Two different approaches have been used to calculate turbulent flow along a long thin cylinder where the flow is aligned with the cylinder. A boundary-layer code is used to predict the mean flow for very long cylinders (length to radius ratio of up to O(106)), with the effects of the turbulence estimated through a turbulence model. Detailed comparison with experimental results shows that the mean properties of the flow are predicted within experimental accuracy. The boundary-layer model predicts that, sufficiently far downstream, the surface shear stress will be (almost) constant. This is consistent with experimental results from long cylinders in the form of sonar arrays. A periodic Navier–Stokes problem is formulated, and solutions generated for Reynolds number from 300 to 5×104. The results are in agreement with those from the boundary-layer model and experiments. Strongly turbulent flow occurs only near the surface of the cylinder, with relatively weak turbulence over most of the boundary layer. For a thick boundary layer with the boundary-layer thickness much larger than the cylinder radius, the mean flow is effectively constant near the surface, in both temporal and spatial frameworks, while the outer flow continues to develop in time or space. Calculations of the circumferentially averaged surface pressure spectrum show that, in physical terms, as the radius of the cylinder decreases, the surface noise from the turbulence increases, with the maximum noise at a Reynolds number of O(103). An increase in noise with a decrease in radius (Reynolds number) is consistent with experimental results.

The turbulent motion of buoyant plumes released into turbulently convecting environments is studied. By assuming that the turbulent environment removes fluid from the plume at a rate proportional to a characteristic environmental velocity scale, we derive a model describing the fluid behaviour. For the example of pure buoyancy plumes, entrainment dominates near the source and the plume radius increases with distance, while further from the source removal, or extrainment, of plume material dominates, and the plume radius decreases to zero. Theoretical predictions are consistent with laboratory experiments, a major feature of which is the natural variability of the convection. We extend the study to include the evolution of a finite confined environment, the end-member regimes of which are a well-mixed environment at all times (high convective velocities), and a ‘filling-box’ model similar to that of Baines & Turner (1969) (low convective velocities). These regimes, and the motion of the interface in a ‘filling-box’ experiment, match experimental observations. We find that the convecting filling box is not stable indefinitely, but that the density stratification will eventually be overcome by thermal convection. The model presented here has important applications in volcanology, ventilation studies and environmental science.

An experimental investigation of an idealized rapidly sheared granular flow was performed to test the predictions of a model based on the kinetic theory of dry granular media. Glass ballotini beads were placed in an annular shear cell and the lower boundary rotated to induce a shearing motion in the bed. A single particle was tracked using the positron emission particle tracking (PEPT) technique, a method that determines the location of a particle through the triangulation of gamma photons emitted by a radioactive tracer particle. The packing fraction and velocity fields within the three-dimensional flow were measured and compared to the predictions of a model developed using the conservation and balance equations applicable to dissipative systems, and solved incorporating constitutive relations derived from kinetic theory. The comparison showed that kinetic theory is able to capture the general features of a rapid shear flow reasonably well over a wide range of shear rates and confining pressures.

The normal-mode linear analysis of an axisymmetric infinite capillary jet is generalized to account for arbitrary initial conditions. An exhaustive study of the dispersion relation reveals the parametric behaviour of all eigenvalues and their corresponding normal modes. The two capillary modes (dominant and subdominant) are found to be necessary and sufficient to describe any possible non-recirculating initial conditions. An infinite set of other modes accounts for initial conditions with recirculating velocity field. The predictions of the normal-mode analysis are contrasted against previous computations of the initial-value problem, previous experiments, and our own one-dimensional numerical simulations. Contrary to the claim of some authors, the normal-mode analysis accurately predicts the initial transient with non-exponential growth of the disturbance amplitude observed in previous works. Simple and accurate formulae for the duration of the initial transient are deduced, with emphasis on improving the growth-rate measurement.

The formation of soap films by vertical withdrawal from a bath is typically described by Frankel's law, which assumes rigid film ‘walls’ and shear-like dynamics. Since most soap films have interfaces that are not rigid, and as the flow in the withdrawal of thin free films is typically extensional, we reconsider the theory of soap film formation. By assuming extensional flow dominated by surface viscous stresses we find that the film thickness scales as the two-thirds power of the withdrawal speed U. This speed dependence is also predicted by Frankel's law; the difference lies in the origin of the viscous resistance which sets the pre-factor. When bulk viscous stresses are important the speed dependence can vary between U2/3 and U2.

We study the mixing properties of two systems: (i) a half-filled quasi-two-dimensional circular drum whose rotation rate is switched between two values and which can be analysed in terms of the existing mathematical formalism of linked twist maps; and (ii) a half-filled three-dimensional spherical tumbler rotated about two orthogonal axes bisecting the equator and with a rotational protocol switching between two rates on each axis, a system which we call a three-dimensional linked twist map, and for which there is no existing mathematical formalism. The mathematics of the three-dimensional case is considerably more involved. Moreover, as opposed to the two-dimensional case where the mathematical foundations are firm, most of the necessary mathematical results for the case of three-dimensional linked twist maps remain to be developed though some analytical results, some expressible as theorems, are possible and are presented in this work. Companion experiments in two-dimensional and three-dimensional systems are presented to demonstrate the validity of the flow used to construct the maps. In the quasi-two-dimensional circular drum, bidisperse (size-varying or density-varying) mixtures segregate to form lobes of small or dense particles that coincide with the locations of islands in computational Poincaré sections generated from the flow model. In the 3d spherical tumbler, patterns formed by tracer particles reveal the dynamics predicted by the flow model.

Direct numerical simulations (DNS) of laminar separation bubbles on a NACA-0012 airfoil at Rec=5×104 and incidence 5° are presented. Initially volume forcing is introduced in order to promote transition to turbulence. After obtaining sufficient data from this forced case, the explicitly added disturbances are removed and the simulation run further. With no forcing the turbulence is observed to self-sustain, with increased turbulence intensity in the reattachment region. A comparison of the forced and unforced cases shows that the forcing improves the aerodynamic performance whilst requiring little energy input. Classical linear stability analysis is performed upon the time-averaged flow field; however no absolute instability is observed that could explain the presence of self-sustaining turbulence. Finally, a series of simplified DNS are presented that illustrate a three-dimensional absolute instability of the two-dimensional vortex shedding that occurs naturally. Three-dimensional perturbations are amplified in the braid region of developing vortices, and subsequently convected upstream by local regions of reverse flow, within which the upstream velocity magnitude greatly exceeds that of the time-average. The perturbations are convected into the braid region of the next developing vortex, where they are amplified further, hence the cycle repeats with increasing amplitude. The fact that this transition process is independent of upstream disturbances has implications for modelling separation bubbles.

We use direct numerical simulation to study turbulent drag reduction by rigid polymer additives, referred to as fibres. The simulations agree with experimental data from the literature in terms of friction factor dependence on Reynolds number and fibre concentration. An expression for drag reduction is derived by adopting the concept of the elastic layer.

A statically stable stratified water column that also contains horizontal property contrasts (either of passive tracer alone or of two dynamically active solutes) is generated and continuously maintained for a long period by releasing two turbulent buoyant plumes of equal buoyancy fluxes into opposite ends of a long channel of water. The bottom outflows from the plumes also continuously excite internal gravity waves that produce a series of counter-flowing quasi-horizontal shear layers which are quasi-stationary relative to the box but whose phase propagates downward through the upward-moving water column. We report that the flow further involves an oscillation associated with the internal waves that gives rise to a sequence of interleaving intrusions across the horizontal gradient region. The wave-driven intrusions are advected upward with the ‘filling-box’ circulation and have the appearance of a spatially growing instability. The intrusions are examined in cases having no horizontal property differences other than a passive tracer. In further experiments where one plume is salt solution and the other is sugar solution, there is vigorous double-diffusive convection on the interleaving intrusions, including salt fingering and diffusive density interfaces, but this convection has only a weak influence on the intrusion thicknesses and velocities. We conclude that under all conditions attained in these experiments, the interleaving is driven by internal waves and not by the property gradients, and we infer that the wave-generated intrusions enhance double-diffusive buoyancy fluxes.

The stability of a hydraulically driven sill flow in a rotating channel with smoothly varying cross-section is considered. The smooth topography forces the thickness of the moving layer to vanish at its two edges. The basic flow is assumed to have zero potential vorticity, as is the case in elementary models of the hydraulic behaviour of deep ocean straits. Such flows are found to always satisfy Ripa's necessary condition for instability. Direct calculation of the linear growth rates and numerical simulation of finite-amplitude behaviour suggests that the flows are, in fact, always unstable. The growth rates and nonlinear evolution depend largely on the dimensionless channel curvature κ=2αg′/f2, where 2α is the dimensional curvature, g′ is the reduced gravity, and f is the Coriolis parameter. Very small positive (or negative) values of κ correspond to dynamically wide channels and are associated with strong instability and the breakup of the basic flow into a train of eddies. For moderate or large values of κ, the instability widens the flow and increases its potential vorticity but does not destroy its character as a coherent stream. Ripa's condition for stability suggests a theory for the final width and potential vorticity that works moderately well. The observed and predicted growth in these quantities are minimal for κ≥1, suggesting that the zero-potential-vorticity approximation holds when the channel is narrower than a Rossby radius based on the initial maximum depth. The instability results from a resonant interaction between two waves trapped on opposite edges of the stream. Interactions can occur between two Kelvin-like frontal waves, between two inertia–gravity waves, or between one wave of each type. The growing disturbance has zero energy and extracts zero energy from the mean. At the same time, there is an overall conversion of kinetic energy to potential energy for κ>0, with the reverse occurring for κ<0. When it acts on a hydraulically controlled basic state, the instability tends to eliminate the band of counterflow that is predicted by hydraulic theory and that confounds hydraulic-based estimates of volume fluxes in the field. Eddy generation downstream of the controlling sill occurs if the downstream value of κ is sufficiently small.

Unsteady compressible flows resulting from the incomplete burst of the shock tube diaphragm are investigated both experimentally and numerically for different initial pressure ratios and opening diameters. The intensity of the shock wave is found to be lower than that corresponding to a complete opening. A heuristic relation is proposed to compute the shock strength as a function of the relative area of the open portion of the diaphragm. Strong pressure oscillations past the shock front are also observed. These multi-dimensional disturbances are generated when the initially normal shock wave diffracts from the diaphragm edges and reflects on the shock tube walls, resulting in a complex unsteady flow field behind the leading shock wave. The limiting local frequency of the pressure oscillations is found to be very close to the ratio of acoustic wave speed in the perturbed region to the shock tube diameter. The power associated with these pressure oscillations decreases with increasing distance from the diaphragm since the diffracted and reflected shocks partially coalesce into a single normal shock front. A simple analytical model is devised to explain the reduction of the local frequency of the disturbances as the distance from the leading shock increases.

In the limit of vanishing viscosity, ν→0, Kolmogorov's two-thirds, 〈(Δυ)2〉~ε2/3r2/3, and five-thirds, E~ε2/3k−5/3, laws are formally equivalent. (Here 〈(Δυ)2〉 is the second-order structure function, ε the dissipation rate, r the separation in physical space, E the three-dimensional energy spectrum, and k the wavenumber.) However, for the Reynolds numbers encountered in terrestrial experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. We ask why this should be. To this end, we create artificial fields of isotropic turbulence composed of a random sea of Gaussian eddies whose size and energy distribution can be controlled. We choose the energy of eddies of scale, s, to vary as s2/3, in accordance with Kolmogorov's 1941 law, and vary the range of scales, γ=smax/smin, in any one realization from γ=25 to γ=800. This is equivalent to varying the Reynolds number in an experiment from Rλ=60 to Rλ=600. We find that, while there is some evidence of a five-thirds law for γ>50; (Rλ>100), the two-thirds law only starts to become apparent when γ approaches 200 (Rλ~240). The reason for this discrepancy is that the second-order structure function is a poor filter, mixing information about energy and enstrophy, and from scales larger and smaller than r. In particular, in the inertial range, 〈(Δυ)2〉 takes the form of a mixed power law, a1 + a2r2 + a3r2/3, where a2r2 tracks the variation in enstrophy and a3r2/3 the variation in energy. These findings are shown to be consistent with experimental data where the ‘pollution’ of the r2/3 law by the enstrophy contribution, a2r2, is clearly evident. We show that higher-order structure functions (of even order) suffer from a similar deficiency.

A general reformulation of the Reynolds stresses created by two-dimensional waves breaking a translational or a rotational invariance is described. This reformulation emphasizes the importance of a geometrical factor: the slope of the separatrices of the wave flow. Its physical relevance is illustrated by two model systems: waves destabilizing open shear flows; and thermal Rossby waves in spherical shell convection with rotation. In the case of shear-flow waves, a new expression of the Reynolds–Orr amplification mechanism is obtained, and a good understanding of the form of the mean pressure and velocity fields created by weakly nonlinear waves is gained. In the case of thermal Rossby waves, results of a three-dimensional code using no-slip boundary conditions are presented in the nonlinear regime, and compared with those of a two-dimensional quasi-geostrophic model. A semi-quantitative agreement is obtained on the flow amplitudes, but discrepancies are observed concerning the nonlinear frequency shifts. With the quasi-geostrophic model we also revisit a geometrical formula proposed by Zhang to interpret the form of the zonal flow created by the waves, and explore the very low Ekman-number regime. A change in the nature of the wave bifurcation, from supercritical to subcritical, is found.

A study of streamwise oriented vortical structures embedded in turbulent boundary layers is performed by investigating an experimental database acquired by stereoscopic particle image velocimetry (SPIV) in a plane normal both to the mean flow and the wall. The characteristics of the experimental data allow us to focus on the spatial organization within the logarithmic region for Reynolds numbers Reθ up to 15000. On the basis of the now accepted hairpin model, relationships and interaction between streamwise vortices are first investigated via computation of two-point spatial correlations and the use of linear stochastic estimation (LSE). These analyses confirm that the shape of the most probable coherent structures corresponds to an asymmetric one-legged hairpin vortex. Moreover, two regions of different dynamics can be distinguished: the near-wall region below y+=150, densely populated with strongly interacting vortices; and the region above y+=150 where interactions between eddies happen less frequently. Characteristics of the detected eddies, such as probability density functions of their radius and intensity, are then studied. It appears that Reynolds number as well as wall-normal independences of these quantities are achieved when scaling with the local Kolmogorov scales. The most probable size of the detected vortices is found to be about 10 times the Kolmogorov length scale. These results lead us to revisit the equation for the mean square vorticity fluctuations, and to propose a new balance of this equation in the near-wall region. This analysis and the above results allow us to propose a new description of the near-wall region, leading to a new scaling which seems to have a good universality in the Reynolds-number range investigated. The possibility of reaching a universal scaling at high enough Reynolds number, based on the external velocity and the Kolmogorov length scale is suggested.

The dominant non-dimensional parameter for isodensity transverse jet flow is the mean jet-to-crossflow velocity ratio, R. In Part 1 (Megerian et al., J. Fluid Mech., vol. 593, 2007, p. 93), experimental results are presented for the behaviour of transverse-jet near-field shear-layer instabilities for velocity ratios in the range 1 < R ≤ 10. A local linear stability analysis is presented in this paper for the subrange R>4, using two different base flows for the transverse jet. The first analysis assumes the flow field to be described by a modified version of the potential flow solution of Coelho & Hunt (J. Fluid Mech., vol. 200, 1989, p. 95), in which the jet is enclosed by a vortex sheet. The second analysis assumes a continuous velocity model based on the same inviscid base flow; this analysis is valid for the larger values of Strouhal number expected to be typical of the most unstable disturbances, and allows prediction of a maximum spatial growth rate for the disturbances. In both approaches, results are obtained by expanding in inverse powers of R so that the free-jet results are obtained as R→∞. The results from both approaches agree in the moderately low-frequency regime. Maximum spatial growth rates and associated Strouhal numbers extracted from the second approach both increase with decreasing velocity ratio R, in agreement with the experimental results from Part 1 in the range 4<R≤10. The nominally axisymmetric mode is found to be the most unstable mode in the transverse-jet shear-layer near-field region, upstream of the end of the potential core. The overall agreement of theoretical and experimental results suggests that convective instability occurs in the transverse-jet shear layer for jet-to-crossflow velocity ratios above 4, and that the instability is strengthened as R is decreased.

We consider the stability of small perturbations to a uniform inviscid compressible flow within a cylindrical linear-elastic thin shell. The thin shell is modelled using Flügge's equations, and is forced from the inside by the fluid, and from the outside by damping and spring forces. In addition to acoustic waves within the fluid, the system supports surface waves, which are strongly coupled to the thin shell. Stability is analysed using the Briggs–Bers criterion, and the system is found to be either stable or absolutely unstable, with absolute instability occurring for sufficiently small shell thicknesses. This is significantly different from the stability of a thin shell containing incompressible fluid, even for parameters for which the fluid would otherwise be expected to behave incompressibly (for example, water within a steel thin shell). Asymptotic expressions are derived for the shell thickness separating stable and unstable behaviour.

We then consider the scattering of waves by a sudden change in the duct boundary from rigid to thin shell, using the Wiener–Hopf technique. For the scattering of an inbound acoustic wave in the rigid-wall section, the surface waves are found to play an important role close to the sudden boundary change. The solution is given analytically as a sum of duct modes.

The results in this paper add to the understanding of the stability of surface waves in models of acoustic linings in aeroengine ducts. The oft-used mass–spring–damper model is regularized by the shell bending terms, and even when these terms are very small, the stability and scattering results are quite different from what has been claimed for the mass–spring–damper model. The scattering results derived here are exact, unique and causal, without the need to apply a Kutta-like condition or to include an instability wave. A movie is available with the online version of the paper.

We study the viscous compressible flow through micro-channels of non-uniform cross-section. A lubrication approximation is applied to analyse the flow through shallow configurations whose gap width is small in comparison with the other characteristic dimensions. Focusing on channels with a symmetric constriction (or cavity) we obtain the solution to the problem by means of a Schwarz–Christoffel transformation. This analytic solution is verified by examining the convergence of numerical simulations with diminishing Reynolds number and gap width. Explicit closed-form expressions for the pressure-head and mass-flow-rate losses in terms of the geometrical parameters characterizing the constriction are presented and discussed in the context of experimental data existing in the literature.