Boundary-forced stratified turbulence is studied in the prototypical case of turbulent channel flow subject to stable stratification. The large-eddy simulation approach is used with a mixed subgrid model that involves a dynamic eddy viscosity component and a scale-similarity component. After an initial transient, the flow reaches a new balanced state corresponding to active wall-bounded turbulence with reduced vertical transport which, for the cases in our study with moderate-to-large levels of stratification, coexists with internal wave activity in the core of the channel. A systematic reduction of turbulence levels, density fluctuations and associated vertical transport with increasing stratification is observed. Countergradient buoyancy flux is observed in the outer region for sufficiently high stratification.

Mixing of the density field in stratified channel flow results from turbulent events generated near the boundaries that couple with the outer, more stable flow. The vertical density structure is thus of interest for analogous boundary-forced mixing situations in geophysical flows. It is found that, with increasing stratification, the mean density profile becomes sharper in the central region between the two turbulent layers at the upper and lower walls, similar to observations in field measurements as well as laboratory experiments with analogous density-mixing situations.

Channel flow is strongly inhomogeneous with alternative choices for the Richardson number. In spite of these complications, the gradient Richardson number, Rig, appears to be the important local determinant of buoyancy effects. All simulated cases show that correlation coefficients associated with vertical transport collapse from their nominal unstratified values over a narrow range, 0.15 < Rig < 0.25. The vertical turbulent Froude number, Frw, has an O(1) value across most of the channel. It is remarkable that stratified channel flow, with such a large variation of overall density difference (factor of 26) between cases, shows a relatively universal behaviour of correlation coefficients and vertical Froude number when plotted as a function of Rig. The visualizations show wavy motion in the core region where the gradient Richardson number, Rig, is large and low-speed streaks in the near-wall region, typical of unstratified channel flow, where Rig is small. It appears from the visualizations that, with increasing stratification, the region with wavy motion progressively encroaches into the zone with active turbulence; the location of Rig ≃ 0.2 roughly corresponds to the boundary between the two zones.

We examine a spiralling slender inviscid liquid jet which emerges from a rapidly rotating orifice. The trajectory of this jet is determined using asymptotic methods, and the stability using a multiple scales approach. It is found that the trajectory of the jet becomes more tightly coiled as the Weber number is decreased. Unstable travelling wave modes are found to grow along the jet. The breakup length of the jet is calculated, showing good agreement with experiments.

Measurements have been made in nominally two-dimensional turbulent wakes generated by five different bluff bodies. Each wake has a different level of large-scale organization which is reflected in different amounts of large-scale anisotropy. Structure functions of streamwise (u) and lateral (v) velocity fluctuations at approximately the same value of Rλ, the Taylor microscale Reynolds number, indicate that inertial-range scales are significantly affected by the large-scale anisotropy. The effect is greater on v than u and more pronounced for the porous-body wakes than the solid-body wakes. In particular, ‘relative’ values of the scaling (or power-law) exponents indicate that the magnitude of the transverse exponents can exceed that of the longitudinal ones in the porous-body wakes. This is supported by the inertial-range behaviour of the spectra of u and v. The difference between the transverse and longitudinal exponents appears to depend on the large-scale anisotropy of the flow, as measured by the ratio of the variances of v and u and ratio of the integral length scales of v and u. The spanwise vorticity spectra are much less affected by the anisotropy than the spectra of u and v.

In this paper, we present relationships between the intrinsic radial distribution function (RDF) for a three-dimensional, isotropic system of particles and the lower-dimensional RDFs obtained experimentally from either two-dimensional or one-dimensional sampling of the data. The lower-dimensional RDFs are shown to be equivalent to integrals of the three-dimensional function, and as such contain less information than their three-dimensional counterpart. An important consequence is that the lower-dimensional RDFs are attenuated at separation distances below the characteristic length scale of the measurement. In addition, the inverse problem (calculating the three-dimensional RDF from the lower-dimensional measurements) is not well posed. However, recent results from direct numerical simulations (Reade & Collins 2000) showed that the three-dimensional RDF for aerosol particles in a turbulent flow field obeys a power-law dependence on r for r [Lt ] η, where η is the Kolmogorov scale of the turbulence. In this case, the inverse problem is well posed and it is possible to obtain the prefactor and exponent of the power law from one- or two-dimensional measurements. A procedure for inverting the data is given. All of the relationships derived in this paper have been validated by data derived from direct numerical simulations.

Electro-osmotic flow is a convenient mechanism for transporting fluid in microfluidic devices. The flow is generated through the application of an external electric field that acts on the free charges that exist in a thin Debye layer at the channel walls. The charge on the wall is due to the particular chemistry of the solid–fluid interface and can vary along the channel either by design or because of various unavoidable inhomogeneities of the wall material or because of contamination of the wall by chemicals contained in the fluid stream. The channel cross-section could also vary in shape and area. The effect of such variability on the flow through microfluidic channels is of interest in the design of devices that use electro-osmotic flow. The problem of electro-osmotic flow in a straight microfluidic channel of arbitrary cross-sectional geometry and distribution of wall charge is solved in the lubrication approximation, which is justified when the characteristic length scales for axial variation of the wall charge and cross-section are both large compared to a characteristic width of the channel. It is thereby shown that the volume flux of fluid through such a microchannel is a linear function of the applied pressure drop and electric potential drop across it, the coefficients of which may be calculated explicitly in terms of the geometry and charge distribution on the wall. These coefficients characterize the ‘fluidic resistance’ of each segment of a microfluidic network in analogy to the electrical ‘resistance’ in a microelectronic circuit. A consequence of the axial variation in channel properties is the appearance of an induced pressure gradient and an associated secondary flow that leads to increased Taylor dispersion limiting the resolution of electrophoretic separations. The lubrication theory presented here offers a simple way of calculating the distortion of the flow profile in general geometries and could be useful in studies of dispersion induced by inhomogeneities in microfluidic channels.

The hypothesis of the small scales being passively swept along by the large-scale motions in turbulent flow is extended to passive scalars in isotropic turbulence. A theory based on strong mutual cancellation between local and advective derivatives and other assumptions is shown to capture the Reynolds and Schmidt number dependence of time scales characterizing Eulerian and Lagrangian rates of change. Agreement with direct numerical simulation data improves systematically with increasing Reynolds number. In accordance with the physics of random sweeping, the Eulerian frequency spectrum is very similar in shape to the wavenumber spectrum, but is broadened at higher frequencies compared to its Lagrangian counterpart. Overall the hypothesis appears to be even more valid for transported scalars than for the velocity field, which gives support to the use of Lagrangian approaches in the study of turbulent mixing.

A new model for the transport equation for the turbulence energy dissipation rate ε and for the anisotropy of the dissipation rate tensor εij, consistent with the near-wall limits, is derived following the term-by-term approach and using results of direct numerical simulations (DNS) for several generic wall-bounded flows. Based on the two-point velocity covariance analysis of Jovanović, Ye & Durst (1995) and reinterpretation of the viscous term, the transport equation is derived in terms of the ‘homogeneous’ part εh of the energy dissipation rate. The algebraic expression for the components of εij was then reformulated in terms of εh, which makes it possible to satisfy the exact wall limits without using any wall-configuration parameters. Each term in the new equation is modelled separately using DNS information. The rational vorticity transport theory of Bernard (1990) was used to close the mean curvature term appearing in the dissipation equation. A priori evaluation of εij, as well as solving the new dissipation equation as a whole using DNS data for quantities other than εij, for flows in a pipe, plane channel, constant-pressure boundary layer, behind a backward-facing step and in an axially rotating pipe, all show good near-wall behaviour of all terms. Computations of the same flows with the full model in conjunction with the low-Reynolds number transport equation for (uiui) All Overbar, using εh instead of ε, agree well with the direct numerical simulations.

Configurations of shock wave reflection in steady supersonic flows have been experimentally investigated using a combination of two wedges. It has been experimentally proved by a symmetric arrangement that both regular and Mach reflections are possible in the dual-solution domain for various aspect ratio models. In the arrangement for the purpose of clarifying the influence of the wedge three-dimensionality, the transition from regular to Mach reflection can happen at any inlet aspect ratio, both when the inlet aspect ratio is increased and when it is reduced. The inlet aspect ratio has no effect on the transition provided it is high enough for the regular reflection point at the spanwise centre to be free from information from wedge edges. Flow visualization data produced using the vapour screen technique indicate that, in a region influenced by information from wedge edges, the three-dimensionality of experimental models promotes regular reflection rather than Mach reflection. To study the criteria for the transition between regular and Mach reflections, an asymmetric arrangement of two wedges has been used, and a hysteresis effect is clearly evident. The transition from regular to Mach reflection, however, occurs significantly below the detachment condition, and moreover, the repeatability of the transition angle is not satisfactorily achieved. These experimental results imply that wind tunnel disturbances may dominate the transition in the dual-solution domain. The stability of regular reflection in the dual-solution domain is discussed, and effects of free-stream disturbances are experimentally examined by producing water vapour in the free stream as an artificial disturbance.

Fluid flow in microdevices consisting of pairs of microchannels in series was studied. The dimensions of the channels are about 40 μm × 1 μm × 2000 μm for the wide and about 20 μm × 1 μm × 2000 μm for the narrow channels. Pairs of wide and narrow channels, with integrated pressure sensors, are connected via transition sections with included angles varying from 5° to 180°. Minor pressure losses (not due to friction) were studied by passing nitrogen through the channels under inlet pressures up to 60 p.s.i. Each device was tested in the contraction mode, flow from wide to narrow channel, and in the opposite expansion mode, flow from narrow to wide channel. Mass flow rate was first measured as a function of the overall pressure drop. The detailed pressure distribution along the straight segments and around the transition section was then measured in order to understand the flow pattern. The Reynolds number for these flows is less than 1, suggesting the flow to be of the Hele-Shaw type with no separation such that the results for all the devices should be similar. However, the flow rate was found to decrease and the pressure loss to increase significantly with increasing included angle of the transition section, regardless of the flow direction. Flow separation due to the transition sections, if indeed there is any, cannot explain the large pressure drop since the kinetic energy is negligible.

We consider the flow of instantaneous releases of a finite volume of viscous fluid in a narrow vertical fracture or Hele-Shaw cell, when there is a still narrower vertical crack in the horizontal base of the cell. The predominant motion is over the horizontal surface, but fluid also drains through the crack, progressively diminishing the volume of the current in the fracture. When the crack is shallow on the scale of the current, it saturates immediately with the draining fluid. In this case, we obtain an exact analytical solution for the motion. When the crack is deeper and does not saturate immediately, we calculate numerically the motion of the fluid in both the fracture and the crack. In each case the current advances to a finite run-out length and then retreats: we describe both phases of the motion and characterize the run-out length in terms of the controlling parameters.

The mechanism of turbulence development in periodic Klebanoff transition in a boundary layer has been studied experimentally and in a direct numerical simulation (DNS) with controlled disturbance excitation. In order to compare the results quantitatively, the flow parameters were matched in both methods, thus providing complementary data with which the origin of turbulence in the transition process could be explained. Good agreement was found for the development of the amplitude and shape of typical disturbance structures, the Λ-vortices, including the development of ring-like vortices and spikes in the time traces. The origin and the spatial development of random velocity perturbations were measured in the experiment, and are shown together with the evolution of local high-shear layers. Since the DNS is capable of providing the complete velocity and vorticity fields, further conclusions are drawn based on the numerical data. The mechanisms involved in the flow randomization process are presented in detail. It is shown how the random perturbations which initially develop at the spike-positions in the outer part of the boundary layer influence the flow randomization process close to the wall. As an additional effect, the interaction of vortical structures and high-shear layers of different disturbance periods was found to be responsible for accelerating the transition to a fully developed turbulent flow. These interactions lead to a rapid intensification of a high-shear layer very close to the wall that quickly breaks down because of the modulation it experiences through interactions with vortex structures from the outer part of the boundary layer. The final breakdown process will be shown to be dominated by locally appearing vortical structures and shear layers.

We use time-of-flight and energy analysis techniques to measure in a vacuum the charge, specific charge and stopping potential of primary and satellite droplets generated by electrosprays of tributyl phosphate solutions. This information, of interest in itself, is subsequently analysed to obtain the following relevant parameters of the jet emanating from the Taylor cone: the velocity of the fluid at the breakup point, the voltage difference between the liquid cone and jet breakup location, and the most probable wavelength for varicose breakup. A large fraction of the electrospray needle voltage is used to accelerate the jet. Indeed, for the solutions of lowest electrical conductivities studied here, the voltage difference between electrospray needle and jet breakup location becomes approximately 90% of the needle voltage. In addition, the pressure of the jet fluid at the breakup point is negligible compared to its specific kinetic energy. The specific charge distribution function of the main droplets produced in the varicose breakup is remarkably narrow. Hence, the limiting and commonly accepted case of varicose breakup at constant electric potential is not consistent with this experimental observation. On the other hand, a scenario in which the electric charge is bound to the jet surface seems to be a good approximation to simulate the effect of charge on capillary breakup. It is also found that the effect of viscosity on the formation of droplets is paramount in electrosprays of moderate and high electrical conductivity. We expect that these measurements will guide the analytical modelling of cone-jets.

Capillary waves, like other surface waves on water, generate a rectified, or time-averaged, vorticity field extending beyond the oscillatory (Stokes) layer at the surface. This vorticity field ω is particularly interesting in relation to the parasitic capillary waves found on the forward slopes of steep gravity waves. Longuet-Higgins (1992) suggested that the rectified vorticity from the parasitic capillaries might contribute significantly to the vorticity observed beneath the crest of the gravity wave. The basic calculations by Longuet-Higgins (1992) were only of the horizontally averaged values of ω. Here we extend his theory by calculating, for pure capillary waves, the space variation of ω, to second order in the steepness of the capillary waves. Thus, the vorticity, and hence velocity, fields are calculated in the oscillatory Stokes layer and just beyond it, to the second order. Good agreement is found both with numerical simulations and with experimental measurements.

The surface-tension-driven motion of a surfactant-coated liquid thread in inviscid surroundings is investigated using linear stability theory as well as one-dimensional nonlinear approximations to the governing Navier–Stokes equations. Examination of analytic limits of the linear dispersion relationship demonstrates that surfactant acts as a distinct mechanism for long-wavelength cut-off, instead of inertia, if the surfactant effects exceed a critical value, β = ½, where β is a dimensionless surface-tension gradient. Two different long-wavelength regimes can be identified, depending on the degree of tangential stress, with β = 1 characterizing a transition from extensionally dominated inertial flow to shear-dominated viscous flow. One-dimensional nonlinear models are formulated which capture the changes in behaviour with variation of β by inclusion of the necessary high-order terms. Scaling close to breakup shows that surfactant is swept away from the pinching region and then has little effect.

The flux Richardson number Rf (also known as the mixing efficiency) for the stably stratified atmospheric boundary layer is investigated as a function of the gradient Richardson number Rig using data taken during two field studies: the Vertical Transport and Mixing Experiment (VTMX) in Salt Lake City, Utah (October 2000), and a long-term rural field data set from Technical Area 6 (TA-6) at Los Alamos National Laboratory, New Mexico. The results show the existence of a maximum Rf (0.4–0.5) at a gradient Richardson number of approximately unity. These large-Reynolds-number results agree well with recent laboratory stratified shear layer measurements, but are at odds with some commonly used Rf parameterizations, particularly under high-Rig conditions. The observed variations in buoyancy flux and turbulent kinetic energy production are consistent with the concept of global intermittency of the atmospheric stable boundary layer.

The dynamics of dry granular flows down a vertical glass pipe of small diameter have been studied experimentally. Simultaneous measurements of pressure profiles, air and grain flow rates and volume fractions of particles have been realized together with spatio-temporal diagrams of the grain distribution down the tube. At large grain flow rates, one observes a stationary flow characterized by high particle velocities, low particle fractions and a downflow of air resulting in an underpressure in the upper part of the pipe. A simple model assuming a free fall of the particles slowed down by air friction and taking into account finite particle fraction effects through Richardson–Zaki's law has been developed: it reproduces pressure and particle fraction variations with distance and estimates friction forces with the wall. At lower flow rates, sequences of high-density plugs separated by low-density bubbles moving down at a constant velocity are observed. The pressure is larger than outside the tube and its gradient reflects closely the weight of the grains. Writing mass and momentum conservation equations for the air and for the grains allows one to estimate the wall friction, which is less than 10% of the weight for grains with a clean smooth surface but up to 30% for grains with a rougher surface. At lower flow rates, oscillating-wave regimes resulting in large pressure fluctuations are observed and their frequency is predicted.

Time-dependent swirling flows inside an enclosed cylindrical rotor–stator cavity with aspect ratio H/R = 4, larger than the ones usually considered in the literature, are studied. Within a certain range of governing parameters, vortex breakdown phenomena can arise along the axis. Very recent papers exhibiting some particular three-dimensional effects have stimulated new interest in this topic. The study is carried out by a numerical resolution of the three-dimensional Navier–Stokes equations, based on high-order spectral approximations in order to ensure very high accuracy of the solutions.

The first transition to an oscillatory regime occurs through an axisymmetric bifurcation (a supercritical Hopf bifurcation) at Re = 3500. The oscillatory regime is caused by an axisymmetric mode of centrifugal instability of the vertical boundary layer and the vortex breakdown is axisymmetric, being composed of two stationary bubbles. For Reynolds numbers up to Re = 3500, different three-dimensional solutions are identified. At Re = 4000, the flow supports the k = 5 mode of centrifugal instability. By increasing the rotation speed to Re = 4500, the vortex breakdown evolves to an S-shaped type after a long computational time. The structure is asymmetric and gyrates around the axis inducing a new time-dependent regime. At Re = 5500, the structure of the vortex breakdown is more complex: the upper part of the structure takes a spiral form. The maximum rotation speed is reached at Re = 10000 and the flow behaviour is now chaotic. The upper structure of the breakdown can be related to the spiral-type. Asymmetric flow separation on the container wall in the form of spiral arms of different angles is also prominent.

The effects of viscosity on the instability properties of the Batchelor vortex are investigated. The characteristics of spatially amplified branches are first documented in the convectively unstable regime for different values of the swirl parameter q and the co-flow parameter a at several Reynolds numbers Re. The absolute–convective instability transition curves, determined by the Briggs–Bers zero-group velocity criterion, are delineated in the (a, q)-parameter plane as a function of Re. The azimuthal wavenumber m of the critical transitional mode is found to depend on the magnitude of the swirl q and on the jet (a > −0.5) or wake (a < −0.5) nature of the axial flow. At large Reynolds numbers, the inviscid results of Olendraru et al. (1999) are recovered. As the Reynolds number decreases, the pocket of absolute instability in the (a, q)-plane is found to shrink gradually. At Re = 667; the critical transitional modes for swirling jets are m = −2 or m = −3 and absolute instability prevails at moderate swirl values even in the absence of counterflow. For higher swirl levels, the bending mode m = −1 becomes critical. The results are in good overall agreement with those obtained by Delbende et al. (1998) at the same Reynolds number. However, a bending (m = +1) viscous mode is found to partake in the outer absolute–convective instability transition for jets at very low positive levels of swirl. This asymmetric branch is the spatial counterpart of the temporal viscous mode isolated by Khorrami (1991) and Mayer & Powell (1992). At Re = 100, the critical transitional mode for swirling jets is m = −2 at moderate and high swirl values and, in order to trigger an absolute instability, a slight counterflow is always required. A bending (m = +1) viscous mode again becomes critical at very low swirl values. For wakes (a < −0.5) the critical transitional mode is always found to be the bending mode m = −1, whatever the Reynolds number. However, above q = 1.5, near-neutral centre modes are found to define a tongue of weak absolute instability in the (a, q)-plane. Such modes had been analytically predicted by Stewartson & Brown (1985) in a strictly temporal inviscid framework.

The development of a second-order integral model for a round turbulent buoyant jet is reported based on new experimental data on turbulent mass and momentum transport. The mean and turbulent characteristics of a round vertical buoyant jet covering the full range from jets to plumes were investigated using a recently developed combined digital particle image velocimetry (DPIV) and planar laser-induced fluorescence (PLIF) system. The system couples the two well-known techniques to enable synchronized planar measurements of flow velocities and concentrations in a study area. The experimental results conserved the mass and momentum fluxes introduced at the source accurately with closure errors of less than 5%. The momentum flux contributed by turbulence and streamwise pressure gradient was determined to be about 10% of the local mean momentum flux in both jets and plumes. The turbulent mass flux, on the other hand, was measured to be about 7.6% and 15% of the mean mass flux for jets and plumes respectively. While the velocity spread rate was shown to be independent of the flow regime, the concentration-to-velocity width ratio λ varied from 1.23 to 1.04 during the transition from jet to plume. Based on the experimental results, a refined second-order integral model for buoyant jets that achieves the conservation of total mass and momentum fluxes is proposed. The model employs the widely used entrainment assumption with the entrainment coefficient taken to be a function of the local Richardson number. Improved prediction is achieved by taking into account the variation of turbulent mass and momentum fluxes. The variation of turbulent mass flux is modelled as a function of the local Richardson number. The turbulent momentum flux, on the other hand, is treated as a fixed percentage of the local mean momentum flux. In addition, unlike most existing integral models that assume a constant concentration-to-velocity width ratio, the present model adopts a more accurate approach with the ratio expressed as a function of the local Richardson number. As a result, smooth transition of all relevant mean and turbulent characteristics from jet to plume is predicted, which is in line with the underlying physical processes.

A simple spectral model is used to examine what is required to determine the energy and integral scale in homogeneous isotropic turbulence. The problem is that these are determined in part by the largest scales of the turbulence which are either not simulated at all by DNS or experiments, or cannot be estimated because of an insufficient statistical sample. The absence of scales an order of magnitude below the peak in the energy spectrum is shown to affect the determination significantly. Since this energy peak shifts to lower wavenumbers as the flow evolves, the problem becomes progressively worse during decay. It is suggested that almost all reported integral scales for isotropic decaying turbulence are questionable, and that the power laws fitted to them are seriously in error. Approximate correction using the spectral model shows that recent DNS data which decay as u2 ∝ tn with constant n, are also consistent with L ∝ t1/2.