We use direct numerical simulation to study the mixing behaviour of pulsed jets in crossflow. The pulse is a square wave and the simulations consider several jet velocity ratios and pulse conditions. Our objective is to study the effects of pulsing and to explain the wide range of optimal pulsing conditions found in experimental studies of the problem. The central theme is that pulsing generates vortex rings; the effect of pulsing on transverse jets can therefore be explained by the behaviour of vortex rings in crossflow. Sau & Mahesh (J. Fluid Mech., vol. 604, 2008, pp. 389–409) show that vortex rings in crossflow exhibit three distinct flow regimes depending on stroke and ring velocity ratios. The simulations of pulsed transverse jets in this paper show that at high velocity ratios, optimal pulse conditions correspond to the transition of the vortex rings produced by pulsing between the different regimes. At low velocity ratios, optimal pulsing conditions are related to the natural time scale on which hairpin vortices form. An optimal curve in the space of stroke and velocity ratios is presented. Data from various experiments are interpreted in terms of the properties of the equivalent vortex rings and shown to collapse on the optimal curve. The proposed regime map allows the effects of experimental parameters such as pulse frequency, duty cycle, modulation and pulse energy all to be predicted by determining their effect on the equivalent stroke and velocity ratios.

This paper presents some new velocity correlation measurements in a two-dimensional zero-pressure-gradient turbulent boundary layer; these extend the work of Grant (1958). The most detailed of the new measurements concern: correlations between velocity fluctuations in the mean flow direction and fluctuations perpendicular to the wall at a different point, usually but not always in the same plane parallel to the wall; correlations between velocity fluctuations in the mean flow direction and fluctuations perpendicular to this but parallel to the wall, usually but not always close to the wall with the separation in the same direction as the latter velocity fluctuation; and various correlations with a fixed separation normal to the wall and a simultaneous variable separation in the mean flow direction.

The results are not consistent with any of the various models of the large eddy structure that have been proposed. The features of the present work most relevant to the formulation of a substitute are brought together in §9. Included are suggestions that the similarities between the wall and outer regions are more marked than the differences, and that the description of the large eddies in the wall region as a coherent eruption from the viscous sublayer is unsatisfactory.

The new experiments are considered in connexion with three questions posed by the previous work. These concern the large eddy contribution to the Reynolds stress; the asymmetry between upstream and downstream separations when there is also a separation normal to the wall; and the similarity between boundary layers and channel flow. A complicated situation surrounds the first point; there are arguments suggesting that the large eddy contribution might in some places be of opposite sign to the total Reynolds stress, but this no longer seems so likely. The upstream-downstream asymmetry is also more complicated than had been previously supposed, but the results can be systematically interpreted if the peaks of the correlation curves are considered in connexion with continuity and the tails of the curves are considered in connexion with the shearing action of the mean flow. Regarding the third point, the present experiments give no indication of any difference in the large eddy structure in the wall regions of boundary layer turbulence and channel flow turbulence.

Section 5 gives a brief account of some experiments with a vibrating ribbon in the turbulent boundary layer.

The subharmonic resonance phenomenon is studied using hot-wire measurements and flow visualization in an initially laminar shear layer forced with two-frequencies for various choices of the fundamental frequency f and its subharmonic f/2 with controlled initial phase difference ϕin between them. We explore the effects of the controlling parameters, namely: (i) forcing frequencies and their initial amplitudes, (ii) initial phase difference ϕin, and (iii) detuning (i.e. when the second forcing frequency is slightly different from f/2). While several of our experimental observations support predictions based on weakly nonlinear theory, others do not. We explain our data in terms of vortex dynamics concepts.

A uniform-mean-gradient shear flow was produced using a ten-layer closed-loop water channel, providing long enough dimensionless flow development times (τ = (x/Ū) (∂ Ū/∂z)) for the turbulence to grow. The rate of growth of the turbulence compares well with similar measurements in wind-tunnel-generated uniform shear flows for which the mean shears and centreline velocities are larger by an order of magnitude. Preliminary investigations were undertaken to study the growth of the turbulent intensity as functions of the mean shear, centreline velocity, and initial disturbance lengthscales. Initial disturbance lengthscales were varied by using grids of different mesh sizes.

Turbulent intensities were found to increase nearly linearly with τ. Differences in grid mesh size produce different offsets in the turbulent intensity level, with a larger grid mesh producing a higher positive offset. This offset persists throughout the growth of the turbulent intensity. These observations provide valuable insight in interpreting previous wind-tunnel measurements, in particular the high-shear experiments of Karnik & Tavoularis (1983). Comparison with the theoretical predictions of Tavoularis (1985) allows for an improved universal characterization of evolving turbulence in a uniform mean shear.

We consider the free boundary problem for the evolution of a nearly straight slender fibre of viscous fluid. The motion is driven by prescribing the velocity of the ends of the fibre, and the free surface evolves under the action of surface tension, inertia and gravity. The three-dimensional Navier–Stokes equations and free-surface boundary conditions are analysed asymptotically, using the fact that the inverse aspect ratio, defined to be the ratio between a typical fibre radius and the initial fibre length, is small. This first part of the paper follows earlier work on the stretching of a slender viscous fibre with negligible surface tension effects. The inclusion of surface tension seriously complicates the problem for the evolution of the shape of the cross-section. We adapt ideas applied previously to two-dimensional Stokes flow to show that the shape of the cross-section can be described by means of a conformal map which depends on time and distance along the fibre axis. We give some examples of suitable relevant conformal maps and present numerical solutions of the resulting equations. We also use analytic methods to examine the coupling between stretching and the evolution of the cross-section shape.

This study is concerned with the effects of particle–particle collisions and the two-way coupling on the dispersed and carrier phase turbulence fluctuations in a channel flow. The time history of the instantaneous turbulent velocity vector was generated by the two-way coupled direct numerical simulation of the Navier–Stokes equations via a pseudo-spectral method. The particle equation of motion included the wall-corrected nonlinear drag force and the wall-induced and shear-induced lift force. The effect of particles on the flow was included in the analysis via a feedback force that acted on the computational grid points. Several simulations for different particle relaxation times and particle mass loadings were performed, and the effects of particle–particle collisions, particle feedback force and inter-particle interactions on the particle deposition velocity, fluid and particle fluctuating velocities, and particle concentration profiles were determined. The effect of particle aerodynamic interactions was also examined for certain cases.

The simulation results indicated that when particle–particle collisions were included in the computation but two-way coupling effects were ignored, the particle normal fluctuating velocity increased in the wall region causing an increase in the particle deposition velocity. When the particle collisions were neglected but the particle–fluid two-way coupling effects were accounted for, the two-way coupling and the particle normal fluctuating velocity decreased near the wall causing a decrease in the particle deposition velocity. In the case of the four-way coupling in which both inter-particle collisions and two-way coupling effects were present, it was found that the particle deposition velocity increased compared with the one-way coupling case. When the particle aerodynamic interactions were added to the four-way coupled case (termed six-way coupled case), no significant changes in the mean fluid and particle velocities and the fluid and particle fluctuating velocities were obtained.

The results for the particle concentration profile indicated that the inclusion of two-way coupling or inter-particle collisions into the computation reduced the accumulation of particles near the wall. It was also observed that particle–particle collisions and two-way coupling weakened the preferential distribution of particles.

The incompressible fluid flow in a rectangular container driven by two facing sidewalls which move steadily in anti-parallel directions is investigated experimentally for Reynolds numbers up to 1200. The moving sidewalls are realized by two rotating cylinders of large radii tightly closing the cavity. The distance between the moving walls relative to the height of the cavity (aspect ratio) is Γ = 1.96. Laser-Doppler and hot-film techniques are employed to measure steady and time-dependent vortex flows. Beyond a first threshold robust, steady, three-dimensional cells bifurcate supercritically out of the basic flow state. Through a further instability the cellular flow becomes unstable to oscillations in the form of standing waves with the same wavelength as the underlying cellular flow. If both sidewalls move with the same velocity (symmetrical driving), the oscillatory instability is found to be tricritical. The dependence on two sidewall Reynolds numbers of the ranges of existence of steady and oscillatory cellular flows is explored. Flow symmetries and quantitative velocity measurements are presented for representative cases.

The stabilities of salt-finger and plume convection, two major flows characterizing the fluid dynamics of NH4Cl solutions cooling from below, are investigated by theoretical and experimental approaches. A linear stability analysis is implemented to study theoretically the onset of salt-finger convection. Special emphasis is placed on the competition between different instability modes. It is found that in most of the cases considered, the neutral curve consists of two separated monotonic branches with a Hopf bifurcation branch in between; the right-hand monotonic branch corresponding to the boundary-layer-mode convection is more unstable than the left-hand monotonic branch corresponding to the mushy-layer mode. We also conducted a series of experiments covering wide ranges of bulk fluid concentration C∞ and bottom temperature TB to study the stability characteristics of plume convection. From the measurement of both temperature and concentration of the interstitial fluid in the mushy layer, we verify that during the progress of solidification the melt in the mush is in a thermodynamic equilibrium state except at the melt/mush interface where most of the solidification occurs. The critical Rayleigh number of the onset of plume convection is found to be Rccm = 1.1 × 107Π* (see (22)), where Π* is the permeability of the mush. This relation is believed to be valid up to supereutectic NH4Cl solutions.

The mean position of the cylinder is in the mean free surface, and the cylinder is oscillating vertically (heaving) with a simple harmonic motion of small amplitude. The problem is studied both theoretically and experimentally. The present paper may be regarded as a sequel to an earlier paper (Ursell, Dean & Yu 1959) on the waves due to a piston wave-maker.

The breakup of a free thin liquid film subjected to an impulsive acceleration is investigated. A soap film is stretched on a frame at the exit of a shock tube. As the shock impacts the film, the film accelerates within a very short time and detaches from the frame at a constant velocity function of the shock strength. The liquid thickness modulations amplify and eventually the film is perforated with a number of holes, subsequently growing in radius and connecting to each other. The initially connex film is left in the form of a web of liquid ligaments which break into droplets. Both the hole density and formation time depend on the film velocity. We analyse these observations with an impulsive Rayleigh–Taylor instability incorporating liquid surface tension. It is shown to account for both the mode selection and its associated time of growth, providing a criterion for the film bursting time and hole density.

Streaks have been found to be an important part of wall-turbulence dynamics. In this paper, we extend the analysis for unbounded shear flows, in particular a Mach 0.4 round jet, using measurements taken using dual-plane, time-resolved, stereoscopic particle image velocimetry (PIV) taken at pairs of jet cross-sections, allowing the evaluation of the cross-spectral density of streamwise velocity fluctuations resolved into azimuthal Fourier modes. From the streamwise velocity results, two analyses are performed: the evaluation of wavenumber spectra (assuming Taylor’s hypothesis for the streamwise coordinate) and a spectral proper orthogonal decomposition (SPOD) of the velocity field using PIV planes in several axial stations. The methods complement each other, leading to the conclusion that large-scale streaky structures are also present in turbulent jets where they experience large growth in the streamwise direction, energetic structures extending up to eight diameters from the nozzle exit. Leading SPOD modes highlight the large-scale, streaky shape of the structures, whose aspect ratio (streamwise over azimuthal length) is approximately 15. The data were further analysed using SPOD, resolvent and transient growth analyses, good agreement being observed between the models and the leading SPOD mode for the wavenumbers considered. The models also indicate that the lift-up mechanism is active in turbulent jets, with streamwise vortices leading to streaks. The results show that large-scale streaks are a relevant part of the jet dynamics.

Kinetic energy and reactive scalar spectra in turbulent premixed flames are studied from compressible three-dimensional direct numerical simulations (DNS) of a temporally evolving rectangular slot-jet premixed flame, a statistically one-dimensional configuration. The flames correspond to a lean premixed hydrogen–air mixture at an equivalence ratio of 0.7, preheated to 700 K and at 1 atm, and three DNS are considered with a fixed jet Reynolds number of 10 000 and a jet Damköhler number varying between 0.13 and 0.54. For the study of spectra, motivated by the need to account for density change, which can be locally strong in premixed flames, a new density-weighted definition for two-point velocity/scalar correlations is proposed. The density-weighted two-point correlation tensor retains the essential properties of its constant-density (incompressible) counterpart and recovers the density-weighted Reynolds stress tensor in the limit of zero separation. The density weighting also allows the derivation of balance equations for velocity and scalar spectrum functions in the wavenumber space that illuminate physics unique to combusting flows. Pressure–dilatation correlation is a source of kinetic energy at high wavenumbers and, analogously, reaction rate–scalar fluctuation correlation is a high-wavenumber source of scalar energy. These results are verified by the spectra constructed from the DNS data. The kinetic energy spectra show a distinct inertial range with a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}-5/3$ scaling followed by a ‘diffusive–reactive’ range at higher wavenumbers. The exponential drop-off in this range shows a distinct inflection in the vicinity of the wavenumber corresponding to a laminar flame thickness, $\delta _L$, and this is attributed to the contribution from the pressure–dilatation term in the energy balance in wavenumber space. Likewise, a clear spike in spectra of major reactant species (hydrogen) arising from the reaction-rate term is observed at wavenumbers close to $\delta _L$. It appears that in the inertial range classical scaling laws for the spectra involving the Kolmogorov scale are applicable, but in the high-wavenumber range where chemical reactions have a strong signature the laminar flame thickness produces a better collapse. It is suggested that a full scaling should perhaps involve the Kolmogorov scale, laminar flame thickness, Damköhler number and Karlovitz number.

Intermittent bursting events, similar to those characterizing the dynamics of near-wall turbulence, have been observed in a low-dimensional dynamical model (Aubry et al. 1988) built from eigenfunctions of the proper orthogonal decomposition (Lumley 1967). In the present work, we investigate the persistency of the intermittent behaviour in higher - but still of relatively low-dimensional dynamical systems. In particular, streamwise variations which were not accounted for in an explicit way in Aubry et al.'s model are now considered. Intermittent behaviour persists but can be of a different nature. Specifically, the non-zero streamwise modes become excited during the eruptive events so that rolls burst downstream into smaller scales. When structures have a finite length, they travel at a convection speed approximately equal to the mean velocity at the top of the layer (y+ ≈ 40). In all cases, intermittency seems to be due to homoclinic cycles connecting hyperbolic fixed points or more complex (apparently chaotic) limit sets. While these sets lie in the zero streamwise modes invariant subspace, the connecting orbits consist of nonzero streamwise modes travelling downstream. Chaotic limit sets connected by quasi-travelling waves have also been observed in a spatio-temporal chaotic regime of the Kuramoto–Sivashinsky equation (Aubry & Lian 1992a). When the limit sets lose their steadiness, the elongated rolls become randomly active, as they probably are in the real flow. A coherent structure study in our resulting flow fields is performed in order to relate our findings to experimental observations. It is shown that streaks, streamwise rolls, horseshoe vortical structures and shear layers, present in our models, are all connected to each other. Finally, criteria to determine a realistic value of the eddy viscosity parameter are developed.

Many physical problems containing rotating symmetry exhibit azimuthal waves, from electromagnetic waves in nanophotonic crystals to seismic waves in giant stars. When this symmetry is broken, clockwise (CW) and counter-clockwise (CCW) waves are split into two distinct modes which can become unstable. This paper focuses on a theoretical study of symmetry breaking in annular cavities containing a number $N$ of flames prone to azimuthal thermo-acoustic instabilities. A general dispersion relation for non-perfectly-axisymmetric cavities is obtained and analytically solved to provide an explicit expression for the frequencies and growth rates of all azimuthal modes of the configuration. This analytical study unveils two parameters affecting the stability of the mode: (i) a coupling strength corresponding to the cumulative effects of the $N$ flames and (ii) a splitting strength due to the symmetry breaking when the flames are different. This theory has been validated using a 3D Helmholtz solver and good agreement is found. When only two types of flames are introduced into the annular cavity, the splitting strength is found to depend on two parameters: the difference between the two burner types and the pattern used to distribute the flames along the azimuthal direction. To first order, this theory suggests that the most stable configuration is obtained for a perfectly axisymmetric configuration. Therefore, breaking the symmetry by mixing different flames cannot improve the stability of an annular combustor independently of the flame distribution pattern.

In this study the evolution of initially homogeneous and isotropic turbulence in the presence of a free surface was investigated. The Navier–Stokes equations were solved via direct pseudo-spectral simulation with a resolution of 963. The Reynolds number based on the volume-averaged turbulence kinetic energy and dissipation rate was 147. Periodic boundary conditions were used in two dimensions, and the top and bottom sides of the domain were flat and shear-free. A random, divergence-free velocity field with a prescribed spectrum was used as the initial condition. An ensemble of sixteen separate simulations was used to calculate statistics.

Near the surface, the Reynolds stresses are anisotropic and the anisotropy extends a distance from the surface roughly equal to the turbulent lengthscale. The tangential vorticity fluctuations also vanish near the surface, owing to the no-shear condition, causing a corresponding decrease in the fluctuating enstrophy. The thickness of the region in which the surface affects the vorticity distribution is roughly one-tenth the turbulent lengthscale. The stress anisotropy near the surface appears to be maintained by reduced dissipation for the tangential velocity fluctuations, reduced pressure–strain transfer from the tangential to surface-normal velocity fluctuations, and rapid decay of the surface-normal velocity fluctuations due to dissipation. The turbulence kinetic energy rises in the near-surface region owing to a decrease in dissipation at the surface. This decrease in dissipation results from the local reduction in enstrophy owing to the vanishing of the tangential vorticity fluctuations at the surface. At the free surface, the mean pressure rises. This is also due to the reduction in enstrophy.

While the tangential vorticity must vanish at the free surface, the flow is fully three-dimensional up to the surface and the production of surface-normal vorticity by vortex stretching attains a maximum at the free surface. The contribution to the total enstrophy by the surface-normal vorticity fluctuations remains relatively constant over depth. The production of the surface-normal enstrophy component due to vortex stretching is roughly balanced by turbulent transport of enstrophy away from the surface. Near the surface, there are elevated levels of production of tangential vorticity by both vortex-stretching and fluctuating shear strains.

A laboratory experiment was conducted to investigate the characteristics of turbulencegenerated by an internal wave ray breaking on a sloping bed. The width of the incident waveray was small compared to the bed length, so that an isolated turbulent patch was generatedby the breaking process, a configuration unique to the present study. The parameter rangecovered subcritical, critical and supercritical frequencies. Flow visualization and velocitymeasurements revealed that near critical conditions the flow was confined to a narrow regionabove the bed and, contrary to expectations, critical waves showed a weak turbulence field.Subcritical and supercritical reflection resembled wave–wave interaction between theincident and the reflected waves and showed comparable centred displacement lengthscales. Asthe incident waves became progressively supercritical instabilities were first initiatedaway from the bed. For supercritical waves the centred displacement lengthscale and theturbulent Reynolds number both increased steadily up to about γ≈2, after which they startedto decrease (γ=ω/ωc, where ω is the frequency of the incidentwave and ωc=Nsinβ is the critical frequency foran ambient uniform stratification of magnitude N and a bed angle of β). Forsubcritical waves an increase in the centred displacement lengthscale and the turbulentReynolds number was also observed. The mixed fluid generated at the boundary collapsed intothe fluid interior in the form of a horizontal two-dimensional viscous–buoyancy intrusion:the efficiency of mixing was, however, very small and no measurable change in the meandensity gradient was observed over the duration of the experiments.

The stratified shear flow over a two-dimensional obstacle of semi-elliptical crosssection is considered. The shear flow is assumed to be inviscid with constant upstream values of the density gradient and dynamic pressure (Long's model). Two complete sets of lee-wave functions, each of which satisfies the condition of no upstream reflexion, are determined in elliptic co-ordinates for ε ≤ 1 and ε ≥ 1, where ε is the ratio of height to half-width of the obstacle. These functions are used to determine the lee-wave field produced by, and the consequent drag on, a semi-elliptical obstacle as functions of ε and the reduced frequency (reciprocal Froude number) within the range of stable flow. The reduced frequency at which static instability first occurs is calculated as a function of ε.

A mathematical model for the unsteady fluid-dynamic response of the semicircular canals is developed. The endolymph is assumed to be an incompressible Newtonian fluid and the presence and effects of both the utricle and the cupula are specifically accounted for. A first approximate solution is obtained using a singular perturbation method. It is shown that the canal can be modelled as a heavily damped, second-order system which behaves as an angular-velocity meter. A comparison of the model response with experimental results is made; fairly good agreement is found.

The evolution of a stratified shear layer with mean shear in the horizontal direction, orthogonal to gravity, is numerically investigated with focus on the structural organization of the vorticity and density fields. Although the Reynolds number of the flow increases with time, facilitating instabilities and turbulence, the bulk Richardson number signifying the level of stratification also increases. Remarkably rich dynamics is found: turbulence; the emergence of coherent core/braid regions from turbulence; formation of a lattice of dislocated vortex cores connected by thin horizontal sheets of collapsed density and vorticity; density-driven intrusions at the edges of the shear layer; and internal wave generation and propagation. Stratification introduces significant vertical variability although it inhibits the vertical velocity. The molecular dissipation of turbulent kinetic energy and of turbulent potential energy are both found to be substantial even in the case with highest stratification, and primarily concentrated in thin horizontal sheets. The simulation data are used to help explain how buoyancy induces the emergence of columnar vortex cores from turbulence and then dislocates these cores to eventually form a lattice of ‘pancake’ eddies connected by thin sheets with large vertical shear (horizontal vorticity) and density gradient.