Experimental results obtained during rapid shearing of several dry, coarse, granular materials in an annular shear cell are described. The main purpose of the tests was to obtain information that could be used to guide the theoretical development of constitutive equations suitable for the rapid flow of cohesionless bulk solids at low stress levels. The shear-cell apparatus consists of two concentric disk assemblies mounted on a fixed shaft. Granular material was contained in an annular trough in the bottom disk and capped by a lipped annular ring on the top disk. The bottom disk can be rotated at specified rates, while the top disk is loaded vertically and is restrained from rotating by a torque arm connected to a force transducer. The apparatus was thus designed to determine the shear and normal stresses as functions of solids volume fraction and shear rate.

Tests were performed with spherical glass and polystyrene beads of nearly uniform diameters, spherical polystyrene beads having a bimodal size distribution and with angular particles of crushed walnut shells. The particles ranged from about ½ to 2 mm in size. At the lower concentrations and high shear rates the stresses are generated primarily by collisional transfer of momentum and energy. Under these conditions, both normal and shear stresses were found to be proportional to the particle density, and the squares of the shear rate and particle diameter. At higher concentrations and lower shear rates, dry friction between particles becomes increasingly important, and the stresses are proportional to the shear rate raised to a power less than two. All tests showed strong increases in stresses with increases in solids concentrations. The ratio of shear to normal stresses showed only a weak dependence upon shear rate, but it increased with decreasing concentration. At the very highest concentrations with narrow shear gaps, finite-particle-size effects became dominant and differences in stresses of as much as an order of magnitude were observed for the same shear rate and solids concentration.

We use silicon strip detectors (originally developed for the CLEO III high-energy particle physics experiment) to measure fluid particle trajectories in turbulence with temporal resolution of up to 70000 frames per second. This high frame rate allows the Kolmogorov time scale of a turbulent water flow to be fully resolved for 140 [ges ] Rλ [ges ] 970. Particle trajectories exhibiting accelerations up to 16000 m s −2 (40 times the r.m.s. value) are routinely observed. The probability density function of the acceleration is found to have Reynolds-number-dependent stretched exponential tails. The moments of the acceleration distribution are calculated. The scaling of the acceleration component variance with the energy dissipation is found to be consistent with the results for low-Reynolds-number direct numerical simulations, and with the K41-based Heisenberg–Yaglom prediction for Rλ [ges ] 500. The acceleration flatness is found to increase with Reynolds number, and to exceed 60 at Rλ = 970. The coupling of the acceleration to the large-scale anisotropy is found to be large at low Reynolds number and to decrease as the Reynolds number increases, but to persist at all Reynolds numbers measured. The dependence of the acceleration variance on the size and density of the tracer particles is measured. The autocorrelation function of an acceleration component is measured, and is found to scale with the Kolmogorov time τη.

This paper studies the two-dimensional convective motion in a rectangular cavity, the two vertical sides of which are maintained at different temperatures. This system is studied for the special case in which the temperature difference ΔT between the two vertical walls is so large that the transfer of heat from one vertical wall to the other is achieved almost entirely by convection. Heat transfer by conduction is assumed to be of importance only in thin boundary layers adjoining the walls. For a cavity of height H, the boundary layers on the two vertical walls are found to have thickness proportional to [ell ], where [ell ]4 = κνH/γgΔT, and the condition for the boundary-layer regime to be established is that [ell ] be small compared with the width of the cavity. An approximate solution of the problem is obtained for the case of large values of the Prandtl number ν/κ, and found to be in satisfactory agreement with experimental results obtained by Elder (1965).

The mean flow and turbulent statistics obtained from the numerical simulation of the fully developed turbulent flow through a straight duct of square cross-section are reported. The Reynolds number based on the bulk velocity and hydraulic diameter is 4410. Spatial and temporal approximations of the equations of motion were derived from standard finite-difference techniques. To achieve sufficient spatial resolution 16.1 × 106 grid nodes were employed. Turbulent statistics along the wall bisectors show good agreement with plane channel data despite the influence of the sidewalls in the former flow. The mean secondary flow field consists of two counter-rotating cells symmetrically placed about the corner bisectors with their common flow towards each corner with strong evidence for the existence of a smaller and much weaker pair situated about the wall bisectors. The mean streamwise vorticity of each corner cell is found to be associated with a stronger vorticity distribution of the opposite sign having an absolute maximum on the nearest duct wall.

Measurements in a zero-pressure-gradient turbulent boundary layer over a mesh-screen rough wall indicate several differences, in both inner and outer regions, in comparison to a smooth-wall boundary layer. The mean velocity distribution indicates that, apart from the expected k-type roughness function shift in the inner region, the strength of the rough-wall outer region ‘wake’ is larger than on a smooth wall. Normalizing on the wall shear stress, there is a significant increase in the normal turbulence intensity and a moderate increase in the Reynolds shear stress over the rough wall. The longitudinal turbulence intensity distribution is essentially the same for both surfaces. Normalized contributions to the Reynolds shear stress from the second (Q2) and fourth (Q4) quadrants are greater over the rough wall. The data indicate that not only are Q2 and Q4 events stronger on the rough wall but their frequency of occurrence is nearly twice as large for the rough wall as for the smooth wall. Comparison between smooth- and rough-wall spectra of the normal velocity fluctuation suggests that the strength of the active motion may depend on the nature of the surface.

We study the linearized Navier–Stokes (LNS) equations in channel flows from an input–output point of view by analysing their spatio-temporal frequency responses. Spatially distributed and temporally varying body force fields are considered as inputs, and components of the resulting velocity fields are considered as outputs into these equations. We show how the roles of Tollmien–Schlichting (TS) waves, oblique waves, and streamwise vortices and streaks in subcritical transition can be explained as input–output resonances of the spatio-temporal frequency responses. On the one hand, we demonstrate the effectiveness of input field components, and on the other, the energy content of velocity perturbation components. We establish that wall-normal and spanwise forces have much stronger influence on the velocity field than streamwise force, and that the impact of these forces is most powerful on the streamwise velocity component. We show this using the relative scaling of the different input–output system components with the Reynolds number. We further demonstrate that for the streamwise constant perturbations, the spanwise force localized near the lower wall has, by far, the strongest effect on the evolution of the velocity field.

This paper presents the in-line force coefficients for circular cylinders in planar oscillatory flows of small amplitude. The results are compared with the theoretical predictions of Stokes (1851) and Wang (1968). For two-dimensional, attached- and laminar-flow conditions the data are, as expected, in good agreement with the Stokes–Wang analysis. The oscillatory viscous flow becomes unstable to axially periodic vortices above a critical Keulegan–Carpenter number K (K = UmT/D, Um = the maximum velocity in a cycle, T = the period of flow oscillation, and D = the diameter of the circular cylinder) for a given β (β = Re/K = D2/vT, Re = UmD/v, and v = the kinematic viscosity of fluid) as shown experimentally by Honji (1981) and theoretically by Hall (1984). The present investigation has shown that the Keulegan—Carpenter number at which the drag coefficient Cd deviates rather abruptly from the Stokes—Wang prediction nearly corresponds to the critical K at which the vortical instability occurs.

Analytical and numerical analyses have been made of the physical behaviour of a collapsing bubble in a liquid. The mathematical formulation takes into account the effects of compressibility of the liquid, non-equilibrium condensation of the vapour, heat conduction and the temperature discontinuity at the phase interface. Numerical solutions for the collapse of the bubble are obtained beyond the time when the bubble reaches its minimum radius up to the stage when a pressure wave forms and propagates outward into the liquid. The numerical results indicate that evaporation and condensation strongly influence the dynamical behaviour of the bubble.

In addition, the propagation of the stress wave, both in a solid and a liquid, due to the collapse of the bubble has been observed by means of the dynamic photoelasticity. It is clearly demonstrated that the stress wave in a photoelastic specimen is caused by impact of the pressure wave radiated from the bubble.

With a transverse array of channels of equal widths but differing resistances, we have generated an improved approximation to spatially homogeneous turbulent shear flow. The scales continue to grow with downstream distance, even in a region where the mean velocity gradient and one-point turbulence moments (component energies and shear stress) have attained essentially constant values. This implies asymptotic non-stationarity in the basic Eulerian frame convected with the mean flow, behaviour which seems to be inherent to homogeneous turbulent shear flow.

Two-point velocity correlations with space separation and with space-time separation yield characteristic departures from isotropy, including clear ‘upstream–downstream’ unsymmetries which cannot be classified simply as axis tilting of ellipse-like iso-correlation contours.

The high wave-number structure is roughly locally isotropic although the turbulence Reynolds number based on Taylor ‘microscale’ and r.m.s. turbulent velocity is only 130. Departures from isotropy in the turbulent velocity gradient moments are measurable.

The approximation to homogeneity permits direct estimation of all components of the turbulent pressure/velocity-gradient tensor, which accounts for inter-component energy transfer and helps to regulate the turbulent shear stress. It is found that its principal axes are aligned with those of the Reynolds stress tensor. Finally, the Rotta (1951, 1962) linear hypothesis for intercomponent energy transfer rate is roughly confirmed.

A scenario of transition to turbulence likely to occur during the development of natural disturbances in a flat-plate boundary layer is studied. The perturbations at the leading edge of the flat plate that show the highest potential for transient energy amplification consist of streamwise aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. Direct numerical simulations are used to follow the nonlinear evolution of these streaks and to verify secondary instability calculations. The theory is based on a linear Floquet expansion and focuses on the temporal, inviscid instability of these flow structures. The procedure requires integration in the complex plane, in the coordinate direction normal to the wall, to properly identify neutral modes belonging to the discrete spectrum. The streak critical amplitude, beyond which streamwise travelling waves are excited, is about 26% of the free-stream velocity. The sinuous instability mode (either the fundamental or the subharmonic, depending on the streak amplitude) represents the most dangerous disturbance. Varicose waves are more stable, and are characterized by a critical amplitude of about 37%. Stability calculations of streamwise streaks employing the shape assumption, carried out in a parallel investigation, are compared to the results obtained here using the nonlinearly modified mean fields; the need to consider a base flow which includes mean flow modification and harmonics of the fundamental streak is clearly demonstrated.

We consider here the manner in which the state of rigid rotation of a contained viscous fluid is established. It is found that the motion consists of three distinct phases, namely, the development of the Ekman layer, the inviscid fluid spin-up, and the viscous decay of residual oscillations. The Ekman layer plays the significant role in the transient process by inducing a small circulatory secondary flow. Low angular momentum fluid entering the layer from the geostrophic interior is replaced by fluid with greater angular momentum convected inward to conserve mass. The rotational velocity in the interior increases as a consequence of conservation of angular momentum, and the vorticity is increased through the stretching of vortex lines. The spin-up time is $T = (L^2|v \Omega)^{\frac {1}{2}}.$ Boundary-layer theory is used to study the phenomenon in the case of general axially symmetric container configuration and explicit formulas are deduced which exhibit the effect of geometry in spin-up. The special case of cylindrical side walls is also investigated by this method. The results of very simple experiments confirm the theoretical predictions.

The experimental results of Saffman & Taylor (1958) and Pitts (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the interface. Using free streamline techniques, the shape of the free surface is expressed as the solution of a nonlinear integro-differential equation. The equation is solved numerically and the solutions are compared with experimental results. The shapes of the profiles are very well predicted, but the dependence of finger width on surface tension is not quantitatively accurate, although the qualitative behaviour is correct. A conflict between the numerics and a formal singular perturbation analysis is noted but not resolved. The stability of the steady finger to small disturbances is also examined. Linearized stability analysis indicates that the two-dimensional fingers are not stabilized by the surface-tension effect, which disagrees with the experimental observations. A possible reason for the discrepancy between theory and experiment is suggested.

Measurements of fluctuating pressure and velocity, together with instantaneous smoke-flow visualizations, are presented in order to reveal the unsteady structure of a separated and reattaching flow. It is shown that throughout the separation bubble a low-frequency motion can be detected which appears to be similar to that found in other studies of separation. This effect is most significant close to separation, where it leads to a weak flapping of the shear layer. Lateral correlation scales of this low-frequency motion are less than the reattachment length, however; it appears that its timescale is about equal to the characteristic timescale for the shear layer and bubble to change between various shedding phases. These phases were defined by the following observations: shedding of pseudoperiodic trains of vortical structures from the reattachment zone, with a characteristic spacing between structures of typically 60% to 80% of the bubble length; a large-scale but irregular shedding of vorticity; and a relatively quiescent phase with the absence of any large-scale shedding structures and a significant ‘necking’ of the shear layer downstream of reattachment.

Spanwise correlations of velocity in the shear layer show on average an almost linear growth of spanwise scale up to reattachment. It appears that the shear layer reaches a fully three-dimensional state soon after separation. The reattachment process does not itself appear to impose an immediate extra three-dimensionalizing effect upon the large-scale structures.

Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

This paper deals with the steady flow in a rectangular cavity where the motion is driven by the uniform translation of the top wall. Creeping flow solutions for cavities having aspect ratios from ¼ to 5 were obtained numerically by a relaxation technique and were shown to compare favourably with Dean & Montagnon's (1949) similarity solution, as extended by Moffatt (1964), in the region near the bottom corners of a square cavity as well as throughout the major portion of a cavity with aspect ratio equal to 5. In addition, for a Reynolds number range from 20 to 4000, flow patterns were determined experimentally by means of a photographic technique for finite cavities, as well as for cavities of effectively infinite depth. These experimental results suggest that, within finite cavities, the high Reynolds number steady flow should consist essentially of a single inviscid core of uniform vorticity with viscous effects being confined to thin shear layers near the boundaries, while, for cavities of infinite depth, the viscous and inertia forces should remain of comparable magnitude throughout the whole domain even in the limit of very large Reynolds number R.

Detailed instantaneous velocity fields of a jet in crossflow have been measured with stereoscopic particle image velocimetry (PIV). The jet originated from a fully developed turbulent pipe flow and entered a crossflow with a turbulent boundary layer. The Reynolds number based on crossflow velocity and pipe diameter was 2400 and the jet to crossflow velocity ratios were R=3.3 and R=1.3. The experimental data have been analysed by proper orthogonal decomposition (POD). For R=3.3, the results in several different planes indicate that the wake vortices are the dominant dynamic flow structures and that they interact strongly with the jet core. The analysis identifies jet shear-layer vortices and finds that these vortical structures are more local and thus less dominant. For R=1.3, on the other hand, jet shear-layer vortices are the most dominant, while the wake vortices are much less important. For both cases, the analysis finds that the shear-layer vortices are not coupled to the dynamics of the wake vortices. Finally, the hanging vortices are identified and their contribution to the counter-rotating vortex pair (CVP) and interaction with the newly created wake vortices are described.

Particle velocity autocorrelations of single spherical beads (46·5 μhollow glass, 87 μ glass, 87 μ corn pollen, and 46·5 μ copper) were measured in a grid-generated turbulence. The hollow glass beads were small and light enough to behave like fluid points; the other types had significant inertia and ‘crossing trajectories’ effects. The autocorrelations decreased much faster for heavier particles, in contradiction to previous experimental results. The integral scale for the copper beads was 1/3 of that for the hollow glass beads. The particle velocity correlations and the Eulerian spatial correlation were coincident within experimental error when the separation was non-dimensionalized by the respective integral scale. The data generated by the hollow glass beads can be used to estimate Lagrangian fluid properities. The Lagrangian time integral scale is approximated by L/u′, where L is the Eulerian integral scale and u′ is the turbulence intensity.

Notwithstanding the fact that the instability of the separated shear layer in the cylinder wake has been extensively studied, there remains some uncertainty regarding not only the critical Reynolds number at which the instability manifests itself, but also the variation of its characteristic frequency with Reynolds number (Re). A large disparity exists in the literature in the precise value of the critical Reynolds number, with quoted values ranging from Re = 350 to Re = 3000. In the present paper, we demonstrate that the spanwise end conditions which control the primary mode of vortex shedding significantly affect the shear-layer instability. For parallel shedding conditions, shear-layer instability manifests itself at Re ≈ 1200, whereas for oblique shedding conditions it is inhibited until a significantly higher Re ≈ 2600, implying that even in the absence of a variation in free-stream turbulence level, the oblique angle of primary vortex shedding influences the onset of shear-layer instability, and contributes to the large disparity in quoted values of the critical Reynolds number. We confirm the existence of intermittency in shear-layer fluctuations and show that it is not related to the transverse motion of the shear layers past a fixed probe, as suggested previously. Such fluctuations are due to an intermittent streamwise movement of the transition point, or the location at which fluctuations develop rapidly in the shear layer.

Following the original suggestion of Bloor (1964), it has generally been assumed in previous studies that the shear-layer frequency (normalized by the primary vortex shedding frequency) scales with Re1/2, although a careful examination of the actual data points from these studies does not support such a variation. We have reanalysed all of the actual data points from previous investigations and include our own measurements, to find that none of these studies yields a relationship which is close to Re1/2. A least-squares analysis which includes all of the previously available data produces a variation of the form Re0·67. Based on simple physical arguments that account for the variation of the characteristic velocity and length scales of the shear layer, we predict a variation for the normalized shear-layer frequency of the form Re0·7, which is in good agreement with the experimental measurements.

In this paper the flow behind a pair of bluff bodies placed side by side in a stream is studied using a variety of flow-visualization methods. Above a critical gap size between the bodies, vortex-shedding synchronization occurs, either in phase or in antiphase. It has previously been assumed that such synchronization forms a wake comprising two parallel vortex streets in phase and in antiphase respectively. In the present paper we find that two antiphase streets are indeed formed, although in-phase shedding leads to the development of a single large-scale wake. The vortices which are formed simultaneously at the cylinders rotate around one another downstream, each pair forming a ‘binary vortex’. The combined wake comprises a street of such vortices, which we term a binary vortex street. Below a critical gap size between the bluff bodies the flow becomes asymmetric. We observe in this regime certain harmonic modes of vortex shedding whereby the shedding frequency on one side of the wake is a multiple of that on the other. Again, a large-scale wake is formed downstream. The present observations lead to a new interpretation of hot-wire-frequency data from other studies in terms of the harmonic modes.