Surface granular flow, comprising granular material flowing on the surface of a heap of the same material, occurs in several industrial and natural systems. The rheology of such a flow was investigated by means of measurements of velocity and number-density profiles in a quasi-two-dimensional rotating cylinder, half-filled with a model granular material – monosize spherical stainless-steel particles. The measurements were made at the centre of the cylinder, where the flow is fully developed, using streakline photography and image analysis. The stress profile was computed from the number-density profile using a force balance which takes into account wall friction. Mean-velocity and root-mean-square (r.m.s.)-velocity profiles are reported for different particle sizes and cylinder rotation speeds. The profiles for the mean velocity superimpose when distance is scaled by the particle diameter d and velocity by a characteristic shear rate and the particle diameter, where βm is the maximum dynamic angle of repose and βs is the static angle of repose. The maximum dynamic angle of repose is found to vary with the local flow rate. The scaling is also found to work for the r.m.s. velocity profiles. The mean velocity is found to decay exponentially with depth in the bed, with decay length λ = 1.1d. The r.m.s. velocity shows similar behaviour but with λ = 1.7d. The r.m.s. velocity profile shows two regimes: near the free surface the r.m.s. velocity is nearly constant and below a transition point it decays linearly with depth. The shear rate, obtained by numerical differentiation of the velocity profile, is not constant anywhere in the layer and has a maximum which occurs at the same depth as the transition in the r.m.s. velocity profile. Above the transition point the velocity distributions are Gaussian and below the transition point the velocity distributions gradually approach a Poisson distribution. The shear stress increases roughly linearly with depth. The variation in the apparent viscosity η with r.m.s. velocity u shows a relatively sharp transition at the shear-rate maximum, and in the region below this point the apparent viscosity η ∼ u−1.5. The measurements indicate that the flow comprises two layers: an upper low-viscosity layer with a nearly constant r.m.s. velocity and a lower layer of increasing viscosity with a decreasing r.m.s. velocity. The thickness of the upper layer depends on the local flow rate and is independent of particle diameter while the reverse is found to hold for the lower-layer thickness. The experimental data is compared with the predictions of three models for granular flow.

The behaviour of the interface between stratified thin liquid films bounded by parallel solid surfaces and subject to van der Waals forces which drive dewetting is studied in this work. Chemically homogeneous surfaces are considered first; this is followed by an investigation of chemically heterogeneous surfaces. The lubrication approximation is applied to obtain a single nonlinear evolution equation which describes the interfacial behaviour, and both the linear stability and nonlinear development of the interface are examined. The sensitivity of the interfacial rupture time to problem parameters such as the viscosity ratio, initial interfacial height, interfacial tension, and magnitude of the van der Waals forces is characterized in detail for the homogeneous case. This serves as a basis for a study of the heterogeneous case, where the strong dependence of the rupture time on the length scale of the heterogeneity is found to be relatively independent of changes in the remaining problem parameters. The mechanisms underlying the rupture-time behaviour are also explored in detail. The results suggest a route by which one liquid can become emulsified in the other, and may be beneficial to industrial processes such as lithographic printing which are based on wettability phenomena.

We consider the onset of thermosolutal (double-diffusive) convection of a binary fluid in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the bounding surfaces, in the case when the solubility of the dissolved component depends on temperature. We use a linear stability analysis to investigate how the dissolution or precipitation of this component affects the onset of convection and the selection of an unstable wavenumber; we extend this analysis using a Galerkin method to predict the structure of the initial bifurcation and compare our analytical results with numerical integration of the full nonlinear equations. We find that the reactive term may be stabilizing or destabilizing, with subtle effects particularly when the thermal gradient is destabilizing but the solutal gradient is stabilizing. The preferred spatial wavelength of convective cells at onset may also be substantially increased or reduced, and strongly reactive systems tend to prefer direct to subcritical bifurcation. These results have implications for geothermal-reservoir management and ore prospecting.

It is generally assumed that flows around wall-mounted sharp-edged bluff bodies submerged in thick turbulent boundary layers are essentially independent of the Reynolds number Re, provided that this exceeds some (2–3) × 104. (Re is based on the body height and upstream velocity at that height.) This is a particularization of the general principle of Reynolds-number similarity and it has important implications, most notably that it allows model scale testing in wind tunnels of, for example, atmospheric flows around buildings. A significant part of the literature on wind engineering thus describes work which implicitly rests on the validity of this assumption. This paper presents new wind-tunnel data obtained in the ‘classical’ case of thick fully turbulent boundary-layer flow over a surface-mounted cube, covering an Re range of well over an order of magnitude (that is, a factor of 22). The results are also compared with new field data, providing a further order of magnitude increase in Re. It is demonstrated that if on the one hand the flow around the obstacle does not contain strong concentrated-vortex motions (like the delta-wing-type motions present for a cube oriented at 45° to the oncoming flow), Re effects only appear on fluctuating quantities such as the r.m.s. fluctuating surface pressures. If, on the other hand, the flow is characterized by the presence of such vortex motions, Re effects are significant even on mean-flow quantities such as the mean surface pressures or the mean velocities near the surfaces. It is thus concluded that although, in certain circumstances and for some quantities, the Reynolds-number-independency assumption is valid, there are other important quantities and circumstances for which it is not.

Owing to radiation losses, resonances in open systems, i.e. solution domains which extend to infinity in at least one direction, are generally complex valued. However, near symmetric centred objects in ducted domains, or in periodic arrays, so-called trapped modes can exist below the cut-off frequency of the first non-trivial duct mode. These trapped modes have no radiation loss and correspond to real-valued resonances. Above the first cut-off frequency isolated trapped modes exist only for specific parameter combinations. These isolated trapped modes are termed embedded, because their corresponding eigenvalues are embedded in the continuous spectrum of an appropriate differential operator. Trapped modes are of considerable importance in applications because at these parameters the system can be excited easily by external forcing. In the present paper directly computed embedded trapped modes are compared with numerically obtained resonances for several model configurations. Acoustic resonances are also computed in two-dimensional models of a butterfly and a ball-type valve as examples of more complicated geometries.

Buoyancy inhomogeneities on sloping surfaces arise in numerous situations, for example, from variations in snow/ice cover, cloud cover, topographic shading, soil moisture, vegetation type, and land use. In this paper, the classical Prandtl model for one-dimensional flow of a viscous stably stratified fluid along a uniformly cooled sloping planar surface is extended to include the simplest type of surface inhomogeneity – a surface buoyancy that varies linearly down the slope. The inhomogeneity gives rise to acceleration, vertical motions associated with low-level convergence, and horizontal and vertical advection of perturbation buoyancy. Such processes are not accounted for in the classical Prandtl model. A similarity hypothesis appropriate for this inhomogeneous flow removes the along-slope dependence from the problem, and, in the steady state, reduces the Boussinesq equations of motion and thermodynamic energy to a set of coupled nonlinear ordinary differential equations. Asymptotic solutions for the velocity and buoyancy variables in the steady state, valid for large values of the slope-normal coordinate, are obtained for a Prandtl number of unity for pure katabatic flow with no ambient wind or externally imposed pressure gradient. The undetermined parameters in these solutions are adjusted to conform to lower boundary conditions of no-slip, impermeability and specified buoyancy. These solutions yield formulae for the boundary-layer thickness and slope-normal velocity component at the top of the boundary layer, and provide an upper bound of the along-slope surface-buoyancy gradient beyond which steady-state solutions do not exist. Although strictly valid for flow above the boundary layer, the steady asymptotic solutions are found to be in very good agreement with the terminal state of the numerical solution of an initial-value problem (suddenly applied surface buoyancy) throughout the flow domain. The numerical results also show that solution non-existence is associated with self-excitation of growing low-frequency gravity waves.

The purpose of this experimental study is to explore the effect of buoyancy on vortex breakdown (VB) in swirling jets. Three non-dimensional parameters govern the flow: the jet exit Reynolds number, the swirl ratio and the Richardson number (buoyancy). The experimental apparatus consists of a vertical swirling water jet which discharges into a large tank. Moderate values of the Reynolds number are used, in the range 150 ≤ Re ≤ 600. Swirl is imparted to the jet in a rotating chamber whereas the temperature difference between the jet and its surroundings is established by passing the jet through a heat exchanger, immersed in a circulating water bath with a controlled temperature. Vector maps of the vertical mid-plane and horizontal cross-sections are obtained by particle image velocimetry (PIV) measurements. It is demonstrated that VB can be effectively suppressed (enhanced) when there is a negative (positive) temperature difference between the jet core and its surrounding fluid. The experimental critical swirl ratio for the appearance of VB is found to be in good agreement with a simple criterion, originally derived by Billant, Chomaz & Huerre (J. Fluid Mech., vol. 376, 1998, p. 183) for isothermal swirling jets and extended here to include buoyancy effects. The transition of VB from a closed bubble to an open cone configuration is mapped in terms of the Reynolds and Richardson numbers. Finally, the effect of the upstream velocity field on the critical rotation rate for the onset of VB and its configuration is exhibited using two different interchangeable rotating chambers.

We report on theoretical studies of molecularly thin Langmuir films on the surface of a quiescent subfluid and qualitatively compare the results to both new and previous experiments. The film covers the entire fluid surface, but domains of different phases are observed. In the absence of external forcing, the compact domains tend to relax to circles, driven by a line tension at the phase boundaries. When stretched (by a transient applied stagnation-point flow or by stirring), a compact domain elongates, creating a bola consisting of two roughly circular reservoirs connected by a thin tether. This shape will then relax slowly to the minimum-energy configuration of a circular domain. The tether is never observed to rupture, even when it is more than a hundred times as long as it is wide. We model these experiments by taking previous descriptions of the full hydrodynamics, identifying the dominant effects via dimensional analysis, and reducing the system to a more tractable form. The result is a free boundary problem for an inviscid Langmuir film whose motion is driven by the line tension of the domain and damped by the viscosity of the subfluid. Using this model we derive relaxation rates for perturbations of a uniform strip and a circular patch. We also derive a boundary integral formulation which allows an efficient numerical solution of the problem. Numerically this model replicates the formation of a bola and the subsequent relaxation observed in the experiments. Finally, we suggest physical properties of the system (such as line tension) that can be deduced by comparison of the theory and numerical simulations to the experiment. Two movies are available with the online version of the paper.

A direct numerical simulation of the incompressible Navier–Stokes equations of the flow over a bump shows a stationary longitudinal instability at a Reynolds number of Re = 400. A three-dimensional global mode linear analysis is used to interpret these results and shows that the most unstable eigenmode is steady and localized in the recirculation bubble, with spanwise wavelength of approximately ten bump heights. An inviscid geometrical optics analysis along closed streamlines is then proposed and modified to account for viscous effects. This motivates a final discussion regarding the physical origin of the observed instability.

In this study, we performed simulations of turbulent flow over rectangular ribs transversely mounted on one side of a plane in a channel, with the other side being smooth. The separation between ribs is large enough to avoid forming stable vortices in the spacing, which exhibits k-type, or sand-grain roughness. The Reynolds number Reτ of our representative direct numerical simulation case is 460 based on the smooth-wall friction velocity and the channel half-width. The roughness height h is estimated as 110 wall units based on the rough-wall friction velocity. The velocity profile and kinetic energy budget verify the presence of an equilibrium, logarithmic layer at y≳2h. In the roughness sublayer, however, a significant turbulent energy flux was observed. A high-energy region is formed by the irregular motions just above the roughness. Visualizations of vortical streaks, disrupted in all three directions in the roughness sublayer, indicate that the three-dimensional flow structure of sand-grain roughness is replicated by the two-dimensional roughness, and that this vortical structure is responsible for the high energy production. The difference in turbulence structure between smooth- and rough-wall layers can also be seen in other flow properties, such as anisotropy and turbulence length scales.

The turbulent flow generated by an oscillating pressure gradient close to an infinite plate is studied by means of numerical simulations of the Navier–Stokes equations to analyse the characteristics of the steady streaming generated within the boundary layer. When the pressure gradient that drives the flow is given by a single harmonic component, the time average over a cycle of the flow rate in the boundary layer takes both positive and negative values and the steady streaming computed by averaging the flow over n cycles tends to zero as n tends to infinity. On the other hand, when the pressure gradient is given by the sum of two harmonic components, with angular frequencies ω1 and ω2 = 2ω1, the time average over a cycle of the flow rate does not change sign. In this case steady streaming is generated within the boundary layer and it persists in the irrotational region. It is shown both theoretically and numerically that in spite of the presence of steady streaming, the time average over n cycles of the hydrodynamic force, acting per unit area of the plate, vanishes as n tends to infinity.

The coherent structure in the near-field of an axisymmetric turbulent jet at a Reynolds number of 3.8 × 105 and Mach number of 0.3 is experimentally characterized by a vector implementation of the proper orthogonal decomposition (POD). The POD eigenfunctions and associated eigenvalues are extracted at several selected streamwise locations in the initial region. The focus on the near-field is motivated by its importance in numerous technical applications. Results show a rapid energy convergence with POD mode number. Examination of the relative energy contained in the combined azimuthal and radial components of the POD modes reveals that it is comparable to that in the streamwise component. The streamwise evolution of the eigenvalue spectra is characterized by a remarkable variation in the azimuthal mode number energy distribution, leading to the dominance of azimuthal mode m = 1 beyond the end of the jet core. In contrast, a scalar implementation using only the streamwise component shows the dominance of mode m = 2 which is consistent with previous scalar implementations of the POD. For a given azimuthal mode number, the eigenvalue spectra exhibit a broad peak which occurs at a constant value of Strouhal number based on local shear layer momentum thickness and local jet maximum velocity. The phase information required for a local reconstruction of the jet structure is obtained by projecting the POD eigenmodes onto instantaneous realizations of the flow at fixed streamwise locations. The instantaneous realizations are obtained by utilizing cross-stream arrays of multi-sensor probes in conjunction with linear stochastic estimation (LSE). Results clearly show the local dynamic behaviour of each component of the jet structure.

A study of the morphology of the vortical skeleton behind a flapping NACA0030 wing with a finite aspect ratio of 3, is undertaken. The motivation for this work originates with the proposal that thrust can be efficiently produced by flapping aerofoils. The test condition corresponds to a Strouhal number of 0.35, Reynolds number, based on aerofoil chord, of 600 and an amplitude of flapping, equal to the chord length of the wing. This test condition corresponds to the optimal thrust-producing case in infinite-span flapping wings. This study investigates the effect of wing three-dimensionality on the structure of the wake-flow. This is accomplished here, by quantitatively describing the spatio-temporal variations in the velocity, vorticity and Reynolds stresses for the finite-span-wing case.

Preliminary flow visualizations suggest that the presence of wingtip vortices for the three-dimensional-wing case, create a different vortical structure to the two-dimensional-wing case. In the case of a two-dimensional-wing, the flow is characterized by the interaction of leading- and trailing-edge vorticity, resulting in the formation of a clear reverse Kármán vortex street at the selected test condition. In the case of a three-dimensional-wing, the flow exhibits a high degree of complexity and three-dimensionality, particularly in the midspan region. Using phase-averaged particle image velocimetry measurements of the forced oscillatory flow, a quantitative analysis in the plane of symmetry of the flapping aerofoil was undertaken. Using a triple decomposition of the measured velocities, the morphological characteristics of the spanwise vorticity is found to be phase correlated with the aerofoil kinematics. Reynolds stresses in the direction of oscillation are the dominant dissipative mechanism. The mean velocity profiles resemble a jet, indicative of thrust production. Pairs of strong counter-rotating vortices from the leading- and trailing-edge of the aerofoil are shed into the flow at each half-cycle. The large-scale structure of the flow is characterized by constructive merging of spanwise vorticity. The midspan region is populated by cross-sections of interconnected vortex rings.

In this paper we investigate the role played by viscous heating in extensional flows of viscous threads with temperature-dependent viscosity. We show that there exists an interesting interplay between the effects of viscous heating, which accelerates thinning, and inertia, which prevents pinch-off. We first consider steady drawing of a thread that is fed through a fixed aperture at given speed and pulled with a constant force at a fixed downstream location. For pulling forces above a critical value, we show that inertialess solutions cannot exist and inertia is crucial in controlling the dynamics. We also consider the unsteady stretching of a thread that is fixed at one end and pulled with a constant force at the other end. In contrast to the case in which inertia is neglected, the thread will always pinch at the end where the force is applied. Our results show that viscous heating can have a profound effect on the final shape and total extension at pinching.

We consider the stability of a thin flexible cylinder considered as a beam, when subjected to axial flow and fixed at the upstream end only. A linear stability analysis of transverse motion aims at determining the risk of flutter as a function of the governing control parameters such as the flow velocity or the length of the cylinder. Stability is analysed applying a finite-difference scheme in space to the equation of motion expressed in the frequency domain. It is found that, contrary to previous predictions based on simplified theories, flutter may exist for very long cylinders, provided that the free downstream end of the cylinder is well-streamlined. More generally, a limit regime is found where the length of the cylinder does not affect the characteristics of the instability, and the deformation is confined to a finite region close to the downstream end. These results are found complementary to solutions derived for shorter cylinders and are confirmed by linear and nonlinear computations using a Galerkin method. A link is established to similar results on long hanging cantilevered systems with internal or external flow. The limit case of vanishing bending stiffness, where the cylinder is modelled as a string, is analysed and related to previous results. Comparison is also made to existing experimental data, and a simple model for the behaviour of long cylinders is proposed.

The laws governing the spread of a cluster of particles in homogeneous isotropic turbulence are derived using a theoretical approach based on inertial subrange scaling and statistical diffusion theory. The equations for the mean square dispersion of a puff admit an analytical solution in the inertial subrange and at large scales. The solution is consistent with Taylor's theory of absolute dispersion. An analytical derivation of the Richardson–Obukhov constant of relative dispersion is presented. A time scale for relative dispersion is identified, as well as relations between Lagrangian and Eulerian structure functions. The results are extended to turbulence at finite Reynolds number. A closure assumption for the relative kinetic energy, based on Taylor's theory, is presented. Comparisons with direct numerical simulations and laboratory experiments are reported.

The two-dimensional problem of wave diffraction by a strip of arbitrary width is investigated here in the context of linearized theory of water waves by reducing it to a pair of Carleman-type singular integral equations. These integral equations have been solved earlier by an iterative process which is valid only for a sufficiently wide strip. A new method is described here by which solutions of these integral equations are determined by solving a set of four Fredholm integral equations of the second kind, and the process is valid for a strip of arbitrary width. Numerical solutions of these Fredholm integral equations are utilized to obtain fairly accurate numerical estimates for the reflection and transmission coefficients. Previous numerical results for a wide strip are recovered from the present analysis. Additional results for the reflection coefficient are presented graphically for moderate values of the strip width which exhibit a less oscillatory nature of the curve than the case of a wide strip.

The motion of small air bubbles in a horizontal solid-body rotating flow is investigated experimentally. Bubbles with a typical radius of 1 mm are released in a liquid-filled horizontally rotating cylinder. We measure the transient motion of the bubbles in solid-body rotation and their final equilibrium position from which we compute drag and lift coefficients for a wide range of dimensionless shear rates 0.1<Sr<2 (Sr is the velocity difference over one bubble diameter divided by the slip velocity of the bubble) and Reynolds numbers 0.01<Re<500 (Re is based on the slip velocity and bubble diameter). For large Sr, we find that the drag force is increased by the shear rate. The lift force shows strong dependence on viscous effects. In particular, for Re<5, we measure negative lift forces, in line with theoretical predictions.

The normal field instability in magnetic liquids is investigated experimentally by means of a radioscopic technique which allows a precise measurement of the surface topography. The dependence of the topography on the magnetic field is compared to results obtained by numerical simulations via the finite-element method. Quantitative agreement has been found for the critical field of the instability, the scaling of the pattern amplitude and the detailed shape of the magnetic spikes. The fundamental Fourier mode approximates the shape to within 10% accuracy for a range of up to 40% of the bifurcation parameter of this subcritical bifurcation. The measured control parameter dependence of the wavenumber differs qualitatively from analytical predictions obtained by minimization of the free energy.

When a reactive and miscible less-viscous liquid displaces a more-viscous liquid in a Hele-Shaw cell, reactive miscible viscous fingering takes place. We succeed in showing experimentally how a reactive miscible viscous fingering pattern in a radial Hele-Shaw cell changes when the viscosity of the more-viscous liquid is varied owing to variation in chemical species concentration induced by an instantaneous chemical reaction. This is done by making use of a polymer solution's dependence of viscosity on pH. When the viscosity is increased by the chemical reaction, the shielding effect is suppressed and the fingers are widened. As a result, the ratio of the area occupied by the fingering pattern in a circle whose radius is the length of the longest finger is larger in the reactive case than in the non-reactive case. When the viscosity is decreased by the chemical reaction, in contrast, the shielding effect is enhanced and the fingers are narrowed. These lead to the area ratio being smaller in the reactive case than in the non-reactive case. A physical model to explain this change in the fingering pattern caused by the chemical reaction is proposed.