The scattering of surface gravity waves by a circular dock is considered in order to determine the horizontal and vertical forces and torque on the dock. An incident plane wave is expanded in Bessel functions, and for each mode the problem is formulated in terms of the potential on the cylindrical surface containing the dock and extending to the bottom. The solution is shown to have phase independent of depth and so may be obtained from an infinite set of real equations, which are solved numerically by Galerkin's method. The convergence of the solution is discussed, and some numerical results are presented.

This problem has been investigated previously by Miles & Gilbert (1968) by a different method, but their work contained errors.

The rate of mixing across a density interface between two layers of liquid has been measured in a laboratory experiment which allows a direct comparison between heat and salinity transports over the same range of density differences. Low Reynolds number turbulence was produced by stirring mechanically at a fixed distance from the interface, either in one or in both layers, and the results for these two sets of experiments are also compared. The measurements cover a factor of two in stirring rate and twenty in density. Over this range of conditions the ratio of entrainment velocity to stirring velocity can be expressed as functions of an overall Richardson number Ri, and in this form the results of the one and two stirred layer experiments are indistinguishable from one another. For density differences produced by heat alone, the functional dependence is close to Ri−1 except at small values of Ri where it approaches a finite limit. For experiments with a salinity difference across the interface, the mixing rate is the same as in the heat experiments at low values of Ri, but falls progressively below this as Ri is increased, with the approximate form $Ri^{\frac{3}{2}} $.

An interpretation of these results has been attempted, using a dimensional analysis and qualitative mechanistic arguments about the nature of the motion. The Ri−1 dependence implies a rate of change of potential energy proportional to the rate of working by the stirrer. The decreased mixing rates for salt have been attributed to a slower rate of incorporation of an entrained element into its surroundings by diffusion, which increases the tendency for it to return to the interface and dissipate energy in wave-like motions.

In order to study the roles of stress anisotropy and of elasticity in the mechanism of drag reduction by polymer additives we investigate a turbulent pipe flow of a dilute polymer solution. The investigation is carried out by means of direct numerical simulation (DNS) and laser Doppler velocimetry (LDV). In our DNS two different models are used to describe the effects of polymers on the flow. The first is a constitutive equation based on Batchelor's theory of elongated particles suspended in a Newtonian solvent which models the viscous anisotropic effects caused by the polymer orientation. The second is an extension of the first model with an elastic component, and can be interpreted as an anisotropic Maxwell model. The LDV experiments have been carried out in a recirculating pipe flow facility in which we have used a solution of water and 20 w.p.p.m. Superfloc A110. Turbulence statistics up to the fourth moment, as well as power spectra of various velocity components, have been measured. The results of the drag-reduced flow are first compared with those of a standard turbulent pipe flow of water at the same friction velocity at a Reynolds number of Reτ≈1035. Next the results of the numerical simulation and of the measurements are compared in order to elucidate the role of polymers in the phenomenon of drag reduction. For the case of the viscous anisotropic polymer model, almost all turbulence statistics and power spectra calculated agree in a qualitative sense with the measurements. The addition of elastic effects, on the other hand, has an adverse effect on the drag reduction, i.e. the viscoelastic polymer model shows less drag reduction than the anisotropic model without elasticity. Moreover, for the case of the viscoelastic model not all turbulence statistics show the right behaviour. On the basis of these results, we propose that the viscous anisotropic stresses introduced by extended polymers play a key role in the mechanism of drag reduction by polymer additives.

This paper is concerned with the theoretical behaviour of the boundary-layer flow over a disk rotating in otherwise still fluid. The flow is excited impulsively at a certain radius at time t = 0. This paper analyses the inviscid stability of the flow and the stability with viscous, Coriolis and streamline curvature effects included. In both cases, within a specific range of the parameter space, it is shown that the flow is absolutely unstable, i.e. disturbances grow in time at every fixed point in space. Outside this range, the flow is convectively unstable or stable. The absolute or convective nature of the instabilities is determined by examining the branch-point singularities of the dispersion relation. Absolute instability is found for Reynolds numbers above 510. Experimentally observed values for the onset of transition from laminar to turbulent flow have an average value of 513. It is suggested that absolute instability may cause the onset of transition to turbulent flow. The results from the inviscid analysis show that the absolute instability is not caused by Coriolis effects nor by streamline curvature effects. This indicates that this mechanism may be possible on swept wings, where Coriolis effects are not present but the boundary layers are otherwise similar.

The large-scale structures that occur in a forced turbulent mixing layer at moderately high Reynolds numbers have been modelled by linear inviscid stability theory incorporating first-order corrections for slow spatial variations of the mean flow. The perturbation stream function for a spatially growing time-periodic travelling wave has been numerically evaluated for the measured linearly diverging mean flow. In an accompanying experiment periodic oscillations were imposed on the turbulent mixing layer by the motion of a small flap at the trailing edge of the splitter plate that separated the two uniform streams of different velocity. The results of the numerical computations are compared with experimental measurements.

When the comparison between experimental data and the computational model was made on a purely local basis, agreement in both the amplitude and phase distribution across the mixing layer was excellent. Comparisons on a global scale revealed, not unexpectedly, less good accuracy in predicting the overall amplification.

Published results on transition in a Stokes layer indicate a wide range of transition Reynolds numbers. As thermal effects in a resonance tube (Merkli & Thomann 1975) depend on the state of the boundary layer, the transition Reynolds number was determined, and a critical Reynolds number Ac ≈ 400 was found. The observations were made with hot wires and with flow visualization by means of smoke, and provide new details on turbulence in a Stokes layer. With this knowledge an explanation of the large discrepancies between some stability theories and the experiments is suggested. The main point is that turbulence occurs in the form of periodic bursts which are followed by relaminarimtion in the same cycle and do not lead to turbulent flow during the whole cycle.

A further, unexpected result of the present investigation is the discovery of vortex patterns superimposed on the normal laminar acoustic motion.

Three-dimensional homogeneous isotropic turbulence at very high Reynolds number R is studied using a variant of the Markovian eddy-damped quasi-normal theory. In the case without helicity, numerical calculations indicate the development of a $k^{-\frac{5}{3}}$ inertial range in the energy spectrum and an onset of significant energy dissipation at a time t* which appears to be independent of the viscosity v as v → 0; analytical arguments having a bearing on this behaviour, described as an ‘energy catastrophe’, are also discussed. The skewness factor (for t > t*), which increases with R, tends to 0·495 when R → ∞. When helicity is present, the existence of simultaneous energy and helicity cascades is demonstrated numerically. It is also shown that the helicity cascade inhibits the energy transfer towards large wavenumbers, in agreement with preliminary low Reynolds number results of Herring and with the conclusion of Kraichnan (1973) based on analysis of the interaction between two helicity waves. This inhibition implies a delay of the onset of energy dissipation at zero viscosity. It is shown that, whatever the relative rate of helicity and energy injection, a regime is attained at large wavenumbers k where the relative helicity tends to zero (with increasing k) and helicity is carried along locally and linearly by the energy cascade like a passive scalar. In practice, the linear regime is attained when the relative helicity is less than about 10%. The Kolmogorov constants of energy and helicity in the inertial range are determined. The impossibility of pure helicity cascades of a type conjectured by Brissaud et al. (1973a) is demonstrated. Finally it is shown that, because of dissipation and non-positive-definiteness of the helicity spectrum, non-zero total helicity may appear in the decay of unforced turbulence with zero total initial helicity, if the helicity spectrum is not initially identically zero.

Lubrication theory and similarity methods are used to determine the spreading rate of a localized monolayer of insoluble surfactant on the surface of a thin viscous film, in the limit of weak capillarity and weak surface diffusion. If the total mass of surfactant increases as tα, then at early times, when spreading is driven predominantly by Marangoni forces, a planar (axisymmetric) region of surfactant is shown to spread as t(1 + α)/3 (t(1 + α)/4). A shock exists at the leading edge of the monolayer; asymptotic methods are used to show that a wavetrain due to capillary forces exists ahead of the shock at small times, but that after a finite time it is swamped by diffusive effects. For α < ½ (α < 1), the diffusive lengthscale at the shock grows faster than the length of the monolayer, ultimately destroying the shock; subsequently, spreading is driven by diffusion, and proceeds as t½. The asymptotic results are shown to be good approximations of numerical solutions of the governing partial differential equations in the appropriate limits. Additional forces are also considered: weak vertical gravity can also destroy the shock in finite time, while effects usually neglected from lubrication theory are important only early in spreading. Experiments have shown that the severe thinning of the film behind the shock can cause it to rupture: the dryout process is modelled by introducing van der Waals forces.

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss & Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core.

The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.

Using a variety of flow-visualization techniques, the flow behind a circular cylinder has been studied. The results obtained have provided a new insight into the vortex-shedding process. Using time-exposure photography of the motion of aluminium particles, a sequence of instantaneous streamline patterns of the flow behind a cylinder has been obtained. These streamline patterns show that during the starting flow the cavity behind the cylinder is closed. However, once the vortex-shedding process begins, this so-called ‘closed’ cavity becomes open, and instantaneous ‘alleyways’ of fluid are formed which penetrate the cavity. In addition, dye experiments also show how layers of dye and hence vorticity are convected into the cavity behind the cylinder, and how they are eventually squeezed out.

We employ a three-dimensional, nonlinear inviscid numerical method, in conjunction with experimental data from live fish and from a fish-like robotic mechanism, to establish the three-dimensional features of the flow around a fish-like body swimming in a straight line, and to identify the principal mechanisms of vorticity control employed in fish-like swimming. The computations contain no structural model for the fish and hence no recoil correction. First, we show the near-body flow structure produced by the travelling-wave undulations of the bodies of a tuna and a giant danio. As revealed in cross-sectional planes, for tuna the flow contains dominant features resembling the flow around a two-dimensional oscillating plate over most of the length of the fish body. For the giant danio, on the other hand, a mixed longitudinal–transverse structure appears along the hind part of the body. We also investigate the interaction of the body-generated vortices with the oscillating caudal fin and with tail-generated vorticity. Two distinct vorticity interaction modes are identified: the first mode results in high thrust and is generated by constructive pairing of body-generated vorticity with same-sign tail-generated vorticity, resulting in the formation of a strong thrust wake; the second corresponds to high propulsive efficiency and is generated by destructive pairing of body-generated vorticity with opposite-sign tail-generated vorticity, resulting in the formation of a weak thrust wake.

We investigate the stability and dynamics of three-dimensional limit-cycle states in flow past a circular cylinder using low-dimensional modelling. High-resolution direct numerical simulations are employed to obtain flow snapshots from which the most energetic modes are extracted using proper orthogonal decomposition. We show that the limit cycle is reproduced very accurately with only twenty three-dimensional modes. The addition of two-dimensional modes to the Karhunen–Loeve expansion basis improves the ability of the model to capture the three-dimensional bifurcation, including the discontinuity in the Strouhal number discovered experimentally.

Results of calculations and experiments on the flow of a viscous liquid through an axisymmetric conduit expansion are reported. The streamlines and vorticity contours are presented as functions of the Reynolds number of the flow. The dynamic interaction between the main flow and the captive eddy between it and the walls is analysed, and it is concluded that, for laminar flow, the main role of the eddy is that of shaping the flow with a rather small energy exchange.

Optimal and robust control theories are used to determine effective, estimator-based feedback control rules for laminar plane channel flows that effectively stabilize linearly unstable flow perturbations at Re=10 000 and linearly stable flow perturbations, characterized by mechanisms for very large disturbance amplification, at Re=5000. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control, and the flow measurement is derived from the wall skin friction. The control objective, beyond simply stabilizing any unstable eigenvalues (which is relatively easy to accomplish), is to minimize the energy of the flow perturbations created by external disturbance forcing. This is important because, when mechanisms for large disturbance amplification are present, small-amplitude external disturbance forcing may excite flow perturbations with sufficiently large amplitude to induce nonlinear flow instability.

The control algorithms used in the present work account for system disturbances and measurement noise in a rigorous fashion by application of modern linear control techniques to the discretized linear stability problem. The disturbances are accounted for both as uncorrelated white Gaussian processes ([Hscr ]2 or ‘optimal’ control) and as finite ‘worst case’ inputs which are maximally detrimental to the control objective ([Hscr ]∞ or ‘robust’ control). Root loci and transient energy growth analyses are shown to be inadequate measures to characterize overall system performance. Instead, appropriately defined transfer function norms are used to characterize all systems considered in a consistent and relevant manner. In order to make a parametric study tractable in this high-dimensional system, a convenient new scaling to the estimation problem is introduced such that three scalar parameters {γ, α, [lscr ]} may be individually adjusted to achieve desired closed-loop characteristics of the resulting systems. These scalar parameters may be intuitively explained, and are defined such that the resulting control equations retain the natural dual structure between the control parameter, [lscr ], and the estimation parameter, α. The performance of the present systems with respect to these parameters is thoroughly investigated, and comparisons are made to simple proportional schemes where appropriate.

The problem of natural convection in a cavity of small aspect ratio with differentially heated end walls is considered. It is shown by use of matched asymptotic expansions that the flow consists of two distinct regimes: a parallel flow in the core region and a second, non-parallel flow near the ends of the cavity. A solution valid at all orders in the aspect ratio A is found for the core region, while the first several terms of the appropriate asymptotic expansion are obtained for the end regions. Parametric limits of validity for the parallel flow structure are discussed. Asymptotic expressions for the Nusselt number and the single free parameter of the parallel flow solution, valid in the limit as A → 0, are derived.

The rate of heat and mass transfer at the surface of acoustically levitated pure liquid droplets is predicted theoretically for the case where an acoustic boundary layer appears near the droplet surface resulting in an acoustic streaming. The theory is based on the computation of the acoustic field and squeezed droplet shape by means of the boundary element method developed in Yarin, Pfaffenlehner & Tropea (1998). Given the acoustic field around the levitated droplet, the acoustic streaming near the droplet surface was calculated. This allowed calculation of the Sherwood and Nusselt number distributions over the droplet surface, as well as their average values. Then, the mass balance was used to calculate the evolution of the equivalent droplet radius in time.

The theory is applicable to droplets of arbitrary size relative to the sound wavelength λ, including those of the order of λ, when the compressible character of the gas flow is important. Also, the deformation of the droplets by the acoustic field is accounted for, as well as a displacement of the droplet centre from the pressure node. The effect of the internal circulation of liquid in the droplet sustained by the acoustic streaming in the gas is estimated. The distribution of the time-average heat and mass transfer rate over the droplet surface is found to have a maximum at the droplet equator and minima at its poles. The time and surface average of the Sherwood number was shown to be described by the expression Sh = KB/√ω[Dscr ]0, where B = A0e/(ρ0c0) is a scale of the velocity in the sound wave (A0e is the amplitude of the incident sound wave, ρ0 is the unperturbed air density, c0 is the sound velocity in air, ω is the angular frequency in the ultrasonic range, [Dscr ]0 is the mass diffusion coefficient of liquid vapour in air, which should be replaced by the thermal diffusivity of air in the computation of the Nusselt number). The coefficient K depends on the governing parameters (the acoustic field, the liquid properties), as well as on the current equivalent droplet radius a.

For small spherical droplets with a[Lt ]λ, K = (45/4π)1/2 = 1.89, if A0e is found from the sound pressure level (SPL) defined using A0e. On the other hand, if A0e is found from the same value of the SPL, but defined using the root-mean-square pressure amplitude (prms = A0e/√2), then Sh = KrmsBrms/ √ω[Dscr ]0, with Brms = √2B and Krms = K/√2 = 1.336. For large droplets squeezed significantly by the acoustic field, K appears always to be greater than 1.89. The evolution of an evaporating droplet in time is predicted and compared with the present experiments and existing data from the literature. The agreement is found to be rather good.

We also study and discuss the effect of an additional blowing (a gas jet impinging on a droplet) on the evaporation rate, as well as the enrichment of gas at the outer boundary of the acoustic bondary layer by liquid vapour. We show that, even at relatively high rates of blowing, the droplet evaporation is still governed by the acoustic streaming in the relatively strong acoustic fields we use. This makes it impossible to study forced convective heat and mass transfer under the present conditions using droplets levitated in strong acoustic fields.

Conditions leading to the onset of air-flow separation over a mobile air-water interface are discussed. It is argued that, in a frame of reference in which the interfacial boundary assumes a steady shape, the occurrence of separation requires a stagnation point on the interface. In the case of air blowing over water waves, this corresponds to the onset of wave breaking. These arguments are strongly supported by flow visualization and pressure measurements carried out in a laboratory wind-wave flume. Furthermore, the pressure measurements show a greatly enhanced interfacial shear stress for a breaking wave compared with that over an unbroken wave of the same wavelength. The implications of these findings for wind-wave generation are discussed.

The motion of instantaneous and maintained releases of buoyant fluid through shallow permeable layers of large horizontal extent is described by a nonlinear advection–diffusion equation. This equation admits similarity solutions which describe the release of one fluid into a horizontal porous layer initially saturated with a second immiscible fluid of different density. Asymptotically, a finite volume of fluid spreads as t1/3. On an inclined surface, in a layer of uniform permeability, a finite volume of fluid propagates steadily alongslope under gravity, and spreads diffusively owing to the gravitational acceleration normal to the boundary, as on a horizontal boundary. However, if the permeability varies in this cross-slope direction, then, in the moving frame, the spreading of the current eventually becomes dominated by the variation in speed with depth, and the current length increases as t1/2. Shocks develop either at the leading or trailing edge of the flows depending upon whether the permeability increases or decreases away from the sloping boundary. Finally we consider the transient and steady exchange of fluids of different densities between reservoirs connected by a shallow long porous channel. Similarity solutions in a steadily migrating frame describe the initial stages of the exchange process. In the final steady state, there is a continuum of possible solutions, which may include flow in either one or both layers of fluid. The maximal exchange flow between the reservoirs involves motion in one layer only. We confirm some of our analysis with analogue laboratory experiments using a Hele-Shaw cell.

In an attempt to explain the mechanics of turbulence suppression previously observed by us in jets under controlled excitation, the near fields of four circular jets, a plane jet and a plane mixing layer, all with laminar efflux boundary layers, have been explored through hot-wire measurements. It is shown that controlled excitation, induced acoustically as well as by vibrating ribbons, can reduce turbulence intensities in all these flows. Reduction by as much as 80% is observed, the maximum decrease occurring at about 400θe downstream from the exit; θe is the initial shear-layer momentum thickness. The suppression effect is a maximum for excitation at the Strouhal number Stθ(≡ fθe/Ue) of about 0.017. In the jets, the turbulence suppression is observed over the range 0·75 ≤ x/D ≤ 8, while for the plane mixing layer it could be detected as far downstream as x ≅ 6000θe.

The flow-fields with and without excitation for a typical case of turbulence suppression have been studied in detail. Spectra of the u signal and time-averaged field data indicate that excitation at Stθ ≅ 0·017 suppresses the formation of naturally occurring energetic vortices - an observation confirmed by flow-visualization experiments and by study of the large-scale coherent structures of the shear layer, educed through conditional-sampling measurements. Excitation at Stθ ≅ 0·017 produces a rapid growth of the shear layer instability mode, and consequently, its saturation, roll-up and breakdown occurs much earlier in x than is found to occur naturally (at Stθ ≅; 0·012). The suppression effect is apparently a consequence of earlier transition of the shear layer vortices, which otherwise naturally grow to larger sizes and survive for larger x, as well as being due to the prevention of successive pairing of these structures.

The phenomenon of bursting, in which streaks in turbulent boundary layers oscillate and then eject low-speed fluid away from the wall, has been studied experimentally, theoretically and computationally for more than 50 years because of its importance to the three-dimensional structure of turbulent boundary layers. Five new three-dimensional solutions of turbulent plane Couette flow are produced, one of which is periodic while the other four are relative periodic. Each of these five solutions demonstrates the breakup and re-formation of near-wall coherent structures. Four of our solutions are periodic, but with drifts in the streamwise direction. More surprisingly, two of our solutions are periodic, but with drifts in the spanwise direction, a possibility that does not seem to have been considered in the literature. It is argued that a considerable part of the streakiness observed experimentally in the near-wall region could be due to spanwise drifts that accompany the breakup and re-formation of coherent structures. A new periodic solution of plane Couette flow is also computed that could be related to transition to turbulence.

The violent nature of the bursting phenomenon implies the need for good resolution in the computation of periodic and relative periodic solutions within turbulent shear flows. This computationally demanding requirement is addressed with a new algorithm for computing relative periodic solutions one of whose features is a combination of two well-known ideas – namely the Newton–Krylov iteration and the locally constrained optimal hook step. Each of the six solutions is accompanied by an error estimate.

Dynamical principles are discussed that suggest that the bursting phenomenon, and more generally fluid turbulence, can be understood in terms of periodic and relative periodic solutions of the Navier–Stokes equation.