At high jet velocity the aerodynamic interaction between a capillary jet and the surrounding medium leads to an enhanced growth rate of axisymmetric disturbances. The available theories which account for this effect fail to agree with experimental observations. The difference is attributed, in part, to the relaxation of the velocity profile in jets formed by fully developed laminar pipe flow. The profile relaxation has a destabilizing effect just as does the aerodynamic interaction. In the absence of velocity-profile relaxation it is shown that the available theories overestimate the aerodynamic effect. A consideration of the viscosity of the ambient fluid yields a semi-empirical modification to the theory which shows good agreement with experimental values.

Taylor (1953, 1954a) showed that, when a cloud of solute is injected into a pipe through which a solvent is flowing, it spreads out, so that the distribution of concentration C is eventually a Gaussian function of distance along the pipe axis. This paper is concerned with the approach to this final form. An asymptotic series is derived for the distribution of concentration based on the assumption that the diffusion of solute obeys Fick's law. The first term is the Gaussian function, and succeeding terms describe the asymmetries and other deviations from normality observed in practice. The theory is applied to Poiseuille flow in a pipe of radius a and it is concluded that three terms of the series describe C satisfactorily if Dt/a2 > 0·2 (where D is the coefficient of molecular diffusion), and that the initial distribution of C has little effect on the approach to normality in most cases of practical importance. The predictions of the theory are compared with numerical work by Sayre (1968) for a simple model of turbulent open channel flow and show excellent agreement. The final section of the paper presents a second series derived from the first which involves only quantities which can be determined directly by integration from the observed values of C without knowledge of the velocity distribution or diffusivity. The latter series can be derived independently of the rest of the paper provided the cumulants of C tend to zero fast enough as t → ∞, and it is suggested, therefore, that the latter series may be valid in flows for which Fick's law does not hold.

Slow flow of a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions. The main result is an integral equation for the force per unit length exerted on the body by the fluid. The novelty is that the body is permitted to twist and dilate in addition to undergoing the translating, bending and stretching, which have been considered by others. The method of derivation is relatively simple, and the resulting integral equation does not involve the limiting processes which occur in the previous work.

This paper reports an experimental investigation of the near wake of a finite-length square cylinder, with one end mounted on a flat plate and the other free. The cylinder aspect ratio or height-to-width ratio H/d ranges from 3 to 7. Measurements were carried out mainly in a closed-loop low-speed wind tunnel at a Reynolds number Red, based on d and the free-stream velocity of 9300 using hot-wire anemometry, laser Doppler anemometry and particle image velocimetry (PIV). The planar PIV measurements were performed in the three orthogonal planes of the three-dimensional cylinder wake, along with flow visualization conducted simultaneously in two orthogonal planes (Red = 221). Three types of vortices, i.e. the tip, base and spanwise vortices were observed and the near wake is characterized by the interactions of these vortices. Both flow visualization and two-point correlation point to an inherent connection between the three types of vortices. A model is proposed for the three-dimensional flow structure based on the present measurements, which is distinct from previously proposed models. The instantaneous flow structure around the cylinder is arch-type, regardless of H/d, consisting of two spanwise vortical ‘legs’, one on each side of the cylinder, and their connection or ‘bridge’ near the free end. Both tip and base vortices are the streamwise projections of the arch-type structure in the (y, z) plane, associated with the free-end downwash flow and upwash flow from the wall, respectively. Other issues such as the topological characteristics, spatial arrangement and interactions among the vortical structures are also addressed.

The connection between fluid flow and optical flow is explored in typical flow visualizations to provide a rational foundation for application of the optical flow method to image-based fluid velocity measurements. The projected-motion equations are derived, and the physics-based optical flow equation is given. In general, the optical flow is proportional to the path-averaged velocity of fluid or particles weighted with a relevant field quantity. The variational formulation and the corresponding Euler–Lagrange equation are given for optical flow computation. An error analysis for optical flow computation is provided, which is quantitatively examined by simulations on synthetic grid images. Direct comparisons between the optical flow method and the correlation-based method are made in simulations on synthetic particle images and experiments in a strongly excited turbulent jet.

The paper examines how a passive contaminant disperses along the axis of a tube in which the flow is driven by a longitudinal pressure gradient varying harmonically with time. This problem is of intrinsic interest and is relevant to some important practical problems. Two examples are dispersion in estuaries and in the blood stream. By means both of statistical arguments and an analysis like that used by Taylor (1953) in the case of a steady pressure gradient it is shown that eventually the mean distribution of concentration satisfies a diffusion equation (and is therefore a Gaussian function of distance along the axis) with an effective longitudinal diffusion coefficient K(t) which is a harmonic function of time with a period equal to one half of that of the imposed pressure gradient. Contrary to the supposition made in most previous work on this problem it is shown by examining some special cases that the harmonic terms in K(t) may have a noticeable effect on the dispersion of the contaminant and in particular on the rate at which it is spreading axially. The size of the effect depends on both the frequency and the Schmidt number and is particularly large at low frequencies. The paper concludes with an analysis of a model of dispersion in estuaries which has been used frequently and it is concluded that here too oscillatory effects may often be noticeable.

The linear stability of a quiescent, three-dimensional rectangular box of fluid heated from below is considered. It is found that finite rolls (cells with two non-zero velocity components dependent on all three spatial variables) with axes parallel to the shorter side are predicted. When the depth is the shortest dimension, the cross-sections of these finite rolls are near-square, but otherwise (in wafer-shaped boxes) narrower cells appear. The value of the critical Rayleigh number and preferred wave-number (number of finite rolls) for a given size box is determined for boxes with horizontal dimensions h, ¼ ≤ h/d ≤ 6, where d is the depth.

Arguments are presented to show that the concept of gradient diffusion is inapplicable to mixing in turbulent shear layers. A new model is proposed for treating molecular mixing and chemical reaction in such flows at high Reynolds number. It is based upon the experimental observations that revealed the presence of coherent structures and that showed that fluid elements from the two streams are distributed unmixed throughout the layer by large-scale inviscid motions. The model incorporates features of the strained flame model and makes use of the Kolmogorov cascade in scales. Several model predictions differ markedly from those of diffusion models and suggest experiments for testing the two approaches.

The theoretical treatments given by earlier authors are classified, reviewed and where necessary extended; then the predictions of twenty of these theories are evaluated and compared with all available experimental data, the root-meansquare error being computed for each theory. The theory of van Driest-II gives the lowest root-mean-square error (11.0%).

A new calculation procedure is developed from the postulate that a unique relation exists between cfFc and RFR where cf is the drag coefficient, R is the Reynolds number, and Fc and FR are functions of Mach number and temperature ratio alone. The experimental data are found to be too scanty for both Fc and FR to be deduced empirically, so Fc is calculated by means of mixing-length theory and FR is found semi-empirically. Tables and charts of values of Fc and FR are presented for a wide range of MG and TS/TG. When compared with all experimental data, the predictions of the new procedure give a root-mean-square error of 9.9%.

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

The first flow visualization experimental results of transition in plane Couette flow are reported. The Couette flow water channel was of an infinite-belt type with counter-moving walls. The belt and channel walls were transparent making it possible to visualize the flow pattern in the streamwise-spanwise plane by utilizing fluid-suspended reflective flakes. Transition was triggered by a high-amplitude pointwise disturbance. The transitional Reynolds number, i.e. the lowest Reynolds number for which turbulence can be sustained, was determined to be 360 ± 10, based on half-channel height and half the velocity difference between the walls. For Reynolds numbers above this value a large enough amplitude of the initial disturbance gave rise to a growing turbulent spot. Its shape and spreading rate was determined for Reynolds numbers up to 1000.

Previous work has shown that aspects of the evolution of large-scale structures, particularly in forced and transitional mixing layers and jets, can be described by linear and nonlinear stability theories. However, questions persist as to the choice of the basic (steady) flow field to perturb, and the extent to which disturbances in natural (unforced), initially turbulent jets may be modelled with the theory. For unforced jets, identification is made difficult by the lack of a phase reference that would permit a portion of the signal associated with the instability wave to be isolated from other, uncorrelated fluctuations. In this paper, we investigate the extent to which pressure and velocity fluctuations in subsonic, turbulent round jets can be described as linear perturbations to the mean flow field. The disturbances are expanded about the experimentally measured jet mean flow field, and evolved using linear parabolized stability equations (PSE) that account, in an approximate way, for the weakly non-parallel jet mean flow field. We utilize data from an extensive microphone array that measures pressure fluctuations just outside the jet shear layer to show that, up to an unknown initial disturbance spectrum, the phase, wavelength, and amplitude envelope of convecting wavepackets agree well with PSE solutions at frequencies and azimuthal wavenumbers that can be accurately measured with the array. We next apply the proper orthogonal decomposition to near-field velocity fluctuations measured with particle image velocimetry, and show that the structure of the most energetic modes is also similar to eigenfunctions from the linear theory. Importantly, the amplitudes of the modes inferred from the velocity fluctuations are in reasonable agreement with those identified from the microphone array. The results therefore suggest that, to predict, with reasonable accuracy, the evolution of the largest-scale structures that comprise the most energetic portion of the turbulent spectrum of natural jets, nonlinear effects need only be indirectly accounted for by considering perturbations to the mean turbulent flow field, while neglecting any non-zero frequency disturbance interactions.

A hyperbolic two-phase flow model involving five partial differential equations is constructed for liquid–gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves in the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast at specific locations only, typically at evaporation fronts. Thus, metastable states are involved for locations far from these fronts. Computational results are compared to the experimental ones. Computed and measured front speeds are of the same order of magnitude and the same tendency of increasing front speed with initial temperature is reported. Moreover, the limit case of evaporation fronts propagating in highly metastable liquids with the Chapman–Jouguet speed is recovered as an expansion wave of the present model in the limit of stiff thermal and chemical relaxation.

Shock waves, dead zones and particle-free regions form when a thin surface avalanche of granular material flows around an obstacle or over a change in the bed topography. Understanding and modelling these flows is of considerable practical interest for industrial processes, as well as for the design of defences to protect buildings, structures and people from snow avalanches, debris flows and rockfalls. These flow phenomena also yield useful constitutive information that can be used to improve existing avalanche models. In this paper a simple hydraulic theory, first suggested in the Russian literature, is generalized to model quasi-two-dimensional flows around obstacles. Exact and numerical solutions are then compared with laboratory experiments. These indicate that the theory is adequate to quantitatively describe the formation of normal shocks, oblique shocks, dead zones and granular vacua. Such features are generated by the flow around a pyramidal obstacle, which is typical of some of the defensive structures in use today.

The turbulent boundary layer along a compression ramp with a deflection angle of 18° at a free-stream Mach number of M = 3 and a Reynolds number of Reθ = 1685 with respect to free-stream quantities and mean momentum thickness at inflow is studied by direct numerical simulation. The conservation equations for mass, momentum, and energy are solved in generalized coordinates using a 5th-order hybrid compact- finite-difference-ENO scheme for the spatial discretization of the convective fluxes and 6th-order central compact finite differences for the diffusive fluxes. For time advancement a 3rd-order Runge–Kutta scheme is used. The computational domain is discretized with about 15 × 106 grid points. Turbulent inflow data are provided by a separate zero-pressure-gradient boundary-layer simulation. For statistical analysis, the flow is sampled 600 times over about 385 characteristic timescales δ0/U∞, defined by the mean boundary-layer thickness at inflow and the free-stream velocity. Diagnostics show that the numerical representation of the flow field is sufficiently well resolved.

Near the corner, a small area of separated flow develops. The shock motion is limited to less than about 10% of the mean boundary-layer thickness. The shock oscillates slightly around its mean location with a frequency of similar magnitude to the bursting frequency of the incoming boundary layer. Turbulent fluctuations are significantly amplified owing to the shock–boundary-layer interaction. Reynolds-stress maxima are amplified by a factor of about 4. Turbulent normal and shear stresses are amplified differently, resulting in a change of the structure parameter. Compressibility affects the turbulence structure in the interaction area around the corner and during the relaxation after reattachment downstream of the corner. Correlations involving pressure fluctuations are significantly enhanced in these regions. The strong Reynolds analogy which suggests a perfect correlation between velocity and temperature fluctuations is found to be invalid in the interaction area.

The three-dimensional development of a plane free shear layer subjected to small sinusoidal perturbations periodically placed along the span is experimentally studied. Both laser induced fluorescence and direct interface visualization are used to monitor the interface between the two fluids. The development of the different flow stabilities is obtained through analysis of the temporal and spatial evolution of the interface separating the two streams. It is shown that the characteristic time of growth of the two-dimensional shear instability is much shorter than that of the three-dimensional instability. The primary Kelvin-Helmholtz instability develops first, leading to the formation of an almost two-dimensional array of spanwise vortex tubes. Under the effect of the strain field created by the evolving spanwise vortices, the perturbed vorticity existing on the braids undergoes axial stretching, resulting in the formation of vortex tubes whose axes are aligned with the principal direction of the positive strain field. During the formation of these streamwise vortex tubes, the spanwise vortices maintain, to a great extent, their two-dimensionality, suggesting an almost uncoupled development of both instabilities. The vortex tubes formed through the three-dimensional instability of the braids further undergo nonlinear interactions with the spanwise vortices inducing on their cores a wavy undulation of the same wavelength, but 180° phase shifted with respect to the perturbation. In addition, it is shown that owing to the nature of the three-dimensional instability, the effect of vertical and axial perturbations are coupled. Finally, the influence of the amplitude and wavelength of the perturbation on the development of the two- and three-dimensional instabilities is described.

A multi-scale asymptotic theory is derived for the evolution and interaction of currents and surface gravity waves in water of finite depth, under conditions typical of coastal shelf waters outside the surf zone. The theory provides a practical and useful model with which wave–current coupling may be explored without the necessity of resolving features of the flow on space and time scales of the primary gravity-wave oscillations. The essential nature of the dynamical interaction is currents modulating the slowly evolving phase of the wave field and waves providing both phase-averaged forcing of long infra-gravity waves and wave-averaged vortex and Bernoulli-head forces and hydrostatic static set-up for the low-frequency current and sea-level evolution equations. Analogous relations are derived for wave-averaged material tracers and density stratification that include advection by horizontal Stokes drift and by a vertical Stokes pseudo-velocity that is the incompressible companion to the horizontal Stokes velocity. Illustrative solutions are analysed for the special case of depth-independent currents and tracers associated with an incident surface wave field and a vortex with O(1) Rossby number above continental shelf topography.

A three-dimensional computer simulation of a concentrated emulsion in shear flow has been developed for low-Reynolds-number finite-capillary-number conditions. Numerical results have been obtained using an efficient boundary integral formulation with periodic boundary conditions and up to twelve drops in each periodically replicated unit cell. Calculations have been performed over a range of capillary numbers where drop deformation is significant up to the value where drop breakup is imminent. Results have been obtained for dispersed-phase volume fractions up to 30% and dispersed- to continuous-phase viscosity ratios in the range of 0 to 5. The results reveal a complex rheology with pronounced shear thinning and large normal stresses that is associated with an anisotropic microstructure that results from the alignment of deformed drops in the flow direction. The viscosity of an emulsion is only a moderately increasing function of the dispersed-phase volume fraction, in contrast to suspensions of rigid particles or undeformed drops. Unlike rigid particles, deformable drops do not form large clusters.

The velocity fields of a turbulent wake behind a flat plate obtained from the direct numerical simulations of Moser et al. (1998) are used to study the structure of the flow in the intermittent zone where there are, alternately, regions of fully turbulent flow and non-turbulent velocity fluctuations on either side of a thin randomly moving interface. Comparisons are made with a wake that is ‘forced’ by amplifying initial velocity fluctuations. A temperature field T, with constant values of 1.0 and 0 above and below the wake, is transported across the wake as a passive scalar. The value of the Reynolds number based on the centreplane mean velocity defect and half-width b of the wake is Re ≈ 2000.

The thickness of the continuous interface is about 0.07b, whereas the amplitude of fluctuations of the instantaneous interface displacement yI(t) is an order of magnitude larger, being about 0.5b. This explains why the mean statistics of vorticity in the intermittent zone can be calculated in terms of the probability distribution of yI and the instantaneous discontinuity in vorticity across the interface. When plotted as functions of y−yI the conditional mean velocity 〈U〉 and temperature 〈T〉 profiles show sharp jumps at the interface adjacent to a thick zone where 〈U〉 and 〈T〉 vary much more slowly.

Statistics for the conditional vorticity and velocity variances, available in such detail only from DNS data, show how streamwise and spanwise components of vorticity are generated by vortex stretching in the bulges of the interface. While mean Reynolds stresses (in the fixed reference frame) decrease gradually in the intermittent zone, conditional stresses are roughly constant and then decrease sharply towards zero at the interface. Flow fields around the interface, analysed in terms of the local streamline pattern, confirm and explain previous results that the advancement of the vortical interface into the irrotational flow is driven by large-scale eddy motion.

Terms used in one-point turbulence models are evaluated both conventionally and conditionally in the interface region, and the current practice in statistical models of approximating entrainment by a diffusion process is assessed.

The problem of determining the steady axially symmetrical motion induced by a sphere rotating with constant angular velocity about a diameter in an incompressible viscous fluid which is at rest at large distances from it is considered. The basic independent variables are the polar co-ordinates (r, θ) in a plane through the axis of rotation and with origin at the centre of the sphere. The equations of motion are reduced to three sets of nonlinear second-order ordinary differential equations in the radial variable by expanding the flow variables as series of orthogonal Gegenbauer functions with argument μ = cosθ. Numerical solutions of the finite set of equations obtained by truncating the series after a given number of terms are obtained. The calculations are carried out for Reynolds numbers in the range R = 1 to R = 100, and the results are compared with various other theoretical results and with experimental observations.

The torque exerted by the fluid on the sphere is found to be in good agreement with theory at low Reynolds numbers and appears to tend towards the results of steady boundary-layer theory for increasing Reynolds number. There is excellent agreement with experimental results over the range considered. A region of inflow to the sphere near the poles is balanced by a region of outflow near the equator and as the Reynolds number increases the inflow region increases and the region of outflow becomes narrower. The radial velocity increases with Reynolds number at the equator, indicating the formation of a radial jet over the narrowing region of outflow. There is no evidence of any separation of the flow from the surface of the sphere near the equator over the range of Reynolds numbers considered.