Motivated by our interest in unsteady aerodynamics of insect flight, we devise a computational tool to solve the Navier–Stokes equation around a two-dimensional moving wing, which mimics biological locomotion. The focus of the present work is frequency selection in forward flapping flight. We investigate the time scales associated with the shedding of the trailing- and leading-edge vortices, as well as the corresponding time-dependent forces. We present a generic mechanism of the frequency selection as a result of unsteady aerodynamics.

Existing data on the condensation of steam and moist air in supersonic nozzles are compared with predictions based on nucleation and drop-growth theory. It is concluded that, if the surface tension is assumed independent of curvature, and the classical liquid-drop theory (based on a stationary liquid drop) is used, the theory is in general agreement with the data. The effects of uncertainties in cluster surface energy and also of the large corrections to nucleation theory due to the ‘gasification’ concept are examined. The gasification correction is in accord with experimental data only if the surface tension is considered to rise significantly with curvature. In neither case can the Tolman or Kirkwood–Buff equations be supported. A review of existing data shows that there is some question as to the appropriate value of the condensation coefficient but this is of little consequence as long as the accommodation coefficient for the liquid–vapour surface is taken to be unity. The usefulness of the nozzle experiments for testing the validity of nucleation theory is demonstrated.

A model is developed for the adjustment of the spatially averaged time-mean flow of a deep turbulent boundary layer over small roughness elements to a canopy of larger three-dimensional roughness elements. Scaling arguments identify three stages of the adjustment. First, the drag and the finite volumes of the canopy elements decelerate air parcels; the associated pressure gradient decelerates the flow within an impact region upwind of the canopy. Secondly, within an adjustment region of length of order $L_c$ downwind of the leading edge of the canopy, the flow within the canopy decelerates substantially until it comes into a local balance between downward transport of momentum by turbulent stresses and removal of momentum by the drag of the canopy elements. The adjustment length, $L_c$, is proportional to (i) the reciprocal of the roughness density (defined to be the frontal area of canopy elements per unit floor area) and (ii) the drag coefficient of individual canopy elements. Further downstream, within a roughness-change region, the canopy is shown to affect the flow above as if it were a change in roughness length, leading to the development of an internal boundary layer. A quantitative model for the adjustment of the flow is developed by calculating analytically small perturbations to a logarithmic turbulent velocity profile induced by the drag due to a sparse canopy with $L/L_c \ll 1$, where $L$ is the length of the canopy. These linearized solutions are then evaluated numerically with a nonlinear correction to account for the drag varying with the velocity. A further correction is derived to account for the finite volume of the canopy elements. The calculations are shown to agree with experimental measurements in a fine-scale vegetation canopy, when the drag is more important than the finite volume effects, and a canopy of coarse-scale cuboids, when the finite volume effects are of comparable importance to the drag in the impact region. An expression is derived showing how the effective roughness length of the canopy, $\z0eff$, is related to the drag in the canopy. The value of $\z0eff$ varies smoothly with fetch through the adjustment region from the roughness length of the upstream surface to the equilibrium roughness length of the canopy. Hence, the analysis shows how to resolve the unphysical flow singularities obtained with previous models of flow over sudden changes in surface roughness.

Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/non-turbulent interface a self-similar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constant-property surfaces (such as the iso-velocity and iso-concentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence - for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).

This paper deals with an experimental and numerical study of the dynamics of an underwater explosion and its associated fluid–structure interaction. Experimental studies of the complex fluid–structure interaction phenomena were carried out in a specially designed test pond. The pond is equipped with a high-speed camera and pressure and displacement sensors. The high-speed camera was used to capture the expansion and collapse of the gas bubble created by the explosion. Several different structures were used in the experiments, including both rigid and resilient plates of circular shape. The deformation of the plate was measured with a non-contact laser telemetry device. The numerical simulations of the explosion bubble interacting with a submerged resilient structure were performed using a three-dimensional bubble dynamics code in conjunction with a structural code. The bubble code is based on the boundary-element method (BEM) and has been coupled to a structural finite-element code (PAM-CRASH$^{\rm TM})$. The experimental results were compared against the numerical results for different bubble–structure configurations and orientations. Several physical phenomena that have been observed, such as bubble jetting and bubble migration towards the structure are discussed.

We consider the flapping stability and response of a thin two-dimensional flag of high extensional rigidity and low bending rigidity. The three relevant non-dimensional parameters governing the problem are the structure-to-fluid mass ratio, μ = ρsh/(ρfL); the Reynolds number, Rey = VL/ν; and the non-dimensional bending rigidity, KB = EI/(ρfV2L3). The soft cloth of a flag is represented by very low bending rigidity and the subsequent dominance of flow-induced tension as the main structural restoring force. We first perform linear analysis to help understand the relevant mechanisms of the problem and guide the computational investigation. To study the nonlinear stability and response, we develop a fluid–structure direct simulation (FSDS) capability, coupling a direct numerical simulation of the Navier–Stokes equations to a solver for thin-membrane dynamics of arbitrarily large motion. With the flow grid fitted to the structural boundary, external forcing to the structure is calculated from the boundary fluid dynamics. Using a systematic series of FSDS runs, we pursue a detailed analysis of the response as a function of mass ratio for the case of very low bending rigidity (KB = 10−4) and relatively high Reynolds number (Rey = 103). We discover three distinct regimes of response as a function of mass ratio μ: (I) a small μ regime of fixed-point stability; (II) an intermediate μ regime of period-one limit-cycle flapping with amplitude increasing with increasing μ; and (III) a large μ regime of chaotic flapping. Parametric stability dependencies predicted by the linear analysis are confirmed by the nonlinear FSDS, and hysteresis in stability is explained with a nonlinear softening spring model. The chaotic flapping response shows up as a breaking of the limit cycle by inclusion of the 3/2 superharmonic. This occurs as the increased flapping amplitude yields a flapping Strouhal number (St = 2Af/V) in the neighbourhood of the natural vortex wake Strouhal number, St ≃ 0.2. The limit-cycle von Kármán vortex wake transitions in chaos to a wake with clusters of higher intensity vortices. For the largest mass ratios, strong vortex pairs are distributed away from the wake centreline during intermittent violent snapping events, characterized by rapid changes in tension and dynamic buckling.

We study the deformation and conditions for breakup of a liquid drop of viscosity λμ freely suspended in another liquid of viscosity μ with which it is immiscible and which is being sheared. The problem at zero Reynolds number is formulated exactly as an integral equation for the unknown surface velocity, which is shown to reduce to a particularly simple form when Δ = 1. This equation is then solved numerically, for the case in which the impressed shear is a radially symmetric extensional flow, by an improved version of the technique used, for Δ = 0, by Youngren & Acrivos (1976) so that we model the time-dependent distortion of an initially spherical drop. It is shown that, for a given Δ, a steady shape is attained only if the dimensionless group Ω ≡4πGμa/γ lies below a critical value Ωc(Δ), where G refers to the strength of the shear field, a is the radius of the initial spherical drop and γ is the interfacial tension. On the other hand, when Ω > Ωc the drop extends indefinitely along its long axis. The numerical results for Δ = 0·3, 0·5, 1, 2, 10 and 100 are in good agreement with the predictions of the small deformation analysis by Taylor (1932) and Barthès-Biesel & Acrivos (1973) and, at the smaller Δ, with those of slender-body theory (Taylor 1964; Acrivos & Lo 1978).

We investigate the dynamics of the near wake in turbulent flow past a circular cylinder up to ten cylinder diameters downstream. The very near wake (up to three diameters) is dominated by the shear layer dynamics and is very sensitive to disturbances and cylinder aspect ratio. We perform systematic spectral direct (DNS) and large-eddy simulations (LES) at Reynolds number (Re) between 500 and 5000 with resolution ranging from 200 000 to 100 000 000 degrees of freedom. In this paper, we analyse in detail results at Re = 3900 and compare them to several sets of experiments. Two converged states emerge that correspond to a U-shape and a V-shape mean velocity profile at about one diameter behind the cylinder. This finding is consistent with the experimental data and other published LES. Farther downstream, the flow is dominated by the vortex shedding dynamics and is not as sensitive to the aforementioned factors. We also examine the development of a turbulent state and the inertial subrange of the corresponding energy spectrum in the near wake. We find that an inertial range exists that spans more than half a decade of wavenumber, in agreement with the experimental results. In contrast, very low-resolution spectral simulation as well as other dissipative LES do not describe accurately the inertial range although they predict low-order statistics relatively accurately. This finding is analysed in the context of coherent structures using a phase averaging technique and a procedure to extract the most energetic (on the average) eigenmodes of the flow. The results suggest that a dynamical model would require of the order of twenty modes to describe the vortex shedding dynamics with reasonable accuracy.

The interaction of a laser-induced cavitation bubble with an elastic boundary and its dependence on the distance between bubble and boundary are investigated experimentally. The elastic boundary consists of a transparent polyacrylamide (PAA) gel with 80% water concentration with elastic modulus E = 0.25 MPa. At this E-value, the deformation and rebound of the boundary is very pronounced providing particularly interesting features of bubble dynamics. It is shown by means of high-speed photography with up to 5 million frames s−1 that bubble splitting, formation of liquid jets away from and towards the boundary, and jet-like ejection of the boundary material into the liquid are the main features of this interaction. The maximum liquid jet velocity measured was 960 m s−1. Such high-velocity jets penetrate the elastic boundary even through a water layer of 0.35 mm thickness. The jetting behaviour arises from the interaction between the counteracting forces induced by the rebound of the elastic boundary and the Bjerknes attraction force towards the boundary. General principles of the formation of annular and axial jets are discussed which allow the interpretation of the complex dynamics. The concept of the Kelvin impulse is examined with regard to bubble migration and jet formation. The results are discussed with respect to cavitation erosion, collateral damage in laser surgery, and cavitation-mediated enhancement of pulsed laser ablation of tissue.

A theory is derived for the class of long two-dimensional waves, comprising solitary and periodic cnoidal waves, that can propagate with unchanging form in heterogeneous fluids. The treatment is generalized to the extent that the waves are supposed to arise on a horizontal stream of incompressible fluid whose density and velocity are arbitrary functions of height, and the upper surface of the fluid is allowed either to be free or to be fixed in a horizontal plane. Explicit formulae for the wave properties and a general interpretation of the physical conditions for the occurrence of the waves are achieved without need to specify particular physical models; but in a later part of the paper, §4, the results are applied to three examples that have been worked out by other means and so provide checks on the present theory. These general results are also shown to accord nicely with the principle of ‘conjugate-flow pairs’ which was explained by Benjamin (1962b) with reference to swirling flows along cylindrical ducts, but which is known to apply equally well to flow systems of the kind in question here.

The theory reveals certain physical peculiarities of a type of flow model often used in theoretical studies of internal-wave phenomena, being specified so as to make the equation for the stream-function linear. In an appendix, some observations are also made regarding the ‘Boussinesq approximation’, which too is often used as a simplifying assumption in this field. It is shown, adding to a recent discussion by Long (1965), that finite internal waves may depend crucially on small effects neglected in this approximation.

Aspects of the nonlinear dynamics of waves shoaling between 9 and 1 m water depths are elucidated via the bispectrum. Bispectral-signal levels are generally high, indicating significant nonlinear coupling. In 9 m depth, the biphases of interactions involving frequencies at, and higher than, the peak of the energy spectra are suggestive of Stokes-like nonlinearities (Hasselman, Munk & MacDonald 1963). Further shoaling gradually modifies these biphases to values consistent with a wave profile that is pitched shoreward, relative to a vertical axis. Bicoherence and biphase observations with a double-peaked (swell and wind-wave) power spectrum provide evidence for excitation of modes at intermediate frequencies via difference interactions, as well as the sum interactions responsible for harmonic growth. Shoreward-propagating low-frequency (surf-beat) energy is shown to have statistically significant coupling to higher-frequency modes within the power-spectral peak. In 18 m depth, the biphase of these interactions is close to 180°, a value consistent with the classical concept of bound long waves. In shallower water, however, substantial biphase evolution occurs, and there is no longer a unique phase relationship between surf beat and the envelope of high-frequency waves. The contributions to sea-surface-elevation skewness and asymmetry (with respect to a vertical axis) from interactions among various wave triads are given by the real and imaginary parts of the bispectrum, respectively. In very shallow water, coupling between surf beat and higher-frequency waves results in a skewness with sign opposite to, and about 40% of the magnitude of, the skewness resulting from interactions between the power-spectral-peak frequency and higher frequencies.

Results are presented of a series of experiments which were carried out with great care to produce accurate, comparable results in which the relatively small changes in the flow past a circular cylinder associated with varying blockage ratios can be clearly distinguished. The experiments cover a range of blockage from 2 to 16% and of aspect ratios from 4 to 10. End plates were fitted to the cylinders in all experiments.

For blockage ratios less than 6%, it is shown that the effects of blockage on pressure distribution and the drag coefficient are small and that the Strouhal number is unaffected by blockage. For blockage ratios in the range 6–16%, there is considerable distortion of the flow due to blockage and the effects are complex. The pressure distribution is of a different form and the Strouhal number changes. Hoevever, conflicting influences result in a blocked drag coefficient which is not very different from that at no blockage. Reduction in aspect ratio has effects on drag coefficient and on base pressure coefficient which are similar to those associated with increase in blockage ratio.

It is concluded that blockage correction procedures based on the momentum method and on the image method are unsatisfactory in their prediction of the unblocked drag coefficient but the momentum method predicts the unblocked base pressure coefficient quite well.

Experimental measurements of the frequencies of discrete tones induced by flow over rectangular cavities were carried out over a range of low subsonic Mach numbers to provide a reliable data base for (aircraft wheel well) cavity noise consideration. A mathematical model of the cavity tones and pressure oscillation phenomenon based on the coupling between shear layer instabilities and acoustic feedback is developed to help in understanding the tone generation mechanism. Good agreement is found between discrete tone frequencies predicted by the model and experimental measurements over a wide range of Mach numbers. Evidence of tones generated by the cavity normal mode resonance mechanism at very low subsonic Mach numbers is also presented.

The form of the momentum equation for one-dimensional (vertical) unsteady mean motion of solid particles in a fluidized bed or a sedimenting dispersion is established from physical arguments. In the case of a fluidized bed that is slightly non-uniform this equation contains two dependent variables, the local mean particle velocity V and the local concentration ϕ, and several statistical parameters of the particle motion in a uniform bed. All these parameters are functions of ϕ with clear physical meanings, and the important ones are measurable. It is a novel feature of the equation that it contains two explicit contributions to the bulk modulus of elasticity of the particle configuration, one arising from the transfer of particle momentum by velocity fluctuations and one arising from the effective repulsive force exerted between particles in random motion. This latter contribution, which proves to be the more important of the two, is related to the gradient diffusivity of the particles, a key quantity in the new theory.

The equation of mean motion of the particles and the equation of particle conservation are sufficient to determine the behaviour of a small disturbance with sinusoidal variation of V and ϕ in the vertical direction. Particle inertia forces in such a propagating wavy disturbance may promote amplitude growth, whereas particle diffusion tends to suppress it, and instability occurs when the particle Froude number exceeds a critical value. Rough estimates of the relevant parameters allow the criterion for instability to be put in approximate numerical from for both gas-fluidized beds (for which the flow Reynolds number at marginal stability is small) and liquid-fluidized beds of solid spherical particles (for which the Reynolds number is well above unity), although more information about the particle diffusivity in particular is needed. The predictions of the theory appear to be in qualitative accord with the available observational data on instability of gas- and liquid-fluidized beds.

Fully developed incompressible turbulent pipe flow at bulk-velocity- and pipe-diameter-based Reynolds number ReD=44000 was simulated with second-order finite-difference methods on 630 million grid points. The corresponding Kármán number R+, based on pipe radius R, is 1142, and the computational domain length is 15R. The computed mean flow statistics agree well with Princeton Superpipe data at ReD=41727 and at ReD=74000. Second-order turbulence statistics show good agreement with experimental data at ReD=38000. Near the wall the gradient of with respect to ln(1−r)+ varies with radius except for a narrow region, 70 < (1−r)+ < 120, within which the gradient is approximately 0.149. The gradient of with respect to ln{(1−r)++a+} at the present relatively low Reynolds number of ReD=44000 is not consistent with the proposition that the mean axial velocity is logarithmic with respect to the sum of the wall distance (1−r)+ and an additive constant a+ within a mesolayer below 300 wall units. For the standard case of a+=0 within the narrow region from (1−r)+=50 to 90, the gradient of with respect to ln{(1−r)++a+} is approximately 2.35. Computational results at the lower Reynolds number ReD=5300 also agree well with existing data. The gradient of with respect to 1−r at ReD=44000 is approximately equal to that at ReD=5300 for the region of 1−r > 0.4. For 5300 < ReD < 44000, bulk-velocity-normalized mean velocity defect profiles from the present DNS and from previous experiments collapse within the same radial range of 1−r > 0.4. A rationale based on the curvature of mean velocity gradient profile is proposed to understand the perplexing existence of logarithmic mean velocity profile in very-low-Reynolds-number pipe flows. Beyond ReD=44000, axial turbulence intensity varies linearly with radius within the range of 0.15 < 1−r < 0.7. Flow visualizations and two-point correlations reveal large-scale structures with comparable near-wall azimuthal dimensions at ReD=44000 and 5300 when measured in wall units. When normalized in outer units, streamwise coherence and azimuthal dimension of the large-scale structures in the pipe core away from the wall are also comparable at these two Reynolds numbers.

The goal of this study is to characterize the various breakdown states taking place in a swirling water jet as the swirl ratio S and Reynolds number Re are varied. A pressure-driven water jet discharges into a large tank, swirl being imparted by means of a motor which sets into rotation a honeycomb within a settling chamber. The experiments are conducted for two distinct jet diameters by varying the swirl ratio S while maintaining the Reynolds number Re fixed in the range 300<Re<1200. Breakdown is observed to occur when S reaches a well defined threshold Sc≈1.3–1.4 which is independent of Re and nozzle diameter used. This critical value is found to be in good agreement with a simple criterion derived in the same spirit as the first stage of Escudier & Keller's (1983) theory. Four distinct forms of vortex breakdown are identified: the well documented bubble state, a new cone configuration in which the vortex takes the form of an open conical sheet, and two associated asymmetric bubble and asymmetric cone states, which are only observed at large Reynolds numbers. The two latter configurations differ from the former by the precession of the stagnation point around the jet axis in a co-rotating direction with respect to the upstream vortex flow. The two flow configurations, bubble or cone, are observed to coexist above the threshold Sc at the same values of the Reynolds number Re and swirl parameter S. The selection of breakdown state is extremely sensitive to small temperature inhomogeneities present in the apparatus. When S reaches Sc, breakdown gradually sets in, a stagnation point appearing in the downstream turbulent region of the flow and slowly moving upstream until it reaches an equilibrium location. In an intermediate range of Reynolds numbers, the breakdown threshold displays hysteresis lying in the ability of the breakdown state to remain stable for S<Sc once it has taken place. Below the onset of breakdown, i.e. when 0<S<Sc, the swirling jet is highly asymmetric and takes the shape of a steady helix. By contrast above breakdown onset, cross-section visualizations indicate that the cone and the bubble are axisymmetric. The cone is observed to undergo slow oscillations induced by secondary recirculating motions that are independent of confinement effects.

A quantitative experimental study of the two-dimensional inverse energy cascade is presented. The flow is electrically driven in a horizontal layer of mercury and three-dimensional perturbations are suppressed by means of a uniform magnetic field, so that the flow can be well approximated by a two-dimensional Navier–Stokes equation with a steady forcing term and a linear friction due to the Hartmann layer. Turbulence is produced by the instability of a periodic square network of 36 electrically driven alternating vortices. The inverse cascade is limited at large scales, either by the linear friction or by the finite size of the domain, depending on the experimental parameters. In the first case, $k^{-\frac{5}{3}$ spectra are measured and the corresponding two-dimensional Kolmogorov constant is in the range 3–7. In the second case, a condensation of the turbulent energy in the lowest mode, corresponding to a spontaneous mean global rotation, is observed. Such a condensation was predicted by Kraichnan (1967) from statistical thermodynamics arguments, but without the symmetry breaking. Random reversals of the rotation sense, owing to turbulent fluctuations, are more and more sparse as friction is decreased. The lowest mode fluctuations and the small scales are statistically independent.

One basic effect of turbulence in turbulent premixed combustion is for the fluctuating velocity field to wrinkle the flame and greatly increase its surface area. In the flamelet theory, this effect is described by the flame surface density. An exact evolution equation for the flame surface density, called the Σ-equation, may be written, where basic physical mechanisms like production by hydrodynamic straining and destruction by propagation effects are described explicitly. Direct numerical simulation (DNS) is used in this paper to estimate the different terms appearing in the Σ-equation. The numerical configuration corresponds to three-dimensional premixed flames in isotropic turbulent flow. The simulations are performed for various mixture Lewis numbers in order to modify the strength and nature of the flame-flow coupling. The DNS-based analysis provides much information relevant to flamelet models. In particular, the flame surface density, and the source and sink terms for the flame surface density, are resolved spatially across the turbulent flame brush. The geometry as well as the dynamics of the flame differ quite significantly from one end of the reaction zone to the other. For instance, contrary to the intuitive idea that flame propagation effects merely counteract the wrinkling due to the turbulence, the role of flame propagation is not constant across the turbulent brush and switches from flame surface production at the front to flame surface dissipation at the back. Direct comparisons with flamelet models are also performed. The Bray-Moss-Libby assumption that the flame surface density is proportional to the flamelet crossing frequency, a quantity that can be measured in experiments, is found to be valid. Major uncertainties remain, however, over an appropriate description of the flamelet crossing frequency. In comparison, the coherent flame model of Marble & Broadwell achieves closure at the level of the Σ-equation and provides a more promising physically based description of the flame surface dynamics. Some areas where the model needs improvement are identified.

In this paper we examine some general features of the time-dependent dynamics of drop deformation and breakup at low Reynolds numbers. The first aspect of our study is a detailed numerical investigation of the ‘end-pinching’ behaviour reported in a previous experimental study. The numerics illustrate the effects of viscosity ratio and initial drop shape on the relaxation and/or breakup of highly elongated droplets in an otherwise quiescent fluid. In addition, the numerical procedure is used to study the simultaneous development of capillary-wave instabilities at the fluid-fluid interface of a very long, cylindrically shaped droplet with bulbous ends. Initially small disturbances evolve to finite amplitude and produce very regular drop breakup. The formation of satellite droplets, a nonlinear phenomenon, is also observed.

Theory and lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at small Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is determined at small but finite Reynolds numbers, at solid volume fractions up to the close-packed limits of the arrays. For small solid volume fraction, the simulations are compared to theory, showing that the first inertial contribution to the drag force, when scaled with the Stokes drag force on a single sphere in an unbounded fluid, is proportional to the square of the Reynolds number. The simulations show that this scaling persists at solid volume fractions up to the close-packed limits of the arrays, and that the first inertial contribution to the drag force relative to the Stokes-flow drag force decreases with increasing solid volume fraction. The temporal evolution of the spatially averaged velocity and the drag force is examined when the fluid is accelerated from rest by a constant average pressure gradient toward a steady Stokes flow. Theory for the short- and long-time behaviour is in good agreement with simulations, showing that the unsteady force is dominated by quasi-steady drag and added-mass forces. The short- and long-time added-mass coefficients are obtained from potential-flow and quasi-steady viscous-flow approximations, respectively.