A fully developed turbulent boundary layer with a zero pressure gradient was explored by using temperature as a passive contaminant in order to study the large-scale structure. The temperature tracer was introduced into the flow field by heating the entire wall to approximately 12°C above the free-stream temperature. The most interesting observation was the existence of a sharp internal temperature front, characterized by a rapid decrease in temperature, that extended throughout the entire boundary layer. In the outer, intermittent region, the internal temperature front was always associated with the upstream side of the turbulent bulges, i.e. the ‘backs’. It extended across the entire logarithmic region and was related to the sharp acceleration associated with the bursting phenomenon near the wall. Conditional averages of the velocities measured with the temperature front revealed that it was associated with an internal shear layer. The results suggest that this shear layer provides a dynamical relationship between the large structures in the outer, intermittent region and the bursting phenomenon near the wall.

A model probability density equation is obtained by approximating the convective and diffusive terms in a single-point density formulation of homogeneous turbulent scalar transport, with first-order reaction, in a plume. The equation, which includes the intermittency factor of the scalar field explicitly, is then shown to support similarity solutions under some constraining assumptions. Comparison of the similarity solutions with field measurements of conditioned concentrations shows that they can reproduce the general features of the data for both low intermittency and high intermittency measurement regimes. On the basis of these asymptotic results a speculative modelling of the terms representing entrainment at the plume interface is proposed and a class of similarity solutions for the intermittency factor is obtained by numerical integration.

A necessary condition for linear stability of steady inviscid helical gas flows is found by the generalized progressing-wave expansion method. The criterion obtained is compared with the known Richardson number criteria giving sufficient conditions for stability.

An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.

The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.

The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

This paper is concerned with understanding the dynamic behaviour of cavitating hydraulic machines during unsteady or transient operation. The linear transfer matrices which relate the small fluctuating pressures and mass flow rates at inlet and discharge are functions not only of the frequency but also of the mean operating state of the machine, especially the degree of cavitation. The recent experimental transfer matrices obtained by Ng & Brennen (1978) for some axial flow pumps revealed some dynamic characteristics which were unaccounted for by any existing theoretical analysis; their visual observations suggested that the bubbly cavitating flow in the blade passages could be responsible for these effects.

A theoretical model of the dynamic response of this bubbly blade-passage flow is described in the present paper. Void-fraction fluctuations in this flow result not only from pressure fluctuations but also because the fluctuating angle of attack causes fluctuations in the rate of production of bubbles near the leading edge. The latter causes kinematic waves which interact through the boundary conditions with the dynamic waves caused by pressure fluctuation. The resulting theoretical transfer functions which result are in good qualitative agreement with the experiments; with appropriate choices of two parameters (the practical values of which are difficult to assess) good quantitative agreement is also obtained. The theoretical model also provides one possible explanation of the observation that the pump changes from an essentially passive dynamic element in the absence of cavitation to a progressively more active element as the extent of cavitation increases.

Late transitional and turbulent flows in the mixing-layer region of a round jet are investigated for a range of Reynolds numbers by using flow-visualization and hotwire techniques. Attention is focused on the vortices in the transition region and the large eddies in the turbulent region. The interaction and coalescence of vortex rings in the transition region are described. The transition region is characterized by a growth of three-dimensional flow due to a wave instability of the cores of the vortex rings. The merging of these distorted vortices produces large eddies which can remain coherent up to the end of the potential-core region of the jet. A conditional sampling technique is used to measure eddies moving near the jet centre-line. These eddies differ significantly from the ring vortices as they are three-dimensional and contain irregular small-scale turbulence. However, when averaged, their structure is similar in cross-section to that of a vortex ring. These sampled eddies contribute greatly to local velocity fluctuations and statistical correlations. The experiments indicate a need for careful consideration of the meanings of terms such as ‘vortex’, ‘eddy’ and ‘turbulent flow’. In particular care must be taken to discriminate between the orderly, easily visualized, vortices in the transition regions of free shear flows and the less clearly visualized, but strong, large eddies in the fully developed turbulent regions.

The flow around a circular cylinder placed at various heights above a plane boundary has been investigated experimentally. The cylinder spanned the test section of a wind tunnel and was aligned with its axis parallel to a long plate and normal to the free stream. It was placed 36 diameters downstream of the leading edge of the plate and its height above the plate was varied from zero, the cylinder lying on the surface, to 3·5 cylinder diameters. The thickness of the turbulent boundary layer on the plate at the cylinder position, but with it removed from the tunnel, was equal to 0·8 of the cylinder diameter. Distributions of mean pressure around the cylinder and along the plate were measured at a Reynolds number, based on cylinder diameter, of 4·5 × 104. Spectral analysis of hot-wire signals demonstrated that regular vortex shedding was suppressed for all gaps less than about 0·3 cylinder diameters. For gaps greater than 0·3 the Strouhal number was found to be remarkably constant and the only influence of the plate on vortex shedding was to make it a more highly tuned process as the gap was reduced. Flow-visualization experiments in a smoke tunnel revealed the wake structure at various gap-to-diameter ratios.

Conservable quantities measuring ‘wave activity’ are discussed. The equation for the most fundamental such quantity, wave-action, is derived in a simple but very general form which does not depend on the approximations of slow amplitude modulation, linearization, or conservative motion. The derivation is elementary, in the sense that a variational formulation of the equations of fluid motion is not used. The result depends, however, on a description of the disturbance in terms of particle displacements rather than velocities. A corollary is an elementary but general derivation of the approximate form of the wave-action equation found by Bretherton & Garrett (1968) for slowlyvarying, linear waves.

The sense in which the general wave-action equation follows from the classical ‘energy-momentum-tensor’ formalism is discussed, bringing in the concepts of pseudomomentum and pseudoenergy, which in turn are related to special cases such as Blokhintsev's conservation law in acoustics. Wave-action, pseudomomentum and pseudoenergy are the appropriate conservable measures of wave activity when ‘waves’ are defined respectively as departures from ensemble-, space- and time-averaged flows.

The relationship between the wave drag on a moving boundary and the fluxes of momentum and pseudomomentum is discussed.

Two kinds of experiment were made in the wake of a cylinder at Reynolds numbers ranging between 20 and 150. One was a close look at the structure of the vortex street with a stationary cylinder at Reynolds numbers greater than 48. The other experiment was made at lower Reynolds numbers with a cylinder vibrating normal to the flow direction. In this case an artificially induced small-amplitude fluctuation grows exponentially with the rate predicted by the stability theory. Because of the similarity between the two kinds of wake, we postulate that the shedding of the vortex at low Reynolds numbers is initiated by the linear growth, namely, the fluctuation with the frequency of maximum linear growth rate develops into vortex streets. By using the measured width of the wake at the stagnation point in the wake and the result of the stability theory, we could calculate the Strouhal number for Reynolds numbers ranging from 48 to 120. The predicted Strouhal numbers agree well with the values from direct measurements.

This paper is concerned with small amplitude vortical and entropic unsteady motions imposed on steady potential flows. Its main purpose is to show that, even in this unsteady compressible and vortical flow, the perturbations in pressure p’ and velocity u can be written as p’ = ρ0D0ϕ/Dt and u = ϕ + u(I) respectively, where D0/Dt is the convective derivative relative to the mean potential flow, u(I) is a known function of the imposed upstream disturbance and ϕ is a solution to the linear inhomogeneous wave equation \[ \frac{D_0}{Dt}\bigg(\frac{1}{c^2_0}\frac{D_0\phi}{Dt}\bigg)-\frac{1}{\rho_0}\nabla\cdot(\rho_0\nabla\phi)=\frac{1}{\rho_0}\nabla\cdot\rho_0{\bf u}^{(I)} \] with a dipole source term ρ0−1 [xdtri ]ρ0u(I) whose strength ρ0u(I) is a known function of the imposed upstream distortion field. (Here c0 and ρ0 denote the speed of sound and density of the background potential flow.) This equation is used to extend Hunt's (1973) generalization of the ‘rapid-distortion’ theory of turbulence developed by Batchelor & Proudman (1954) and Ribner & Tucker (1953). These theories predict changes occurring in weakly turbulent flows that are distorted (by solid obstacles and other external influences) in a time short relative to the Lagrangian integral scale.

The theory is applied to the unsteady supersonic flow around a corner and a closed-form analytical solution is obtained. Detailed calculations are carried out to show how the expansion at the corner affects a turbulent incident stream.

The problem considered is the diffusion of a passive scalar in a ‘fluid’ in random motion when the fluid velocity field is Gaussian and statistically homogeneous, isotropic and stationary. A self-consistent expansion for the effective long-time diffusivity is obtained and the approximations derived from this series by retaining up to three terms are explicitly calculated for simple idealized forms of the velocity correlation function for which numerical simulations are available for comparison for zero molecular diffusivity. The dependence of the effective diffusivity on the molecular diffusivity is determined within this idealization. The results support Saffman's contention that the molecular and turbulent diffusion processes interfere destructively, in the sense that the total effective diffusivity about a fixed point is less than that which would be obtained if the two diffusion processes acted independently.

Quantitative velocity measurements of unsteady convection in a Bénard cell suddenly cooled from above to a supercritical Rayleigh number are reported. The results are obtained by a novel application of laser speckle photography adapted for the study of fluid dynamics. This technique allows two-dimensional full-field velocity information to be obtained solely from a selected plane within the volume of moving liquid at any given instant in time. The convective rolls are found to be regular and oriented parallel to the short side of the cell. The number of rolls within the cell is approximately twice as great as the number reported for steady conditions. The maximum vertical velocity distribution is observed to be a sinusoidal function of the horizontal distance with a detectable third-harmonic component.

A visual study of a turbulent boundary-layer flow was conducted by photographing the motions of small tracer particles using a stereoscopic medium-speed camera system moving with the flow. In some experiments, dye injection at the leading edge of the flat plate helped to delineate the outer edge of the boundary layer. The technique allowed the three-dimensional aspects of the flow to be studied in some detail, and in particular allowed axial vortex motions in the wall region to be identified.

The flow was found to exhibit three characteristic regions which can be roughly divided into the wall and outer regions of the boundary layer and an irrotational region, unmarked by dye, outside the instantaneous edge of the boundary layer. Briefly, the outer region of the boundary layer was dominated by transverse vortex motions that formed as a result of an interaction between low-speed and high-speed (sweep) fluid elements in that region. The present results clearly show that bulges in the edge of the boundary layer are associated with transverse vortex motions. In addition, the transverse vortex motions appear to induce massive inflows of fluid from the irrotational region deep into the outer region of the boundary layer. The outer edge of the boundary layer thus becomes further contorted, contributing to the intermittency of the region. Furthermore, the outer-region motions give rise to the conditions necessary for the dominant wall-region activity of ejections and axial vortex motions. It is not the energetic wall-region ejections that move to the outer region and give rise to the contorted edge of the boundary layer as has been suggested by others.

The wall-region axial vortex motions were intense and lasted for a time short compared with the lifetime of outer-region transverse vortex motions. The present results strongly suggest that wall-region vortex motions are a result of interaction between the incoming higher-speed fluid from the outer region of the boundary layer and the outflowing low-speed wall-region fluid. This is in direct contrast to all models that suggest that axial vortex pairs in the wall region are the factor that gives rise to the outflow of low-speed fluid trapped between.

Although all the elements necessary to make up a horseshoe vortex structure riding along the wall were present, such a composite was not observed. However, this could be visualized as a possible model to represent the ensemble average of the flow.

Finally, the massive inflows from the irrotational region were observed to precede the appearance of low- and high-speed fluid elements in the boundary layer, thus completing the deterministic cycle of individual coherent events.

The geometry of large-scale structures in the turbulent mixing layer of a moderate Reynolds number jet is deduced from measurements of the fluctuating pressure in the hydrodynamic near field. The structures are rings of concentrated vorticity that distort with downstream distance until statistical axisymmetry disappears. The rings are spaced quasi-periodically and coalesce with each other, producing larger spacings. Statistical and flow-visualization techniques are applied to free and forced jets over a range of Reynolds numbers from 5000 to 50000 to demonstrate that rings of a given spacing do not coalesce with each other until they are far enough downstream that the local mixing layer has attained some critical thickness which scales with the wavelength of the vortex pair. Wave dispersion is evaluated as a plausible mechanism for localizing the coalescences. The central feature of the model is the observation that a shear layer is dispersive to wavelengths much longer than its thickness and non-dispersive to shorter waves.

The self-similar solution of the gasdynamic equations of a strong cylindrical shock wave moving through an ideal gas, with γ = cp/cv, is considered. These equations are greatly simplified following the transformation of the reduced velocity \[ U_1(\xi)\rightarrow U_1 = {\textstyle\frac{1}{2}}(\gamma + 1)(U+\xi). \] The requirement of a single maximum pressure, dζP = 0, leads to an analytical determination of the self-similarity exponent α(γ). For gases with γ < 2 + 3½ the slight maximum pressure occurs behind the shock front, nearing it as γ increases. For γ < 2 + 3½, this maximum ensues right at the shock front and the pressure distribution then decreases monotonically. The postulate of analyticity by Gelfand and Butler is shown to concur with the requirement dζP = 0. The saturated density of the gas left in the wake of the shock is computed and − U is shown to be the reduced velocity of sound at P = Pm.

The temperature dependence of the density of both pure and saline water, even to very high salinity and pressure levels, decreases at decreasing temperature toward an extremum. The nature of this variation precludes approximating the buoyancy-force density difference linearly with a temperature difference. This peculiar density variation of water has very significant effects, even at environmental temperature levels. A new equation has appeared which relates density to temperature, salinity and pressure with very high accuracy. Its form is especially suited to the analysis of convective motions. We consider here vertical boundary-layer flows. Analysis of flows arising from thermal buoyancy and from combined buoyancy effects shows the simplicity of the formulation. Relatively few new parameters arise. Extensive calculations for thermally buoyant flows show the large magnitude of the effects of the complicated density variation on transport. Buoyancy-force reversals and convective inversions are predicted. The latter are in close agreement with past experiments. A new Grashof number arises which is an accurate indication of the actual local flow vigour. The effects of specific temperature conditions are given in detail. The appreciable effect of the Prandtl number is calculated. Transport parameters are given for salinities and pressures up to 40 p.p.t. and 1000 bars, respectively.

Measurements have been made of the time-mean velocity, all three fluctuating velocity components and the turbulence shear stress in the plane mixing layer produced in an open wind tunnel between two nominally parallel air streams with fairly low turbulence levels. The hot-wire measuring system is described, the evaluation equations derived and the principal results presented.

The theory presented here describes the motion of a large gas bubble rising through upward-flowing liquid in a tube. The basis of the theory is that the liquid motion round the bubble is inviscid, with an initial distribution of vorticity which depends on the velocity profile in the liquid above the bubble. Approximate solutions are given for both laminar and turbulent velocity profiles and have the form \begin{equation} U_s = U_c+(gD)^{\frac{1}{2}}\phi(U_c/(gD)^{\frac{1}{2}}), \end{equation}Us being the bubble velocity, Uc the liquid velocity at the tube axis, g the acceleration due to gravity, and D the tube diameter; ϕ indicates a functional relationship the form of which depends upon the shape of the velocity profile. With a turbulent velocity profile, a good approximation to (1) which is suitable for many practical purposes is \begin{equation} U_s = U_s + U_{s0}, \end{equation}Us0 being the bubble velocity in stagnant liquid. Published data for turbulent flow are known to agree with (2), so that in this case the theory supports a well-known empirical result. Our laminar flow experiments confirm the validity of (1) for low liquid velocities.

This paper describes the results of our experiments with shock waves refracting at a CO2/CH4 interface. The refraction is slow-fast because the speed of sound in the incident gas (CO2) is less than that in the transmitting gas (CH4). We found three phenomena which apparently have not been reported before and which all have free precursor shocks in their wave systems; schlieren photographs of them are presented. As a result of the present and earlier work, we can assert that there exist at least four different free precursor refractions. Theoretical studies suggest that the slow-fast phenomena can be conveniently classified into three groups characterized by different ranges of values of the inverse strength ζi of the incident shock i. The classification may be an exhaustive list of the phenomena, at least when the gases are nearly perfect, but we cannot be sure. We present experimental data on all the phenomena in each group, including data on the transition conditions from one wave system to another both within and across the groups.

We study the stability of the cylindrical Couette flow in nematics when the director is parallel to the rotation axis. The contribution of the inertial coupling of velocity fluctuations (responsible for the Taylor instability in isotropic liquids) is shown to be destabilizing when the inner cylinder rotates faster than the outer one. However, the instability remains driven by the mechanisms first discovered by Pieranski & Guyon for the plane shear case and quite specific to nematics. This mechanism couples the different orientation fluctuations via viscous torques and the corresponding threshold is given by \[ s\tau_0\sim 1, \] where τ0 is the time constant for the diffusion of orientation fluctuations. The contribution of inertia terms is measured by 2ωm τv, where τv is the time constant for the diffusion of velocity fluctuations. In usual nematics one has τv/τ0 ∼ 10−5 so that corrections due to rotation are small in general. At different stages of the discussion differences between the case of nematics and that of isotropic liquids are pointed out. We also study the possibility of an oscillatory instability when α3 is positive and large, where no stationary instability can occur.