Swirling flow through a pipe is a highly complex turbulent flow and is still challenging to predict. An experimental investigation is performed to obtain systematic data about the flow and to understand its physics. A free-vortex-type swirling flow is introduced in a long straight circular pipe. The swirling component decays downstream as a result of wall friction. The velocity distributions are continuously changing as they approach fully developed parallel flow. The swirl intensity Ω, defined as a non-dimensional angular momentum flux, decays exponentially. The decay coefficients, however, are not constant as conventionally assumed, but depend on the swirl intensity. The wall shear stresses are measured by a direct method and, except in a short inlet region, are a function only of the swirl intensity and the Reynolds number. The velocity distributions and all Reynolds stress components are measured at various axial positions in the pipe. The structure of the tangential velocity profile is classified into three regions: core, annular and wall regions. The core region is characterized by a forced vortex motion and the flow is dependent upon the upstream conditions. In the annular region, the skewness of the velocity vector is noticeable and highly anisotropic so that the turbulent viscosity model does not work well here. The tangential velocity is expressed as a sum of free and forced vortex motion. In the wall region the skewness of the flow becomes weak, and the wall law modified by the Monin–Oboukhov formula is applicable. Data on the microscale and the spectrum are also presented and show quite different turbulence structures in the core and the outer regions.

The mean circulation of planetary fluids tends to develop uniform potential vorticity q in regions where closed time-mean streamlines or closed isolines of mean potential vorticity exist. This state is established in statistically steady flows by geostrophic turbulence or by wave-induced potential-vorticity flux. At the outer edge of the closed contours the expelled gradients of q are concentrated. Beyond this transition lies motionless fluid, or a different flow regime in which the planetary gradient of q may be dominant. The homogenized regions occur where direct forcing by external stress or heating within the closed isoline is negligible, upon the potential-density surface under consideration. In the stably stratified ocean such regions are found at depths greater than those of direct wind-induced stress or penetrative cooling. In ‘channel’ models of the atmosphere we again find constant q when mesoscale eddies cause the dominant potential-vorticity flux. In the real atmosphere the results here can apply only where internal heating is negligible. The derivations given here build upon the Prandtl–Batchelor theorem, which applies to non-rotating, steady two-dimensional flow. Supporting evidence is found in numerical circulation models and oceanic observations.

By generalizing the theory of ‘rapid distortion’ of turbulence developed by Batchelor & Proudman (1954) it is shown in this paper that the turbulent velocity around a bluff body placed in a turbulent flow can be calculated outside and upstream of the regions of separated flow, if the incident turbulent flow satisfies the following conditions: (i) if a/Lx [Lt ] 1 or = O(1), Re−1$u^{\prime}_{\infty}/\overline{u}_{\infty} $ [Lt ] 1 [Lt ] Re½; (ii) if a/Lx [Gt ] 1, Re−1 [Lt ] $u^{\prime}_{\infty}$ [Lt ] 1/(a/Lx) and Re [Gt ] (a/Lx)2, where $Re = \overline{u}_{\infty} a/ν$, is the mean uniform incident velocity, $u^{\prime}_{\infty}$ is the r.m.s. velocity of the homogeneous incident turbulence, a is a transverse dimension of the body (the radius in the case of a circular cylinder), Lx is the integral scale of the incident turbulence and v is the kinematic viscosity.

Detailed calculations are given for the flow around a circular cylinder with particular emphasis on the turbulence very close to the surface. (The results can be generalized to other cylindrical bodies.) Mean-square values and spectra of velocity have been found only in the limiting situations where the turbulence scale is very much larger or smaller than the size of the body, i.e. Lx [Gt ] a or Lx [Lt ] a. But, whatever the value of a/Lx, if the frequency is sufficiently large the results for spectra tend to those of the limiting situation where Lx [Lt ] a. The reason why the turbulence velocities have not been calculated for intermediate values of a/Lx is that closed-form solutions cannot be found and that the computing time then required is quite excessive. However, some computed results are used in the paper to suggest the qualitative behaviour of the turbulence when Lx is of order a. An important result of the theory is that it illuminates and distinguishes between the governing physical processes of distortion of the turbulence by the mean flow, the direct ‘blocking’ of the turbulence by the body, and concentration of vortex lines at the body's surface.

The results of the theory have many applications, for example in calculating turbulent dispersion and fluctuating pressures on the body, as shown elsewhere by Hunt & Mulhearn (1973) and Hunt (1973).

In conclusion the theoretical results are briefly compared with experimental measurements of turbulent flows round non-circular cylinders. A detailed comparison with measurements round circular cylinders will be published later by Petty (1974).

The fine structure of the summer thermocline in the Mediterranean Sea around Malta, investigated by a new temperature-gradient meter and by photographs of dye tracers, is summarized. The principal internal feature of the thermocline is a series of thin, laminar-flow sheets of high static stability, separated by weakly turbulent layers of only moderate density gradient and a few metres thick. A mechanism for generating the patches of turbulence observed on these thermocline sheets is established by comparing dye photographs with a theory by Miles & Howard (1964).

A methodology termed the ‘filtered mass density function’ (FMDF) is developed and implemented for large-eddy simulation (LES) of variable-density chemically reacting turbulent flows at low Mach numbers. This methodology is based on the extension of the ‘filtered density function’ (FDF) scheme recently proposed by Colucci et al. (1998) for LES of constant-density reacting flows. The FMDF represents the joint probability density function of the subgrid-scale (SGS) scalar quantities and is obtained by solution of its modelled transport equation. In this equation, the effect of chemical reactions appears in a closed form and the influences of SGS mixing and convection are modelled. The stochastic differential equations (SDEs) which yield statistically equivalent results to those of the FMDF transport equation are derived and are solved via a Lagrangian Monte Carlo scheme. The consistency, convergence, and accuracy of the FMDF and the Monte Carlo solution of its equivalent SDEs are assessed. In non-reacting flows, it is shown that the filtered results via the FMDF agree well with those obtained by the ‘conventional’ LES in which the finite difference solution of the transport equations of these filtered quantities is obtained. The advantage of the FMDF is demonstrated in LES of reacting shear flows with non-premixed reactants. The FMDF results are appraised by comparisons with data generated by direct numerical simulation (DNS) and with experimental measurements. In the absence of a closure for the SGS scalar correlations, the results based on the conventional LES are significantly different from those obtained by DNS. The FMDF results show a closer agreement with DNS. These results also agree favourably with laboratory data of exothermic reacting turbulent shear flows, and portray several of the features observed experimentally.

The analysis of two-layer exchange flow through contractions with a barotropic component treated by Armi & Farmer (1986) is extended to include exchange flows over sills and through a combination of a sill and contraction. It is shown that exchange over a sill is fundamentally different from exchange through a contraction. Control at the sill crest acts primarily through the deeper layer into which the sill projects and only indirectly controls the surface layer. This asymmetry in the control results in asymmetrical flows. The interface depth above the crest is not one half the total depth, as assumed in other studies by analogy with flow through contractions, but is somewhat deeper; the maximal exchange rate is less than for flow through a contraction of equal depth. When both a sill and a contraction are present, the contraction influences control at the sill crest only if it lies between the sill and the source of denser water. The response to barotropic flow is also asymmetrical: the transition to single-layer flow occurs at much lower speeds for a barotropic component in one direction than the other.

Results of the analysis are applied to exchange flow through the Strait of Gibraltar, which includes both a sill and a contraction. It is shown that maximal exchange conditions apply throughout part of the tidal cycle, and observations illustrate several of the analytical predictions for barotropic flows, including the formation of fronts, single-layer flow, submaximal exchange and reverse flow.

We report the results of physical experiments that demonstrate the strong influence of the thermal conductivity of the substrate on the evaporation of a pinned droplet. We show that this behaviour can be captured by a mathematical model including the variation of the saturation concentration with temperature, and hence coupling the problems for the vapour concentration in the atmosphere and the temperature in the liquid and the substrate. Furthermore, we show that including two ad hoc improvements to the model, namely a Newton's law of cooling on the unwetted surface of the substrate and the buoyancy of water vapour in the atmosphere, give excellent quantitative agreement for all of the combinations of liquid and substrate considered.

This paper is concerned with unsteady laminar boundary layers on solid bodies in the presence of a fluctuating external flow of small amplitude. Any containing enclosures are assumed to be at infinity. Compressibility is ignored and conditions are given under which this and other approximations are valid. Special attention is focussed on the phenomenon of the formation of a steady-streaming flow, induced by the Reynolds stresses in the oscillatory boundary layer: it is shown that, if the characteristic Reynolds number of the steady streaming is large, there is an outer boundary layer within which the steady-streaming velocity decays to zero. The thickness of this outer layer is large compared with that of the inner (oscillatory) layer, but small compared with a typical dimension of the body.

The partial differential equation for the flow in the outer layer is solved in a typical case by a generalization of a series-expansion method due to Fettis. Similarity solutions of the equation are also described.

The theory is applied specifically to the case of flow generated by a circular cylinder oscillating along a diameter in an infinite fluid. Qualitative agreement is obtained with experiments performed by Schlichting.

An experimental investigation of flow over an axisymmetric cavity shows that self-sustained, periodic oscillations of the cavity shear layer are associated with low cavity drag. In this low-drag mode the flow regulates itself to fix the mean-shear-layer stagnation point at the downstream corner. Above a critical value of the cavity width-to-depth ratio there is an abrupt and large increase of drag due to the onset of the ‘wake mode’ of instability. It is also shown by measurement of the momentum balance how the drag of the cavity is related to the state of the shear layer, as defined by the mean momentum transport $\rho\overline{u}\overline{v}$ and the Reynolds stress $\rho\overline{u^{\prime}v^{\prime}}$, and how these are related to the amplifying oscillations in the shear layer. The cavity shear layer is found to be different, in several respects, from a free shear layer.

The rapid shearing of a mixture of cohesionless glass spheres and air or water was studied in an annular, parallel-plate shear cell designed after Savage (1978). Two types of flow were observed. In the first type of flow the entire mass of the granular material was mobilized. At high shear rates the shear and normal stresses were found to be quadratically dependent upon the mean shear rate (at constant volume concentration), in general agreement with the observations of Bagnold (1954) and Savage & Sayed (1984), and the ‘kinetic’ theory of Jenkins & Savage (1983). The stresses were found to be weakly dependent on the volume concentration up to approximately 0.5, and strongly dependent above this concentration. For flows in which water was the interstitial fluid, the ratio of the shear stress to the normal stress was slightly higher (than in air), and the stresses at lower shear rates were found to be more nearly linearly related to the shear rate. It is suggested that these effects are contributed to by the viscous dampening of grain motions by the water. The second type of flow was distinguished by the existence of an internal boundary above which the granular material deformed rapidly, but below which the granular material remained rigidly locked in place. The thickness of the shearing layer was measured to be between 5 and 15 grain diameters. The stress ratio at the bottom of the shearing layer was found to be nearly constant, suggesting the internal boundary is a consequence of the immersed weight of the shearing grains, and may be described by a Coulomb yield criterion. A scaled concentration is proposed to compare similar data obtained using different-sized materials or different apparatus. An intercomparison of the two types of flow studied, along with a comparison between the present experiments and those of Bagnold (1954) and Savage & Sayed (1984), suggests that the nature of the boundaries can have a significant effect upon the dynamics of the entire flow.

A systematic study has been undertaken to quantify the effect of jet temperature on the noise radiated by subsonic jets. Nozzles of different diameters were tested to uncover the effects of Reynolds number. All the tests were carried out at Boeing's Low Speed Aeroacoustic Facility, with simultaneous measurement of thrust and noise. It is concluded that the change in spectral shape at high jet temperatures, normally attributed to the contribution from dipoles, is due to Reynolds number effects and not dipoles. This effect has not been identified before. A critical value of the Reynolds number that would need to be maintained to avoid the effects associated with low Reynolds number has been estimated to be ${\sim}$400 000. It is well-known that large-scale structures are the dominant generators of noise in the peak radiation direction for high-speed jets. Experimental evidence is presented that shows the spectral shape at angles close to the jet axis from unheated low subsonic jets to be the same as from heated supersonic jets. A possible mechanism for the observed trend is proposed. When a subsonic jet is heated with the Mach number held constant, there is a broadening of the angular sector in which peak radiation occurs. Furthermore, there is a broadening of the spectral peak. Similar trends have been observed at supersonic Mach numbers. The spectral shapes in the forward quadrant and in the near-normal angles from unheated and heated subsonic jets also conform to the universal shape obtained from supersonic jet data. Just as for unheated jets, the peak frequency at angles close to the jet axis is independent of jet velocity as long as the acoustic Mach number is less than unity. The extensive database generated in the current test programme is intended to provide test cases with high-quality data that could be used for the evaluation of theoretical/semi-theoretical jet noise prediction methodologies.

The late-stage demixing following spinodal decomposition of a three-dimensional symmetric binary fluid mixture is studied numerically, using a thermodynamically consistent lattice Boltzmann method. We combine results from simulations with different numerical parameters to obtain an unprecedented range of length and time scales when expressed in reduced physical units. (These are the length and time units derived from fluid density, viscosity, and interfacial tension.) Using eight large (2563) runs, the resulting composite graph of reduced domain size l against reduced time t covers 1 [lsim ] l [lsim ] 105, 10 [lsim ] t [lsim ] 108. Our data are consistent with the dynamical scaling hypothesis that l(t) is a universal scaling curve. We give the first detailed statistical analysis of fluid motion, rather than just domain evolution, in simulations of this kind, and introduce scaling plots for several quantities derived from the fluid velocity and velocity gradient fields. Using the conventional definition of Reynolds number for this problem, Reϕ = ldl/dt, we attain values approaching 350. At Reϕ [gsim ] 100 (which requires t [gsim ] 106) we find clear evidence of Furukawa's inertial scaling (l ∼ t2/3), although the crossover from the viscous regime (l ∼ t) is both broad and late (102 [lsim ] t [lsim ] 106). Though it cannot be ruled out, we find no indication that Reϕ is self-limiting (l ∼ t1/2) at late times, as recently proposed by Grant & Elder. Detailed study of the velocity fields confirms that, for our most inertial runs, the RMS ratio of nonlinear to viscous terms in the Navier–Stokes equation, R2, is of order 10, with the fluid mixture showing incipient turbulent characteristics. However, we cannot go far enough into the inertial regime to obtain a clear length separation of domain size, Taylor microscale, and Kolmogorov scale, as would be needed to test a recent ‘extended’ scaling theory of Kendon (in which R2 is self-limiting but Reϕ not). Obtaining our results has required careful steering of several numerical control parameters so as to maintain adequate algorithmic stability, efficiency and isotropy, while eliminating unwanted residual diffusion. (We argue that the latter affects some studies in the literature which report l ∼ t2/3 for t [lsim ] 104.) We analyse the various sources of error and find them just within acceptable levels (a few percent each) in most of our datasets. To bring these under significantly better control, or to go much further into the inertial regime, would require much larger computational resources and/or a breakthrough in algorithm design.

We depict and analyse the complete evolution of an air bubble formed in a water bulk, from the time it emerges at the liquid surface, up to its fragmentation into dispersed drops. To this end, experiments describing the drainage of the bubble cap film, its puncture and the resulting bursting dynamics determining the aerosol formation are conducted on tapwater bubbles. We discover that the mechanism of marginal pinching at the bubble foot and associated convection motions in the bubble cap, known as marginal regeneration, both drive the bubble cap drainage rate, and are responsible for its puncture. The resulting original film thickness evolution law in time, supplemented with considerations about the nucleation of holes piercing the film together culminate in a determination of the cap film thickness at bursting , where is the bubble cap radius of curvature, and a length which we determine. Subsequent to a hole nucleation event, the cap bursting dynamics conditions the resulting spray. The latter depends both on the bubble shape prescribed by , where is the capillary length based on gravity, and on . The mean drop size , the number of drops generated per bubble and the drop size distribution are derived, comparing well with measurements. Combined with known bubble production rates over the ocean, our findings offer an adjustable parameter-free prediction for the aerosol flux and spray structure caused by bubble bursting in this precise context.

Partial cavitation in the separated region forming from the apex of a wedge is examined to reveal the flow mechanism responsible for the transition from stable sheet cavity to periodically shedding cloud cavitation. High-speed visualization and time-resolved X-ray densitometry measurements are used to examine the cavity dynamics, including the time-resolved void-fraction fields within the cavity. The experimentally observed time-averaged void-fraction profiles are compared to an analytical model employing free-streamline theory. From the instantaneous void-fraction flow fields, two distinct shedding mechanisms are identified. The classically described re-entrant flow in the cavity closure is confirmed as a mechanism for vapour entrainment and detachment that leads to intermittent shedding of smaller-scale cavities. But, with a sufficient reduction in cavitation number, large-scale periodic cloud shedding is associated with the formation and propagation of a bubbly shock within the high void-fraction bubbly mixture in the separated cavity flow. When the shock front impinges on flow at the wedge apex, a large cloud is pinched off. For periodic shedding, the speed of the front in the laboratory frame is of the order of half the free-stream speed. The features of the observed condensation shocks are related to the average and dynamic pressure and void fraction using classical one-dimensional jump conditions. The sound speed of the bubbly mixture is estimated to determine the Mach number of the cavity flow. The transition from intermittent to transitional to strongly periodic shedding occurs when the average Mach number of the cavity flow exceeds that required for the generation of strong shocks.

Results are presented for a numerical simulation of vortex-induced vibrations of a circular cylinder of low non-dimensional mass (m* = 10) in the laminar flow regime (60 < Re < 200). The natural structural frequency of the oscillator, fN, matches the vortex shedding frequency for a stationary cylinder at Re = 100. This corresponds to fND2/ν = 16.6, where D is the diameter of the cylinder and ν the coefficient of viscosity of the fluid. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables form in two dimensions. Unlike at high Re, where the cylinder response is known to be associated with three branches, at low Re only two branches are identified: ‘initial’ and ‘lower’. For a blockage of 2.5% and less the onset of synchronization, in the lower Re range, is accompanied by an intermittent switching between two modes with vortex shedding occurring at different frequencies. With higher blockage the jump from the initial to lower branch is hysteretic. Results from free vibrations are compared to the data from experiments for forced vibrations reported earlier. Excellent agreement is observed for the critical amplitude required for the onset of synchronization. The comparison brings out the possibility of hysteresis in forced vibrations. The phase difference between the lift force and transverse displacement shows a jump of almost 180° at, approximately, the middle of the synchronization region. This jump is not hysteretic and it is not associated with any radical change in the vortex shedding pattern. Instead, it is caused by changes in the location and value of the maximum suction on the lower and upper surface of the cylinder. This is observed clearly by comparing the time-averaged flow for a vibrating cylinder for different Re. While the mean flow for Re beyond the phase jump is similar to that for a stationary cylinder, it is associated with a pair of counter-rotating vortices in the near wake for Re prior to the phase jump. The phase jump appears to be one of the mechanisms of the oscillator to self-limit its vibration amplitude.

Direct numerical simulation data of a Mach 2.9, 24○ compression ramp configuration are used to analyse the shock motion. The motion can be observed from the animated DNS data available with the online version of the paper and from wall-pressure and mass-flux signals measured in the free stream. The characteristic low frequency is in the range of (0.007–0.013) U∞/δ, as found previously. The shock motion also exhibits high-frequency, of O(U∞/δ), small-amplitude spanwise wrinkling, which is mainly caused by the spanwise non-uniformity of turbulent structures in the incoming boundary layer. In studying the low-frequency streamwise oscillation, conditional statistics show that there is no significant difference in the properties of the incoming boundary layer when the shock location is upstream or downstream. The spanwise-mean separation point also undergoes a low-frequency motion and is found to be highly correlated with the shock motion. A small correlation is found between the low-momentum structures in the incoming boundary layer and the separation point. Correlations among the spanwise-mean separation point, reattachment point and the shock location indicate that the low-frequency shock unsteadiness is influenced by the downstream flow. Movies are available with the online version of the paper.

An approximate integral-equation approach is used to model the growth and collapse of a vapour cavity in close proximity to an initially plane free surface. By comparison with experiment, it is shown to predict all the salient features of the bubble and freesurface interaction, provided that the complete nonlinear Bernoulli pressure condition is applied on both surfaces. Features observed and predicted include the formation of an accelerating liquid jet in the bubble and a pronounced spike in the free surface during the collapse phase of the bubble's life. If the bubble is initially sufficiently close to the free surface, it will become ‘entrained’ in the raised free surface with a veneer of liquid separating the two free surfaces.

The motion of turbulent jets of heavy salt solution injected upwards into a tank of fresh water has been compared with that of plumes which are initially buoyant but become heavy as they mix with the environment. The reversal of buoyancy in the latter case is produced by using fluids having a non-linear density change on mixing, a laboratory analogue of the density changes occurring at the top of a cumulus cloud due to evaporation. The behaviour in the two cases is quite different; salt jets reach a steady height about which only small fluctuations occur, while the plumes with reversing buoyancy exhibit violent regular oscillations. This phenomenon, which is clearly a property of the ‘evaporation’ and not just of the geometry, is suggested as a likely explanation of the observed oscillation of the tops of cumulus towers. Dimensional arguments have been used to relate the experimental results to the volume, momentum and buoyancy fluxes at the source. An application of one of the deduced relations to the atmosphere gives realistic periods for the cloud-top oscillations.

Low-dimensional empirical Galerkin models are developed for spatially evolving laminar and transitional shear layers, based on a Karhunen–Loève decomposition of incompressible two- and three-dimensional Navier–Stokes simulations. It is shown that the key to an accurate Galerkin model is a novel analytical pressure-term representation. The effect of the pressure term is elucidated by a modal energy-flow analysis in a mixing layer, which generalizes the framework developed by Rempfer (1991). In convectively unstable shear layers, it is shown in particular that neglecting small energy terms leads to large amplitude errors in the Galerkin model. The effect of the pressure term and small neglected energy flows is very important for a two-dimensional mixing layer, is less pronounced for the three-dimensional analogue, and can be considered as small in an absolutely unstable wake flow.

A solution of decaying two-dimensional turbulence at large Reynolds number is analysed by means of an automated vortex census. The census identifies the flow structures which approximately conform to the idealized shape of an isolated, coherent vortex. It also determines vortex characteristics, such as amplitude, size, radial profile, and deformation from the ideal axisymmetric shape. The distributions of these characteristics within the vortex population are examined, as are their time evolutions. Interpretation of these distributions is made with reference to both the random initial conditions for the solution and the dynamical processes of vortex emergence, survival, and interaction.