We present the results of an experimental and numerical investigation into Taylor-Couette flow with gap-length to width ratios (Γ = l/d) ranging from 0.3 to 1.4. Laser-Doppler-velocimetry is used to obtain quantitative information on the bifurcation set experimentally, and novel flow phenomena are uncovered. These results are compared with those obtained using numerical bifurcation techniques applied to a finite-element discretization of the Navier-Stokes equations. In general, the agreement is good and most of the observations are satisfactorily explained.

The instabilities of an acoustically excited coflowing jet are investigated by using a time series of cross-sections of large-scale structures present in the flow and by a fully three-dimensional numerical simulation via the vortex-element method. These structures are first revealed by a flow-visualization technique based on smoke seeding the inner jet flow and observing the flow cross-sections as the structures are convected past a stationary sheet of laser light. Usage of image-processing techniques along with computer graphics allows the full three-dimensional reconstruction of the structure. The general morphology of the flow is determined from the reconstructed views and shows that the vorticity is mainly concentrated into a single large-scale vortex ring. Hypotheses regarding the nature and origin of the longitudinal instabilities observed in the flow can be drawn from the reconstructed views, and are supported by the numerical simulations. Quantitative values such as the surface-to-volume ratio and the entrainment level are also obtained from the experimental data. The onset of stremwise vorticity and its distribution are derived from the numerical calculations as well. Three-dimensional views and a holographic display of a charactristic eddy are also presented.

The laminar flow around a prolate spheroid at 6° angle of attack has been determined by the numerical solution of steady, three-dimensional boundary-layer equations with the external-pressure distribution obtained from an analytic solution of the inviscid-flow equations. The flow is shown to comprise a region of positive crossflow, followed by a substantial region of negative crossflow, a separation line and two terminal lines beyond which solutions of the boundary-layer equations could not be obtained. The separation line defines one boundary of a region of open separation and accords with the argument of Lighthill in that separation of three-dimensional boundary-layer flows is defined by a skin-friction line. A procedure is described that permits the identification of this skin-friction line and requires that it passes through the first location at which the longitudinal component of the wall shear is zero and the circumferential component negative. The numerical tests show that the finite-difference scheme based on the characteristic box allows calculations against the circumferential flow and with an accuracy equal to that of the regular box provided that a stability criterion is used to choose the grid intervals. This stability criterion is shown to be essential for accurate solutions in the vicinity of the separation and terminal lines and implies the need for extremely fine grids. It is evident that similar numerical constraints will apply to calculations performed with an interactive boundary-layer procedure or with higher-order forms of the Navier-Stokes equations.

Atherosclerotic lesions in mammalian blood vessels show a definite spatial pattern, and it has been proposed that lesions occur preferentially in regions with a low wall shear stress. Near the entrance to an intercostal artery, lesions on the wall of the aorta occur, initially at least, downstream of the entrance. We model the flow in this region by a linear shear flow along a flat wall (the wall of the main tube/aorta) past an infinitely deep circular hole perpendicular to the wall (the side-branch/intercostal artery), with fluid being sucked into the hole. By assuming Stokes flow, the three-dimensional model problem is reduced to two independent problems on a two-dimensional domain. By addition, the solutions for any particular side-branch flow rate. We find that the wall shear stress in the main tube is elevated upstream of the side-branch entrance, and downstream as well is the side-branch flow rate is small. However, if the flow rate in the side branch is large enough, there will be regions of both elevated and reduced wall shear stress in the main tube downstream of the side-branch entrance, including a stagnation point. The wall shear is lower downsteam than upstream unless there is no net flow into the side branch.

The solutions given apply to the case with flow out of as well as into the hole. Also, the asymptotic structure for the flow in the hole when there is no net flow into the hole, and the analysis of three-dimensional flow near a sharp corner, are given.

Turbulent shear layers generated from a common splitter plate as well as from a half-frame screen are investigated experimentally in the developing regime and in the asymptotic regime. The phase-averaged means with time delay of velocity fluctuations are analysed in terms of Fourier modes in the frequency domain to give both the amplitude and frequency of the local fundamental mode due to the presence of the large-scale organized motion. The amplitudes of both the streamwise and the transverse components tend to relax to asymptotic values that are independent of the velocity ratio as well as the shear-layer apparatus. The Strouhal number St, defined as fL/Ua, where f is the local fundamental mode frequency, L is the shearlayer width and Ua, is the average convection velocity of the structures is found to be initially dependent on the local Reynolds number. In the asymptotic regime, the orientation of the large-scale structures is tilted backward toward the higher-speed side in all cases.

The effects of concave curvature on turbulent boundary-layer structure are investigated, using flow visualization and two-component laser-Doppler anemometry. Destabilizing curvature amplifies large-scale motions normal to the wall. When the boundary layer entering the curve is free of spanwise non-uniformities, the resulting large-eddy structure does not consist of distinct longitudinal vortices, as suggested by some previous studies. Rather, the visualized flow is dominated by large eddies (inflows and outflows) that have a streamwise extent of only a few boundary-layer thicknesses, are quite unsteady, and do not cause significant spanwise variations in the mean properties of the boundary layer. Mixing across the boundary layer is enhanced by the new eddy structure, bringing high-momentum fluid closer to the wall than in a normal, flat boundary layer and causing a significant increase in skin friction. Spectral results show that increases in turbulence intensities and Reynolds shear stress across the outer layer are due almost entirely to increased energy in low-frequency, large-scale fluctuations.

Flow visualization suggests that the large-scale inflows and outflows have strong influence on flow structure in the near-wall region. However, when the local value of the friction velocity, uτ, is used for scaling, near-wall profiles of Reynolds-averaged quantities show relatively minor differences between the flat and concave cases.

The response of the boundary layer to the sudden onset of concave curvature is found to involve two overlapping stages. First, a centrifugal mechanism causes higher-velocity eddies near the start of curvature to migrate toward the wall, while lower-velocity eddies migrate away from the wall. These negatively correlated motions produce an increase in the magnitude of the correlation coefficient within a few initial boundary-layer thicknesses (δ0) from the start of curvature. The further development of the layer requires the slower growth and amplification of the large-scale inflows and outflows. This development of the new large-scale eddy structure continues for at least 20 δ0.

Concave curvature has a destabilizing effect on a turbulent boundary layer that causes the formation of large-scale inflow and outflow regions. These structures are larger and more energetic than large eddies in a flat boundary layer, particularly in terms of velocity fluctuations normal to the wall. Flow visualization has suggested that the large-scale inflows and outflows have a strong influence on turbulence structure in the near-wall region. However, near-wall profiles of Reynolds-averaged quantities in the concave boundary layer do not indicate major structural changes. In this paper, the effects of concave curvature on near-wall structure are investigated in two flow cases: (i) the natural concave boundary layer, where the large-scale eddies do not have preferred spanwise locations and the layer remains nearly two-dimensional in the means; and (ii) a case in which vortex generators are used to induce a fixed array of longitudinal roll cells, allowing measurements to be made under stationary inflow and outflow regions. Burst frequencies obtained using an extension of the uv-quadrant method confirm the visual impression that inflows suppress the bursting process, while outflows enhance it. Reynolds-averaged measurements show that turbulence intensity and the uv correlation coefficient are also suppressed under the inflows. Based on these results, a conceptual model for the effects of large-scale eddies on near-wall flow and skin friction in a concave layer is proposed.

Cylinders of various cross-section were subjected to controlled oscillations in a direction transverse to the incident flow. Excitation was at frequency fe, relative to the formation frequency f*0 of large-scale vortices from the corresponding stationary cylinder, and at Reynolds numbers in the range 584 [les ] Re [les ] 1300. Modifications of the near wake were characterized by visualization of the instantaneous flow structure in conjunction with body displacement-flow velocity correlations.

At fe/f*0 = ½, corresponding to subharmonic excitation, as well as at fe/f*0 = 1, the near wake structure is phase-locked (synchronized) to the cylinder motion. However, the synchronization mechanism is distinctly different in these two regimes. Near or at fe/f*0 = 1, the phase of the shed vortex with respect to the cylinder displacement switches by approximately π. Characteristics of this phase switch are related to cylinder geometry. It does not occur if the cylinder has significant afterbody.

Over a wide range of fe/f*0, the perturbed near wake rapidly recovers to a largescale antisymmetrical mode similar in form to the well-known Kármán vortex street. The mechanisms of small-scale (fe) vortex interaction leading to recovery of the large-scale (f0) vortices are highly ordered and repeatable, though distinctly different, for superharmonic excitation (fe/f*0 = n = 2, 3, 4) and non-harmonic excitation (non-integer values of fe/f*0).

The frequency f0 of the recovered vortex street downstream of the body shows substantial departure from the shedding frequency f*0 from the corresponding stationary body. It locks-on to resonant modes corresponding to f0/fe = 1/n. This wake response involves strictly hydrodynamic phenomena. It shows, however, a resonant behaviour analogous to that of coupled flow-acoustic systems where the shear layer is convectively unstable

A circular cylinder subjected to forced oscillations at angle α with respect to the free stream shows a number of admissible modes of vortex formation synchronized with the body motion. These modes can be categorized into two basic groups: symmetrical vortex formation; and antisymmetrical vortex formation. Whereas there is a single symmetrical mode, there are four basic antisymmetrical modes. Three of these antisymmetrical modes show period doubling relative to the classical Kármán mode. This doubling arises from the symmetrical perturbation component induced by the cylinder motion at α ± 90°. Synchronization, i.e. phase-locking of the vortex shedding with the cylinder motion, is possible for all of these modes. It occurs even when streamwise (α = 0°) motion induces an antisymmetrical mode.

When synchronization does not occur, there is competition between the symmetrical and antisymmetrical modes; the near-wake structure successively locks-on to each mode over a defined number of cycles, abruptly switching between modes. The number of occurrences of each mode is a well-defined function of excitation frequency and angle α.

If, in contrast to steady-state motion of the cylinder, there is an initial transient motion, the transition between symmetrical and antisymmetrical modes has a markedly different character, emphasizing the importance of initial conditions. Abrupt onset of sinusoidal motion produces an initially synchronized symmetrical mode, which gradually decays to an antisymmetrical mode. The number of excitation cycles to onset of decay to antisymmetrical mode is highly repeatable. Moreover, the mechanism of decay of the near wake from the symmetrical to antisymmetrical mode can occur deterministically over a defined number of cycles.

An experimental investigation was made of convection in a fluid with a strongly temperature-dependent viscosity. The determination of the critical Rayleigh number, Rc, using the appearance of convection to define onset, was complicated by the occurrence of subcritical instabilities initiated by horizontal temperature gradients at the side boundaries. The increase in Rc over the expected value was less than predicted by linear theory, probably owing to the effect of finite conductivity boundaries and the temperature dependence of other fluid properties.

The stability of various convective planforms was studied as a function of Rayleigh number, wavenumber and viscosity variation using controlled initial conditions to specify the wavenumber and pattern, Rayleigh numbers of up to 63000 and viscosity variations of up to 1000. In addition to the rolls and hexagons seen in constant- and weakly temperature-dependent-viscosity fluid, a new planform of squares was observed at large viscosity variations.

Experiments with viscosity variations of 50 and 1000 showed that hexagons and squares were stable at Rayleigh numbers less than 25000 over a limited range of wavenumbers, which was shifted to higher values with increasing viscosity variation. Temperature profiles through the layer revealed that this shift in wavenumber was associated with the development of a thick, stagnant, cold boundary layer which reduced the effective depth of the layer.

Experiments with a fixed wavenumber showed that rolls were unstable at all Rayleigh numbers for a viscosity contrast greater than 40, whereas squares did not become stable until the viscosity contrast exceeded 6. At low viscosity variations and high Rayleigh numbers rolls became unstable to a bimodal pattern, but at high viscosity variations and a Rayleigh number of 25000 squares broke down into the spoke pattern, a convective flow not observed until Rayleigh numbers of around 100000 in a constant-viscosity fluid.

There is a close relationship between the symmetry changes in the plane and space groups of convective systems and the type of the bifurcation. To explore this relationship the plane- and space-group symmetry of many convective circulations is first classified using standard crystallographic notation. The transitions that occur between these patterns can be described by the loss of either a translational or a point-group symmetry element. These are referred to as klassengleiche and translationengleiche or k and t, transitions respectively. Any transition can be decomposed into a series of k and t transitions. The symmetry of the governing differential equations is most easily discussed when these are written in terms of potentials, and allows transitions to be classified as pitchfork, transcritical or Hopf bifurcations. Such classification can be carried out from symmetry alone, without any consideration of the functional form of the solutions, the Rayleigh number or the importance of the nonlinear terms. For this purpose it is convenient to define a factor group, the irreducible representations of which transform as do the variables and differential operators in the equations.

Analysis of planform transitions observed in convective flows when the viscosity is temperature dependent using plane groups shows that all except the transition from the conductive solution to hexagons are pitchfork bifurcations. The factor groups involved are the cyclic group Z2, and the dihedral groups D3 and D5. When the viscosity is constant, space groups are needed, and symmetry arguments show that all except the Eckhaus instability are pitchfork bifurcations, including that from the conductive solution. Hopf transitions to solutions periodic in time, in double-diffusive, in small- and in large-Prandtl-number convection, all involve loss of reflection symmetry in time and the factor group is D2. The same approach suggests how transitions to circulations that are not periodic in either space or time may occur by period doubling in space or in time or in both.

A weakly nonlinear Hamiltonian model is derived from the exact water wave equations to study the time evolution of spatially periodic wavetrains. The model assumes that the spatial spectrum of the wavetrain is formed by only three free waves, i.e. a carrier and two side bands. The model has the same symmetries and invariances as the exact equations. As a result, it is found that not only the permanent form travelling waves and their stability are important in describing the time evolution of the waves, but also a new kind of family of solutions which has two basic frequencies plays a crucial role in the dynamics of the waves. It is also shown that three is the minimum number of free waves which is necessary to have chaotic behaviour of water waves.

The barotropic potential vorticity equation or topographic wave equation is not linear in topography: the solution structure for topography formed from a sum of obstacles is not the sum of solutions for the obstacles in isolation, even when these individual solutions have identical frequencies. This paper considers the mechanism by which normal modes of oscillation above one mountain are modified by interactions with its neighbours. Exact explicit solutions for the normal modes above a pair of circular seamountains show that the interactions between the mountains rapidly approaches the large-separation approximation obtained by considering solely the first reflection of the disturbance of one mountain at the other. For mountains of one diameter separation at the closest point, the approximation is accurate to within 1%. Perhaps surprisingly, coupling between two identical mountains is weak and resonance occurs between mountains and dales of equal and opposite height.

The accurate approximate solutions enable consideration of the effects on a mountain of an infinite set of randomly distributed neighbours. The ensembleaveraged frequency for a mountain of given height is determined in terms of the area fraction of the other mountains. The idea of an effective topography is introduced for the ensemble-averaged stream function: it is that (non-random) topography generating a stream function identical to the ensemble-averaged stream function. This differs markedly from the ensemble-averaged topography. The explicit form of the effective topography is derived for a set of right circular cylinders.

Parametric studies have been made of devices introduced into the outer region of a low-Reynolds-number turbulent boundary layer (Reθ = 1000-3500) with a view to understanding the manner in which such ‘manipulators’ reduce the surface drag. The devices considered were single flat plates, a cylinder with the same drag as one of these, and two plates stacked, staggered, and in tandem, with chord Reynolds numbers Rec in the range 1000-100000. Direct measurements of local skin friction using a floating-plate drag balance are reported together with the results of laser- sheet smoke flow visualization. The skin-friction results are in good agreement with other floating-element data while the visual and photographic studies in both stationary and convected frames completely support the hairpin description of the boundary-layer structure, and reveal that the wake of the device may play a more important role than has previously been suggested. A picture is presented of the interaction of the devices and their wakes with the hairpin eddy structure which could explain the magnitudes and shapes of the skin-friction distributions observed downstream; their dependence on the height, thickness, and if appropriate, length and spacing of the device(s); the optimum values for these parameters; and the existence of a preferred, tandem plate configuration. The results suggest that such plates do not act primarily as large-eddy break-up systems (LEBUs). Instead several ‘active’ mechanisms are identified which supplement the ‘passive’ effect of the imposed momentum defect. The suppression of large-scale motions is one of these, but, a t least for our relatively thick devices, it would appear that it is an interaction between the vortices, introduced into the layer via the wake, and the near-wall structure that provides the principal mechanism for reducing the skin friction. The observation that the maximum skin-friction reduction always occurred close to the position where these vortices reached the sublayer provides strong evidence for such a view.

This paper describes the dissipative phase of internal-wave run-up on uniform slopes of 0.030 and 0.054 rad as observed in a series of laboratory experiments. The waves were generated continuously at the interface of two miscible layers of differing density. As each wave in the perodic train propagated onto the slope, it steepened and developed into a solitary-like wave before finally overturning. Surrounding fluid was engulfed into the wave as it overturned and the resulting gravitational instability produced considerable turbulence and mixing. The broken wave took the form of a discrete bolus of dense fluid which propagated for some distance up the slope. Bulk parameters which characterize the nature of the bolus were defined and the dependence of these on the incident wave parameters and their behaviour during the run-up phase were examined.

The effect of surface curvature on the low-wavenumber-frequency spectrum of turbulence-induced surface pressure fluctuations is considered and an estimate for the associated flow noise is obtained. The form of the low-wavenumber cross-spectral density of pressure on the surface of an infinitely long, rigid circular cylinder of radius a due to a statistically stationary turbulent boundary-layer flow at a low Mach number is determined. Viscous effects are ignored. It is shown that, in contrast to the case of an infinite plane surface, the pressure spectrum is finite everywhere in the wavenumber plane (k, n/a) except for a logarithmic, integrable, singularity at the acoustic wavenumber corresponding to the axisymmetric mode (n/a = 0); k and n/a being, respectively, the downstream and circumferential wavenumber. For non-axisymmetric modes (|n/a| > 0), the spectrum has two finite peaks in the radiative domain |k| < ω/c; ω being the frequency and c being the sound speed. For ωa/c large, the peaks occur in the vicinity of the total acoustic surface wavenumbers κ = ± ω/c (κ = (k2 + n2 / a2)½) and the principal contribution which determines the peak characteristics can be identified as being due to creeping rays emanating from turbulent sources on the cylinder out of the line of sight of the associated receiver point. For large value of ωa/c, the point pressure spectrum and the associated radiated sound vary logarithmically with $(\omega a/c)^{\frac{2}{3}}$; corresponding estimates for cylinders of moderate and small radius are also obtained. For an almost plane cylinder, it is shown that the effect of curvature may be included by a suitable simple modification of the form of the pressure spectrum for an infinite plane surface.

In a previous study using finite-amplitude techniques (Maslowe 1977), a strong instability mechanism was discovered that takes effect at the Richardson numbers consistent with turbulence observations in the atmosphere and oceans. The mechanism involves second harmonic resonance of two neutral or nearly neutral modes at a Richardson number of roughly 0.22. In the present investigation, the nonlinear Boussinesq equations have been solved numerically to further explore this instability and to assess the limits of validity of the theory. Qualitative agreement between the theory and numerical simulations was satisfactory as the most significant numerical results were predicted by the theory. In particular, the wave interaction leads to impressive instabilities at Richardson numbers large enough that a single linearly unstable wave would amplify only weakly. At a Richardson number of 0.14, for example, the saturation amplitude of the long wave in the two-wave interacting case was 15 times as large as the amplitude of the linearly most unstable wave (evolving by itself) at the same Richardson number.

We consider the effects of strong critical-layer nonlinearity on the spatially growing instabilities of a shear layer between two parallel streams. A composite expansion technique is used to obtain a single formula that accounts for both shear-layer spreading and nonlinear critical-layer effects. Nonlinearity causes the instability to saturate well upstream of the linear neutral stability point. It also produces vorticity roll-up that cannot be predicted by linear theory.

The centrifugal separation of an initially homogeneous mixture of particles and fluid is considered in ‘long’ containers for which end-wall effects are negligible. The vorticity produced in the mixture region during separation is shown to be a function of time only. As a consequence, the ‘full’ nonlinear theory in such containers, of arbitrary cross-section, can be reduced to the determination of an appropriate analytic function. A problem of technological interest is discussed.

Coupled nonlinear equations are derived for the amplitudes of two small-amplitude resonantly interacting gravity waves in water of non-uniform depth. Such resonance is possible only for wavelengths long compared to the depth. It is shown that the same equations are obtained from the exact Euler equations, from the nonlinear shallow water theory, and from the Boussinesq equations.