This paper presents laser-Doppler measurements of the mean velocity and statistical moments of turbulent velocity fluctuations in the near-wall region of a fully developed pipe flow at low Reynolds numbers. A refractive-index-matched fluid was used in a Duran-glass test section to permit access to the near-wall region without distortion of the laser beams. All measurements were corrected for the influence of the finite size of measuring control volume. Measurements of long-time statistical averages of all three fluctuating velocity components in the near-wall region are presented. It is shown that the turbulence intensities in the wall region do not scale with inner variables. However, the limiting behaviour of the intensity components very close to the wall show only small variations with the Reynolds number. Measurements of higher-order statistical moments, the skewness and flatness factors, of axial and tangential velocity components confirm the limiting behaviour of these quantities obtained from direct numerical simulations of turbulent channel flow. The comparison of measured data with those obtained from direct numerical simulations reveals that noticeable discrepancies exist between them only with regard to the flatness factor of the radial velocity component near the wall. The measured v’ flatness factor does not show the steep rise close to the wall indicated by numerical simulations. Analysis of the measured data in the near-wall region reveals significant discrepancies between the present LDA measurements and experimental results obtained using the hot-wire anemometry.

The linear stability of a shear layer of an inviscid compressible fluid is considered. It is shown that there is instability of two-dimensional disturbances at all values of the Mach number, contrary to previous results for a vortex sheet. The difference arises from the discovery of a second unstable mode. This mode is supersonic, decays weakly with distance from the shear layer, and is not governed by the principle of exchange of stabilities. Detailed numerical and asymptotic results are given for the hyperbolic-tangent shear layer.

This investigation presents a unique and elaborate set of experiments relating the generation of noise to the evolution of large-scale turbulence structures within an ideally expanded, Mach 1.28, high-Reynolds-number $(1.03\,{\times}\,10^{6})$ jet. The results appear to indicate many similarities between the noise generation processes of high-speed low-Reynolds-number and high-speed high-Reynolds-number jets. Similar to the rapid changes observed in theregion of noise generation in low-Reynolds-number jets in previous experimental and computational work, a series of robust flow features formed approximately one convective time scale before noise emission and then rapidly disintegrated shortly before the estimated moment of noise emission. Coincident with the disintegration, a positive image intensity fluctuation formed at the jet centreline in a region that is immediately past the endof the potential core. This indicates mixed fluid had reached the jet core. These results are consistent with the formation of large-scale structures within the shear layer, which entrain ambient air into the jet, and their eventual interaction and disintegration apparently result in noise generation. These results are quite different from the evolution of the jet during prolonged periods that lacked significant sound emission. The observations presented in this work were made through the use of well-established technique that were brought together in an unconventional fashion. The sources of large-amplitude sound waves were estimated in time and three-dimensional space using a novel microphone array/beamforming algorithm while the noise-generation region of the mixing layer was simultaneously visualized on two orthogonal planes (one of which was temporally resolved). The flow images were conditionally sampled based on whether or not a sound wave wascreated within the region of the flow while it was being imaged and a series of images was compiled that was roughly phase-locked onto the moment of sound emission. Another set of images was gathered based on a lack of sound waves reaching the microphone array over several convective time scales. Proper orthogonal decomposition (POD) was then used tocreate a basis for the flow images and this basis was used to reconstruct the evolution of the jet.

An experimental study of the structure of the internal layer which grows down-stream from a rough-to-smooth surface change shows it to be essentially different from that studied by Antonia & Luxton (1971 b) for the case of a smooth-to-rough perturbation. The rate of growth of the internal layer is less than that for the smooth-to-rough step and it appears that the more intense initial rough-wall flow dictates the rate of diffusion of the disturbance for a considerable distance. Inside the internal layer the mixing length I is increased relative to the equilibrium distribution I = KY. A turbulent energy budget shows that the advection is comparable with the production or dissipation, whilst there seems to be some diffusion of energy into the internal-layer region close to the wall. The boundary layer, as a whole, recovers much more slowly following a rough-to-smooth change than following a smooth-to-rough change, and at the last measuring station (16 boundary-layer thicknesses from the start of the smooth surface) the distributions of mean velocity and Reynolds shear stress are far from self-preserving.

The flow about a body placed inside a channel differs from its unbounded counterpart because of the effects of wall confinement, shear in the incoming velocity profile, and separation of vorticity from the channel walls. The case of a circular cylinder placed between two parallel walls is here studied numerically with a finite element method based on the vorticity–streamfunction formulation for values of the Reynolds number consistent with a two-dimensional assumption.

The transition from steady flow to a periodic vortex shedding regime has been analysed: transition is delayed as the body approaches one wall because the interaction between the cylinder wake and the wall boundary layer vorticity constrains the separating shear layer, reducing its oscillations. The results confirm previous observations of the inhibition of vortex shedding for a body placed near one wall. The unsteady vortex shedding regime changes, from a pattern similar to the von Kármán street (with some differences) when the body is in about the centre of the channel, to a single row of same-sign vortices as the body approaches one wall. The separated vortex dynamics leading to this topological modification is presented.

The mean drag coefficients, once they have been normalized properly, are comparable when the cylinder is placed at different distances from one wall, down to gaps less than one cylinder diameter. At smaller gaps the body behaves similarly to a surface-mounted obstacle. The lift force is given by a repulsive component plus an attractive one. The former, well known from literature, is given by the deviation of the wake behind the body. Evidence of the latter, which is a consequence of the shear in front of the body, is given.

A novel data-driven modal decomposition of fluid flow is proposed, comprising key features of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The first mode is the normalized real or imaginary part of the DMD mode that minimizes the time-averaged residual. The $N$th mode is defined recursively in an analogous manner based on the residual of an expansion using the first $N-1$ modes. The resulting recursive DMD (RDMD) modes are orthogonal by construction, retain pure frequency content and aim at low residual. Recursive DMD is applied to transient cylinder wake data and is benchmarked against POD and optimized DMD (Chen et al., J. Nonlinear Sci., vol. 22, 2012, pp. 887–915) for the same snapshot sequence. Unlike POD modes, RDMD structures are shown to have purer frequency content while retaining a residual of comparable order to POD. In contrast to DMD, with exponentially growing or decaying oscillatory amplitudes, RDMD clearly identifies initial, maximum and final fluctuation levels. Intriguingly, RDMD outperforms both POD and DMD in the limit-cycle resolution from the same snapshots. Robustness of these observations is demonstrated for other parameters of the cylinder wake and for a more complex wake behind three rotating cylinders. Recursive DMD is proposed as an attractive alternative to POD and DMD for empirical Galerkin models, in particular for nonlinear transient dynamics.

A direct numerical simulation was carried out of plane turbulent Couette flow at a Reynolds number of 750, based on half the velocity difference between the walls and half the channel width. Particular attention was paid to choosing a computational box that is large enough to accommodate even the largest scales of the turbulence. In the central region of the channel very large elongated structures were observed, in accordance with earlier findings. The study is focused on the properties of these structures, but is also aimed at obtaining accurate turbulence statistics. Terms in the energy budget were evaluated and discussed. Also, the limiting values of various quantities were determined and their relevance in high Reynolds number flows discussed. The large structures were shown to be very sensitive to an imposed system rotation. They could be essentially eliminated with a stabilizing system rotation (around the spanwise axis) small enough for only minor damping of the rest of the scales. Despite the fact that the large structures dominate the appearance of the flow field their energy content was shown to be relatively small, on the order of 10% of the total turbulent kinetic energy.

Direct numerical simulation is used to examine the interaction of turbulence with a wall in the absence of mean shear. The influence of a solid wall on turbulence is analysed by first considering two ‘simpler’ types of boundaries. The first boundary is an idealized permeable wall. This boundary isolates and elucidates the viscous effects created by the wall. The second boundary is an idealized free surface. This boundary complements the first by allowing one to isolate and investigate the kinematic effects that occur near boundaries. The knowledge gained from these two simpler flows is then used to understand how turbulence is influenced by solid walls where both viscous and kinematic effects occur in combination.

Examination of the instantaneous flow fields confirms the presence of previously hypothesized structures (splats), and reveals an additional class of structures (antisplats). Statistical analysis of the Reynolds stresses and Reynolds stress transport equations indicates the relative importance of dissipation, intercomponent energy transfer, and energy transport. It is found that it is not the structures themselves, but the imbalance between structures which leads to intercomponent energy transfer. Remarkably, this imbalance (and hence near-wall intercomponent energy transfer) is controlled by viscous processes such as dissipation and diffusion. The analysis presented herein is a departure from past notions of how boundaries influence turbulence. The efficacy of these qualitative physical concepts is demonstrated in Part 2 where improved near-wall turbulence models are derived based on these ideas.

The consolidation or concentration of suspended particulate solids under the influence of gravitational forces is a problem of widespread practical and theoretical interest. The literature, which is scattered over several fields, contains most of the elements necessary for a complete understanding of gravity settling, but considerable controversy and confusion persists about their synthesis. Here we propose to construct a quantitative theory covering the full range of processes from transient settling of large, stable particles to the slow consolidation of flocculated suspensions of submicron particles. Conditions for the existence of shocks are identified and the basic equations describing the phenomena are solved numerically for several Péclet numbers.

Detailed velocity measurements and numerical predictions are presented for the flow through a plane nominally two-dimensional duct with a Symmetric sudden expansion of area ratio 1:2. Both the experiments and the predictions confirm a symmetry-breaking bifurcation of the flow leading to one long and one short Separation zone for channel Reynolds numbers above 125, based on the upstream channel height and the maximum flow velocity upstream. With increasing Reynolds numbers above this value, the short separated region remains approximately constant in length whereas the long region increases in length.

The experimental data were obtained using a one-component laser-Doppler anemometer at many Reynolds number values, with more extensive measurements being performed for the three Reynolds numbers 70, 300 and 610. Predictions were made using a finite volume method and an explicit quadratic Leith type of temporal discretization. In general, good agreement was found between measured and predicted velocity profiles for all Reynolds numbers investigated.

We present numerical simulations of stably stratified, vortically forced turbulence at a wide range of Froude numbers. Large-scale vortical forcing was chosen to represent geophysical vortices which break down at small scales where Coriolis effects are weak. The resulting vortical energy spectra are much steeper in the horizontal direction and shallower in the vertical than typical observations in the atmosphere and ocean, as noted in previous studies. We interpret these spectra in terms of the vertical decoupling which emerges in the strongly stratified limit. We show that this decoupling breaks down at a vertical scale of $U/N$, where $N$ is the Brunt–Väisälä frequency and $U$ is a characteristic horizontal velocity, confirming previous scaling arguments. The transfer of vortical energy to wave energy is most efficient at this vertical scale; vertical spectra of wave energy are correspondingly peaked at small scales, as observed in past work. The equilibrium statistical mechanics of the inviscid unforced truncated problem qualitatively predicts the nature of the forced–dissipative solutions, and confirms the lack of an inverse cascade of vortical energy.

Turbulent flow in a rectangular channel is investigated to determine the scale and pattern of the eddies that contribute most to the total turbulent kinetic energy and the Reynolds shear stress. Instantaneous, two-dimensional particle image velocimeter measurements in the streamwise-wall-normal plane at Reynolds numbers Reh = 5378 and 29 935 are used to form two-point spatial correlation functions, from which the proper orthogonal modes are determined. Large-scale motions – having length scales of the order of the channel width and represented by a small set of low-order eigenmodes – contain a large fraction of the kinetic energy of the streamwise velocity component and a small fraction of the kinetic energy of the wall-normal velocities. Surprisingly, the set of large-scale modes that contains half of the total turbulent kinetic energy in the channel, also contains two-thirds to three-quarters of the total Reynolds shear stress in the outer region. Thus, it is the large-scale motions, rather than the main turbulent motions, that dominate turbulent transport in all parts of the channel except the buffer layer. Samples of the large-scale structures associated with the dominant eigenfunctions are found by projecting individual realizations onto the dominant modes. In the streamwise wall-normal plane their patterns often consist of an inclined region of second quadrant vectors separated from an upstream region of fourth quadrant vectors by a stagnation point/shear layer. The inclined Q4/shear layer/Q2 region of the largest motions extends beyond the centreline of the channel and lies under a region of fluid that rotates about the spanwise direction. This pattern is very similar to the signature of a hairpin vortex. Reynolds number similarity of the large structures is demonstrated, approximately, by comparing the two-dimensional correlation coefficients and the eigenvalues of the different modes at the two Reynolds numbers.

A non-orthogonal helical co-ordinate system is introduced to study the effect of curvature and torsion on the flow in a helical pipe. It is found that both curvature and torsion induce non-negligible effects when the Reynolds number is less than about 40. When the Reynolds number is of order unity, torsion induces a secondary flow consisting of one single recirculating cell while curvature causes an increased flow rate. These effects are quite different from the two recirculating cells and decreased flow rate at high Reynolds numbers.

We present a mathematical model of the unidirectional solidification of a suspension of hard-sphere colloids. Similarity solutions are obtained for the volume fraction and temperature profiles ahead of a planar solidification front. The highly nonlinear functional dependence of the diffusion coefficient on the volume fraction gives rise to a range of behaviours. For small particles, Brownian diffusion dominates and the system behaviour is reminiscent of binary-alloy solidification. Constitutional supercooling occurs at the interface under certain conditions, leading potentially to an instability in the shape of the interface. For larger particles, Brownian diffusion is weak and the particles form a porous layer above the interface. In this case constitutional supercooling reaches a maximum near the surface of the layer, and the porous medium itself is potentially unstable. In stable systems there exists the possibility of secondary nucleation of ice.

In this article we present numerical solutions to multicellular natural convection in a vertical enclosure. The calculated streamlines faithfully represent what has been seen in the laboratory by smoke traces in air and particle traces in oils. The calculated isotherms for air correspond to reported interferometric patterns. Solutions exhibiting travelling waves for water were calculated near conditions where they should occur according to linear stability theory. Heat-transfer results for air are given and their dependence on the aspect ratio of the enclosure exhibited.

In Part 1 an extension of the attached eddy hypothesis was developed and applied to equilibrium pressure gradient turbulent boundary layers. In this paper the formulation is applied to data measured by the authors from non-equilibrium layers and agreement with the extended theory is encouraging. Also power spectra of the Reynolds stresses as developed from the extended theory compare favourably with experiment. The experimental data include a check of cone-angle effects by using a flying hot wire.

In this article, we considered experimentally the deformation and breakup of Newtonian and non-Newtonian conducting drops in surrounding fluid subjected to a uniform electric field. First, we examined three distinctive cases of Newtonian-fluid pairs with different relative conductivities, namely highly conducting drops, conducting drops and slightly conducting drops. The results on the Newtonian fluids demonstrated that when the conductivity of the drop is very large relative to that of the surrounding fluid, the deformation response of such highly conducting drops is described well by the electrohydrostatic theory, especially with regard to the prediction of the critical point. Specifically, when the ratio of drop to continuous-phase resistivity, R, was less than 10−5, the electrohydrostatic theory was quite satisfactory. Then, the non-Newtonian effect on the drop deformation and breakup was studied for highly conducting drops which satisfied the condition R < O(10−5). The highly conducting drop became stable in a weak or moderate field strength when either the drop or the continuous phase was non-Newtonian. On the other hand, when both the phases were non-Newtonian, more complicated responses were observed depending on the ratio of zero-shear-rate viscosities. Although the effects of the rheological properties are minimal on all features away from the critical conditions for breakup or prior to the instability, the non-Newtonian properties have a significant influence during drop burst, which is accompanied by large velocities and velocity gradients. In particular, when the ratio of the zero-shear-rate viscosity of the drop to that of the ambient fluid was much larger than unity, non-Newtonian properties of the drop phase enhanced the drop stability. Conversely, the elasticity of the continuous phase deteriorated the drop stability. Meanwhile if the zero-shear-rate viscosity ratio was much smaller than unity, the elasticity of the continuous phase produced a stabilizing effect. The effects of resistivity and viscosity ratios on the breakup modes were also investigated. When at least one of the two contiguous phases possessed considerable non-Newtonian properties, tip streaming appeared.

In this study we examine the impact of the strength of the large-scale motions on the amplitude and frequency of the small scales in high-Reynolds-number turbulent boundary layers. Time series of hot-wire data are decomposed into large- and small-scale components, and the impact of the large scale on the amplitude and frequency of the small scales is considered. The amplitude modulation effect is examined by conditionally averaging the small-scale intensity (${ u}_{S}^{2} $) for various values of the large-scale fluctuation (${u}_{L} $). It is shown that ${ u}_{S}^{2} $ increases with increasing value of ${u}_{L} $ in the near-wall region, whereas, farther away from the wall, ${ u}_{S}^{2} $ decreases with increasing ${u}_{L} $. The rate of increase in small-scale intensity with the strength of the large-scale signal is neither symmetric (about ${u}_{L} = 0$) nor linear. The extent of the frequency modulation is examined by counting the number of occurrences of local maxima or minima in the small-scale signal. It is shown that the frequency modulation effect is confined to the near-wall region and its extent diminishes rapidly beyond ${y}^{+ } = 100$. A phase lag between the large- and small-scale fluctuations, in terms of amplitude modulation, has also been identified, which is in agreement with previous studies. The phase lag between large- and small-scale fluctuations for frequency modulation is comparable to that of amplitude modulation in the near-wall region. The combined effect of both amplitude and frequency modulation is also examined by computing conditional spectra of the small-scale signal conditioned on the large scales. In the near-wall region, the results indicate that the peak value of pre-multiplied spectra increases with increasing value of ${u}_{L} $, indicating amplitude modulation, while the frequency at which this peak occurs also increases with increasing value of ${u}_{L} $, revealing frequency modulation. The overall trends observed from the conditional spectra are consistent with the results obtained through statistical analyses. Finally, a physical mechanism that can capture most of the above observations is also presented.

This paper reports an experimental investigation, using shadowgraphs and pressure measurements, of the detailed behaviour of converging weak shock waves near three different kinds of focus. Shocks are brought to a focus by reflecting initially plane fronts from concave end walls in a large shock tube. The reflectors are shaped to generate perfect foci, arêtes and caustics. It is found that, near the focus of a shock discontinuity, a complex wave field develops, which always has the same basic character, and which is always essentially nonlinear. A diffracted wave field forms behind the non-uniform converging shock; its compressive portions steepen to form diffraction shocks, while diffracted expansion waves overtake and weaken the diffraction shocks. The diffraction shocks participate in a Mach reflexion process near the focus, whose development is determined by competition between the convergence of the sides of the focusing front and acceleration of its central portion. In fact, depending on the aperture of the convergence and the strength of the initial wave, the three-shock intersections of the Mach reflexions either cross on a surface of symmetry or remain uncrossed. In the former case, which is observed if the shock wave is relatively weak, the wavefronts emerge from focus crossed and folded, in accordance with the predictions of geometrical acoustics theory. In the latter, the strong-shock case, the fronts beyond focus are uncrossed, as predicted by the theory of shock dynamics. It is emphasized that in both cases the behaviour at the focus is nonlinear. The overtaking of the diffraction shocks by the diffracted expansions limits the amplitude of the converging wave near focus, and is the mechanism by which the maximum amplification factor observed at focus is determined. In all cases, maximum pressures are limited to rather low values.

A straight tube with a smooth axisymmetric constriction is an idealized representation of a stenosed artery. We examine the three-dimensional instabilities and transition to turbulence of steady flow, steady flow plus an oscillatory component, and an idealized vascular pulsatile flow in a tube with a smooth 75 % stenosis using both linear stability analysis and direct numerical simulation. Steady flow undergoes a weak Coanda-type wall attachment and turbulent transition through a subcritical bifurcation, leading to hysteretic behaviour with respect to changes in Reynolds number. The pulsatile flows become unstable through a subcritical period-doubling bifurcation involving alternating tilting of the vortex rings that are ejected from the throat with each pulse. These tilted vortex rings rapidly break down through a self-induction mechanism within the confines of the tube. While the linear instability modes for pulsatile flow have maximum energy well downstream of the stenosis, we have established using direct numerical simulation that breakdown can gradually propagate upstream until it occurs within a few tube diameters of the constriction, in agreement with previous experimental observations. At the Reynolds numbers employed in the present study, transition is localized, with relaminarization occurring further downstream. A non-exhaustive investigation has also been undertaken into the receptivity of the axisymmetric shear layer in the idealized physiological pulsatile flow, with the results suggesting it has localized convective instability over part of the pulse cycle.