When a viscous liquid is displaced by a long air bubble in a capillary, it leaves behind a wetting liquid film. A lubrication analysis by Bretherton (1961), which assumes a mobile surface, underpredicts the film thickness at low bubble speeds. In this investigation, the Marangoni effect of small amounts of impurities is shown to be capable of explaining this discrepancy. We carry out an asymptotic analysis for different convective, diffusive and kinetic timescales and show that, if transport in the film is mass-transfer limited such that a bulk concentration gradient exists in the film, the film thickness increases by a maximum factor of 4 2/3; over Bretherton's mobile result at low bubble speeds. Moreover, at large bubble speeds, Bretherton's mobile prediction is approached for all ranges of timescales. For intermediate bubble speeds, the film thickness varies with respect to the bubble speed with an exponent smaller than 2/3 of the mobile theory. These results are favourably compared to literature data on film thickness.

Unsteady flow over a stationary sphere with a small fluctuation in the free-stream velocity is considered at small Reynolds number, Re. A matched asymptotic solution is obtained for the frequency-dependent (or the acceleration-dependent) part of the unsteady flow at very small frequency, ω, under the restriction St [Lt ] Re [Lt ] 1, where St is the Strouhal number. The acceleration-dependent part of the unsteady drag is found to be proportional to St ∼ ω instead of the ω½ dependence predicted by Stokes’ solution. Consequently, the expression for the Basset history force is incorrect for large time even for very small Reynolds numbers. Present results compare well with the previous numerical results of Mei, Lawrence & Adrian (1991) using a finite-difference method for the same unsteady flow at small Reynolds number. Using the principle of causality, the present analytical results at small Re, the numerical results at finite Re for low frequency, and Stokes’ results for high frequency, a modified expression for the history force is proposed in the time domain. It is confirmed by comparing with the finite-difference results at arbitrary frequency through Fourier transformation. The modified history force has an integration kernel that decays as t−2, instead of t½, at large time for both small and finite Reynolds numbers.

A two-dimensional stability analysis of the flow in a straight alluvial channel has been carried out, using the vorticity transport equation. In the analysis an attempt has been made to account for the influence of gravity on bed-load transport, and this turned out to change the stability quite significantly.

In the case of instability, the further growth of the dunes has been investigated using a second-order approximation, This nonlinear theory explains the experimental fact that the dunes very soon become asymmetric.

Helium mass flow rates in a microchannel were measured, for a wide Knudsen-number range, in isothermal steady conditions. The flow Knudsen numbers, considered here, cover the range from continuum slip regime to the near free molecular regime. We used a single-channel system involved in an experimental platform more powerful than those previously used. The experimental errors and uncertainties were accurately investigated and estimated. In the continuum slip regime, it was found that the first-order approach is pertinent for Knudsen number between 0.03 and 0.3. Moreover, the slip coefficient was deduced by comparing the experiments with the theoretical first-order slip continuum approach. For Knudsen number between 0.03 and 0.7, a polynomial second-power form is proposed for the mass flow rate expression. Otherwise, the experimental results on the mass flow rate were compared with theoretical values calculated from kinetic approaches over the 0.03–50 Knudsen number range, and an overall agreement appears through the comparison. It was also found, when the Knudsen number increased, that the wall influence on measurement occurred first through the accommodation process in the transition regime followed by the wall influence through the aspect ratio in the free molecular regime.

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.

An aerofoil leading-edge profile based on wavy (sinusoidal) protuberances/tubercles is investigated to understand the mechanisms by which they are able to reduce the noise produced through the interaction with turbulent mean flow. Numerical simulations are performed for non-lifting flat-plate aerofoils with straight and wavy leading edges (denoted by SLE and WLE, respectively) subjected to impinging turbulence that is synthetically generated in the upstream zone (free-stream Mach number of 0.24). Full three-dimensional Euler (inviscid) solutions are computed for this study thereby eliminating self-noise components. A high-order accurate finite-difference method and artefact-free boundary conditions are used in the current simulations. Various statistical analysis methods, including frequency spectra, are implemented to aid the understanding of the noise-reduction mechanisms. It is found with WLEs, unlike the SLE, that the surface pressure fluctuations along the leading edge exhibit a significant source-cutoff effect due to geometric obliqueness which leads to reduced levels of radiated sound pressure. It is also found that there exists a phase interference effect particularly prevalent between the peak and the hill centre of the WLE geometry, which contributes to the noise reduction in the mid- to high-frequency range.

The influence of surface roughness on the slip behaviour of a Newtonian liquid in steady planar shear is investigated using three different approaches, namely Stokes flow calculations, molecular dynamics (MD) simulations and a statistical mechanical model for the friction coefficient between a corrugated wall and the first liquid layer. These approaches are used to probe the behaviour of the slip length as a function of the slope parameter $ka\,{=}\,2 \pi a/\lambda$, where $a$ and $\lambda$ represent the amplitude and wavelength characterizing the periodic corrugation of the bounding surface. The molecular and continuum approaches both confirm a monotonic decay in the slip length with increasing $ka$ but the rate of decay as well as the magnitude of the slip length obtained from the Stokes flow solutions exceed the MD predictions as the wall feature sizes approach the liquid molecular dimensions. In the limit of molecular-scale wall corrugation, a Green–Kubo analysis based on the fluctuation–dissipation theorem accurately reproduces the MD results for the behaviour of the slip length as a function of $a$. In combination, these three approaches provide a detailed picture of the influence of periodic roughness on the slip length which spans multiple length scales ranging from molecular to macroscopic dimensions.

Details of the shape of the liquid bridge joining a nascent water drop to its parent body are presented for times before, after and at the instant of bifurcation when the drop is created and also when the secondary droplet is formed. After the instant of bifurcation there is ‘unbalanced’ surface tension which gives an impulse to the rest of the fluid causing strong surface deformations. The major point of this work is to draw attention to the strong up–down asymmetry at each bifurcation point. The geometric similarity at each bifurcation instant supports the conjecture that the flow converges to just one similarity solution of the type described by Keller & Miksis (1983) in which only surface tension and inertia are important. Features of the flow before and after bifurcation are discussed.

New and unexpected results are presented regarding the nonlinear interactions between a wavepacket and a vortical mean flow, with an eye towards internal wave dynamics in the atmosphere and oceans and the problem of ‘missing forces’ in atmospheric gravity-wave parametrizations. The present results centre around a pre-wave-breaking scenario termed ‘wave capture’, which differs significantly from the standard such scenarios associated with critical layers or mean density decay with altitude. We focus on the peculiar wave–mean interactions that accompany wave capture. Examples of these interactions are presented for layerwise-two-dimensional, layerwise-non-divergent flows in a three-dimensional Boussinesq system, in the strong-stratification limit.

The nature of the interactions can be summarized in the phrase ‘wave–vortex duality’, whose key points are firstly that wavepackets behave in some respects like vortex pairs, as originally shown in the pioneering work of Bretherton (1969), and secondly that a collection of interacting wavepackets and vortices satisfies a conservation theorem for the sum of wave pseudomomentum and vortex impulse, provided that the impulse is defined appropriately. It must be defined as the rotated dipole moment of the Lagrangian-mean potential vorticity (PV). This PV differs crucially from the PV evaluated from the curl of either the Lagrangian-mean or the Eulerian-mean velocity. The results are established here in the strong-stratification limit for rotating (quasi-geostrophic) as well as for non-rotating systems. The concomitant momentum budgets can be expected to be relatively complicated, and to involve far-field recoil effects in the sense discussed in Bühler & McIntyre (2003). The results underline the three-way distinction between impulse, pseudomomentum, and momentum. While momentum involves the total velocity field, impulse and pseudomomentum involve, in different ways, only the vortical part of the velocity field.

Shear-induced migration of 2 $\mu$m diameter spherical colloidal particles flowing through rectangular channels (50 $\mu$m $\times$ 500 $\mu$m cross-section) is studied by confocal microscopy. The confocal microscope allows imaging of the flowing particles far from the walls of the channel, at particle velocities up to 8000 $\mu$m s$^{-1}$. The particle volume fraction is varied from $\phi = 0.05$ to 0.34, and the flow rate is also varied, which results in a bulk Péclet number (${\hbox{\it Pe}}_{\rm B}$) which varies by two orders of magnitude. Concentration profiles are measured across the narrow dimension of the channel; particles at the larger volume fractions migrate toward the centreline, with the migration progressively stronger as ${\hbox{\it Pe}}_{\rm B}$ increases. The flow has been analysed using an existing mixture flow model under the assumption of fully developed flow and a proposed constitutive law which describes the suspension normal stresses as a function of both $\phi$ and the local Péclet number, ${\hbox{\it Pe}}$, the latter being defined as a variable quantity through the local shear rate. Shear thinning and shear thickening are not included. Comparisons made with the experimental data indicate that the dependence of the extent of migration upon ${\hbox{\it Pe}}_{\rm B}$ is captured well but discrepancies arise, at least in part because the assumption of full development is not valid for these experiments.

We discuss a series of numerical experiments on the dispersion of neutrally buoyant particles in two-dimensional turbulent flows. The topology of two-dimensional turbulence is parametrized in terms of the relative dominance of deformation or rotation; this leads to a segmentation of the turbulent field into hyperbolic and elliptic domains. We show that some of the characteristic structural domains of two-dimensional turbulent flows, namely coherent structures and circulation cells, generate particle traps and peculiar accelerations which induce several complex properties of the particle dispersion processes at intermediate times. In general, passive particles are progressively pushed from the coherent structures and tend to concentrate in highly hyperbolic regions in the proximity of the isolines of zero vorticity. For large dispersion times, the background turbulent field is a privileged domain of particle richness; there is however a permanent particle exchange between the background field and the energetic circulation cells which surround the coherent structures. At intermediate times, an anomalous dispersion regime may appear, depending upon the relative weight of the different topological domains active in two-dimensional turbulence. The use of appropriate conditional averages allows the basic topology of two-dimensional turbulence to be characterized from a Lagrangian point of view. In particular, an intermediate $t^{\frac{5}{4}}$ anomalous dispersion law is shown to be associated with the action of hyperbolic regions where deformation dominates rotation; the motion of the advected particles in strongly elliptic regions where rotation dominates over deformation is shown to be associated with a $<e1>t<sup>t^{\frac{5}{3}}$ dispersion law. Because neutral particles concentrate on average in hyperbolic regions, the $<e1>t^{\frac{5}{4}}$ dispersion law is quite robust and it can be observed under very general circumstances.

Properties of turbulent thermal convection were measured in air between horizontal plates maintained at constant temperatures. Rayleigh numbers of 6·3 × 105, 2·5 × 106 and 1·0 × 107 were studied with a convection chamber designed to allow measurements to be taken along a horizontal path. Vertical profiles are presented of horizontally averaged temperature; r.m.s. fluctuations of temperature, horizontal velocity and vertical velocity; total heat flux; the correlation coefficient between vertical velocity and temperature; all terms in the thermal variance equation; and of terms in the turbulent kinetic energy balance.

One-dimensional spectra of temperature, horizontal velocity and vertical velocity at two different heights all disclose large intensity for waves of length 5 to 15 times the plate separation. Secondary peaks occur for scales of from 0·7 to 1·7 times the separation height.

Many of the observations are consistent with a thermal structure dominated by plumes extending most or all of the distance between plates.

High-resolution measurements have been performed of the convective heat current as a function of time when a Rayleigh-Bénard cell is swept through its threshold with a specified time-dependent heat input. The results are interpreted in terms of the amplitude equation which exactly describes the slow variations in space and time of hydrodynamic quantities near the threshold. A phenomenological forcing field is added to this equation, and its form and magnitude are fitted to the onset time of the convective heat current. A deterministic model in which the field is an adjustable constant yields a good fit to the data for both a step and a linear ramp in the heat input. An alternative stochastic model, in which the field is a Gaussian variable with zero mean and a white-noise spectrum, is adequate for the ramp experiments, but cannot fit the step data for any value of the mean-square field. The systematics of the field and onset time versus ramp rate are studied in both the deterministic and stochastic models, and attempts are made to interpret the field in terms of physical mechanisms. When the data for long times are analysed in terms of the amplitude equation, it is found that the state first excited at onset is not the roll pattern which is stable in steady state. Instead, the system goes first to an intermediate state, which we tentatively identify as a hexagonal configuration. The decay of this state is governed by a further adjustable field in the amplitude equation.

The mean flow development in an initially turbulent boundary layer subjected to a large favourable pressure gradient beginning at a point x0 is examined through analyses expected a priori to be valid on either side of relaminarization. The ‘quasi-laminar’ flow in the later stages of reversion, where the Reynolds stresses have by definition no significant effect on the mean flow, is described by an asymptotic theory constructed for large values of a pressure-gradient parameter Λ, scaled on a characteristic Reynolds stress gradient. The limiting flow consists of an inner laminar boundary layer and a matching inviscid (but rotational) outer layer. There is consequently no entrainment to lowest order in Λ−1, and the boundary layer thins down to conserve outer vorticity. In fact, the predictions of the theory for the common measures of boundary-layer thickness are in excellent agreement with experimental results, almost all the way from x0. On the other hand the development of wall parameters like the skin friction suggests the presence of a short bubble-shaped reverse-transitional region on the wall, where neither turbulent nor quasi-laminar calculations are valid. The random velocity fluctuations inherited from the original turbulence decay with distance, in the inner layer, according to inverse-power laws characteristic of quasi-steady perturbations on a laminar flow. In the outer layer, there is evidence that the dominant physical mechanism is a rapid distortion of the turbulence, with viscous and inertia forces playing a secondary role. All the observations available suggest that final retransition to turbulence quickly follows the onset of instability in the inner layer.

It is concluded that reversion in highly accelerated flows is essentially due to domination of pressure forces over the slowly responding Reynolds stresses in an originally turbulent flow, accompanied by the generation of a new laminar boundary layer stabilized by the favourable pressure gradient.

We present a numerical method that allows us to solve the Navier-Stokes equation with boundary conditions for the viscous flow between two concentrically rotating cylinders as an initial-value problem. We use a pseudospectral code in which all of the time-splitting errors are removed by using a set of Green functions (capacitance matrix) that allows us to satisfy the inviscid boundary conditions exactly. For this geometry we find that a small time-splitting error can produce large errors in the computed velocity field. We test the code by comparing our numerically determined growth rates and wave speeds with linear theory and by comparing our computed torques with experimentally measured values and with the values that appear in other published numerical simulations. We find good agreement in all of our tests of the numerical calculation of wavy vortex flows. A test that is more sensitive than the comparison of torques is the comparison of the numerically computed wave speed with the experimentally observed wave speed. The agreements between the simulated and measured wave speeds are within the experimental uncertainties; the best-measured speeds have fractional uncertainties of less than 0.2%.

Research on the abstract properties of the Navier–Stokes equations in three dimensions has cast a new light on the time-asymptotic approximate solutions of those equations. Here heuristic arguments, based on the rigorous results of that research, are used to show the intimate relationship between the sufficient number of degrees of freedom describing fluid flow and the bound on the fractal dimension of the Navier–Stokes attractor. In particular it is demonstrated how the conventional estimate of the number of degrees of freedom, based on purely physical and dimensional arguments, can be obtained from the properties of the Navier–Stokes equation. Also the Reynolds-number dependence of the sufficient number of degrees of freedom and of the dimension of the attractor in function space is elucidated.

This paper describes an attempt to verify experimentally the wavemaker theory for a piston-type wavemaker. The theory is based upon the usual assumptions of classical hydrodynamics, i.e. that the fluid is inviscid, of uniform density, that motion starts from rest, and that non-linear terms are neglected. If the water depth, wavelength, wave period, and wavemaker stroke (of a harmonically oscillating wavemaker) are known, then the wavemaker theory predicts the wave motion everywhere, and in particular the wave height a few depths away from the wavemaker.

The experiments were conducted in a 100 ft. wave channel, and the wave-height envelope was measured with a combination hook-and-point gauge. A plane beach (sloping 1:15) to absorb the wave energy was located at the far end of the channel. The amplitude-reflexion coefficient was usually less than 10%. Unless this reflexion effect is corrected for, it imposes one of the most serious limitations upon experimental accuracy. In the analysis of the present set of measurements, the reflexion effect is taken into account.

The first series of tests was concerned with verifying the wavemaker theory for waves of small steepness (0.002 ≤ H/L ≤ 0.03). For this range of wave steepnesses, the measured wave heights were found to be on the average 3.4% below the height predicted by theory. The experimental error, as measured by the scatter about aline 3.4% below the theory, was of the order of 3%. The systematic deviation of 3.4% is believed to be partly due to finite-amplitude effects and possibly to imperfections in the wavemaker motion.

The second series of tests was concerned with determining the effects of finite amplitude. For therange of wave steepnesses 0.045 ≤ H/L ≤ 0.048, themeasured wave heights were found to be on the average 10% below the heightspredictedfrom the small-amplitude theory. The experimental error was again of the order of 3%.

It is considered that these measurements confirm the validity of the small-amplitude wave theory. No confirmation of this accuracy has hitherto been given for forced motions.

The analysis and experiments in this paper are restricted to the flow between two coaxial, infinite disks, one rotating and one stationary. The results of numerical calculations show that many solutions can exist for a given Reynolds number Ωl2/v (Ω is the angular velocity of the rotating disk and I is the spacing between the two disks). Out of a greater number of possible solutions, three solution branches have been identified; the branches correspond to one-, two- and three-flow cells in the meridional plane.

The one-cell branch has been accorded detailed treatment. Within this branch there are two subbranches. The first, now well documented in the literature, includes solutions from zero to infinite Reynolds number. The latter limiting case is characterized by an inward-flowing boundary layer on the stationary disk and an outward-flowing boundary layer on the rotating disk. In between is a core flow rotating with a constant angular velocity. The second sub-branch of the single-cell flows, apparently unknown heretofore, begins with an infinite Reynolds number, decreases to a minimum and then increases to an infinite Reynolds number again. The first infinite Reynolds number limit again corresponds to two boundary-layer flows separated by a core flow with constantangular velocity opposite in direction to the angular velocity of the rotating disk. The second limiting case of infinite Reynolds number is the free-disk solution of von Kármán (1921). Asymptotic solutions have been obtained which more fully describe the nature of this flow as the Reynolds number increases.

The second part of the paper presents experimental measurements corresponding to the Reynolds number range 0–100. Profiles were measured with a hot-wire anemometer. The measurements are in agreement with the first, one-cell branch of solutions. A semi-quantitative evaluation of edge effects is obtained.

An experimental study is performed on the vortex induced vibrations of a rigid flexibly mounted circular cylinder placed in a crossflow. The cylinder is allowed to oscillate in combined crossflow and in-line motions, and the ratio of the nominal in-line and transverse natural frequencies is varied systematically. Experiments were conducted on a smooth cylinder at subcritical Reynolds numbers between 15 000 and 60 000 and on a roughened cylinder at supercritical Reynolds numbers between 320 000 and 710 000, with a surface roughness equal to 0.23 % of the cylinder diameter. Strong qualitative and quantitative similarities between the subcritical and supercritical experiments are found, especially when the in-line natural frequency is close to twice the value of the crossflow natural frequency. In both Reynolds number regimes, the test cylinder may exhibit a ‘dual-resonant’ response, resulting in resonant crossflow motion at a frequency fv, near the Strouhal frequency, and resonant in-line motion at 2 fv. This dual resonance is shown to occur over a relatively wide frequency region around the Strouhal frequency, accompanied by stable, highly repeatable figure-eight cylinder orbits, as well as large third-harmonic components of the lift force. Under dual-resonance conditions, both the subcritical and the supercritical response is shown to collapse into a narrow parametric region in which the effective natural-frequency ratio is near the value 2, regardless of the nominal natural-frequency ratio. Some differences are noted in the magnitudes of forces and the cylinder response between the two different Reynolds number regimes, but the dual-resonant response and the resulting force trends are preserved despite the large Reynolds number difference.

The fundamental hypotheses underlying Kolmogorov-Oboukhov (1962) turbulence theory (K62) are examined directly and quantitutivezy by using high-resolution numerical turbulence fields. With the use of massively parallel Connection Machine-5, we have performed direct Navier-Stokes simulations (DNS) at 5123 resolution with Taylor microscale Reynolds number up to 195. Three very different types of flow are considered: free-decaying turbulence, stationary turbulence forced at a few large scales, and a 2563 large-eddy simulation (LES) flow field. Both the forced DNS and LES flow fields show realistic inertial-subrange dynamics. The Kolmogorov constant for the k−5/3 energy spectrum obtained from the 5123 DNS flow is 1.68 ±.15. The probability distribution of the locally averaged disspation rate εr, over a length scale r is nearly log-normal in the inertial subrange, but significant departures are observed for high-order moments. The intermittency parameter p, appearing in Kolmogorov's third hypothesis for the variance of the logarithmic dissipation, is found to be in the range of 0.20 to 0.28. The scaling exponents over both εr, and r for the conditionally averaged velocity increments $\overline{\delta_ru|\epsilon_r}$ are quantified, and the direction of their variations conforms with the refined similarity theory. The dimensionless averaged velocity increments $(\overline{\delta_ru^n|\epsilon_r})/(\epsilon_rr)^{n/3}$ are found to depend on the local Reynolds number Reεr = ε1/3rr4/3/ν in a manner consistent with the refined similarity hypotheses. In the inertial subrange, the probability distribution of δru/(εrr)1/3 is found to be universal. Because the local Reynolds number of K62, Rεr = ε1/3rr4/3/ν, spans a finite range at a given scale r as compared to a single value for the local Reynolds number Rr = ε−1/3r4/3/ν in Kolmogorov's (1941a,b) original theory (K41), the inertial range in the K62 context can be better realized than that in K41 for a given turbulence field at moderate Taylor microscale (global) Reynolds number Rλ. Consequently universal constants in the second refined similarity hypothesis can be determined quite accurately, showing a faster-than-exponential growth of the constants with order n. Finally, some consideration is given to the use of pseudo-dissipation in the context of the K62 theory where it is found that the probability distribution of locally averaged pseudo-dissipation ε′r deviates more from a log-normal model than the full dissipation εr. The velocity increments conditioned on ε′r do not follow the refined similarity hypotheses to the same degree as those conditioned on εr.