Conditionally sampled hot-wire measurements were taken in a pipe at Reynolds numbers corresponding to the onset of turbulence. The pipe was smooth and carefully aligned so that turbulent slugs appeared naturally at Re > 5 × 104. Transition could be initiated at lower Re by introducing disturbances into the inlet. For smooth or only slightly disturbed inlets, transition occurs as a result of instabilities in the boundary layer long before the flow becomes fully developed in the pipe. This type of transition gives rise to turbulent slugs which occupy the entire cross-section of the pipe, and they grow in length as they proceed downstream. The leading and trailing ‘fronts’ of a turbulent slug are clearly defined. A unique relation seems to exist between the velocity of the interface and the velocity of the fluid by which relaminarization of turbulent fluid is prevented. The length of slugs is of the same order of magnitude as the length of the pipe, although the lengths of individual slugs differ at the same flow conditions. The structure of the flow in the interior of a slug is identical to that in a fully developed turbulent pipe flow. Near the interfaces, where the mean motion changes from a laminar to a turbulent state, the velocity profiles develop inflexions. The total turbulent intensity near the interfaces is very high and it may reach 15% of the velocity at the centre of the pipe. A turbulent energy balance was made for the flow near the interfaces. All of the terms contributing to the energy balance must vanish identically somewhere on the interface if that portion of the interface does not entrain non-turbulent fluid. It appears that diffusion which also includes pressure transport is the most likely mechanism by which turbulent energy can be transferred to non-turbulent fluid. The dissipation term at the interface is negligible and increases with increasing turbulent energy towards the interior of the slug.

Mixed laminar and turbulent flows were observed far downstream for \[ 2000 < Re < 2700 \] when a large disturbance was introduced into the inlet. The flow in the vicinity of the inlet, however, was turbulent at much lower Re. The turbulent regions which are convected downstream at a velocity which is slightly smaller than the average velocity in the pipe we shall henceforth call puffs. The leading front of a puff does not have a clearly defined interface and the trailing front is clearly defined only in the vicinity of the centre-line. The length and structure of the puff is independent of the character of the obstruction which created it, provided that the latter is big enough to produce turbulent flow at the inlet. The puff will be discussed in more detail later.

The wake of a circular cylinder is investigated near the oscillation threshold by means of a laser probe. Above the threshold the transient regime is studied and described by a Stuart-Landau law (already found to be relevant in explaining free-oscillating regimes). Below the critical point, impulse and resonant regimes are examined, so the coefficients of the Stuart-Landau equation are determined.

Moreover, in the supercritical regime, the behaviour of the (externally forced) oscillating system is described, varying parameters such as threshold deviation, forcing frequency and amplitude. The different zones of entrainment and desynchronization are given for simple or harmonic frequency.

The stability properties of the flow past an infinitely long circular cylinder are studied in the context of linear theory. An immersed-boundary technique is used to represent the cylinder surface on a Cartesian mesh. The characteristics of both direct and adjoint perturbation modes are studied and the regions of the flow more sensitive to momentum forcing and mass injection are identified. The analysis shows that the maximum of the perturbation envelope amplitude is reached far downstream of the separation bubble, where as the highest receptivity is attained in the near wake of the cylinder, close to the body surface. The large difference between the spatial structure of the two-dimensional direct and adjoint modes suggests that the instability mechanism cannot be identified from the study of either eigenfunctions separately. For this reason a structural stability analysis of the problem is used to analyse the process which gives rise to the self-sustained mode. In particular, the region of maximum coupling among the velocity components is localized by inspecting the spatial distribution of the product between the direct and adjoint modes. Results show that the instability mechanism is located in two lobes placed symmetrically across the separation bubble, confirming the qualitative results obtained through a locally plane-wave analysis. The relevance of this novel technique to the development of effective control strategies for vortex shedding behind bluff bodies is illustrated by comparing the theoretical predictions based on the structural perturbation analysis with the experimental data of Strykowski & Sreenivasan (J. Fluid Mech. vol. 218, 1990, p. 71).

A theoretical method is given for the determination of the shape of a fluid drop in steady and unsteady flows by making an expansion in terms of the drop deformation. Effects of fluid viscosity and interfacial tension are taken into account. Examples given include the determination of the shape of a drop in shear and in hyperbolic flow when each is started impulsively from rest.

A visualization method is used to obtain the main features of the hydrodynamic field for flow past a circular cylinder moving at a uniform speed in a direction perpendicular to its generating lines in a tank filled with a viscous liquid. Photographs are presented to show the particular fineness of the experimental technique. More especially, the closed wake and the velocity distribution behind the obstacle are investigated; the changes in the geometrical parameters describing the eddies with Reynolds number (5 < Re < 40) and with the ratio λ between the diameters of the cylinder and tank are given. A comparison with existing numerical and experimental results is presented and some remarks are made about the calculation techniques proposed up to the present. The limits of the Reynolds-number range for which the twin vortices exist and adhere stably to the cylinder are determined.

The primary objective of this investigation is to determine experimentally the sources of jet mixing noise. In the present study, four different approaches are used. It is reasonable to assume that the characteristics of the noise sources are imprinted on their radiation fields. Under this assumption, it becomes possible to analyse the characteristics of the far-field sound and then infer back to the characteristics of the sources. The first approach is to make use of the spectral and directional information measured by a single microphone in the far field. A detailed analysis of a large collection of far-field noise data has been carried out. The purpose is to identify special characteristics that can be linked directly to those of the sources. The second approach is to measure the coherence of the sound field using two microphones. The autocorrelations and cross-correlations of these measurements offer not only valuable information on the spatial structure of the noise field in the radial and polar angle directions, but also on the sources inside the jet. The third approach involves measuring the correlation between turbulence fluctuations inside a jet and the radiated noise in the far field. This is the most direct and unambiguous way of identifying the sources of jet noise. In the fourth approach, a mirror microphone is used to measure the noise source distribution along the lengths of high-speed jets. Features and trends observed in noise source strength distributions are expected to shed light on the source mechanisms. It will be shown that all four types of data indicate clearly the existence of two distinct noise sources in jets. One source of noise is the fine-scale turbulence and the other source is the large turbulence structures of the jet flow. Some of the salient features of the sound field associated with the two noise sources are reported in this paper.

When a smooth jet of water falls vertically on to a horizontal plane, it spreads out radially in a thin layer bounded by a circular hydraulic jump, outside which the depth is much greater. The motion in the layer is studied here by means of boundary-layer theory, both for laminar and for turbulent flow, and relations are obtained for the radius of the hydraulic jump. These relations are compared with experimental results. The analogous problems of two-dimensional flow are also treated.

Experimental results for the release of a fixed volume of one homogeneous fluid into another of slightly different density are presented. From these results and those obtained by previous experiments, it is argued that the resulting gravity current can pass through three states. There is first a slumping phase, during which the current is retarded by the counterflow in the fluid into which it is issuing. The current remains in this slumping phase until the depth ratio of current to intruded fluid is reduced to less than about 0.075. This may be followed by a (previously investigated) purely inertial phase, wherein the buoyancy force of the intruding fluid is balanced by the inertial force. Motion in the surrounding fluid plays a negligible role in this phase. There then follows a viscous phase, wherein the buoyancy force is balanced by viscous forces. It is argued and confirmed by experiment that the inertial phase is absent if viscous effects become important before the slumping phase has been completed. Relationships between spreading distance and time for each phase are obtained for all three phases for both two-dimensional and axisymmetric geometries. Some consequences of the retardation of the gravity current during the slumping phase are discussed.

The ability of a large-eddy simulation to represent the large-scale motions in the interior of a turbulent flow is well established. However, concerns remain for the behaviour close to rigid surfaces where, with the exception of low-Reynolds-number flows, the large-eddy description must be matched to some description of the flow in which all except the larger-scale ‘inactive’ motions are averaged. The performance of large-eddy simulations in this near-surface region is investigated and it is pointed out that in previous simulations the mean velocity profile in the matching region has not had a logarithmic form. A number of new simulations are conducted with the Smagorinsky (1963) subgrid model. These also show departures from the logarithmic profile and suggest that it may not be possible to eliminate the error by adjustments of the subgrid lengthscale. An obvious defect of the Smagorinsky model is its failure to represent stochastic subgrid stress variations. It is shown that inclusion of these variations leads to a marked improvement in the near-wall flow simulation. The constant of proportionality between the magnitude of the fluctuations in stress and the Smagorinsky stresses has been empirically determined to give an accurate logarithmic flow profile. This value provides an energy backscatter rate slightly larger than the dissipation rate and equal to idealized theoretical predictions (Chasnov 1991).

We report on lattice-Boltzmann simulations of slow fluid flow past mono- and bidisperse random arrays of spheres. We have measured the drag force on the spheres for a range of diameter ratios, mass fractions and packing fractions; in total, we studied 58 different parameter sets. Our simulation data for the permeability agrees well with previous simulation results and the experimental findings. On the basis of our data for the individual drag force, we have formulated new drag force relations for both monodisperse and polydisperse systems, based on the Carman–Kozeny equations; the average deviation of our binary simulation data with the new relation is less than 5%. We expect that these new relations will significantly improve the numerical modelling of gas–solid systems with polydisperse particles, in particular with respect to mixing and segregation phenomena. For binary systems with large diameter ratios (1:4), the individual drag force on a particle, as calculated from our relations, can differ by up to a factor of five compared with predictions presently favoured in chemical engineering.

The non-equilibrium behaviour of concentrated colloidal dispersions is studied using Stokesian Dynamics, a molecular-dynamics-like simulation technique for analysing suspensions of particles immersed in a Newtonian fluid. The simulations are of a monodisperse suspension of Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of hydrodynamic and Brownian forces, over a range of volume fraction 0.316 [les ] ϕ [les ] 0.49. For Pe < 10, Brownian motion dominates the behaviour, the suspension remains well-dispersed, and the viscosity shear thins. The first normal stress difference is positive and the second negative. At higher Pe, hydrodynamics dominate resulting in an increase in the long-time self-diffusivity and the viscosity. The first normal stress difference changes sign when hydrodynamics dominate. Simulation results are shown to agree well with both theory and experiment.

An apparatus is described for determining particle passages through any selected point on a tube cross-section. The method depends on the simultaneous blocking out of two mutually perpendicular light beams by a particle passing through their common region. The coincidence is registered and counted electronically. At higher particle concentrations coincidences are also registered arising from a pair-wise occupation of the light beams by two particles. An analysis is presented showing that these pair coincidences can be allowed for exactly in terms of experimentally measurable quantities.

The reliability and reproducibility of the method is discussed and illustrated by examples from sphere suspensions in Poiseuille flow.

A formal algebraic analogy is drawn between meteorological parameters, such as the Richardson number, and the parameters describing the effect of rotation or streamline curvature on a turbulent flow. The analogy between the phenomena is a good first approximation. Semi-quantitative use of the analogy to apply meteorological data to curved shear layers shows that the effects of curvature on the apparent mixing length are appreciable if the shear-layer thickness exceeds roughly 1/300 of the radius of curvature; larger effects may occur in compressible flow. Application of the Monin-Oboukhov formula considerably improves the agreement between prediction and experiment in boundary layers on curved surfaces.

This paper summarizes experimental studies of incompressible elliptic jets of different aspect ratios and initial conditions, and effects of excitations at selected frequencies and amplitudes. Elliptic jets are quite different from the extensively studied plane and circular jets - owing mainly to the fact that the azimuthal curvature variation of a vortical structure causes its non-uniform self-induction and hence complex three-dimensional deformation. Such deformation, combined with properly selected excitation can substantially alter entrainment and other turbulence phenomena, thus suggesting preference for the elliptic shape in many jet applications. The dominance of coherent structures in the jet far field is evident from the finding that switching over of the cross-section shape continues at least up to 100 equivalent diameters De. The locations and the number of switchovers are strongly dependent on the initial condition, on the aspect ratio, and, when excited, on the Strouhal number and the excitation level. We studied jets with constant exit momentum thickness θe, all around the perimeter, thus separating the effects of azimuthal variations of θe, (typical of elliptic jets) and of the shear-layer curvature. Also investigated are the instability characteristics, and enhanced entrainment caused by bifurcation as well as pairing of vortical structures. We discuss shear-layer and jet- column domains, and find the latter to be characterized by two modes : the preferred mode and the stable pairing mode - similar to those found in circular jets -both modes scaling on the newly-defined lengthscale De. The paper documents some time- average measurements and their comparison with those in circular and plane jets.

Direct numerical solutions of the three-dimensional time-dependent Navier-Stokes equations are presented for the evolution of three-dimensional finite-amplitude disturbances of plane Poiseuille and plane Couette flows. Spectral methods using Fourier series and Chebyshev polynomial series are used. It is found that plane Poiseuille flow can sustain neutrally stable two-dimensional finite-amplitude disturbances at Reynolds numbers larger than about 2800. No neutrally stable two-dimensional finite-amplitude disturbances of plane Couette flow were found.

Three-dimensional disturbances are shown to have a strongly destabilizing effect. It is shown that finite-amplitude disturbances can drive transition to turbulence in both plane Poiseuille flow and plane Couette flow at Reynolds numbers of order 1000. Details of the resulting flow fields are presented. It is also shown that plane Poiseuille flow cannot sustain turbulence at Reynolds numbers below about 500.

The onset of periodic behaviour in two-dimensional laminar flow past bodies of various shapes is examined by means of finite-element simulations. The transition from steady to periodic flow is marked by a Hopf bifurcation, which we locate by solving an appropriate extended set of steady-state equations. The bodies considered are a circular cylinder, triangular prisms of various shapes, and flat plates and elliptical cylinders aligned over a range of angles to the direction of flow. Our results for the circular cylinder are in good agreement with experimental observations and with the results of time-dependent calculations.

Experiments have been performed on the tip vortex trailing from a rectangular NACA 0012 half-wing. Preliminary studies showed the vortex to be insensitive to the introduction of a probe and subject only to small wandering motions. Meaningful velocity measurements could therefore be made using hot-wire probes.

Detailed analysis of the effects of wandering was performed to properly reveal the flow structure in the core region and to give confidence in measurements made outside the core. A theory has been developed to correct mean-velocity profiles for the effects of wandering and to provide complete quantitative estimates of its amplitude and contributions to Reynolds stress fields. Spectral decomposition was found to be the most effective method of separating these contributions from velocity fluctuations due to turbulence.

Outside the core the flow structure is dominated by the remainder of the wing wake which winds into an ever-increasing spiral. There is no large region of axisymmetric turbulence surrounding the core and little sign of turbulence generated by the rotational motion of the vortex. Turbulence stress levels vary along the wake spiral in response to the varying rates of strain imposed by the vortex. Despite this complexity, the shape of the wake spiral and its turbulent structure reach an approximately self-similar form.

On moving from the spiral wake to the core the overall level of velocity fluctuations greatly increases, but none of this increase is directly produced by turbulence. Velocity spectra measured at the vortex centre scale in a manner that implies that the core is laminar and that velocity fluctuations here are a consequence of inactive motion produced as the core is buffeted by turbulence in the surrounding spiral wake. Mean-velocity profiles through the core show evidence of a two-layered structure that dies away with distance downstream.

In most work on the theory of stability of laminar flow, infinitesimal disturbances only have been considered, so that only the initial growth of the disturbance has been determined. It is the object of the present paper to extend the theory to larger amplitudes and to study the mechanics of disturbance growth with the inherent non-linearity of the hydrodynamical system taken into account.

The Reynolds stress (where averages are taken with respect to some suitable space coordinate) is the fundamental consequence of the non-linearity, and its effects can be anticipated as follows. Initially a disturbance grows exponentially with time according to the linear theory, but eventually it reaches such a size that the transport of momentum by the finite fluctuations is appreciable and the associated mean stress (the Reynolds stress) then has an appreciable effect on the mean flow. This distortion of the mean flow modifies the rate of transfer of energy from the mean flow to the disturbance and, since this energy transfer is the cause of the growth of the disturbance, there is a modification of the rate of growth of the latter.

It is suggested that, in many cases, an equilibrium state may be possible in which the rate of transfer of energy from the (distorted) mean flow to the disturbance balances precisely the rate of viscous dissipation of the energy of disturbance. A theory based on certain assumptions about the energy flow is given to describe both the growth of the disturbance and the final equilibrium state, and application is made to the cases of Poiseuille flow between parallel planes and flow between rotating cylinders. The distorted mean flow in the equilibrium state can be calculated and from this, in the latter case, the torque required to maintain the cylinders in motion. Good agreement is obtained with G. I. Taylor's measurements of the torque for the case when the inner cylinder rotates and the outer cylinder is at rest.

Some of the consequences of the conservation of magnetic helicity $\int \rm{a.b}\it{d}^{\rm{3}}\rm{r\qquad (a\; =\; vector\; potential\; of\; magnetic\; field\; b)}$ for incompressible three-dimensional turbulent MHD flows are investigated. Absolute equilibrium spectra for inviscid infinitely conducting flows truncated at lower and upper wavenumbers kmin and kmax are obtained. When the total magnetic helicity approaches an upper limit given by the total energy (kinetic plus magnetic) divided by kmin, the spectra of magnetic energy and helicity are strongly peaked near kmin; in addition, when the cross-correlations between the velocity and magnetic fields are small, the magnetic energy density near kmin greatly exceeds the kinetic energy density. Several arguments are presented in favour of the existence of inverse cascades of magnetic helicity towards small wavenumbers leading to the generation of large-scale magnetic energy.

A conceptual framework for analysing the energetics of density-stratified Boussinesq fluid flows is discussed. The concept of gravitational available potential energy is used to formulate an energy budget in which the evolution of the background potential energy, i.e. the minimum potential energy attainable through adiabatic motions, can be explicitly examined. For closed systems, the background potential energy can change only due to diabatic processes. The rate of change of background potential energy is proportional to the molecular diffusivity. Changes in the background potential energy provide a direct measure of the potential energy changes due to irreversible diapycnal mixing. For open systems, background potential energy can also change due to boundary fluxes, which can be explicitly measured. The analysis is particularly appropriate for evaluation of diabatic mixing rates in numerical simulations of turbulent flows. The energetics of a shear driven mixing layer is used to illustrate the analysis.