A hierarchy of low-dimensional Galerkin models is proposed for the viscous, incompressible flow around a circular cylinder building on the pioneering works of Stuart (1958), Deane et al. (1991), and Ma & Karniadakis (2002). The empirical Galerkin model is based on an eight-dimensional Karhunen–Loève decomposition of a numerical simulation and incorporates a new ‘shift-mode’ representing the mean-field correction. The inclusion of the shift-mode significantly improves the resolution of the transient dynamics from the onset of vortex shedding to the periodic von Kármán vortex street. In addition, the Reynolds-number dependence of the flow can be described with good accuracy. The inclusion of stability eigenmodes further enhances the accuracy of fluctuation dynamics. Mathematical and physical system reduction approaches lead to invariant-manifold and to mean-field models, respectively. The corresponding two-dimensional dynamical systems are further reduced to the Landau amplitude equation.

The characteristic lengths of the oscillating wakes of bluff bodies is discussed; in particular, those used in the universal non-dimensional frequencies proposed by Roshko (1954b) and Goldburg, Washburn & Florsheim (1965). It is concluded that these are equivalent at high Reynolds number. A closer examination leads to the conclusion that there are two simultaneous characteristic lengths; the scale of the formation region, and the width to which the free shear layers diffuse. Discussion of the mechanics of the formation region results in a physical basis for the determination of the frequency by these two characteristic lengths. The ideas developed are applied to the effects of splitter plates in the wake. The possibility of a high-Reynolds-number symmetrical formation region is suggested as an explanation of the very small lift values observed in the absence of free-stream disturbances.

This paper describes an experimental investigation of transport processes in the near wake of a circular cylinder at a Reynolds number of 140000. The flow in the first eight diameters of the wake was measured using X-array hot-wire probes mounted on a pair of whirling arms. This flying-hot-wire technique increases the relative velocity component along the probe axis and thus decreases the relative flow angle to usable values in regions where fluctuations in flow velocity and direction are large. One valuable fringe benefit of the technique is that rotation of the arms in a uniform flow applies a wide range of relative flow angles to the X-arrays, making them inherently self-calibrating in pitch. An analog circuit was used to generate an intermittency signal, and a fast surface-pressure sensor was used to generate a phase signal synchronized with the vortex-shedding process. The phase signal allowed sorting of the velocity data into 16 populations, each having essentially constant phase. An ensemble average for each population yielded a sequence of pictures of the instantaneous mean flow field, with the vortices frozen as they would be in a photograph. In addition to globally averaged data for velocity and stress, the measurements yield non-steady mean data (in the sense of an average a t constant phase) for velocity, intermittency, vorticity, stress and turbulent-energy production as a function of phase for the first eight diameters of the near wake. The stresses were resolved into a contribution from the periodic motion and a contribution from the random motion at constant phase. The two contributions are found to have comparable amplitudes but quite different geometries, and the time average of their sum (the conventional global Reynolds stress) therefore has a quite-complex structure. The non-steady mean-vorticity field is obtained with good resolution as the curl of the non-steady mean-velocity field. Less than half of the shed circulation appears in the vortices, and there is a slow decay of this circulation for each shed vortex as it moves downstream. In the discussion, considerable emphasis is put on the topology of the non-steady mean flow, which emerges as a pattern of centres and saddles in a frame of reference moving with the eddies. The kinematics of the vortex-formation process are described in terms of the formation and evolution of saddle points between vortices in the first few diameters of the near wake. One important conclusion is that a substantial part of the turbulence production is concentrated near the saddles and that the mechanism of turbulence production is probably vortex stretching at intermediate scales. Entrainment is also found to be closely associated with saddles and to be concentrated near the upstream-facing interface of each vortex.

The intrinsic equation governing the curvature K and the torsion τ of an isolated very thin vortex filament without stretching in an incompressible inviscid fluid is reduced to a non-linear Schrödinger equation \[ \frac{{\rm l}}{i}\frac{\partial \psi}{\partial t} = \frac{\partial^2\psi}{\partial s^2}+{\textstyle\frac{1}{2}}(|\psi|^2+A)\psi, \] where t is the time, s the length measured along the filament, ψ is the complex variable \[ \psi = \kappa\exp\left(i\int_0^{s}\tau \,ds\right) \] and is a function oft. It is found that this equation yields a solution describing the propagation of a loop or a hump of helical motion along a line vortex, with a constant velocity 2τ. The relation to the system of intrinsic equations derived by Betchov (1965) is discussed.

The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, has been shown to differ from that in still fluid owing to a bias from the particle inertia (Maxey 1987). Previous numerical results for particles in a random flow field, where the flow dynamics were not considered, showed an increase in the average settling velocity. Direct numerical simulations of the motion of heavy particles in isotropic homogeneous turbulence have been performed where the flow dynamics are included. These show that a significant increase in the average settling velocity can occur for particles with inertial response time and still-fluid terminal velocity comparable to the Kolmogorov scales of the turbulence. This increase may be as much as 50% of the terminal velocity, which is much larger than was previously found. The concentration field of the heavy particles, obtained from direct numerical simulations, shows the importance of the inertial bias with particles tending to collect in elongated sheets on the peripheries of local vortical structures. This is coupled then to a preferential sweeping of the particles in downward moving fluid. Again the importance of Kolmogorov scaling to these processes is demonstrated. Finally, some consideration is given to larger particles that are subject to a nonlinear drag force where it is found that the nonlinearity reduces the net increase in settling velocity.

The most widely used definitions of a vortex are not objective: they identify different structures as vortices in frames that rotate relative to each other. Yet a frame-independent vortex definition is essential for rotating flows and for flows with interacting vortices. Here we define a vortex as a set of fluid trajectories along which the strain acceleration tensor is indefinite over directions of zero strain. Physically, this objective criterion identifies vortices as material tubes in which material elements do not align with directions suggested by the strain eigenvectors. We show using examples how this vortex criterion outperforms earlier frame-dependent criteria. As a side result, we also obtain an objective criterion for hyperbolic Lagrangian structures.

In this paper, we study the transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. We use simultaneous force, displacement and vorticity measurements (using DPIV) for the first time in free vibrations. There exist two distinct types of response in such systems, depending on whether one has a high or low combined mass–damping parameter (m*ζ). In the classical high-(m*ζ) case, an ‘initial’ and ‘lower’ amplitude branch are separated by a discontinuous mode transition, whereas in the case of low (m*ζ), a further higher-amplitude ‘upper’ branch of response appears, and there exist two mode transitions.

To understand the existence of more than one mode transition for low (m*ζ), we employ two distinct formulations of the equation of motion, one of which uses the ‘total force’, while the other uses the ‘vortex force’, which is related only to the dynamics of vorticity. The first mode transition involves a jump in ‘vortex phase’ (between vortex force and displacement), ϕvortex, at which point the frequency of oscillation (f) passes through the natural frequency of the system in the fluid, f ∼ fNwater. This transition is associated with a jump between 2S [harr ] 2P vortex wake modes, and a corresponding switch in vortex shedding timing. Across the second mode transition, there is a jump in ‘total phase’, phis;total , at which point f ∼ fNvacuum. In this case, there is no jump in ϕvortex, since both branches are associated with the 2P mode, and there is therefore no switch in timing of shedding, contrary to previous assumptions. Interestingly, for the high-(m*ζ) case, the vibration frequency jumps across both fNwater and fNvacuum, corresponding to the simultaneous jumps in ϕvortex and ϕtotal. This causes a switch in the timing of shedding, coincident with the ‘total phase’ jump, in agreement with previous assumptions.

For large mass ratios, m* = O(100), the vibration frequency for synchronization lies close to the natural frequency (f* = f/fN ≈ 1.0), but as mass is reduced to m* = O(1), f* can reach remarkably large values. We deduce an expression for the frequency of the lower-branch vibration, as follows:

formula here

which agrees very well with a wide set of experimental data. This frequency equation uncovers the existence of a critical mass ratio, where the frequency f* becomes large: m*crit = 0.54. When m* < m*crit, the lower branch can never be reached and it ceases to exist. The upper-branch large-amplitude vibrations persist for all velocities, no matter how high, and the frequency increases indefinitely with flow velocity. Experiments at m* < m*crit show that the upper-branch vibrations continue to the limits (in flow speed) of our facility.

This paper considers the flow due to a jet spreading out over a plane surface, either radially or in two dimensions. Solutions of the boundary layer equations are sought, according to which the form of the velocity distribution across the jet does not vary along its length. For laminar flow, such a similarity solution is obtained explicitly. For turbulent flow, an eddy viscosity is introduced, and it is eventually seen that complete similarity is not attainable, but that confident predictions can nevertheless be made about the nature of the velocity distribution and the rate of growth of the wall jet.

The presence of the edge of a half plane in a turbulent fluid results in a large increase in the noise generated by that fluid at low Mach numbers. The parameter which is important is the product $2k\overline{r}_0$, where $\overline{r}_0$ is the distance of the centre of an eddy from the edge. Eddies which satisfy the inequality $2k\overline{r}_0\ll 1$ have the sound output of the quadrupoles associated with the fluid motion in a plane normal to the edge increased by the factor $(k\overline{r}_0)^{-3}$. There is no enhancement of the sound from the longitudinal quadrupoles with axes parallel to the edge: the rr, θr and θθ quadrupoles are the dominant sound sources. The far field sound intensity induced by these sources depends upon the fifth power of a typical fluid velocity. The intensity has a directional dependence on cos2 ½θ if the half plane is rigid and sin2 ½θ if it is a pressure release surface, θ = 0 being a direction in the half plane.

If the eddies are far from the edge so that $(k\overline{r}_0)^{\frac{1}{2}} \gg 1$ then the farfield sound has the same features as would be predicted by geometrical acoustics. The edge does not produce any significant sound amplification.

It is shown that a rigid sphere transported along in Poiseuille flow through a tube is subject to radial forces which tend to carry it to a certain equilibrium position at about 0.6 tube radii from the axis, irrespective of the radial position at which the sphere first entered the tube. It is further shown that the trajectories of the particles are portions of one master trajectory and that the origin of the forces causing the radial displacements is in the inertia of the moving fluid. An analysis of the parameters determining the behaviour is presented and a phenomenological description valid at low Reynolds numbers is arrived at in terms of appropriate reduced variables. These phenomena have already been described in a preliminary note (Segré & Silberberg 1961). The present more complete analysis confirms the conclusions, but it appears that the dependence of the effects on the particle radius go with the third and not the fourth power as was then reported.

It is also shown that the description of the phenomena becomes more complicated at tube Reynolds numbers above about 30.

Statistics obtained from seven different direct numerical simulations (DNSs) pertaining to a canonical turbulent boundary layer (TBL) under zero pressure gradient are compiled and compared. The considered data sets include a recent DNS of a TBL with the extended range of Reynolds numbers Reθ = 500–4300. Although all the simulations relate to the same physical flow case, the approaches differ in the applied numerical method, grid resolution and distribution, inflow generation method, boundary conditions and box dimensions. The resulting comparison shows surprisingly large differences not only in both basic integral quantities such as the friction coefficient cf or the shape factor H12, but also in their predictions of mean and fluctuation profiles far into the sublayer. It is thus shown that the numerical simulation of TBLs is, mainly due to the spatial development of the flow, very sensitive to, e.g. proper inflow condition, sufficient settling length and appropriate box dimensions. Thus, a DNS has to be considered as a numerical experiment and should be the subject of the same scrutiny as experimental data. However, if a DNS is set up with the necessary care, it can provide a faithful tool to predict even such notoriously difficult flow cases with great accuracy.

Recent measurements of wave spectra and observations by remote sensing of the sea surface indicate that the author's (1958) conception of an upper-limit asymptote to the spectrum, independent of wind stress, is no longer tenable. The nature of the equilibrium range is reexamined, using the dynamical insights into wave–wave interactions, energy input from the wind and wave-breaking that have been developed since 1960. With the assumption that all three of these processes are important in the equilibrium range, the wavenumber spectrum is found to be of the form $(\cos\theta)^p u_{*} g^{-\frac{1}{2}}k^{-\frac{7}{2}}$, where p ∼ ½ and the frequency spectrum is proportional to u*gσ−4. These forms have been found by Kitaigorodskii (1983) on a quite different dynamical basis; the latter is consistent with the form found empirically by Toba (1973) and later workers. Various derived spectra, such as those of the sea-surface slope and of an instantaneous line traverse of the surface, are also given, as well as directional frequency spectra and frequency spectra of slope.

The theory also provides expressions for the spectral rates of action, energy and momentum loss from the equilibrium range by wave-breaking and for the spectrally integrated rates across the whole range. These indicate that, as a wave field develops with increasing fetch or duration, the momentum flux to the underlying water by wave-breaking increases asymptotically to a large fraction of the total wind stress and that the energy flux to turbulence in the water, occurring over a wide range of scales, increases logarithmically as the extent of the equilibrium range increases. Interrelationships are pointed out among different sets of measurements such as the various spectral levels, the directional distributions, the total mean-square slope and the ratio of downwind to crosswind mean-square slopes.

Finally, some statistical characteristics of the breaking events are deduced, including the expected length of breaking fronts (per unit surface area) with speeds of advance between c and c+dc and the number of such breaking events passing a given point per unit time. These then lead to simple expressions for the density of whitecapping, those breaking events that produce bubbles and trails of foam, the total number of whitecaps passing a given point per unit time and, more tenuously, the whitecap coverage.

We analyse the currently popular vortex identification criteria that are based on point-wise analysis of the velocity gradient tensor. A new measure of spiralling compactness of material orbits in vortices is introduced and using this measure a new local vortex identification criterion and requirements for a vortex core are proposed. The inter-relationships between the different criteria are explored analytically and in a few flow examples, using both zero and non-zero thresholds for the identification parameter. These inter-relationships provide a new interpretation of the various criteria in terms of the local flow kinematics. A canonical turbulent flow example is studied, and it is observed that all the criteria, given the proposed usage of threshold, result in remarkably similar looking vortical structures. A unified interpretation based on local flow kinematics is offered for when similarity or differences can be expected in the vortical structures educed using the different criteria.

It is found that when the average kinetic energies of normal velocity components in decaying, grid-generated turbulence are equilibrated by a symmetric contraction of the wind tunnel, this equality can persist downstream. A second result is further confirmation of the fact that the best power-law fit to the inverse turbulent energy during the early part of decay is near (x–x1)1·25, for both rod grids and disk grids. The Kolmogorov decay law $\sim (t - t_1)^{\frac{10}{7}}$ is re-derived by a spectral method which is essentially equivalent to the original. Finally, a crude theoretical estimate of component energies in the straight duct after a weak contraction seems to support the experiments.

Some preliminary results on the behaviour of controlled wave disturbances introduced artificially into turbulent channel flow are reported. Weak plane-wave disturbances are introduced by vibrating ribbons near each wall. The amplitude and relative phase of the streamwise component of the induced wave is educed from a hot wire signal, allowing the wave speed and attenuation characteristics and the wave shape to be traced downstream. The normal component and wave Reynolds stress have been inferred from these data. It appears that Orr–Sommerfeld theories attempted to date are inadequate for description of these waves.

The mechanisms of displacement of one fluid by another are investigated in an etched network.

Experiments show that both fluids are simultaneously present in a duct, the wetting fluid remaining in the extreme corners of the cross-section. Calculation of displacement pressures are in good agreement with experiments for drainage, imbibition and removal of blobs. The results may be related to some flow behaviour exhibited in porous media.

An experimental study of boundary-layer turbulence in a free surface channel flow is described. Attention is concentrated on the effects of different surface roughness conditions on the turbulence structure in the boundary region. Hydrogen bubble flow tracers and medium high-speed motion photography were used to obtain an instantaneous visual and quantitative description of the flow field. In particular it proved possible to record instantaneous longitudinal and vertical velocity profiles from which distributions of the instantaneous Reynolds stress contribution were computed.

Two well-defined intermittent features of the flow structure were visually identified close to the boundary. These consisted of fluid ejection phases, previously reported by Kline et al. (1967) for smooth boundary flow, and fluid inrush phases. Conditional averaging of the instantaneous velocity data yielded quantitative confirmation that ejection phases corresponded with ejection of low momentum fluid outwards from the boundary whilst inrush phases were associated with the transport of high momentum fluid inwards towards the boundary. Inrush and ejection events were present irrespective of the surface roughness condition.

Conditional averaging also indicated that both inrush and ejection sequences correlate with an extremely high contribution to Reynolds stress and hence turbulence production close to the boundary. Indeed the present results, taken with those from previous studies, suggest that turbulence production is dominated by the joint contribution from the inrush and ejection events. It is emphasized that these structural features are intermittent, forming important linked elements of a randomly repeating cycle of wall-region turbulence production which is apparently driven by some violent three-dimensional instability mechanism.

Whilst the most coherent effects of the observed inrush phases appear to be mainly confined to a region close to the boundary, the influence of the ejection phases is far more extensive. The ejected low momentum fluid elements, drawn from the viscous sublayer and from between the interstices of the roughness elements, travel outwards from the boundary into the body of the flow and give rise to very large positive contributions to Reynolds stress at points remote from the boundary. This effect is sufficiently strong to prompt the suggestion that the ejection process could represent a universal and dominant mode of momentum transport outside the immediate wall region and possibly extending across the entire thickness of the boundary layer.

A structural model based on the present observations is seen to exhibit consistency with many commonly visualized features and recorded average properties of turbulent boundary-layer flows in general.

When a liquid meniscus held at the exit of a metallic capillary tube is charged to a high voltage V, the free surface often takes the form of a cone whose apex emits a steady microjet, and thus injects a certain charge I and liquid volume Q per unit time into the surrounding gas. This work deals with liquids with relatively large conductivities K, for which the jet diameter dj is much smaller than the diameter dn of the capillary tube. In the limit dj/dn → 0, the structure of the jet (dj and I, in particular) becomes independent of electrostatic parameters such as V or the electrode configuration, being governed mostly by the liquid properties and flow rate Q. Furthermore, the measured current is given approximately by I = f(ε) (γQK/ε)½ for a wide variety of liquids and conditions (ε, and γ are, respectively, the dielectric constant of the liquid and the coefficient of interfacial tension; f(ε) is shown in figure 11). The following explanation is proposed for this behaviour. Convection associated with the liquid flow Q transports the net surface charge towards the cone tip. This upsets the electrostatic surface charge distribution slightly at distances r from the apex large compared to a certain charge relaxation length λ, but substantially when r ∼ λ. When the fluid motion is modelled as a sink flow, λ is of the order of r* = (Qεε0/K)$\frac13$ (ε0 is the electrical permittivity of vacuum). If, in addition, the surface charge density is described through Taylor's theory, the corresponding surface current convected towards the apex scales as Is ∼ (γQK/ε)½, as observed for the spray current. The sink flow hypothesis is shown to be realistic for sufficiently small jet Reynolds numbers. In a few photographs of ethylene glycol cone jets, we find the rough scaling dj ∼ 0.4r* for the jet diameter, which shows that the jet forms as soon as charge relaxation effects set in. In the limit ε [Gt ] 1, an upper bound is found for the convected current at the virtual cone apex, which accounts for only one-quarter of the total measured spray current. The rest of the charge must accordingly reach the head of the jet by conduction through the bulk.

Previous research has established that surfaces with tiny ribs (riblets) aligned in the streamwise direction can reduce the turbulent wall-shear stress below that of a smooth surface. Typical skin-friction reductions have been found to be about 5%. The results of the present investigation, however, demonstrate a considerable improvement over this value. This improvement is achieved by a systematic experimental optimization which has been guided by theoretical concepts.

A key feature of our experiments is the utilization of an oil channel. Previous experiments in wind tunnels had to contend with very small riblet dimensions which typically had a lateral rib spacing of about 0.5 mm or less. By contrast, in our oil channel, the ribs can have a lateral spacing of between about 2 and 10 mm. This increased size of the surface structures enables test surfaces to be manufactured with conventional mechanical methods, and it also enables us to build test surfaces with adjustable geometry. In addition, the Berlin oil channel has a novel shear stress balance with an unprecedented accuracy of ±0.3%. This latter feature is a prerequisite for a systematic experimental optimization.

In the present investigation, surfaces with longitudinal ribs and additional slits are studied. The experiments cover a fairly large range of parameters so that the drag reduction potential of a surface with ribs and/or slits is worked out conclusively. A large parameter range is made possible because of the adjustability of the surfaces as well as the automatic operation of the oil channel. In particular, the following tests were run:

(i) Shear stress measurements with conventional riblet configurations, i.e. with triangular and semi-circular grooves, have been carried out. These measurements were necessary in order to establish the connection between our oil channel data and previous data from wind tunnels. As was previously established, we found a drag reduction of about 5%.

(ii) An adjustable surface with longitudinal blade ribs and with slits was built and tested. Both groove depth and slit width could be varied separately and continuously during the experiment. It turned out, that slits in the surface did not contribute to the drag reduction. Nevertheless, these investigations show how perforated surfaces (e.g. for boundary-layer control) can be designed for minimal parasitic drag. On the other hand, with closed slits, an optimal groove depth for the rib surface could be determined, i.e. half of the lateral rib spacing. For this configuration, we found an 8.7% skin-friction reduction. By carefully eliminating deleterious effects (caused by little gaps, etc.), the skin-friction reduction could be improved to a record value of 9.9%.

(iii) A quantitative comparison between theory and experiment was carried out. The theory is based on the assumption that riblets impede the fluctuating turbulent crossflow near the wall. In this way, momentum transfer and shear stress are reduced. The simplified theoretical model proposed by Luchini (1992) is supported by the present experiments.

(iv) For technological applications of riblets, e.g. on long-range commercial aircraft, the above thin-blade ribs are not practical. Therefore, we have devised a surface that combines a significantly improved performance (8.2 %) with a geometry which exhibits better durability and enables previously developed manufacturing methods for plastic riblet film production to be used. Our riblet geometry exhibits trapezoidal grooves with wedge-like ribs. The flat floor of the trapezoidal grooves permits an undistorted visibility through the transparent riblet film which is essential for crack inspection on aircraft.

Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.