A flow circulation in a closed circular-cylindrical container is produced by a rotating lid. After a transient phase from an initial state at rest a steady-flow situation is reached for a certain parameter range. In a subspace of this parameter range an undulating meridional flow occurs that may exhibit at the axis of rotation one or several separation bubbles which are interpreted as vortex breakdown. Numerical calculations on the basis of the Navier-Stokes equations for incompressible homogeneous and Boussinesq fluids enable the study of the influence of various flow parameters on the properties of these separation bubbles. The parameters varied are the Reynolds, Prandtl, Rayleigh, and Eckert numbers together with the ratio of height to radius of the container. The numerical results are in good agreement with experiments performed by Vogel, Ronnenberg, and Escudier. The stability of the fluid motions in these experiments with respect to non-axisymmetric disturbances strongly suggests that the corresponding axisymmetric solutions of the Navier-Stokes equations are stable configurations.

Spanwise vorticity measurements have been performed in zero-pressure-gradient boundary layers over the range 1010 < Rθ < 4850 (Rθ ≡ U∞ θ/ν, where U∞ is the free-stream velocity and θ is the momentum deficit thickness) using a four-wire probe. In addition, experiments quantifying the spatial and temporal resolution required to obtain an accurate statistical representation of the small-scale structure of wall-bounded turbulence were performed. Furthermore, a thorough investigation of statistical convergence for a variety of fluctuating quantities was performed. Comparisons with earlier high-resolution studies indicate that the maximum value of u′/uτ increases with increasing Reynolds number over the given Rθ range (u′ ≡ r.m.s. u, and uτ is the friction velocity). It is suggested that detecting this dependence provides a good measure of probe resolution. In general it was found that statistics of velocity gradients were distinctly more sensitive to finite probe size than velocity statistics. Wire spacing experiments suggest that Wyngaard's (1969) criterion is to a good approximation valid even under anisotropic conditions. Furthermore, it was found that instantaneously spatial averaging of ∂u/∂t caused significant attenuation in the resulting r.m.s., and that this averaging procedure is sensitive to the level of mean shear. A simple method of estimating how noise in the u-velocity signals enters into the ∂u/∂y signals is presented. The convergence study shows that statistical convergence criteria developed from free-shear flows severely underestimates the averaging times required in boundary layers. A table of general convergence criteria is provided.

We examine the problem of determining the particle-phase velocity variance and rhe-ology of sheared gas-solid suspensions at small Reynolds numbers and finite Stokes numbers. Our numerical simulations take into account the Stokes flow interactions among particles except for pairs of particles with a minimum gap width comparable to or smaller than the mean free path of the gas molecules for which the usual lubrication approximation breaks down and particle collisions occur in a finite time. The simulation results are compared to the predictions of two theories. The first is an asymptotic theory for large Stokes number St and nearly elastic collisions, i.e. St [Gt] 1 and 0 ≤ 1 - e [Lt] 1, e being the coefficient of restitution. In this limit, the particle velocity distribution is close to an isotropic Maxwellian and the velocity variance is determined by equating the energy input in shearing the suspension to the energy dissipation by inelastic collisions and viscous effects. The latter are estimated by solving the Stokes equations of motion in suspensions with the hard-sphere equilibrium spatial and velocity distribution while the shear energy input and energy dissipation by inelastic effects are estimated using the standard granular flow theory (i.e. St = ∞). The second is an approximate theory based on Grad's moments method for which St and 1 – e are O(1). The two theories agree well with each other at higher values of volume fraction ϕ of particles over a surprisingly large range of values of St. For smaller ϕ however, the two theories deviate significantly except at sufficiently large St. A detailed comparison shows that the predictions of the approximate theory based on Grad's method are in excellent agreement with the results of numerical simulations.

The first measurement of the influence of the Stokes number on the distributions of particle concentration and velocity at the exit of a long pipe are reported, together with the subsequent influence on the downstream evolution of these distributions through a particle-laden jet in co-flow. The data were obtained by simultaneous particle image velocimetry (PIV) and planar nephelometry (PN), using four cameras to provide high resolution through the first 30 jet diameters and also correction for optical attenuation. These data provide much more detailed information than is available from previous measurements. From them, a new understanding is obtained of how the Stokes number influences the flow at the jet exit plane and how this influence propagates throughout the jet.

A combustion instability in a combustor terminated by a nozzle is analysed and modelled based on a low-order Helmholtz solver. A large eddy simulation (LES) of the corresponding turbulent, compressible and reacting flow is first performed and analysed based on dynamic mode decomposition (DMD). The mode with the highest amplitude shares the same frequency of oscillation as the experiment (approximately 320 Hz) and shows the presence of large entropy spots generated within the combustion chamber and convected down to the exit nozzle. The lowest purely acoustic mode being in the range 700–750 Hz, it is postulated that the instability observed around 320 Hz stems from a mixed entropy–acoustic mode, where the acoustic generation associated with entropy spots being convected throughout the choked nozzle plays a key role. The DMD analysis allows one to extract from the LES results a low-order model that confirms that the mechanism of the low-frequency combustion instability indeed involves both acoustic and convected entropy waves. The delayed entropy coupled boundary condition (DECBC) (Motheau, Selle & Nicoud, J. Sound Vib., vol. 333, 2014, pp. 246–262) is implemented into a numerical Helmholtz solver where the baseline flow is assumed at rest. When fed with appropriate transfer functions to model the entropy generation and convection from the flame to the exit, the Helmholtz/DECBC solver predicts the presence of an unstable mode around 320 Hz, in agreement with both LES and experiments.

Revisiting the classical acoustics problem of rectangular side-branch cavities in a two-dimensional duct of infinite length, we use the finite-element method to numerically compute the acoustic resonances as well as the sound transmission and reflection for an incoming fundamental duct mode. To satisfy the requirement of outgoing waves in the far field, we use two different forms of absorbing boundary conditions, namely the complex scaling method and the Hardy space method. In general, the resonances are damped due to radiation losses, but there also exist various types of localized trapped modes with nominally zero radiation loss. The most common type of trapped mode is antisymmetric about the duct axis and becomes quasi-trapped with very low damping if the symmetry about the duct axis is broken. In this case a Fano resonance results, with resonance and antiresonance features and drastic changes in the sound transmission and reflection coefficients. Two other types of trapped modes, termed embedded trapped modes, result from the interaction of neighbouring modes or Fabry–Pérot interference in multi-cavity systems. These embedded trapped modes occur only for very particular geometry parameters and frequencies and become highly localized quasi-trapped modes as soon as the geometry is perturbed. We show that all three types of trapped modes are possible in duct–cavity systems and that embedded trapped modes continue to exist when a cavity is moved off centre. If several cavities interact, the single-cavity trapped mode splits into several trapped supermodes, which might be useful for the design of low-frequency acoustic filters.

Real turbulent flows are difficult to classify as either spatially homogeneous or isotropic. Nonetheless these idealizations allow the identification of certain universal features associated with the small-scale motions almost invariably observed in a variety of different conditions. The single most significant aspect is a flux of energy through the spectrum of inertial scales related to the phenomenology commonly referred to as the Richardson cascade. Inhomogeneity, inherently present in near-wall turbulence, generates additional energy fluxes of a different nature, corresponding to the spatial redistribution of turbulent kinetic energy. Traditionally the spatial flux is associated with a single-point observable, namely the turbulent kinetic energy density. The flux through the scales is instead classically related to two-point statistics, given in terms of an energy spectrum or, equivalently, in terms of the second-order moment of the velocity increments. In the present paper, starting from a suitably generalized form of the classical Kolmogorov equation, a scale-by-scale balance for the turbulent fluctuations is evaluated by examining in detail how the energy associated with a specific scale of motion – hereafter called the scale energy – is transferred through the spectrum of scales and, simultaneously, how the same scale of motion exchanges energy with a properly defined spatial flux. The analysis is applied to a data set taken from a direct numerical simulation (DNS) of a low-Reynolds-number turbulent channel flow. The detailed scale-by-scale balance is applied to the different regions of the flow in the various ranges of scales, to understand how – i.e. through which mechanisms, at which scales and in which regions of the flow domain – turbulent fluctuations are generated and sustained. A complete and formally precise description of the dynamics of turbulence in the different regions of the channel flow is presented, providing rigorous support for previously proposed conceptual models.

A general solution for Stokes’ equation in bipolar co-ordinates is derived, and then applied to the arbitrary motion of a sphere in the presence of a plane fluid/fluid interface. The drag force and hydrodynamic torque on the sphere are then calculated for four specific motions of the sphere; namely, translation perpendicular and parallel to the interface and rotation about an axis which is perpendicular and parallel, respectively, to the interface. The most significant result of the present work is the comparison between these numerically exact solutions and the approximate solutions from part 1. The latter can be generalized to a variety of particle shapes, and it is thus important to assess their accuracy for this case of spherical particles where an exact solution can be obtained. In addition to comparisons with the approximate solutions, we also examine the predicted changes in the velocity, pressure and vorticity fields due to the presence of the plane interface. One particularly interesting feature of the solutions is the fact that the direction of rotation of a freely suspended sphere moving parallel to the interface can either be the same as for a sphere rolling along the interface (as might be intuitively expected), or opposite depending upon the location of the sphere centre and the ratio of viscosities for the two fluids.

A model of the diffusive interface in double-diffusive convection at high Rayleigh number is proposed. The interface is assumed to have a double structure: two marginally stable boundary layers from which blobs or thermals arise on the outer edges of the interface, separated by a diffusive core across which all transport takes place by molecular diffusion. The model is time-independent and comparison is made with unsteady ‘run-down’ experiments on the assumption that the experiments run down through a sequence of equilibrium states each of which can be considered separately. The model predicts a constant ratio of the buoyancy fluxes of the two components at a value equal to the square root of the ratio of their molecular diffusivities, and individual fluxes in reasonable agreement with the available experimental data. Some time-dependent features of the model are also examined.

An investigation is presented that analyses the energy flows that are connected to the dynamical behaviour of coherent structures in a transitional flat-plate boundary layer. Based on a mathematical description of the three-dimensional coherent structures of this flow as provided by the Karhunen–Loève procedure, energy equations for the coherent structures are derived by Galerkin projection of the Navier–Stokes equations in vorticity transport formulation onto the corresponding basis of eigenfunctions. In a first step, the time-averaged energy balance – showing the energy flows that support the different coherent structures and thus maintain the fluctuations of the velocity field – is considered. In a second step, the instantaneous power budget is investigated for the particularly interesting case of a coherent structure providing a prime contribution to the characteristic spike events of the transitional boundary layer. As this structure shows a strong variation in energy, the question about which mechanisms cause these variations is addressed. Our results show that the occurrence of a spike must be attributed to an autonomous event and cannot be interpreted as just an epiphenomenon of the passage of a Λ-vortex.

A three-dimensional direct numerical simulation (DNS) study of a spatially evolving planar turbulent reacting jet is reported. Combustion of methane with air is modelled using a four-step reduced mechanism in the non-premixed regime. A total of eight chemical species are integrated in time along with the fluid mechanical fields. The solution of the compressible Navier–Stokes equations is obtained numerically for moderately low Mach number. A large computational grid, with 100 million grid points, is required in order to resolve the flame. The cold flow Reynolds number is 3000. The focus of the study is to investigate the dynamics of extinction fronts in three-dimensional turbulent flows. A novel data reduction and identification algorithm was developed to postprocess the large DNS database and extract the shape of the evolving flame surface including its edges and their propagation velocity. The joint probability density function (p.d.f.) of edge velocity and scalar dissipation was obtained and the results indicate that the three-dimensional flame edges propagate with a velocity that is largely controlled by the local rate of scalar dissipation, or equivalently in terms of the local Damköhler number at the flame edge, as predicted by theory. Naturally, the effects of unsteadiness in this flow produce a broad joint p.d.f. The statistics collected also suggest that the mean value of the hydrogen radical reaction rate are very small in the turbulent regions of the flow owing to the functional form of the hydrogen radical reaction rate itself. The consequence of these results in the context of turbulent combustion modelling is discussed. Additional statistical and morphological information of the flame is provided.

The normal impact of a drop of yield-stress fluid on a flat rigid surface is investigated experimentally. Using different model fluids (polymer microgels, clay suspensions) and impacted surfaces (partially wettable, super-hydrophobic), we find a rich variety of impact regimes from irreversible viscoplastic coating to giant elastic spreading and recoil. A minimal model of inertial spreading, taking into account an elasto-viscoplastic rheology, allows explaining in a single framework the different regimes and scaling laws. In addition, semi-quantitative predictions for the spread factor are obtained when the measured rheological parameters of the fluid (elasticity, yield stress, viscosity) are injected into the model. Our study offers a means to probe the short-time rheology of yield-stress fluids and highlights the role of elasticity on the unsteady hydrodynamics of these complex fluids. Movies are available with the online version of the paper (go to journals.cambridge.org/flm).

In this paper the coherent structure in the similarity region of the turbulent planar jet is examined experimentally by application of the proper orthogonal decomposition (POD). In particular, twin cross-stream rakes of X-wire probes are used to take cross-spectral measurements with different spanwise separations between the rakes and at several locations throughout the similarity region. The resulting POD spatial eigenfunctions for each of the three velocity components depend on cross-stream spatial coordinate, Strouhal number, and spanwise wavenumber. Corresponding eigenvalue distributions are obtained in Strouhal number–spanwise wavenumber space. Eigenvalue convergence is demonstrated to be rapid. When properly scaled the eigenfunctions and eigenvalues are shown to exhibit self-similarity though the streamwise location at which this commences depends on the particular velocity component. The results suggest that the flow supports a planar structure aligned in the spanwise direction as well as an essentially three-dimensional structure with asymmetrical shape in the cross-stream direction and pseudo-periodically distributed in the spanwise direction. Comparison of the single- and dual-rake implementations of the POD presented in this paper demonstrate that measurements confined to a single plane are incapable of properly extracting the planar modes. Rather, the single-rake implementation results in modes that appear to be a weighted sum of modes corresponding to different spanwise wavenumbers.

The natural frequencies and damping ratios for surface waves in a circular cylinder are calculated on the assumptions of a fixed contact line, Stokes boundary layers, and either a clean or a fully contaminated surface. These theoretical predictions are compared with the measurements for the first six modes in a brimfull, sharp-edged cylinder of radius 2.77 cm and depth 3.80 cm. The differences between the predicted and observed frequencies were less than 0.5% for all but the fundamental axisymmetric mode with a clean surface. The difference between the predicted and observed damping ratio for the dominant mode with a clean surface was 20%; this difference was significantly larger for the higher modes with a clean surface and for all of the modes with a contaminated surface.

In this paper two aspects of the deepening of a mixed layer in a stratified fluid are examined in the laboratory. The first is the deepening of a layer into a region of constant density gradient. Turbulence is produced by an oscillating grid which generates a horizontally homogeneous field of motion with no significant mean flow. It is found that the rate at which the potential energy of the basic stratification is increased by the mixing does not bear a simple relationship to the rate of energy input by the grid. On the other hand, when allowance is made for the decay of turbulent energy away from the grid and only that portion to reach the bottom of the mixed layer is considered, the rate of potential energy increase is found to be proportional to this available energy. The second aspect to be discussed is the effect of energy radiation by internal waves in the region below the mixed layer. Estimates are made of the possible loss of energy to these waves, which reduces the amount available to deepen the layer. An experimental demonstration of up to 50 % reduction in the mixing rate due to the presence of internal waves is given. Finally, the implications of these results are discussed in the light of current theoretical models of the deepening process.

A series of laboratory experiments and numerical simulations have been performed to investigate the rapid fluid-like flow of a finite mass of granular material down a chute with partial lateral confinement. The chute consists of a section inclined at 40° to the horizontal, which is connected to a plane run-out zone by a smooth transition. The flow is confined on the inclined section by a shallow parabolic cross-slope profile. Photogrammetric techniques have been used to determine the position of the evolving boundary during the flow, and the free-surface height of the stationary granular deposit in the run-out zone. The results of three experiments with different granular materials are presented and shown to be in very good agreement with numerical simulations based on the Savage–Hutter theory for granular avalanches. The basal topography over which the avalanche flows has a strong channelizing effect on the inclined section of the chute. As the avalanche reaches the run-out zone, where the lateral confinement ceases, the head spreads out to give the avalanche a characteristic ‘tadpole’ shape. Sharp gradients in the avalanche thickness and velocity began to develop at the interface between the nose and tail of the avalanche as it came to rest, indicating that a shock wave develops close to the end of the experiments.

Observations of the flow of dense fluid into uniformly density-stratified environments down plane slopes with small inclination to the horizontal ([les ] 20°) are described, and a quantitative model for such flows is presented. In these experiments the dense fluid is released at the top of the slope for a finite period of time. The resulting downslope gravity current, or downflow, has uniform thickness with a distinct upper boundary, until it approaches its level of neutral density where the fluid leaves the proximity of the slope. Turbulent transfers of mass and momentum occur across the upper boundary, causing a continuous loss of fluid from the downflow in most cases, and associated loss of momentum. The flow may be characterized by the local values of the Richardson number Ri, the Reynolds number Re (generally large), and of M = QN3/g′2, where Q is the (two-dimensional) volume flux, N the buoyancy frequency and g′ the (negative) buoyancy of the dense fluid. The model for the downflow describes the turbulent transfers in terms of entrainment, detrainment and drag coefficients, Ee, Ed and k respectively, and the observations enable the determination of these coefficients in terms of the local values of M and Ri. The model may be regarded as an extension of that Ellison & Turner (1959) to stratified environments, describing the consequent substantial changes in mixing and distribution of the inflow. It permits the modelling of the bulk properties of these flows in geophysical situations, including shallow and deep flows in the ocean.

Two conceptual frameworks for the origin of the streaky pattern in near-wall developed turbulent flows are compared. According to the framework that dominated the research over several decades, the pattern of streaks is dictated by the pattern of wall-normal motions via the lift-up mechanism. Various concepts within this framework describe the wall-normal motions as induced by longitudinal vortices, hairpin vortices, vortex packets, etc. According to the newly emerging conceptual framework, the combined action of lift-up of the mean profile, mean shear, and viscous diffusion has its own pattern-forming properties. The pattern of streaks is dictated by these linear effects to a much greater extent than by the pattern of the wall-normal motions. Numerical results supporting the new conceptual framework are presented. An approximate approach for calculating the streak spacing within the new framework is proposed. It is shown to have a significant predictive ability.

This work aims to provide a systematic experimental study on the wake of two tandem cylinders of unequal diameters. The fluid dynamics around a circular cylinder of diameter $D$ placed in the wake of another circular cylinder with a smaller diameter of $d$ is investigated, including the time-mean drag coefficient ( $C_{D}$ ), the fluctuating drag and lift coefficients ( $C_{D}^{\prime }$ and $C_{L}^{\prime }$ ), the Strouhal number ( $St$ ) and the flow structures. The Reynolds number based on $D$ is kept constant at $4.27\times 10^{4}$ . The ratios $d/D$ and $L/d$ vary from 0.2 to 1.0 and 1.0 to 8.0 respectively, where $L$ is the distance from the upstream cylinder centre to the forward stagnation point of the downstream cylinder. The ratios $d/D$ and $L/d$ are found, based on extensive hotwire, particle imaging velocimetry, pressure and flow visualization measurements, to have a marked influence on the wake dynamics behind the cylinders. As such, the flow is classified into the reattachment and co-shedding flow regimes, the latter being further subdivided into the lock-in, subharmonic lock-in and no lock-in regions. It is found that the critical spacing that divides the two regimes is dictated by the upstream-cylinder vortex formation length and becomes larger for smaller $d/D$ . The characteristic flow properties are documented in each regime and subdivided region, including the flow structure, $St$ , wake width, vortex formation length and the lateral width between the two gap shear layers. The variations in $C_{D}$ , $C_{D}^{\prime }$ , $C_{L}^{\prime }$ and the pressure distribution around the downstream cylinder are connected to the flow physics.

Several means for visualizing large-scale vortex structure in a turbulent mixing layer are proposed. Most of the observations are recorded along the low-speed side of the mixing layer, external to the rotational portion of the flow. Conventional correlation measurements in both the streamwise and spanwise directions indicate that the vortex structure becomes independent of the downstream coordinate in a non-dimensional distance of order λx/θi = 300–400, where $\lambda = \Delta U/2\overline{U}$ is the speed ratio and θi is the initial integral thickness. Simultaneous hot-wire measurements at 12 spanwise positions allow computer reconstruction of the velocity field as a function of span and time. These visualizations show the vortex structures to be primarily aligned across the span of the flow, but to contain irregularities. Spanwise correlation lengths are of the order of 3–5δω (δω is the local vorticity thickness). However, the large vortices typically have lengths of order 20δω when the irregularities along the span are ignored.