The flow produced by an infinite rotating disk when the fluid at infinity is in a state of solid rotation is investigated numerically. When the fluid at infinity is rotating in the same sense as the disk, physically acceptable solutions exist in all cases. When the fluid at infinity is rotating in the opposite sense to that of the disk, the only physically acceptable solutions appear to be those in which there is a uniform suction present acting through the disk.

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.

In a horizontal convecting layer several distinct transitions occur before the flow becomes turbulent. These are studied experimentally for several Prandtl numbers from 1 to 104. Cell size plan form, transitions in plan form, transition to time-dependence, as well as the heat flux, are measured for Rayleigh numbers from 103 to 105. The second transition, occurring at around 12 times the critical Rayleigh number, is one from steady two-dimensional rolls to a steady regular cellular pattern. There is associated with this a discrete change of slope of the heat flux curve, coinciding with the second transition observed by Malkus. Transitions to time-dependence will be discussed in part 2.

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.

The linearized stability problem for steady, cellular convection resulting from gradients in surface tension is examined in some detail. Earlier work by Pearson (1958) and Sternling & Scriven (1959, 1964) has been extended by considering the effect of gravity waves. In order to avoid the use of an assumed coupling mechanism at the interface, the relevant dynamical equations were retained for both phases. It is shown that the existence of a critical Marangoni number is assured, and that for many situations this critical value is essentially that which is appropriate to the case of a non-deformable interface. Usually, surface waves are important only at very small wave-numbers, but they are dominant for unusually thin layers of very viscous liquids.

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.

The ‘plug’ flow emerging from a long rotating tube into a large stationary reservoir has been used in an experimental investigation of centrifugally unstable swirling jets. A moderate Reynolds number, $\hbox{\it Re}\,{=}\,1000$, was studied extensively, and swirl numbers, $S$, the ratio of nozzle exit rotating speed to the mean mass axial velocity, were in the range 0–1.1. Four regimes were covered: non-swirling jets with $S\,{=}\,0$, weakly swirling jets in the range $0\,{<}\,S\,{<}\,S_{c1}$, strongly swirling jets before vortex breakdown with $S_{c1}\,{<}\,S\,{<}\,S_{c2}$, and stable vortex breakdown for $S\,{>}\,S_{c2}$, where $S_{c1}\,{=}\,0.6$ and $S_{c2}\,{=}\,0.88$.

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.

Experiments were carried out using models having L/D [les ] 2 and the resulting pressure distributions and vortex shedding characteristics are presented. A simple visualization technique which provides explanations of some of the measured results is described. It is concluded that splitter planes reduce the drag markedly by stabilizing the separation points and produce a wake narrower than that for a plain cylinder, raise the base pressure by as much as 50% and affect the Strouhal number to a lesser degree. Careful measurement techniques have enabled these effects to be presented accurately.

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 effects of vortex generators and periodic excitation on vorticity dynamics and the phenomenon of axis switching in a free asymmetric jet are studied experimentally. Most of the data reported are for a 3:1 rectangular jet at a Reynolds number of 450 000 and a Mach number of 0.31. The vortex generators are in the form of ‘delta tabs’, triangular-shaped protrusions into the flow, placed at the nozzle exit. With suitable placement of the tabs, axis switching could be either stopped or augmented. Two mechanisms are identified governing the phenomenon. One, as described by previous researchers, is due to the difference in induced velocities for different segments of a rolled-up azimuthal vortical structure. The other is due to the induced velocities of streamwise vortex pairs in the flow. While the former mechanism, referred to here as the ωθ-dynamics, is responsible for a rapid axis switching in periodically forced jets, e.g. screeching supersonic jets, the effect of the tabs is governed mainly by the latter mechanism, referred to as the ωx-dynamics. Both dynamics can be active in a natural asymmetric jet; the tendency for axis switching caused by the ωθ-dynamics may be, depending on the streamwise vorticity distribution, either resisted or enhanced by the ωx-dynamics. While this simple framework qualitatively explains the various observations made on axis switching, mechanisms actually in play may be much more complex. The two dynamics are not independent as the flow field is replete with both azimuthal and streamwise vortical structures which continually interact. Phase-averaged measurements for a periodically forced case, over a volume of the flow field, are carried out in an effort to gain insight into the dynamics of these vortical structures. The results are used to examine such processes as the reorientation of the azimuthal vortices, the resultant evolution of streamwise vortex pairs, as well as the redistribution of streamwise vortices originating from secondary flow within the nozzle.

The impact of a drop on the plane surface of the same liquid is studied numerically. The accuracy of the calculation is substantiated by its good agreement with available experimental data. An attempt is made to explain the recent observation that, in a restricted range of drop radii and impact velocities, small air bubbles remain entrained in the liquid. The implications of this process for the underwater sound due to rain are considered. The numerical approach consists of a new formulation of the boundary-element method which is explained in detail. Techniques to stabilize the calculation in the presence of strong surface-tension effects are also described.

We report the results of an experimental study of the transition to turbulence in a pipe under the condition of constant mass flux. The transition behaviour and structures observed in this experiment were qualitatively the same as those described in previous reported studies performed in pressure-driven systems. A variety of jet and suction devices were used to create repeatable disturbances which were then used to test the stability of developed Poiseuille flow. The Reynolds number (Re) and the parameters governing the disturbances were varied and the outcome, whether or not transition occurred some distance downstream of the injection point, was recorded. It was found that a critical amplitude of disturbance was required to cause transition at a given Re and that this amplitude varied in a systematic way with Re. This finite, critical level was found to be a robust feature, and was relatively insensitive to the form of disturbance. We interpret this as evidence for disconnected solutions which may provide a pointer for making progress in this fundamental, and as yet unresolved, problem in fluid mechanics.

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.

We present results of experiments on a turbulent grid flow and a few results on measurements in the outer region of a boundary layer over a smooth plate. The air flow measurements included three velocity components and their nine gradients. This was done by a twelve-wire hot-wire probe (3 arrays × 4 wires), produced for this purpose using specially made equipment (micromanipulators and some other auxiliary special equipment), calibration unit and calibration procedure. The probe had no common prongs and the calibration procedure was based on constructing a calibration function for each combination of three wires in each array (total 12) as a three-dimensional Chebishev polynomial of fourth order. A variety of checks were made in order to estimate the reliability of the results.

Among the results the most prominent are the experimental confirmation of the strong tendency for alignment between vorticity and the intermediate eigenvector of the rate-of-strain tensor, the positiveness of the total enstrophy-generating term ωiωjsij (sij = ½(∂ui/∂xj+∂uj/∂xi), ωi = εijk∂uj/∂xk) even for rather short records and the tendency for alignment in the strict sense between vorticity and the vortex stretching vector Wi = ωjsij. An emphasis is put on the necessity to measure invariant quantities, i.e. independent of the choice of the system of reference (e.g. sijsij and ωiωjsij) as the most appropriate to describe physical processes. From the methodological point of view the main result is that the multi-hot-wire technique can be successfully used for measurements of all the nine velocity derivatives in turbulent flows, at least at moderate Reynolds numbers.

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.

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.

We present novel measurements of the primary instabilities of thin liquid films flowing down an incline. A fluorescence imaging method allows accurate measurements of film thickness h(x, y, t) in real time with a sensitivity of several microns, and laser beam deflection yields local measurements with a sensitivity of less than one micron. We locate the instability with good accuracy despite the fact that it occurs (asymptotically) at zero wavenumber, and determine the critical Reynolds number Rc for the onset of waves as a function of angle β. The measurements of Rc(β) are found to be in good agreement with calculations, as are the growth rates and wave velocities. We show experimentally that the initial instability is convective and that the waves are noisesustained. This means that the waveform and its amplitude are strongly affected by external noise at the source. We investigate the role of noise by varying the level of periodic external forcing. The nonlinear evolution of the waves depends strongly on the initial wavenumber (or the frequency f). A new phase boundary f*s(R) is measured, which separates the regimes of saturated finite amplitude waves (at high f) from multipeaked solitary waves (at low f). This boundary probably corresponds approximately to the sign reversal of the third Landau coefficient in weakly nonlinear theory. Finally, we show that periodic waves are unstable over a wide frequency band with respect to a convective subharmonic instability. This instability leads to disordered two-dimensional waves.

The modal distributions of coherent structures evolving near the nozzle of a circular jet are considered. The effects produced on the instability modes by transverse curvature, flow divergence, inhomogeneous inflow conditions, and the detailed shape of the mean velocity profile, are investigated both theoretically and experimentally. Linear stability analysis applied to a thin shear layer surrounding a large-diameter jet (i.e. a jet whose diameter is large in comparison with a typical width of the shear layer) indicates that many azimuthal modes are equally unstable. An increase in the relative thickness of the shear layer limits the number of unstable modes, and only one helical mode remains unstable at the end of the potential core. The linear model used as a transfer function is capable of predicting the spectral distribution of the velocity perturbations in a jet. This provides a rational explanation for the stepwise behaviour of the predominant frequency resulting from a continuous increase in the jet velocity.

The most effective movements of swimming aquatic animals of almost all sizes appear to have the form of a transverse wave progressing along the body from head to tail. The main features of this undulatory mode of propulsion are discussed for the case of large Reynolds number, based on the principle of energy conservation. The general problem of a two-dimensional flexible plate, swimming at arbitrary, unsteady forward speeds, is solved by applying the linearized in viscid flow theory. The large-time asymptotic behaviour of an initial-value harmonic motion shows the decay of the transient terms. For a flexible plate starting with a constant acceleration from at rest, the small-time solution is evaluated and the initial optimum shape is determined for the maximum thrust under conditions of fixed power and negligible body recoil.