An analytical study was made of the natural convection induced in an enclosure by a small hot spot centrally located on the floor. The enclosure was a circular cylinder, vertically oriented, with height equal to radius. A Prandtl number of 0.7 (air) was assumed; the Grashof number (Gr) was based on cylinder height and hot spot temperature. The equations of fluid flow in axisymmetric cylindrical co-ordinates were simplified with the Boussinesq approximation. The equations were solved numerically with a computationally stable, explicit method. The computation, starting from quiescent conditions, proceeded through the initial transient to the fully developed flow. Solutions were obtained for Gr from 4 × 104 to 4 × 1010. The theoretical flows are in excellent agreement with experimentally observed laminar flows (Gr [lsim ] 1.2 × 109) which are discussed in a companion paper, Torrance, Orloff & Rockett (1969). Turbulence was observed experimentally for Gr [gsim ] 1.2 × 109. When the theoretical calculations were extended to Gr = 4 × 1010, a periodic vortex shedding developed, suggestive of the onset of laminar instability. The theoretical results reveal a √Gr scaling for the initial flow transients and, at large Gr, the velocities and heat transfer rates.

This paper modifies and refines earlier work of Palm (1960) concerning the finiteamplitude steady state of cellular convective motion attained when a horizontal layer of fluid becomes unstable as a result of being heated from below. The two non-linear ordinary differential equations to which the problem was reduced by Palm (under certain conditions) are given in a corrected form, and are then analysed in some detail. The principal conclusions are that, for the model considered, hexagonal convection cell may be the stable equilibrium state only if the variation of kinematic viscosity with temperature is sufficiently great. Under the same circumstances a two-dimensional roll cell is also possible, the initial conditions determining which state actually occurs. Although further work is indicated, it seems probable that in an actual experiment with sufficiently large kinematic-viscosity variation, the hexagonal cells are more likely to appear. The analysis enables conclusions to be drawn concerning the flow direction at the cell centre, and also shows that a disturbance of sufficient magnitude may grow even though the situation is a stable one by linearized theory. Comparison with experiment is discussed.

An experiment with the pressure-driven flow down a long rotating channel is described. For zero rotation the flow is quasi-parabolic, laminar, and one-dimensional up the channel. With slight rotation Ω there is a weak double-vortex secondary circulation aligned with the channel. At intermediate Ω there exists an instability in the form of logitudinal rolls of non-dimensional wavenumber 5. The instability disappears at high rotation rates.

The general stability problem for a rotating zonal flow $\overline{U}(y)$ is considered theoretically. For perturbations independent of the co-ordinate in the direction of the flow, the problem is exactly analogous to the stability problem of a temperature-stratified fluid where the stratification $\overline{T}_z(z)$ corresponds to the quantity \[ (\partial\overline{U}/\partial y)(1/2\Omega)-1. \] This analogy extends to much more general mean fields (e.g. non-linear or time dependent) than does the oft-quoted analogy between thermal convection and cylindrical Couette flow. The instability theory is in qualitative agreement with the experiment.

The phenomena of excitation-induced suppression and amplification of broadband jet noise have been experimentally investigated in an effort to understand the mechanisms, especially in relation to the near flow-field large-scale structure dynamics. Suppression is found to occur only in jets at low speeds with laminar exit boundary layers, the optimum occurring for excitation at Stθ ≈ 0.017, where Stθ is the Strouhal number based on the initial shear-layer momentum thickness. The suppression mechanism is linked to an initial-condition effect on the large-scale structure dynamics. The interaction and evolution of laminar-like structures at low jet speeds produce more (normalized) noise and turbulence, compared to asymptotically lower levels at high speeds when the initial shear layer is no longer laminar. The effect of initial condition has been demonstrated by tripped versus untripped jet data. The excitation at Stθ ≈ 0.017 results in a quick roll-up and transition of the laminar shear-layer vortices, yielding coherent structures which are similar to those at high speeds. Thus, the broadband noise and turbulence are suppressed, but at the most to the asymptotically lower levels. When at the asymptotic level, the broadband jet noise can only be amplified by the excitation; the amplification is found to be maximum for excitation in the StD range of 0.65–0.85, StD being the Strouhal number based on the jet diameter. Excitation in this StD range also produces strongest vortexpairing activity. From spectral analysis of the flow-field and the near sound-pressure field, it is inferred that the pairing process induced by the excitation is at the origin of the broadband noise amplification.

This paper presents experimental measurements of the approach and rebound of a particle colliding obliquely with a wall in a viscous fluid. Steel and glass particles 12.7 mm in diameter were used. The experiments were performed using a thick Zerodur wall (a glass-like material) with various mixtures of glycerol and water. Normal and tangential coefficients of restitution were defined from the ratios of the respective velocity components at the point of contact just prior to and after impact. These coefficients account for losses due to lubrication effects and inelasticity. A third parameter, a coefficient of sliding friction, provides a measure of the tangential force acting on the particle as it slides during a collision.

Oblique collisions in a fluid are qualitatively similar to oblique collisions in a dry system, with a lowered friction coefficient dependent on surface roughness. For smooth surfaces the friction coefficient is drastically reduced due to lubrication effects. A theoretical model that takes into account the dependence of viscosity on pressure is proposed to explain the observed tangential force acting on a smooth sphere during an oblique collision. The model relies on an inferred uniform temperature increase within the lubrication layer, a consequence of viscous heating during impact. The tangential force felt by the particle is expressed as a friction coefficient dependent on the viscosity within the lubrication layer. The viscosity increases owing to pressure effects and decreases owing to thermal effects. For rough surfaces the friction coefficient is comparable to that measured in dry systems, since the surface asperities may interact with each other through the lubrication layer.

We present an experimental study of the formation and stability of small-scale beads deposited onto a solid surface by sweeping a droplet stream over it. We are concerned particularly with beads formed from molten droplets deposited on a cold substrate, as in the work of Gao & Sonin (1994), but some results of isothermal deposition are also shown. We show that a molten bead forms with parallel contact lines which have been arrested by freezing while the bead itself is still largely in a liquid state, and that the still-molten material is stable when the contact angle is less than ½π and unstable when it exceeds ½π, consistent with Davis's (1980) theory. In addition, we present a relatively simple inviscid theory for small-scale (small Bond number) beads which allows us to compute the wavelength associated with the maximum growth rate of the instability, and show that it agrees with the dominant wavelength in the experiments.

Kraichnan's (1967) predictions concerning a simultaneous direct enstrophy cascade and inverse energy cascade for high Reynolds number two-dimensional turbulence are tested numerically using a variant of the eddy-damped quasi-normal approximation. For the initial-value problem, an analytic study using this theory shows that, in the zero-viscosity limit, energy and enstrophy are conserved for arbitrarily long times, contrary to the three-dimensional case, where the energy is conserved for only a finite time, after which it is dissipated. Non-local effects in the enstrophy inertial range, which are difficult to treat by conventional numerical schemes (Leith 1971; Leith & Kraichnan 1972), are shown to be representable by an additional diffusion term in the spectral equation. The resulting equation, including non-local effects, is integrated numerically. When enstrophy and energy are continuously injected at a fixed wavenumber, it is shown numerically that a quasi-steady regime is obtained where enstrophy cascades to large wavenumbers across a k−3 inertial range with zero energy transfer while energy flows indefinitely to small wavenumbers across a $k^{-\frac{5}{3}}$ inertial range with zero enstrophy transfer.

Equations that follow from the Navier–Stokes equation and incompressibility but with no other approximations are ‘exact’. Exact equations relating second- and third- order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: (i) the trace of the structure functions, (ii) DNS that has periodic boundary conditions, and (iii) an average over a sphere in r-space. Case (iii) introduces the average over orientations of r into the structure-function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

The effect of spatial periodicity in grain structure on the average transport properties resulting from flow through porous media are derived from the basic conservation equations. At high Péclet number, the mechanical dispersion that is induced by the stochastic fluid velocity field in disordered media and is independent of the molecular diffusivity is absent in periodic media where the velocity field is deterministic. Instead, the fluid motion enhances diffusion by an amount proportional to U2l2/D when the bulk flow is in certain directions (of which there are an infinite number), and to D otherwise. The non-mechanical dispersion mechanisms associated with the zero velocity of the fixed grains is qualitatively similar in ordered and disordered media.

The primary objective of this paper is to provide a detailed confirmation of the validity of potential-flow theory for describing steep gravity waves produced in an experimental tank. Very high-resolution computations are carried out which use a refined mixed Eulerian-Lagrangian solution scheme under the assumptions of potential flow. The numerical results for a plunging breaker produced by a programmed piston-type wavemaker are found to be in excellent agreement with tank measurements up to and including overturning. The calculated free-surface elevations are almost indistinguishable from measured profiles, even close to where the wave plunges. The horizontal and vertical water-particle velocities measured with a laser anemometer throughout the water depth at two longitudinal stations are also well predicted by the theory. In contrast to the fully nonlinear theory, predictions based on linearized theory become poorer as the wave packet moves down the tank. To allow other investigators to evaluate the computations and experiments, the Fourier amplitudes and phases which completely specify the time history of the wavemaker's velocity are given in Appendix B.

The mean lift coefficient of a circular cylinder executing rotary motions in a uniform flow is investigated. These motions include steady rotation, and rotary oscillations with a net rotation rate. Results for the steadily rotating cylinder show that for a given rotation rate, larger cylinder aspect ratios yield higher lift coefficients. It was also found that the addition of forced rotary oscillations to the steady rotation of the cylinder increases the lift coefficient in the cases where the wake would normally be separated in the steadily rotating case, but decreases it otherwise. In addition, a method for estimating the mean lift of a rotating cylinder is presented. Estimates based on this method compare favourably with similar data published for steadily rotating cylinders.

Results from direct numerical simulation for three-dimensional Rayleigh–Bénard convection in samples of aspect ratio and up to Rayleigh number are presented. The broad range of Prandtl numbers is considered. In contrast to some experiments, we do not see any increase in with increasing , neither due to an increasing , nor due to constant heat flux boundary conditions at the bottom plate instead of constant temperature boundary conditions. Even at these very high , both the thermal and kinetic boundary layer thicknesses obey Prandtl–Blasius scaling.

The early stages of drop impact onto a solid surface are considered. Detailed numerical simulations and detailed asymptotic analysis of the process reveal a self-similar structure both for the velocity field and the pressure field. The latter is shown to exhibit a maximum not near the impact point, but rather at the contact line. The motion of the contact line is furthermore shown to exhibit a ‘tank-treading’ motion. These observations are apprehended with the help of a variant of Wagner theory for liquid impact. This framework offers a simple analogy where the fluid motion within the impacting drop may be viewed as the flow induced by a flat rising expanding disk. The theoretical predictions are found to be in very close agreement both qualitatively and quantitatively with the numerical observations for approximately three decades in time. Interestingly, the inviscid self-similar impact pressure and velocities are shown to depend solely on the self-similar variables $(r/\sqrt{t},z/\sqrt{t})$. The structure of the boundary layer developing along the wet substrate is investigated as well. It is found to be in first approximation analogous to the boundary layer growing in the trail of a shockwave. Interestingly, the corresponding boundary layer structure only depends on the impact self-similar variables. This allows us to construct a seamless uniform analytical approximation encompassing both impact and viscous effects. The depiction of the different dynamical fields enables to quantitatively predict observables of interest, such as the evolution of the integral viscous shearing force and of the net normal force.

Measurements are given of the total pressure and yaw relative to the pipe axis of flow in various pipe bends attached to the end of a long straight pipe, for a pipe Reynolds number of 2·36 × 105. The behaviour of the induced secondary flows is presented in detail. In a 180° bend it is found that, from the start of the bend, the secondary flows increase to a maximum and then decrease to a steady value. This effect is explained by relating the local total pressure gradient to the production of streamwise vorticity. The same flow mechanism is used to predict qualitatively the flow pattern in an S-bend in which it is found that the secondary flows cause a complete interchange of the slow moving wall fluid and the faster central core. Results of a computer program for calculating the secondary flow from a knowledge of the upstream velocity profile are included. Although real fluid effects, other than those causing the upstream velocity profile, are omitted from the computations, the measured flow pattern is reasonably well predicted in a long bend up to a bend angle of about 75°.

Strong interactions of shock waves with boundary layers lead to flow separations and enhanced heat transfer rates. When the approaching boundary layer is hypersonic and transitional the problem is particularly challenging and more reliable data is required in order to assess changes in the flow and the surface heat transfer, and to develop simplified models. The present contribution compares results for transitional interactions on a flat plate at Mach 6 from three different experimental facilities using the same instrumented plate insert. The facilities consist of a Ludwieg tube (RWG), an open-jet wind tunnel (H2K) and a high-enthalpy free-piston-driven reflected shock tunnel (HEG). The experimental measurements include shadowgraph and infrared thermography as well as heat transfer and pressure sensors. Direct numerical simulations (DNS) are carried out to compare with selected experimental flow conditions. The combined approach allows an assessment of the effects of unit Reynolds number, disturbance amplitude, shock impingement location and wall cooling. Measures of intermittency are proposed based on wall heat flux, allowing the peak Stanton number in the reattachment regime to be mapped over a range of intermittency states of the approaching boundary layer, with higher overshoots found for transitional interactions compared with fully turbulent interactions. The transition process is found to develop from second (Mack) mode instabilities superimposed on streamwise streaks.

A dilute dispersion containing drops of one fluid dispersed in a second, immiscible fluid is considered. The drops are sufficiently small that inertia is negligible and that they remain spherical. Two drops of different size are in relative motion due to either Brownian diffusion or gravitational sedimentation. When the drops become close, they interact with each other owing to hydrodynamic disturbances and van der Waals attractions, and, under favourable conditions, they will collide with each other and coalesce. The rate at which two drops collide is predicted by solving the diffusion equation for Brownian coalescence, and by using a trajectory analysis to follow the relative motion of pairs of drops for gravity-induced coalescence.

The emphasis of our analysis is on the effects of drop interactions on their collision rate, and these are described by the collision efficiency. Since the hydrodynamic resistance to the drop relative motion reduces with a decreasing ratio of the viscosities of the drop fluid and the surrounding fluid, the collision efficiency increases with decreasing viscosity ratio. A qualitative difference in the collision behaviour of viscous drops from that of rigid spheres is demonstrated; finite collision rates for drops are predicted even in the absence of attractive forces, provided that drop deformation is negligible, whereas rigid particles with smooth surfaces will not come into contact in a fluid continuum unless an attractive force is present which is able to overcome the lubrication forces resisting the relative motion. Hydrodynamic interactions between two spherical drops are accounted for exactly by determining the two-sphere relative mobility functions from previous solutions for two drops moving along and normal to their line of centres. These solutions are based on the method of reflections for widely separated drops, lubrication theory for drops in near-contact, and bispherical coordinates for general separations. The hydrodynamic interactions have a greater effect on reducing the rate of gravity collisions than the rate of Brownian collisions.

The problem solved concerns the disturbance to a stream of shallow water due to an immersed slender body, with special application to the steady motion of ships in shallow water. Formulae valid to first order in slenderness are given for the wave resistance and vertical forces at both sub- and supercritical speeds. The vertical forces are used to predict sinkage and trim of ships and satisfactory comparisons with model experiments are made.

A numerical investigation is conducted on the flow around and vibration response of an elastic square cylinder (side width $D$) in the wake of a stationary cylinder at Reynolds numbers of $Re=100$ and 200 based on $D$ and the free-stream velocity. The downstream cylinder, referred to as the wake cylinder, is allowed to vibrate in the transverse direction only. The reduced velocity $U_{r}$ is varied from 1 to 30. Cylinder centre-to-centre spacing ratios of $L^{\ast }(=L/D)=2$ and 6 are considered. Simulations are also conducted for a single isolated cylinder, and the results are compared with those for the wake cylinder. The focus is given to vibration response, frequency response, fluctuating lift force, phase relationship between the lift and displacement, work done and the flow structure modification during the cylinder vibration. The results reveal that the dependence of the Strouhal number $St$ on $U_{r}$ can distinguish different branches more appropriately than that of the vibration amplitude on $U_{r}$. The vibration response of the single cylinder at $Re=100$ is characterized by the initial, lower and desynchronization branches. On the other hand, that at $Re=200$ undergoes initial, lower and galloping branches. The galloping involves the characteristics of both the initial and the lower branches or the initial and the desynchronization branches depending on $U_{r}$. For the wake cylinder, the gap flow has a significant impact on the vibration response, leading to (i) the absence of galloping at either $Re$ and $L^{\ast }$, (ii) the presence of an upper branch at $Re=200$, $L^{\ast }=6$ and (iii) an initial branch of different characteristics at $Re=100$, $L^{\ast }=6$. The different facets are discussed in terms of wake structures, work done and phase lag between lift and displacement.

A descriptive experimental study was made in both air and water of the temporally periodic motion that occurs in the vortex whistle and cyclone separator. The motion can be described in terms of an oscillator that derives its energy from hydrodynamic instability of the steady swirling flow and whose frequency is determined by an angular velocity characteristic of this steady flow. The relevant dynamical parameters are the Rossby number and Reynolds number for the steady flow with the addition of the Strouhal number for the time-dependent flow. The results of this study were compared with the vortex breakdown phenomenon over swept-back wings. Breakdown can be described in the same terms as for the other two cases and it appears that all three motions are basically the same.

We have made computational experiments to study the stability and long-time evolution of two-dimensional wakes. We have used the VORTEX code, a finite-difference realization of two-dimensional motions in incompressible inviscid fluids. In the first experiment an initial shear-unstable triangular velocity profile evolves into a non-homogeneous, finite-area, asymmetric vortex array and like-signed regions attract and fuse (or coalesce). Enhanced transport across the profile is due to ‘capture’ and convection of small-scale vortex regions by larger opposite-signed vortex regions. In the following experiments we study the stability of an asymmetric four-vortex finite-area system corresponding to a von Kármán street of point vortices. Here the critical parameter is b/a, the initial transverse-to-longitudinal separation ratio of vortex centres. At \[ b/a = 0.281 \] the four-vortex system is stable and we observe that large-area vortex regions develop elliptical (m = 2), triangular (m = 3), etc. surface modes owing to mutual interactions. At b/a = 0 the measured growth rate is smaller than that for the corresponding von Kármán system and at b/a = 0·6 the measured growth rate is larger. At b/a = 0 one vortex undergoes fission in the high-shear field produced by two nearest-neighbour opposite-signed vortex regions. Heuristic comparisons are made with the two-dimensional tunnel experiments of Taneda and others.