From a series of experiments using simplified mechanical models we suggest certain minor modifications to the Weis-Fogh (1973)–Lighthill (1973) explanation of the so-called ‘clap and fling’ mechanism for the generation of large lift coefficients by insects in hovering flight. Of particular importance is the production and motion of a leading edge, separation vortex that accounts for virtually all of the circulation generated during the initial phase of the ‘fling’ process. The magnitude of this circulation is substantially larger than that calculated using inviscid theory. During the motion that subsequently separates the wings, the vorticity over each of them is convected and combined to become a tip vortex of uniform circulation spanning the space between them. This combined vortex moves downwards as a part of a ring, of large impulse, that is then continuously fed from quasi-steady separation bubbles that move with the wings as they continue to open at a large angle of attack. Such effects are able to account for the large lift forces generated by the insect.

The purpose of this paper is to present numerical solutions for two-dimensional time-dependent flow about rectangles in infinite domains. The numerical method utilizes third-order upwind differencing for convection and a Leith type of temporal differencing. An attempted use of a lower-order scheme and its inadequacies are also described. The Reynolds-number regime investigated is from 100 to 2800. Other parameters that are varied are upstream velocity profile, angle of attack, and rectangle dimensions. The initiation and subsequent development of the vortex-shedding phenomenon is investigated. Passive marker particles provide an exceptional visualization of the evolution of the vortices both during and after they are shed. The properties of these vortices are found to be strongly dependent on Reynolds number, as are lift, drag, and Strouhal number. Computed Strouhal numbers compare well with those obtained from a wind-tunnel test for Reynolds numbers below 1000.

Hot-film anemometer measurements have been carried out in a fully developed turbulent channel flow. An oil channel with a thick viscous sublayer was used, which permitted measurements very close to the wall. In the viscous sublayer between y+ ≃ 0·1 and y+ = 5, the streamwise velocity fluctuations decreased at a higher rate than the mean velocity; in the region y+ [lsim ] 0·1, these fluctuations vanished at the same rate as the mean velocity.

The streamwise velocity fluctuations u observed in the viscous sublayer and the fluctuations (∂u/∂y)0 of the gradient at the wall were almost identical in form, but the fluctuations of the gradient at the wall were found to lag behind the velocity fluctuations with a lag time proportional to the distance from the wall. Probability density distributions of the streamwise velocity fluctuations were measured. Furthermore, measurements of the skewness and flatness factors made by Kreplin (1973) in the same flow channel are discussed. Measurements of the normal velocity fluctuations v at the wall and of the instantaneous Reynolds stress −ρuv were also made. Periods of quiescence in the − ρuv signal were observed in the viscous sublayer as well as very active periods where ratios of peak to mean values as high as 30:1 occurred.

The velocity field of homogeneous isotropic turbulence is simulated by a large number (38–1200) of random Fourier modes varying in space and time over a large number (> 100) of realizations. They are chosen so that the flow field has certain properties, namely (i) it satisfies continuity, (ii) the two-point Eulerian spatial spectra have a known form (e.g. the Kolmogorov inertial subrange), (iii) the time dependence is modelled by dividing the turbulence into large- and small-scales eddies, and by assuming that the large eddies advect the small eddies which also decorrelate as they are advected, (iv) the amplitudes of the large- and small-scale Fourier modes are each statistically independent and each Gaussian. The structure of the velocity field is found to be similar to that computed by direct numerical simulation with the same spectrum, although this simulation underestimates the lengths of tubes of intense vorticity.

Some new results and concepts have been obtained using this kinematic simulation: (a) for the inertial subrange (which cannot yet be simulated by other means) the simulation confirms the form of the Eulerian frequency spectrum $\phi^{\rm E}_{11} = C^{\rm E}\epsilon^{\frac{2}{3}}U^{\frac{2}{3}}_0\omega^{-\frac{5}{3}}$, where ε,U0,ω are the rate of energy dissipation per unit mass, large-scale r.m.s. velocity, and frequency. For isotropic Gaussian large-scale turbulence at very high Reynolds number, CE ≈ 0.78, which is close to the computed value of 0.82; (b) for an observer moving with the large eddies the ‘Eulerian—Lagrangian’ spectrum is ϕEL11 = CELεω−2, where CEL ≈ 0.73; (c) for an observer moving with a fluid particle the Lagrangian spectrum ϕL11 = CLεω−2, where CL ≈ 0.8, a value consistent with the atmospheric turbulence measurements by Hanna (1981) and approximately equal to CEL; (d) the mean-square relative displacement of a pair of particles 〈Δ2〉 tends to the Richardson (1926) and Obukhov (1941) form 〈Δ2〉 = GΔεt3, provided that the subrange extends over four decades in energy, and a suitable origin is chosen for the time t. The constant GΔ is computed and is equal to 0.1 (which is close to Tatarski's 1960 estimate of 0.06); (e) difference statistics (i.e. displacement from the initial trajectory) of single particles are also calculated. The exact result that Y2 = GYεt3 with GY = 2πCL is approximately confirmed (although it requires an even larger inertial subrange than that for 〈Δ2〉). It is found that the ratio [Rscr ]G = 2〈Y2〉/〈Δ2〉≈ 100, whereas in previous estimates [Rscr ]G≈ 1, because for much of the time pairs of particles move together around vortical regions and only separate for the proportion of the time (of O(fc)) they spend in straining regions where streamlines diverge. It is estimated that [Rscr ]G ≈ O(fc−3). Thus relative diffusion is both a ‘structural’ (or ‘topological’) process as well as an intermittent inverse cascade process determined by increasing eddy scales as the particles separate; (f) statistics of large-scale turbulence are also computed, including the Lagrangian timescale, the pressure spectra and correlations, and these agree with predictions of Batchelor (1951), Hinzc (1975) and George et al. (1984).

A theoretical discussion is given of the motion of a fluid contained in a tube forming a closed loop that is heated from below and cooled from above. The fluid is assumed to have uniform temperature over each cross-section, and the heat transfer is assumed proportional to the difference between the local temperatures of the fluid and the tube. The latter temperature is prescribed. The system has one steady solution with warm fluid rising in one branch and cold fluid sinking in the other. This solution may, however, become unstable in an oscillatory manner. A weak instability takes the form of pulsations, the motion being always of one sign, while a strong instability takes the form of oscillations with zero mean motion. These oscillations are irregular and do not repeat themselves even over very long times.

These unstable motions are associated with thermal anomalies in the fluid that are advected materially around the loop. The anomalies amplify through the correlated variations in flow rate. A warm pocket of fluid creates maximum flow rate going through the upper part and minimum flow rate going through the lower part of the loop. Accordingly it passes quicker through the heat sink than through the heat source, and the latter becomes more effective. Similarly, the heat sink acts more effectively on a cold pocket of fluid.

The curve of neutral stability is worked out as a function of the two parameters of the problem, a non-dimensional gravity and a non-dimensional friction coefficient. The instability has also been studied by direct numerical time integration of the model equations.

It is suggested that the mechanism of instability found for this model operates also in more complicated systems, and can explain the pulsative type of motions observed recently in certain convection experiments.

It is common knowledge that relatively small drops or bubbles have a tendency to stick to the surfaces of solids. Two specific problems are investigated: the shape of the largest drop or bubble that can remain attached to an inclined solid surface; and the shape and speed at which it moves along the surface when these conditions are exceeded. The slope of the fluid-fluid interface relative to the surface of the solid is assumed to be small, making it possible to obtain results using analytic techniques. It is shown that from both a physical and mathematical point of view contact-angle hysteresis, i.e. the ability of the position of the contact line to remain fixed as long as the value of the contact angle θ lies within the interval θR [les ] θ [les ] θA, where θA [nequiv ] θR, emerges as the single most important characteristic of the system.

A tensorially consistent near-wall second-order closure model is formulated. Redistributive terms in the Reynolds stress equations are modelled by an elliptic relaxation equation in order to represent strongly non-homogeneous effects produced by the presence of walls; this replaces the quasi-homogeneous algebraic models that are usually employed, and avoids the need for ad hoc damping functions. A quasi-homogeneous model appears as the source term in the elliptic relaxation equation-here we use the simple Rotta return to isotropy and isotropization of production formulae. The formulation of the model equations enables appropriate boundary conditions to be satisfied.

The model is solved for channel flow and boundary layers with zero and adverse pressure gradients. Good predictions of Reynolds stress components, mean flow, skin friction and displacement thickness are obtained in various comparisons to experimental and direct numerical simulation data.

The model is also applied to a boundary layer flowing along a wall with a 90°, constant-radius, convex bend. Because the model is of a general, tensorially invariant form, special modifications for curvature effects are not needed; the equations are simply transformed to curvilinear coordinates. The model predicts many important features of this flow. These include: the abrupt drop of skin friction and Stanton number at the start of the curve, and their more gradual recovery after the bend; the suppression of turbulent intensity in the outer part of the boundary layer; a region of negative (counter-gradient) Reynolds shear stress; and recovery from curvature in the form of a Reynolds stress ‘bore’ propagating out from the surface. A shortcoming of the present model is that it overpredicts the rate of this recovery.

A heat flux model is developed. It is shown that curvature effects on heat transfer can also be accounted for automatically by a tensorially invariant formulation.

We present the results of a combined experimental and theoretical investigation of the normal impact of hydrophobic spheres on a water surface. Particular attention is given to characterizing the shape of the resulting air cavity in the low Bond number limit, where cavity collapse is driven principally by surface tension rather than gravity. A parameter study reveals the dependence of the cavity structure on the governing dimensionless groups. A theoretical description based on the solution to the Rayleigh–Besant problem is developed to describe the evolution of the cavity shape and yields an analytical solution for the pinch-off time in the zero Bond number limit. The sphere's depth at cavity pinch-off is also computed in the low Weber number, quasi-static limit. Theoretical predictions compare favourably with our experimental observations in the low Bond number regime, and also yield new insight into the high Bond number regime considered by previous investigators. Discrepancies are rationalized in terms of the assumed form of the velocity field and neglect of the longitudinal component of curvature, which together preclude an accurate description of the cavity for depths less than the capillary length. Finally, we present a theoretical model for the evolution of the splash curtain formed at high Weber number and couple it with the underlying cavity dynamics.

This paper reports the results of a two-dimensional finite element simulation of the motion of a circular particle in a Couette and a Poiseuille flow. The size of the particle and the Reynolds number are large enough to include fully nonlinear inertial effects and wall effects. Both neutrally buoyant and non-neutrally buoyant particles are studied, and the results are compared with pertinent experimental data and perturbation theories. A neutrally buoyant particle is shown to migrate to the centreline in a Couette flow, and exhibits the Segré-Silberberg effect in a Poiseuille flow. Non-neutrally buoyant particles have more complicated patterns of migration, depending upon the density difference between the fluid and the particle. The driving forces of the migration have been identified as a wall repulsion due to lubrication, an inertial lift related to shear slip, a lift due to particle rotation and, in the case of Poiseuille flow, a lift caused by the velocity profile curvature. These forces are analysed by examining the distributions of pressure and shear stress on the particle. The stagnation pressure on the particle surface are particularly important in determining the direction of migration.

Direct numerical simulation of the incompressible Navier–Stokes equations is used to study flows where laminar boundary-layer separation is followed by turbulent reattachment forming a closed region known as a laminar separation bubble. In the simulations a laminar boundary layer is forced to separate by the action of a suction profile applied as the upper boundary condition. The separated shear layer undergoes transition via oblique modes and Λ-vortex-induced breakdown and reattaches as turbulent flow, slowly recovering to an equilibrium turbulent boundary layer. Compared with classical experiments the computed bubbles may be classified as ‘short’, as the external potential flow is only affected in the immediate vicinity of the bubble. Near reattachment budgets of turbulence kinetic energy are dominated by turbulence events away from the wall. Characteristics of near-wall turbulence only develop several bubble lengths downstream of reattachment. Comparisons are made with two-dimensional simulations which fail to capture many of the detailed features of the full three-dimensional simulations. Stability characteristics of mean flow profiles are computed in the separated flow region for a family of velocity profiles generated using simulation data. Absolute instability is shown to require reverse flows of the order of 15–20%. The three-dimensional bubbles with turbulent reattachment have maximum reverse flows of less than 8% and it is concluded that for these bubbles the basic instability is convective in nature.

The effect of high levels of free-stream turbulence on the transition in a Blasius boundary layer is studied by means of direct numerical simulations, where a synthetic turbulent inflow is obtained as superposition of modes of the continuous spectrum of the Orr–Sommerfeld and Squire operators. In the present bypass scenario the flow in the boundary layer develops streamwise elongated regions of high and low streamwise velocity and it is suggested that the breakdown into turbulent spots is related to local instabilities of the strong shear layers associated with these streaks. Flow structures typical of the spot precursors are presented and these show important similarities with the flow structures observed in previous studies on the secondary instability and breakdown of steady symmetric streaks.

Numerical experiments are performed by varying the energy spectrum of the incoming perturbation. It is shown that the transition location moves to lower Reynolds numbers by increasing the integral length scale of the free-stream turbulence. The receptivity to free-stream turbulence is also analysed and it is found that two distinct physical mechanisms are active depending on the energy content of the external disturbance. If low-frequency modes diffuse into the boundary layer, presumably at the leading edge, the streaks are induced by streamwise vorticity through the linear lift-up effect. If, conversely, the free-stream perturbations are mainly located above the boundary layer a nonlinear process is needed to create streamwise vortices inside the shear layer. The relevance of the two mechanisms is discussed.

Experiments on fluidization with water of spherical particles falling against gravity in columns of rectangular cross-section are described. All of them are dominated by inertial effects associated with wakes. Two local mechanisms are involved: drafting and kissing and tumbling into stable cross-stream arrays. Drafting, kissing and tumbling are rearrangement mechanisms in which one sphere is captured in the wake of the other. The kissing spheres are aligned with the stream. The streamwise alignment is massively unstable and the kissing spheres tumble into more stable cross-stream pairs of doublets which can aggregate into larger relatively stable horizontal arrays. Cross-stream arrays in beds of spheres constrained to move in two dimensions are remarkable. These arrays may even coalesce into aggregations of close-packed spheres separated by regions of clear water. A somewhat weaker form of cooperative motion of cross-stream arrays of rising spheres is found in beds of square cross-section where the spheres may move freely in three dimensions. Horizontal arrays rise where drafting spheres fall because of greater drag. Aggregation of spheres seems to be associated with relatively stable cooperative motions of horizontal arrays of spheres rising in their own wakes.

The characteristics of the unsteady boundary layer and stall events occurring on an oscillating NACA 0012 airfoil were investigated by using closely spaced multiple hot-film sensor arrays at $Re\,{=}\,1.35\,{\times}\,10^{5}$. Aerodynamic forces and pitching moments, integrated from surface pressure measurements, and smoke-flow visualizations were also obtained to supplement the hot-film measurements. Special attention was focused on the behaviour of the spatial-temporal progression of the locations of the boundary-layer transition and separation, and reattachment and relaminarization points, compared to the static values, for a range of oscillation frequency and amplitude both prior to, during and after the stall. The initiation, growth and rearward convection of a leading-edge vortex, and the role of the laminar separation bubble leading to the dynamic stall, as well as the mechanisms responsible for the stall events observed at different test conditions were also characterized. The hot-film measurements were also correlated with the aerodynamic load and pitching moment results to quantify the values of lift increment and stall angle delay as a result of the observed boundary layer and stall events. The results reported here provide an insight into the detailed nature of the unsteady boundary-layer events as well as the stalling mechanisms at work at different stages in the dynamic-stall process.

The three-dimensional resonant interaction of a plane Tollmien-Schlichting wave, having a frequency f1, with a pair of oblique waves having frequencies ½ f1, was observed and studied experimentally. In the initial stages, the interaction proved to be a parametric resonance, resulting in the amplification of small random priming (background) oscillations of frequency ½ f1, and of a packet of low-frequency oscillations. The resonant interaction of waves in a boundary layer was investigated also by introducing a priming oscillation with frequency f’ = ½ f1 + Δf for different values of the frequency detuning Δf. The importance of the discovered wave interaction in boundary-layer transition is demonstrated. Causes of realization of different types of laminar-flow breakdown are discussed.

The structure and growth of the internal boundary layer which forms downstream of a sudden change from a smooth to a rough surface under zero pressure gradient conditions has been studied experimentally. To keep pressure disturbances due to the roughness change small, the level of the rough surface was depressed, so that the crest of the roughness was aligned with the level of the smooth surface. It has been found that, in the region near the change, the structure of the internal layer is largely independent of that in the almost undisturbed outer layer, whilst both the zero time delay and the moving axis integral length scales in the internal layer are significantly reduced below those on the smooth wall. The growth-rate of the internal layer is similar to that of the zero pressure gradient boundary layer, whilst the level of turbulence inside the internal layer is high because of the large turbulent energy production near the rough wall. From the mixing length results, and an analysis of the turbulent energy equation, it is deduced that the internal layer flow near the wall is not in energy equilibrium, and hence the concept of inner layer similarity breaks down. From an initially self-preserving state on the smooth wall, the turbulent boundary layer approaches a second self-preserving state on the rough wall well downstream of the roughness step.

The boundary layer develops along a flat plate with a Reynolds number high enough to sustain turbulence and allow accurate experimental measurements, but low enough to allow a direct numerical simulation. A favourable pressure gradient just downstream of the trip (experiment) or inflow boundary (simulation) helps the turbulence to mature without unduly increasing the Reynolds number. The pressure gradient then reverses, and the β-parameter rises from −0.3 to +2. The wall-pressure distribution and Reynolds number of the simulation are matched to those of the experiment, as are the gross characteristics of the boundary layer at the inflow. This information would be sufficient to calculate the flow by another method. Extensive automation of the experiment allows a large measurement grid with long samples and frequent calibration of the hot wires. The simulation relies on the recent ‘fringe method’ with its numerical advantages and good inflow quality. After an inflow transient good agreement is observed; the differences, of up to 13%, are discussed. Moderate deviations from the law of the wall are found in the velocity profiles of the simulation. They are fully correlated with the pressure gradient, are in fair quantitative agreement with experimental results of Nagano, Tagawa & Tsuji. and are roughly the opposite of uncorrected mixing-length-model predictions. Large deviations from wall scaling are observed for other quantities, notably for the turbulence dissipation rate. The a1 structure parameter drops mildly in the upper layer with adverse pressure gradient.

The mechanism of wake-induced vibrations (WIV) of a pair of cylinders in a tandem arrangement is investigated by experiments. A typical WIV response is characterized by a build-up of amplitude persisting to high reduced velocities; this is different from a typical vortex-induced vibration (VIV) response, which occurs in a limited resonance range. We suggest that WIV of the downstream cylinder is excited by the unsteady vortex–structure interactions between the body and the upstream wake. Coherent vortices interfering with the downstream cylinder induce fluctuations in the fluid force that are not synchronized with the motion. A favourable phase lag between the displacement and the fluid force guarantees that a positive energy transfer from the flow to the structure sustains the oscillations. If the unsteady vortices are removed from the wake of the upstream body then WIV will not be excited. An experiment performed in a steady shear flow turned out to be central to the understanding of the origin of the fluid forces acting on the downstream cylinder.

The development of a two-dimensional viscous incompressible flow generated from a circular cylinder impulsively started into rectilinear motion is studied computationally. An adaptative numerical scheme, based on vortex methods, is used to integrate the vorticity/velocity formulation of the Navier–Stokes equations for a wide range of Reynolds numbers (Re = 40 to 9500). A novel technique is implemented to resolve diffusion effects and enforce the no-slip boundary condition. The Biot–Savart law is employed to compute the velocities, thus eliminating the need for imposing the far-field boundary conditions. An efficient fast summation algorithm was implemented that allows a large number of computational elements, thus producing unprecedented high-resolution simulations. Results are compared to those from other theoretical, experimental and computational works and the relation between the unsteady vorticity field and the forces experienced by the body is discussed.

The characteristics of thermal convection in a fluid whose viscosity varies strongly with temperature are studied in the laboratory. At the start of an experiment, the upper boundary of an isothermal layer of Golden Syrup is cooled rapidly and maintained at a fixed temperature. The fluid layer is insulated at the bottom and cools continuously. Rayleigh numbers calculated with the viscosity of the well-mixed interior are between 106 and 108 and viscosity contrasts are up to 106. Thermal convection develops only in the lower part of the thermal boundary layer, and the upper part remains stagnant. Vertical profiles of temperature are measured with precision, allowing deduction of the thickness of the stagnant lid and the convective heat flux. At the onset of convection, the viscosity contrast across the unstable boundary layer has a value of about 3. In fully developed convection, this viscosity contrast is higher, with a typical value of 10. The heat flux through the top of the layer depends solely on local conditions in the unstable boundary layer and may be written \[Q_{\rm s} = - CK_{\rm m} (\alpha g/\kappa \nu_{\rm m})^{\frac{1}{3}} \Delta T^{\frac{4}{3}}_{\rm v}\], where km and νm are thermal conductivity and kinematic viscosity at the temperature of the well-mixed interior, κ thermal diffusivity, α the coefficient of thermal expansion, g the acceleration due to gravity. ΔTv, is the ‘viscous’ temperature scale defined by \[\Delta T_{\rm v} = - \frac{\mu (T_{\rm m})}{({\rm d}\mu /{\rm d}T)(T_{\rm m})}\] where μ(T) is the fluid viscosity and Tm the temperature of the well-mixed interior. Constant C takes a value of 0.47 ± 0.03. Using these relations, the magnitude of temperature fluctuations and the thickness of the stagnant lid are calculated to be in excellent agreement with the experimental data. One condition for the existence of a stagnant lid is that the applied temperature difference exceeds a threshold value equal to (2ΔTv).

The effect of viscous dissipation in natural convection is appreciable when the induced kinetic energy becomes appreciable compared to the amount of heat transferred. This occurs when either the equivalent body force is large or when the convection region is extensive. Viscous dissipation is considered here for vertical surfaces subject to both isothermal and uniform-flux surface conditions. A perturbation method is used and the first temperature perturbation function is calculated for Prandtl numbers from 10−2 to 104. The magnitude of the effect depends upon a dissipation number, which is not expressible in terms of the Grashof or the Prandtl number.