We address the flutter instability of a flexible plate immersed in an axial flow. This instability is similar to flag flutter and results from the competition between destabilizing pressure forces and stabilizing bending stiffness. In previous experimental studies, the plates have always appeared much more stable than the predictions of two-dimensional models. This discrepancy is discussed and clarified in this paper by examining experimentally and theoretically the effect of the plate aspect ratio on the instability threshold. We show that the two-dimensional limit cannot be achieved experimentally because hysteretical behaviour and three-dimensional effects appear for plates of large aspect ratio. The nature of the instability bifurcation (sub- or supercritical) is also discussed in the light of recent numerical results.

A comparison between the experimental visualization and numerical simulations of the occurrence of vortex breakdown in laminar swirling flows produced by a rotating endwall is presented. The experimental visualizations of Escudier (1984) were the first to detect the presence of multiple recirculation zones and the numerical model presented here, consisting of a numerical solution of the unsteady axisymmetric Navier-Stokes equations, faithfully reproduces these phenomena and all other observed characteristics of the flow. Further, the numerical calculations elucidate the onset of oscillatory flow, an aspect of the flow that was not clearly resolved by the flow visualization experiments. Part 2 of the paper examines the underlying physics of these vortex flows.

Many studies of weakly nonlinear surface waves are based on so-called reduced integrodifferential equations. One of these is the widely used Zakharov four-wave equation for purely gravity waves. But the reduced equations now in use are not Hamiltonian despite the Hamiltonian structure of exact water wave equations. This is entirely due to shortcomings of their derivation. The classical method of canonical transformations, generalized to the continuous case, leads automatically to reduced equations with Hamiltonian structure. In this paper, attention is primarily paid to the Hamiltonian reduced equation describing the combined effects of four- and five-wave weakly nonlinear interactions of purely gravity waves. In this equation, for brevity called five-wave, the non-resonant quadratic, cubic and fourth-order nonlinear terms are eliminated by suitable canonical transformation. The kernels of this equation and the coefficients of the transformation are expressed in explicit form in terms of expansion coefficients of the gravity-wave Hamiltonian in integral-power series in normal variables. For capillary–gravity waves on a fluid of finite depth, expansion of the Hamiltonian in integral-power series in a normal variable with accuracy up to the fifth-order terms is also given.

The flow in the noise-producing region of a circular jet is found to be dominated by a group of large eddies, containing nearly a quarter of the turbulent shear stress in the quasi-plane region of the shear layer: their contribution to the shear stress decreases as the effects of axisymmetry become noticeable at more than about two diameters downstream of the nozzle. These large eddies appear to be almost entirely responsible for the irrotational fluctuations near the nozzle, which, for this and other reasons, are larger relative to the reference dynamic pressure than in other shear flows. As a consequence of this, the convection velocity near the high- and low-velocity edges of the flow is biased towards the mean velocity in the high-intensity region. The dominance of the large eddies therefore explains the measurements of near-field pressure fluctuations by Franklin & Foxwell (1958), and of convection velocity by Davies, Barratt & Fisher (1963) and the present authors. The strength of these large eddies, compared with those in the boundary layer or wake, is remarkable.

The large eddies appear to be mixing-jets similar to those found by Grant (1958) in the wake, but with their projection in the (y, z)-plane inclined at about 45° to the y (radial) axis instead of lying along the y-axis as in the wake.

It is suggested that the augmentation of these large eddies by artificial means could be used to increase the mixing rate and permit the reduction of jet noise by means of acceptably short ejector shrouds.

The medium-scale motion is found to be far from isotropic in scales, although the two scales associated with a given vorticity component are more nearly equal. This phenomenon is also noticeable in the wake.

It is found that the departure from self-preservation, which starts when the shear layer thickness is no longer small compared with the nozzle radius, does not grossly affect the region of high turbulence intensity and maximum noise production until this region itself is no longer small compared with the radius. The maximum shear stress seven diameters downstream of the exit is still 70% of its value near the exit, and the non-dimensional mean velocity gradient is practically unchanged.

Three constitutive laws (Skalak et al.'s law extended to area-compressible interfaces, Hooke's law and the Mooney–Rivlin law) commonly used to describe the mechanics of thin membranes are presented and compared. A small-deformation analysis of the tension–deformation relation for uniaxial extension and for isotropic dilatation allows us to establish a correspondence between the individual material parameters of the laws. A large-deformation analysis indicates that the Mooney–Rivlin law is strain softening, whereas the Skalak et al. law is strain hardening for any value of the membrane dilatation modulus. The large deformation of a capsule suspended in hyperbolic pure straining flow is then computed for several membrane constitutive laws. A capsule with a Mooney–Rivlin membrane bursts through the process of continuous elongation, whereas a capsule with a Skalak et al. membrane always reaches a steady state in the range of parameters considered. The small-deformation analysis of a spherical capsule embedded in a linear shear flow is modified to account for the effect of the membrane dilatation modulus.

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.

A general set of conservation equations and constitutive integrals for the dynamic properties of the rapid flow of a granular material consisting of slightly inelastic and slightly rough spherical particles is derived by following an approach used in the kinetic theory of dense gases. By taking moments of the translational and rotational particle velocities in the general transport moment equation and making the Enskog approximation, the singlet velocity distribution function is determined. As a result, the constitutive relations and coefficients such as stresses, energy fluxes, rates of translational and rotational energy interchanges, shear viscosity, spin viscosity, bulk viscosity and ‘thermal’ conductivities are obtained. The present theory incorporates the kinetic as well as the collisional contributions for stresses and energy fluxes. Thus, it is appropriate for dilute as well as dense concentrations of solids. For the case of simple shear flow, there is favourable agreement between the theoretical predictions of stresses and both the experimental measurements and the results from computer simulations.

By adding minute concentrations of a high-molecular-weight polymer, liquid jets or bridges collapsing under the action of surface tension develop a characteristic shape of uniform threads connecting spherical fluid drops. In this paper, high-precision measurements of this beads-on-string structure are combined with a theoretical analysis of the limiting case of large polymer relaxation times and high polymer extensibilities, for which the evolution can be divided into two distinct regimes. For times smaller than the polymer relaxation time over which the beads-on-string structure develops, we give a simplified local description, which still retains the essential physics of the problem. At times much larger than the relaxation time, we show that the solution consists of exponentially thinning threads connecting almost spherical drops. Both experiment and theoretical analysis of a one-dimensional model equation reveal a self-similar structure of the corner where a thread is attached to the neighbouring drops.

The structure of a premixed flame front propagating in a region of two-dimensional turbulence is investigated using full numerical simulation including heat release, variable properties, and one-step Arrhenius chemistry. The influence of reactant Lewis number (Le = ratio of thermal to species diffusivity) is reported for Le = 0.8, Le = 1.0, and Le = 1.2 flames. Local flame behaviour is described by comparing the local instantaneous turbulent flame structure (local consumption rate of reactants) to the steady one-dimensional laminar flame structure for the same thermochemical parameters. Statistics of flame front strain rates and curvature are calculated and global quantities of interest in modelling (flame surface area, mean reactant consumption rate per unit area of flame, and turbulent flame speed) are reported. Principal findings are: that probability density functions (p.d.f.s) of flame curvature are nearly symmetric about a near-zero mean; that the flame tends to align preferentially with extensive tangential strain rates; that the local flame structure of the non-unity Lewis number flames correlates more strongly with local flame curvature than with tangential strain rate; that the mean consumption rate per unit area is relatively insensitive to curvature and is controlled by the mean tangential strain rate; and, that more flame area is generated for Le < 1 than for Le > 1. Implications of the results for flamelet models of turbulent premixed combustion are discussed.

The results of an experimental investigation dealing with finite-amplitude internal solitary waves in a two-fluid system are presented. Particular attention is paid to characterizing solitons in terms of their shape and amplitude–wavelength scale relationship. Two cases are considered, viz., a shallow- and a deep-water configuration, in order to study the depth effect upon the propagational characteristics of the waves. Comparisons are made between the experimental results and existing internal-wave theories. In addition, discussion is presented describing how these existing theories may be extended to include higher-order nonlinear and viscous effects.

A numerical solution is obtained to describe the behaviour of a uniform bore over a sloping beach and the subsequent run-up and back-wash. The results exhibit features which have only previously been described in a qualitative manner. These include the formation’ of a landward-facing bore in the back-wash. A comprehensive set of results are presented for a typical initial subcritical bore height ratio.

Measurements of the vortex-shedding frequency behind a vertical rectangular prism and a vertical circular cylinder attached to a plane wall are correlated with the characteristics of the smooth-wall turbulent boundary layer in which they are immersed. Experimental data were collected to investigate the effects of (i) the aspect ratio of these bodies and (ii) the boundary-layer characteristics on the vortex-shedding frequency. The Strouhal number for the rectangular prism and the circular cylinder, defined by S = fcw/U0 and fcd/U0 respectively, was found to be expressed by a power function of the aspect ratio h/w (or h/d). Here fc is the vortex-shedding frequency, U0 is the free-stream velocity, h is the height, w is the width and d is the diameter. As the aspect ratio is reduced, the type of vortex shedding behind each of the two bodies was found to change from the Karman-type vortex to the arch-type vortex at the aspect ratio of 2·0 for the rectangular prism and 2·5 for the circular cylinder.

The present study examines the flow past slender bodies possessing finite centre-line curvature in a viscous, incompressible fluid without any appreciable inertia effects. We consider slender bodies having arbitrary centre-line configurations, circular transverse cross-sections, and longitudinal cross-sections which are approximately elliptic close to the body ends (i.e. prolate-spheroidal body ends). The no-slip boundary condition on the body surface is satisfied, using a convenient stepwise procedure, to higher orders in the slenderness parameter (ε) than has previously been possible. In fact, the boundary condition is satisfied up to an error term of O(ε2) by distributing appropriate stokeslets, potential doublets, rotlets, sources, stresslets and quadrupoles on the body centre-line. The methods used here produce an integral equation valid along the entire body length, including the ends, whose solution determines the stokeslet strength or equivalently the force per unit length up to a term of O(ε2). The O(ε2) correction to the stokeslet strength is also found. The theory is used to examine the motion of a partial torus and a helix of finite length. For helical bodies comparisons are made between the present theory and the resistive-force theory using the force coefficients of Gray & Hancock and Lighthill. For the motion considered the Gray & Hancock force coefficients generally underestimate the force per unit length, whereas Lighthill's coefficients provide good agreement except in the vicinity of the body ends.

The steady separated flow past a circular cylinder was investigated experimentally. By artificially stabilizing the steady wake, this system was studied up to Reynolds numbers R considerably larger than any previously attained, thus providing a much clearer insight into the asymptotic character of such flows at high Reynolds numbers. Some of the experimental results were unexpected. It was found that the pressure coefficient at the rear of the cylinder remained unchanged for 25 [les ] R [les ] 177, that the circulation velocity within the wake approached a non-zero limit as the Reynolds number increased, and that the wake length increased in direct proportion to the Reynolds number.

This paper summarizes and extends some mathematical results for a model for a class of water-entry problems characterized by the geometrical property that the impacting body is nearly parallel to the undisturbed water surface and that the impact is so rapid that gravity can be neglected. Explicit solutions for the pressure distributions are given in the case of two-dimensional flow and a variational formulation is described which provides a simple numerical algorithm for three-dimensional flows. We also pose some open questions concerning the well-posedness and physical relevance of the model for exit problems or when there is an air gap between the impacting body and the water.

Brownian motion of particles suspended in a fluid is studied, and expressions derived for the particle diffusivity and velocity autocorrelation function. The theory of thermal noise in a general linear system is applied to both the particles and the fluid in a novel formulation. This enables the recent modification of the Langevin equation to include the effect of fluid inertia to be seen as just a necessary but simple reinterpretation of the original analysis, without introducing the theory of non-Markovian processes.

Direct numerical simulations of the incompressible Navier–Stokes equations are employed to study the turbulent wall-shear stress in a turbulent channel flow forced by lateral sinusoidal oscillations of the walls. The objective is to produce a documented database of numerically computed friction reductions. To this aim, the particular numerical requirements for such simulations, owing for example to the time-varying direction of the skin-friction vector, are considered and appropriately accounted for.

A detailed analysis of the dependence of drag reduction on the oscillatory parameters allows us to address conflicting results hitherto reported in the literature. At the Reynolds number of the present simulations, we compute a maximum drag reduction of 44.7%, and we assess the possibility for the power saved to be higher than the power spent for the movement of the walls (when mechanical losses are neglected). A maximum net energy saving of 7.3% is computed.

Furthermore, the scaling of the amount of drag reduction is addressed. A parameter, which depends on both the maximum wall velocity and the period of the oscillation, is found to be linearly related to drag reduction, as long as the half-period of the oscillation is shorter than a typical lifetime of the turbulent near-wall structures. For longer periods of oscillation, the scaling parameter predicts that drag reduction will decrease to zero more slowly than the numerical data. The same parameter also describes well the optimum period of oscillation for fixed maximum wall displacement, which is smaller than the optimum period for fixed maximum wall velocity, and depends on the maximum displacement itself.

A series of experiments is described in which a jet issues from an orifice at the nose of a body in supersonic flow to oppose the mainstream. An analytical model of the flow is developed which suggests that the aerodynamic features of a steady flow depend primarily on a jet flow-force coefficient, and the Mach number of the jet in its exit plane. A sufficient condition for steady flow is developed. The experiments are found to agree well with predictions based on the flow model. A short account is presented of some previous investigations, and some of their conclusions are re-examined in the light of the present study.

The flow around the three-dimensional blunt geometry presented in the work of Ahmed, Ramm & Faitin (Tech. Rep., 1984) is investigated experimentally at $\mathit{Re}= {U}_{0} H/ \nu = 9. 2\times 1{0}^{4} $ (where ${U}_{0} $ is free-stream velocity, $H$ the height of the body and $\nu $ viscosity). The very large recirculation on the base responsible for the dominant part of the drag is characterized. The analyses of the coherent dynamics of the wake reveal the presence of two very distinctive time scales. At long time scales ${T}_{l} \sim 1{0}^{3} H/ {U}_{0} $, the recirculation region shifts between two preferred reflectional-symmetry-breaking positions leading to a statistically symmetric wake; the sequence of these asymmetric states is random. This bi-stable behaviour is independent of the Reynolds number but occurs only above a critical value of ground clearance. At short time scales ${T}_{s} \sim 5H/ {U}_{0} $, the wake presents weak coherent oscillations in the vertical and lateral directions. They are respectively associated with the interaction of the top/bottom and lateral shear layers; when normalized by the height and width of the body, the Strouhal numbers are close to 0.17. These results suggest an alternate shedding associated with the vertical oscillation and a one-sided vortex shedding in the lateral direction with an orientation linked to the current asymmetric position. Finally, the impact of these coherent wake motions on the base pressure is discussed to motivate further drag reduction strategies.

In the first part of the paper the dimensional laws governing the processes of heat and momentum transport from an infinite rough plane are assembled and their consequences set out. In the second part, the detailed equations for the turbulent energy, the mean square temperature fluctuation and the covariance of temperature and vertical velocity are used, together with some speculative assumptions concerning the dissipative action of the turbulence, to derive a series of relations between the turbulent intensities and the Austausch coefficients. One of these relations indicates that the flux form of the Richardson number cannot exceed a critical value which is about 0·15. It follows that in highly stable conditions the buoyancy forces have little direct effect on the turbulent energy balance, their action being primarily to cause a reduction in the scale of the motion and some change in its structure.