The stability of a magnetic-velocity shear layer, of thickness L, to hydromagnetic-gravity waves of zonal wavenumber k is investigated analytically, within the Boussinesq approximation, in the situation where ε(= kL) is small. It is found that, in addition to the unstable modes of the corresponding sheet, new modes of instability of growth rate of order ε2 are also present provided one critical level exists within the layer. The existence of one critical level also effects over-reflexion of stable modes. Furthermore it is shown that a magnetic shear acting alone can lead to instability as well as effecting over-reflexion of stable modes.

The paper is a study of experimental data in the light of new theories of turbulence recently developed by the first author for a number of problems including flow in a pipe, boundary layer at zero incidence, atmospheric boundary layer, turbulent convection and distribution of energy in wavenumber space in decaying, isotropic turbulence. In each of these, a basic element is a ‘mesolayer’ or ‘mesoregion’ in physical space or wavenumber space which is absent in earlier theories and which intrudes between the inner and outer regions preventing the overlap assumed in the derivation of the classical results, e.g. the logarithmic profile in shear flow. The new and old theories differ both in principle and in the final results: the new ideas replace rather than modify or extend the older ones.

The main purpose of this paper is to bring together accumulated evidence concerning the mesolayer theories. We believe that this evidence provides overwhelming support for the existence of the mesolayer and for its pervasive importance in problems of turbulence.Editorial footnote. Although the referees were not persuaded that the claims for the ‘new theories of turbulence’, made by the authors in the abstract and elsewhere in this paper, are justified, we think that publication in the Journal may serve a useful purpose. The authors have assembled a large body of data for various turbulent flow systems. These data should enable readers to test different aspects of the ‘classical’ and ‘new’ theories for themselves and should stimulate thought about the foundations of the classical ideas and about extensions of these ideas, as well as about the validity of the new theories.

The shapes and terminal velocities of bubbles rising in viscous liquids have been determined. For Morton numbers (M) greater than 4 × 10−3 the drag coefficient and dimensionless bubble shape are functions only of Reynolds number (R). Shape regimes and terminal rise velocities have been correlated. The flow field around a rising bubble was visualized through the hydrogen bubble tracer technique. For M > 4 × 10−3 and R < 110 the bubbles trailed closed, laminar toroidal wakes. For R > 110 the wake was open and unsteady. Streamlines for the flow were obtained by raising a ciné camera at the same speed as the bubble and filming the H2 tracer bubbles. Results are presented for R < 150 and 7·4 × 10−4 < M < 850.

An investigation of the turbulent flow structure over a progressive water wave, as well as the structure of the wave-induced flow field in a transformed wave-following frame, is reported. Experimental results are given for a free-stream velocity of 2·4 m s−1 over a 1 Hz mechanically generated deep-water wave. The velocity components were measured with a cross hot-film probe oscillating in a transformed wave-following frame. The amplitude and phase of the wave-induced velocity components are deduced by correlation to the generated water wave. The mean flow tends to follow the wave form so that the water wave should not be regarded as surface roughness. The mean velocity profile is basically log-linear and is similar to that over a smooth plate, because ripples riding on the waves do not produce sufficient roughness to interfere with the wind field. The wave-induced motion in the free stream is irrotational; but, in the boundary layer, it has strong shear behaviour related to the wave-associated Reynolds stress. The shear stress production as well as the energy production from the mean flow is concentrated near the interface. A phase jump of 180° in the wave-induced turbulent Reynolds stresses in the middle of the boundary layer was observed. The relationships between the induced turbulent Reynolds stresses and the induced velocities are of an eddy-viscosity type.

In a high-speed subsonic jet impinging on a flat plate, the surface pressure fluctuations have a broad spectrum due to the turbulent nature of the high-Reynolds-number jet. However, these pressure fluctuations dramatically change their pattern into almost periodic waves, if the plate is placed close to the nozzle (x0/d < 7·5). In the present study extensive measurements of the near-field pressure provide solid support for the hypothesis that a feedback mechanism is responsible for the sudden change observed in the pressure fluctuations at the onset of resonance. The feedback loop consists of two elements: the downstream-convected coherent structures and upstream-propagating pressure waves generated by the impingement of the coherent structures on the plate. The upstream-propagating waves and the coherent structures are phase-locked at the nozzle exit. The upstream-propagating waves excite the thin shear layer near the nozzle lip and produce periodic coherent structures. The period is determined by the convection speed of the coherent structures, the speed of the upstream-propagating waves as well as the distance between the nozzle and the plate. An instability process, herein referred to as the ‘collective interaction’, was found to be critical in closing the feedback loop near the nozzle lip.

A floating horizontal cylinder of infinite length is performing simple harmonic oscillations of small amplitude in the free surface of a uniform inviscid fluid under gravity. The cylinder intersects the mean free surface at right angles, and the fluid is bounded below by a fixed horizontal plane. Let the corresponding two-dimensional velocity potential be expressed as a distribution of simple wave sources over the boundary of the body. Then it is known that the source density satisfies a Fredholm integral equation of the second kind which has a unique solution except at those frequencies (the irregular frequencies) at which the Fredholm determinant vanishes. The present work is concerned with the irregular frequencies. Let the simple wave sources be replaced by a fundamental solution which consists of a simple wave source together with additional wave singularities inside the cylinder. It is shown how irregular frequencies can be eliminated by an appropriate choice of these singularities.

When a water tap is turned off gradually, a critical point is reached at which the flow changes abruptly from a continuous stream to a series of drops that form at the tap. This problem is studied within the context of a one-dimensional (Cosserat) jet theory. The exact inviscid, steady, nonlinear jet equations are solved and steady draw-down shapes are obtained for various values of Weber and Bond numbers. A critical Weber number is obtained (as a function of Bond number) below which no steady solution is found that satisfies the constraints imposed at the nozzle. The results are compared with classical experiments and appear to explain the observed ‘first critical velocity’.

The stability of a weakly nonlinear wave train on deep water to two- and three-dimensional modulations is investigated using an improved approximation due to Zakharov (1968). The results are expressible in simple analytical forms, and show good quantitative agreement with available experimental data and exact numerical calculations over a broad range of wave steepness in the unidirectional case.

This paper describes mean-flow and turbulence measurements conducted in a round jet over a range of Mach numbers from 0·3 to 1·7 and jet-exit static temperatures from −40 to over 400 °C. It is a continuation of an earlier work, reported by Lau, Morris & Fisher (1979), to try to map the distribution of the various flow characteristics in the jet flow field and to observe the effects of changing jet exit conditions. In the earlier study, the effort was confined to isothermal jets at a limited number of exit Mach numbers, and the laser velocimeter proved to be a particularly useful instrument, especially in situations where the more severe flow conditions made it impossible to extract fluctuating-velocity data by any other means. The present effort capitalizes on this aspect of the velocimeter and also its ability to measure mean velocities accurately; and the extended range and detail of jet conditions chosen for this study is intended to provide a clearer understanding of the effects of systematically changing the jet conditions. Corresponding Pitot and total temperature measurements are also carried out under a representative set of jet conditions specifically to try to shed light on the effect of jet heating. Based on the various axial and radial distributions which are obtained, a picture is constructed of the changing boundaries of the shear layer with changing jet conditions.

An experimental study of an excited jet interrupted normally by a 90° sharp edge is described in this paper. Based on schlieren technique, it is possible to distinguish three flow regions: the main jet flow, the wake flow behind the sharp edge and the redirected flow on the surface. By following the individual articially excited jet vortices, it is possible to observe the structure and to estimate the convection velocity of these vortices in the three flow regions. By comparison with the results of the uninterrupted jet, the effect of the sharp edge is obtained and presented.

Flow patterns near conical stagnation points in supersonic flow have been investigated on the basis of potential flow. Near the conical stagnation point the nonlinear equation for the conical velocity potential reduces to the equation of Laplace. Solutions of the equation of Laplace for incompressible plane flow are then used as a guide to generate conical stagnation-point solutions. Apart from known types of streamline patterns, such as nodes and saddle points, new types are found. Among them are oblique saddle points, saddle-nodes, topological nodes and topological saddle points. They may be used to clarify certain questions in a number of practical conical-flow problems. The oblique saddle point may be used to describe the inviscid flow associated with flow separation and also certain features of the flow over an external corner. The saddlenode, being structurally unstable, may fall apart into a saddle and a node. It may then be used to interpret the lift-off phenomenon of the singularity in the flow around a circular cone at incidence as a bifurcation. Similarly, this may be done for the appearance of a dividing streamline in the same flow at still higher angles of incidence, where a vortex system is formed at the leeward side of the cone.

The convection speed and average inclination of the large structure in the turbulent mixing layer of a plane jet have been obtained using different techniques. Ensemble-averaged shapes of the signatures of velocity and temperature associated with this structure are also presented. An array of cold wires is used to simultaneously observe the temperature fluctuation θ at several points in space. A cold-wire/X-wire combination is used to measure θ, the streamwise u and the normal v velocity fluctuations. The front of the structure moves faster than the back and the average inclination of this structure to the jet axis is approximately 40°. A detection scheme, based on positive peaks of the cubed values of the temperature derivative, is used to identify the large structure and obtain average signatures of θ, u, v, and of the products uv, θ and vθ. These signatures are consistent with a vortex-like description of the large structure. They also reflect the existence of a thin interconnecting region or ‘braid’ between two consecutive vortical structures. This region is identified by the sharp changes in temperature and is interpreted as part of the large structure. In the central part of the mixing layer, at least 80% of the Reynolds shear stress and 50% of the average normal heat flux appear to be contributed by the large structure.

Axisymmetric flows with a two-layer density stratification are produced by releasing either a constant flux of fluid from a point source or a constant volume of fluid into a rotating environment with a different density. In both experiments the density interface intersects one horizontal boundary, forming a front. Transition to non-axisym-metric flow is observed and can be described by two parameters: θ, the square of the ratio of the internal Rossby radius of deformation to the horizontal length scale of the flow, and δ, the fraction of the total fluid depth occupied by the layer inside the front. For θ [Lt ] 1 and δ > 10−1 unstable disturbances obtain most of their energy from the potential energy of the flow, whilst for δ < 10−1 extraction of kinetic energy from the basic shear becomes the dominant driving mechanism. When the front intersects the free surface, n = 2 is the minimum azimuthal wavenumber for an unstable disturbance. At large amplitude of the growing waves, baroclinic and barotropic processes combine to form n vortex dipole structures which entrain buoyant fluid from the original vortex and propagate radially over the free surface. Vortices are also produced by the continuous release of fluid from a confined source at its own density level in a region of constant density gradient. As in the two-layer case the axisymmetric vortex grows to a critical size and then becomes unstable to a disturbance with wavenumber n = 2, producing, at large amplitude, two vortex pairs.

Many algorithms have been proposed and evaluated for dealing with laser-anemometer measurements in low particle densities since the original paper of McLaughlin & Tiederman (1973). All of them describe the effect of making a measurement whenever a particle is visible to the measurement instrument. However, in many circumstances, one wishes to sample the velocity at a constant rate–for instance, in measurement of velocity fluctuation spectra. It is shown here that the measurement statistics for this case are different from those previously discussed in the literature. The product of the particle density and the sample interval is the controlling parameter for the statistical description of the measurements. The asymptotic forms for low and high particle density-sample time products are derived.

An investigation into the magnitude and direction of the convective velocity in a plane air jet was performed. Convective velocities were obtained from cross-correlation measurements. They are defined as the ratio of the spacing between two hot-wire probes and the time delay between their signals to reach maximum correlation. These velocities were larger in magnitude than the local mean velocities for lateral distances greater than the half-width of the jet. Frequency analysis of the convective velocity indicates that the large-scale eddies move slower than the mean flow while the small scales move faster. Based on the convective velocity vector, broadband ‘convection lines’ were defined and found to point outward with respect to the streamlines for all values of y/b [Gt ] 0·5. Likewise, frequency investigation indicates that ‘convection lines’ point outward for all y/b [Lt ] 1·3 and then inward for larger values of y/b.

The mixed convection in a two-dimensional line heat source is studied for the situations where buoyancy effects are favourable or adverse with respect to the oncoming vertical stream. The problem is analysed in terms of two co-ordinate expansions, direct and inverse, valid for small and large values of streamwise distance from the heat source. The solution for the first eleven and seven terms in direct and inverse co-ordinate expansions, respectively, are obtained. The direct expansion, when suitably transformed by Euler transformation and other techniques, predicts the velocity and temperature to two-digit accuracy for all values of streamwise coordinates, with a maximum error of 0·1% for velocity, 0·8% for temperature and 2·2% for displacement thickness far downstream from the source.

A theoretical calculation is made of (an off-diagonal element of) the pressure–strain-rate term ρ−10〈p[∇u + (∇u)T]〉 for a simple turbulent shear flow at high Reynolds number. This calculation is described as follows. (1) An expression for the pressure–strain-rate term is analytically derived in terms of measurable quantities (velocity spectra) - this derivation makes use of a cumulant discard. (2) It is proved that, to the lowest order in the spectral anisotropy, the (nonlinear part of) the pressure-strain-rate term is linearly proportional to the Reynolds stress. (3) A formula is derived for the constant of this proportionality (the Rotta constant) in terms of arbitrary velocity spectra. (4) This formula is used to analytically calculate Rotta's constant, Cxz, for a class of models of velocity spectra (the variation of Rotta's constant caused by variations in the spectral shapes is examined). (5) It is found that Cxz is surprisingly insensitive to the large-wavelength part of the spectrum. This insensitivity suggests that Cxz should not vary much from one turbulence application to another provided that the Reynolds number is very large. However, it is also shown that Cxz is unexpectedly sensitive to the short-wavelength part of the spectrum, and varies with Reynolds number when the latter is less than about 30.

The calculation is based on a straightforward solution of the Navier–Stokes equation to obtain formal expressions for u and p. These expressions are then used to write the pressure–strain-rate in terms of a two-time fourth-order velocity correlation. The latter correlation is evaluated by a standard cumulant discard. Simplifying assumptions of the calculation are that average quantities vary little in space and time, and that the mean flow are unidirectional. These simplifications are made in order to emphasize the method of calculation and its details.

Experimentally measured steady and unsteady forces induced by a cross flow over a smooth circular cylinder are reported in this paper. The measurements were made in the 0·305 m research water tunnel of the Pennsylvania State University. The parameters examined include the Reynolds number, which was varied over a range of 2 × 104 to 2 × 106, free-stream turbulent intensity, integral length scale-to-diameter ratio and active span-to-diameter ratio; however, this paper includes only some of these results.

Among the results reported are the complete mean drag data on all the cylinders tested and some fluctuating force data chosen to illustrate the effects of Reynolds number and active span-to-diameter ratio on the measured forces in uniform and turbulent cross flows. From these results, it can be concluded that the unsteady forces bear a functional relation to Reynolds number in the range tested which is very similar to the well-documented behaviour of the mean drag. Thus the effect of free-stream turbulence on both the steady drag forces and the unsteady forces is to shift the transitional region to a lower Reynolds number. Decreasing the active span-to-diameter ratio increases the buffeting lift coefficient at the same Reynolds number. Finally, it is observed that the Strouhal lift signal at transitional Reynolds numbers is no longer quasi-periodic but rather resembles a narrow-band random signal.

Free oscillations are considered of a fluid rotating with constant angular velocity $\Omega \hat{\rm z}$ in a rigid axisymmetric container. Modes are sought that vary rapidly in an axial (r, z) plane with a length scale O(n−1) times that of the container, where n [Gt ] 1. The azimuthal wavenumber k > 0 is also taken to be large. The modulated wave modes postulated (represented as in (4.1)) prove to have a quiescent zone near the axis. Elsewhere their pressure is of a uniform order of magnitude. Their velocity however is locally magnified by a factor O(n) near the critical circles. It is argued that for k/n [Lt ] 1 the modulated waves eligible as modes in smooth, convex containers are of two kinds; one, which generally occurs for continuous frequency bands, being singular and indeterminate; the other being like the modes in a sphere. Modes of the second kind are determined for eigenfrequencies ω ≃ √2 Ω for containers whose axia lcross-sections are symmetrical about z = 0 and about r = ±z.

The turbulent trailing vortex forming from a rolling-up vortex sheet is considered. The inviscid, asymptotic roll-up of a vortex sheet is briefly reviewed, as are the effects to the sheet of merging by viscous and turbulent diffusion. The merged region is found to rapidly attain a state of equilibrium and similarity variables are used to describe it. The detailed distributions of circulation and Reynolds stress are seen to depend to some extent upon the initial spanwise distribution of circulation on the wing. However, a tiny region which is independent of the wing circulation distribution is found to exist near the point of peak tangential velocity. It is suggested that this region is described by Hoffmann & Joubert's logarithmic relationship. Assuming this to be the limiting form for the distribution of circulation near r1, the radius where the tangential velocity takes its peak value v1, an approximate form for the distribution of circulation is found and this is used to determine the form of the Reynolds-stress distribution. It is found that two modes for the decay of v1 with time are possible: one when r1 is much less than ½s, the wing semi-span, and v1 decays like t−½n; and the other when r1 = O(½s) and v1 may decay like t½(n-2); 0 < n < 1, for elliptic wing loading n ≃ ½.