial jets are found to agree with the results for single jets. This good agreement supports the formally suggested simple approach to the understanding of the complicated flow in coaxial jets.

Following weak plane shock diffraction at a knife-edge situated in a duct, a two-dimensional vortex sheet springs from the salient edge. The method of ‘vortex discretization’ is used, in conjunction with a Schwarz-Christoffel transformation, to develop a two-dimensional potential model for this constrained form of vortex generation. The analysis is independent of empirical parameters and describes, qualitatively, the pattern of streamlines through the orifice.

Flow-visualization photographs are presented which illustrate the spiral shape of the starting vortex. Although of a limited nature, quantitative experimental vortex growth rates have been obtained and are compared with initial growth rates predicted theoretically. The results are discussed together with other aspects of the problem, including the limitations of the theory.

An extension of vortex discretization is developed whereby the pressure distribution remote from the vortex sheet can be calculated. The combination of flow separation and the associated static wall pressure distribution gives theoretical insight into the mechanism of flow through an orifice.

The stress system near a rigid boundary in a suspension of neutrally buoyant spheres is considered under the assumption of small Reynolds number. The suspending fluid is assumed to be Newtonian and incompressible. An ergodic principle is formulated for parallel mean flows, and the bulk stress is expressed as a surface average. Only dilute suspensions are considered and particle interactions are neglected. A uniform shear flow past a plane wall with a single spherical particle is studied first. A series solution is developed and the mean velocity and stress fields are computed for a force-free and couple-free sphere and also for spheres with couples applied by external means. The translational and angular velocities of the particle and the stress distribution on the surface of the particle are calculated. Properties of dilute suspensions of spheres are found by appropriate surface averages over the solutions for a single particle. The mean stress on a plane parallel to the wall is shown to reduce to the Einstein value when the distance from the boundary is sufficiently large. Mean velocity profiles of the suspension for Couette flow and Poiseuille flow are developed. It is shown that in an average sense particles rotate more slowly than the ambient fluid in a region approximately three sphere radii thick adjacent to the plane wall. But for the suspension as a whole, an apparent slip velocity always develops in this region. This results in an apparent viscosity which is less than the infinite-suspension value of Einstein.

It is shown that the use of axes moving with the tide (Shinohara et al. 1969) simplifies the analysis of contaminant dispersion in estuaries. Attention is restricted to estuaries which are small in the sense that cross-sectional mixing is rapid and that the tidal elevation can be taken to be constant along the estuary. In agreement with the work of Fischer (1972a, b) it is found that the dominant mechanism for dispersion is the transverse shear and not the vertical shear. Results are presented to illustrate the dependence of the upstream penetration of salt upon the estuary geometry as well as upon the fresh-water discharge rate.

This paper is the second of a pair describing two-point velocity measurements in fully developed pipe flow. A method of processing hot-film anemometer signals to identify intervals of high energy production (‘bursts’) in wall turbulence is presented. The method uses filtered cross-stream spatial derivatives of the axial velocity fluctuations. It is demonstrated to be more sensitive to ‘bursts’ than several other methods of indentification. The bursts identified in this manner are shown to have similar characteristics to those observed in visual studies.

The technique has been applied to the wall region of turbulent pipe flow. Mean burst rates have been obtained at various distances from the wall for three Reynolds numbers. It is shown that the mean burst rate cannot be reliably obtained from a previously used technique based on the autocorrelation of the axial velocity fluctuations.

On the basis of our experiments, the mean burst rate and the turbulent shear stress have been found to vary similarly with distance from the wall. In the region near the wall where the shear stress is constant the mean burst rate is independent of the kinematic viscosity.

Some characteristics of the velocity fluctuations during burst intervals have been studied. All the bursts began with a relative minimum in the axial velocity fluctuations followed by a peak in the cross-stream spatial derivative. A second peak always occurred midway through the burst. The sequence of events is somewhat similar to that in the last stage of laminar-to-turbulent transition.

Decaying grid turbulence was passed over a wall moving at the stream speed. For the high Reynolds number of the experiment, the field due to the wall constraint on the normal component of the velocity fluctuations is found to extend further into the flow than the influence of the viscous boundary condition on the tangential-component fluctuations. Measurements of the variances, length scales and spectra of the three velocity components of the turbulence are compared with the results of a previous experiment and with the theoretical predictions for an idealization of the flow. A simple model for some departures from the theory is proposed.

A method for solving quite general three-dimensional incompressible flow problems, in particular those described by the Navier–Stokes equations, is presented. The essence of the method is the expression of the velocity in terms of scalar and vector potentials, which are the three-dimensional generalizations of the two-dimensional stream function, and which ensure that the equation of continuity is satisfied automatically. Although the method is not new, a correct but simple and unambiguous procedure for using it has not been presented before.

The stability of time-periodic flows in a circular pipe is investigated. The disturbance is assumed to be axially symmetric and to have a small amplitude, so that the governing differential equation is linear. Calculations are carried out for the first ten modes for a range of values of the frequency of the primary motion, of the wavenumber of the disturbance, and of the Reynolds number of the primary flow. In the ranges of the parameters for which the calculations have been carried out, the flows are found to be stable and, as for Stokes flows (von Kerczek & Davis 1974), it is conjectured that the flows under study here are stable for all frequencies and all Reynolds numbers.

The evolution of long, finite amplitude Rossby waves in a horizontally sheared zonal current is studied. The wave evolution is described by the Korteweg–de Vries equation or the modified Korteweg-de Vries equation depending on the atmospheric stratification. In either case, the cross-stream modal structure of these waves is given by the long-wave limit of the neutral eigensolutions of the barotropic stability equation. Both non-singular and singular eigensolutions are considered and the appropriate analysis is developed to yield a uniformly valid description of the motion in the critical-layer region where the wave speed matches the flow velocity. The analysis demonstrates that coherent, propagating, eddy structures can exist in stable shear flows and that these eddies have peculiar interaction properties quite distinct from the traditional views of turbulent motion.

This paper describes an experimental study of a stratified fluid which is flowing over a smooth two-dimensional obstacle which induces no flow separation and in which effects of viscosity and diffusion are not important. The results are restricted to fluid of finite depth. Various properties of the flow field, in particular the criterion for the onset of gravitational instability in the lee-wave field, are measured and compared with the theoretical predictions of Long's model. The agreement is found to be generally poor, and the consequent inapplicability of Long's model is explained by the failure of Long's hypothesis of no upstream influence, which is demonstrably invalid when stationary lee waves are possible. The obstacle generates upstream motions with fluid velocities which appear to be of first order in the obstacle height. These motions have some of the character of shear fronts or columnar disturbance modes and have the same vertical structure as the corresponding lee-wave modes generated downstream. They result in a reduced fluid velocity upstream below the level of the top of the obstacle, together with a jet of increased fluid velocity above this level which pours down the lee side of the obstacle. This phenomenon becomes more pronounced as the number of modes is increased.

An extension of the turbulence ‘test-field model’ (Kraichnan 1971 a) is given for two-dimensional flow with Rossby-wave propagation. Such a unified treatment of waves and turbulence is necessary for flows in which the relative strength of nonlinear terms depends upon the length scale considered. We treat the geophysically interesting case in which long, fast Rossby waves propagate substantially without interaction while short Rossby waves are thoroughly dominated by advection. We recover the observations of Rhines (1975) that the tendency of two-dimensional flow to organize energy into larger scales of motion is inhibited by Rossby waves and that an initially isotropic flow develops anisotropy preferring zonal motion. The anisotropy evolves to an equilibrium functional dependence on the isotropic part of the flow spectrum. Theoretical results are found to be in quantitative agreement with numerical flow simulations.

In steady axisymmetric flows in a closed swirl chamber one can distinguish between the swirl flow proper, with components normal to the meridian plane, and a secondary flow whose components lie in the meridian plane. One can trace the motion of a particle within the meridian plane. The closed path so obtained will be called a streamline, to be parametrized by a stream function ϕ. One can distinguish between the flow in a boundary layer, where the velocity gradient is large, and a core flow, where the velocity and temperature gradients are relatively small. The present article is concerned with only the core flow. In high Reynolds number flows in which the streamlines are not closed there are three quantities which are constant along the streamlines: the total enthalpy (the right-hand side of Bernoulli's equation), the entropy and the moment of momentum of the particles with respect to the axis of symmetry. These are determined by conditions in the entrance cross-section. In flows with closed streamlines these quantities are ultimately determined by the cumulative effects of viscosity and heat conductivity. Conditions expressing cumulative effects enter the analysis as integrabiliby conditions necessary for the existence of a second approximation in a development of the flow field with respect to the reciprocal of the Reynolds number. They are re-expressed as the balance equations for energy, entropy and angular momentum which are to be satisfied on all surfaces ϕ = constant. One thus obtains an algorithm which leads from expressions for the total enthalpy, the entropy and the angular momentum as functions of ϕ to the residuals in the balance equations, also computed as functions of ϕ. The functions with which this algorithm starts must be chosen such that the residuals in the balance equations become zero. The secondary flow can be arbitrarily slow only if the Prandtl number is ½. At the centre of the secondary motion the balance equations are linearly dependent. This fact introduces an additional free parameter which allows one to compute secondary flows with different speeds. The linearized algorithm has the character of a Fredholm integral equation. This suggests an iterative solution similar to a Neumann series. The particles experience periodic changes of state which can be discussed as thermodynamic cycles. Such an analysis shows that the heat inputs occur on average at lower temperatures than the heat outputs. Besides the work that maintains the swirl motion, which is provided by shear-force components normal to the meridian plane, one therefore needs additional work, provided by the shear-force components within the meridian plane, to maintain a secondary motion. Responsible for this state of affairs is the fact that particles which do not quite maintain adiabatic temperatures move within a field with a large pressure gradient caused by the swirl component. This makes the flow sensitive to disturbances in the energy balance.

The behaviour of a fully rough turbulent boundary layer subjected to favourable pressure gradients both with and without blowing was investigated experimentally using a porous test surface composed of densely packed spheres of uniform size. Measurements of profiles of mean velocity and the components of the Reynolds-stress tensor are reported for both unblown and blown layers. Skin-friction coefficients were determined from measurements of the Reynolds shear stress and mean velocity.

An appropriate acceleration parameter Kr for fully rough layers is defined which is dependent on a characteristic roughness dimension but independent of molecular viscosity. For a constant blowing fraction F greater than or equal to zero, the fully rough turbulent boundary layer reaches an equilibrium state when Kr is held constant. Profiles of the mean velocity and the components of the Reynolds-stress tensor are then similar in the flow direction and the skin-friction coefficient, momentum thickness, boundary-layer shape factor and the Clauser shape factor and pressure-gradient parameter all become constant.

Acceleration of a fully rough layer decreases the normalized turbulent kinetic energy and makes the turbulence field much less isotropic in the inner region (for F equal to zero) compared with zero-pressure-gradient fully rough layers. The values of the Reynolds-shear-stress correlation coefficients, however, are unaffected by acceleration or blowing and are identical with values previously reported for smooth-wall and zero-pressure-gradient rough-wall flows. Increasing values of the roughness Reynolds number with acceleration indicate that the fully rough layer does not tend towards the transitionally rough or smooth-wall state when accelerated.

The return to isotropy of homogeneous turbulence without mean velocity gradients is attacked by considering changes to be slow relative to turbulence time scales. This single assumption permits the problem to be cast as one of finding the form of three invariant functions. Examination of limiting behaviour for large Reynolds number and small anisotropy, as well as small Reynolds number and arbitrary anisotropy, places restrictions on the form of the functions. Realizability conditions (requiring that energies be non-negative) reduce the problem to two functions subject to further restrictions. A convenient interpolation form is found for the functions, satisfying all the restrictions, and it is shown that predictions based on this are in excellent agreement with all available data.

The purpose of the present paper is to reach some general conclusions on the motion of rigid particles in a homogeneous shear flow of a viscoelastic fluid. Under the basic assumption of nearly Newtonian slow flow, the creeping-motion equations for a second-order fluid with characteristic time constants κ0(2) and κ0(11) can be employed. It is shown that the κ0(2) contributions to the hydrodynamic force F and couple G depend upon the hydrodynamic force, couple and stresslet which act upon the particle in a Newtonian fluid (termed F(1), G(1) and S(1), respectively). Since this relation involves time derivatives of F(1) and G(1), a little reflexion is needed to realize that the modification of the classical Stokes law for steady translation in a quiescent fluid can have no κ0(2) term. Since no results of such generality are possible for the κ0(11) contributions we focus attention on transversely isotropic particles. Employing the concept of material tensors, the symmetry of such particles dictates the form these tensors adopt. This alone is sufficient to show that sedimentation in a quiescent fluid is accompanied by a change in orientation until a stable terminal orientation is attained. Depending upon the type of particle only one of the two orientations, axis of symmetry parallel or perpendicular to the external force, is stable. Another result concerns two-dimensional shear flow, for which we show that the symmetry axis has to drift through various Jeffery orbits until an equilibrium orientation is reached. While the orbits C = 0 and C = ∞ are equilibrium orbits for every transversely isotropic particle there may be a third such preferred orbit, which we denote by C*. In order for these orbits to be stable certain restrictions have to hold, showing that the orbits C = 0 and C* cannot both be stable. For the special case of a rigid tridumbbell of axis ratio s the orbit C* does not exist. If s > 1 the drift for this particle is into the orbit C = 0 while for s < 1 it is into the orbit C = ∞. This agrees qualitatively quite well with experimental results obtained for rods and disks. No quantitative comparison is possible; the particle shape influences the result quantitatively owing to its effect on the combination of the fluid parameters κ0(2) and κ0(11).

The purpose of the present study is to assess the possibility of the so-called choking flutter in a transonic cascade operating under choking conditions. In the experiment, measurements of the unsteady aerodynamic moment acting on an aerofoil which is oscillating in pitch about the midchord in a transonic channel flow are performed for channel height-to-chord ratios H/c = 0·5−3 and oscillation frequencies up to 120 Hz. For a large channel height such as H/c = 3, the unsteady aerodynamic behaviour is similar to that of an isolated aerofoil, and the system will be aeroelastically stable in the transonic region as well as in the subsonic one. For smaller channel heights such as H/c = 0·5, however, the flow chokes and the aerodynamic behaviour becomes significantly different from that at large channel heights. As soon as a shock appears, i.e. in the incipient transonic region, the aerodynamic derivatives change discontinuously, so that the subsonic and transonic regions can be clearly separated. In this case, the aerodynamic damping becomes negative in the transonic region.

In order to understand the transonic features, a one-dimensional unsteady analysis is undertaken for small channel heights H/c [les ] 0·5. The numerical results show good qualitative agreement with the experimental ones, predicting in the incipient transonic region a discontinuous variation of the aerodynamic moment which results in system instability.

It is concluded that choking flutter may occur in transonic channel flow with a moderately small channel height when the passage is choked.

Temporal and spatial development of short gravity waves in a linear wind-wave tank has been measured for wind speeds up to 15 m/s using microwave Doppler spectrometry. Surface waves of wavelength 4·1 cm, 9·8 cm, 16·5 cm and 36 cm were observed as a function of fetch, wind speed and wind duration. The waves grew exponentially from inception until they were about 10 dB smaller than their maximum height, and the temporal growth and spectral transport (spatial growth) rates were about equal when the wave amplitude was sufficiently small. The amplitude of a short gravity wave of fixed wavelength was found to decrease substantially at winds, fetches or durations greater than those at which the short gravity wave was approximately the dominant wave; such phenomena are sometimes referred to as overshoot. The dominant short gravity wave was observed to reach a maximum amplitude which depended only on wavelength, showing that wave breaking induced by an augmented wind drift cannot be the primary limitation to the wave height. Waves travelling against the wind were observed for wavelengths of 9·8 cm, 16·5 cm and 36 cm and were shown to be generated by the air flow at low wind speeds.

Measured initial growth rates for 16·5 cm and 36 cm waves were greater than expected, suggesting the existence of a growth mechanism in addition to direct transfer from the wind via linear instability of the boundary-layer flow. Initial temporal growth rates and spectral transport rates were compared to yield an experimental determination of the magnitude of the sum of nonlinear interactions and dissipation in short gravity waves. If the steady-state energy input in the neighbourhood of the dominant wave occurs at the measured initial temporal growth rates, then most of the energy input is locally dissipated; relatively little is advected away. Calculated gravitycapillary nonlinear energy transfer rates match those determined from initial growth rates for 9·8 cm waves and the gravity–capillary wave interaction continues to be significant for waves as long as 16·5 cm. For longer waves the gravity–capillary interaction is too small to bring the short gravity wave to a steady state when it is the dominant wave of the wind-wave system.

Further results from the nonlinear numerical model of the air flow in a deep turbulent boundary layer above water waves described in Gent & Taylor (1976) are presented. The results are calculated with the surface roughness z0 both constant and varying with position along the wave. With the form used when z0 varies, the fractional rate |ζ| of energy transfer per radian advance in phase due to the working of the pressure forces is larger than for z0 constant both when the transfer is from wind to waves and when it is from waves to wind. The latter case occurs when the waves are travelling faster than, or against, the wind. The energy transfer rates are compared with other theoretical predictions and with recent field observations.