This paper describes some experimental observations of free-surface flows arising when fluid is poured over the end of a flat plate into a reservoir well below the plate. This class of flows was found to be highly unstable with over seven hundred qualitatively different flows being observed in the experiment. At very small values of the flux the liquid fell from the plate into the reservoir in the form of droplets which developed periodically at a number of sites on the underside of the plate. As the flux was increased these 'sites’ were able to sustain continuous, unbroken streams, and combinations of drip sites and continuous streams were possible. At still larger values of the flux there was a tendency for the liquid to fall in 'sheets’ and several combinations of’ sheet flows’ and continuous streams were observed, some of which flows exhibited a rather chaotic temporal behaviour. At even larger flux values the flow resembled that of the classic waterfall, but here there were also some unexpected instabilities where the attachment line of the free surface of the liquid with the plate developed corrugations.

This paper presents the in-line force coefficients for circular cylinders in planar oscillatory flows of small amplitude. The results are compared with the theoretical predictions of Stokes (1851) and Wang (1968). For two-dimensional, attached- and laminar-flow conditions the data are, as expected, in good agreement with the Stokes–Wang analysis. The oscillatory viscous flow becomes unstable to axially periodic vortices above a critical Keulegan–Carpenter number K (K = UmT/D, Um = the maximum velocity in a cycle, T = the period of flow oscillation, and D = the diameter of the circular cylinder) for a given β (β = Re/K = D2/vT, Re = UmD/v, and v = the kinematic viscosity of fluid) as shown experimentally by Honji (1981) and theoretically by Hall (1984). The present investigation has shown that the Keulegan—Carpenter number at which the drag coefficient Cd deviates rather abruptly from the Stokes—Wang prediction nearly corresponds to the critical K at which the vortical instability occurs.

For flow visualization o-cresolphthalein was found to be superior to thymol blue as an electrode-activated pH indicator because its slower reverse reaction results in better colour retention; its poor solubility was overcome by using a 50:50 water-ethanol mixture.

The method revealed unknown flow patterns in plate heat-exchanger configurations with the corrugations on opposite walls abutting. The main flow is along the furrows on each wall. The interaction between these criss-crossing flows causes spiralling in the flow along a furrow. At angles between corrugations and the duct axis of at least up to 45°, the furrow flow is reflected only at the sidewalls of the duct, whereas at high angles, e.g. at 80°, there are intermediate ‘reflections’ at the nodes where the crests of opposite walls meet.

The dynamic characteristics of the pressure and velocity fields of the unsteady incompressible laminar wake behind a circular cylinder are investigated numerically and analysed physically. The governing equations, written in a velocity—pressure formulation and in conservative form, are solved by a predictor—corrector pressure method, a finite-volume second-order-accurate scheme and an alternating-direction-implicit (ADI) procedure. The initiation mechanism for vortex shedding and the evaluation of the unsteady body forces are presented for Reynolds-number values of 100, 200 and 1000.

The vortex shedding is generated by a physical perturbation imposed numerically for a short time. The flow transition becomes periodic after a transient time interval. The frequency of the drag and lift oscillations agree well with the experimental data.

The study of the interactions of the unsteady pressure and velocity fields shows the phase relations between the pressure and velocity, and the influence of different factors: the strongly rotational viscous region, the convection of the eddies and the almost inviscid flow.

The interactions among the different scales of structures in the near wake are also studied, and in particular the time-dependent evolution of the secondary eddies in relation to the fully developed primary ones is analysed.

The hydrodynamic pressures acting on the surface of a rigid dam during earthquakes are examined both analytically and numerically by a two-dimensional potential-flow theory. Analytical solutions are obtained for the cases where the inclined upstream dam face has a constant slope and the reservoir has a triangular shape. A general numerical scheme via an integral-equation formulation is also presented for complex geometries. Analytical and numerical solutions are compared with experimental data (Zangar 1953) with good agreement.

A model problem is solved mathematically in order to clarify how an essential singularity, which is an integral part of the boundary-layer solution of a time-dependent convective–diffusive system, is removed by the inclusion of the effect of longitudinal diffusion. The model problem involves a uniform velocity field along a plane boundary at which boundary conditions of a mixed type are prescribed. The problem is solved by means of a method involving the Laplace transform and the Wiener–Hopf technique. An exact solution is presented.

Special attention is given to an asymptotic solution that is valid for large values of the dimensionless time. It is shown that the large-time asymptote and the naïve boundary-layer solution are close approximations of one another, except in the neighbourhood of the location where the latter is singular. Around this point the present solution provides an interlayer which matches smoothly the purely time-dependent Rayleigh-like and the stationary components of the boundary-layer approximation.

In this paper the dimensional-analysis approach to wall turbulence of Perry & Abell (1977) has been extended in a number of directions. Further recent developments of the attached-eddy hypothesis of Townsend (1976) and the model of Perry & Chong (1982) are given, for example, the incorporation of a Kolmogoroff (1941) spectral region. These previous analyses were applicable only to the ‘wall region’ and are extended here to include the whole turbulent region of the flow. The dimensional-analysis approach and the detailed physical modelling are consistent with each other and with new experimental data presented here.

Numerical simulations were performed of the evolution of the Kelvin–Helmholtz instability in planar, free shear layers, resulting from coflow past a splitter plate. The calculations solved the time-dependent inviscid compressible conservation equations. New algorithms were developed and tested for inflow and outflow boundary conditions. Since no turbulence subgrid modelling was included, only the large-scale features of the flow are described. The transition from laminar flow was triggered by transverse pressure gradients and subsequent vorticity fluctuations at the shear layer, near the tip of the splitter plate. The calculations were performed for a range of free-stream velocity ratios and sizes of the chamber enclosing the system. The simulations showed that the resulting mixing layers have more of the faster fluid than the slower fluid entrained in the roll-ups. This effect is in general agreement with the results of recent splitter-plate experiments of Koochesfahani, Dimotakis & Broadwell (1983). The calculated mixing asymmetry is more apparent when the velocity ratio of the two streams is larger, and does not depend significantly on the separation between the walls of the chamber.

Fluid flow through a solid matrix is examined by introducing a two-dimensional phenomenological model of flow through a unit cell. The effects of inter-cell mixing on reductions in the upstream prescribed gradients are studied. The velocity gradient is modelled by allowing flows of different average velocities to enter the cell. The exit conditions are then determined by solving for the flow field. It is shown that the extent of the reduction depends on the geometry, Reynolds number and the magnitude of the gradient. Also, some results for reductions in the temperature gradient and a regime diagram for gases are presented.

The boundary-integral, or panel, method of solution of the plane potential-flow equation for incompressible flow is well established. We extend the method to the fully compressible problem, in subcritical flow conditions. The method is applied to single- and to multi-element configurations.

The instantaneous lift forces on a pair of rigid wings opening by rotation about a common trailing edge (the fling) are measured and related to the unsteady flow field as revealed by simultaneous flow visualization. The effect of altering the wing-opening time history and the initial opening angle of the wing pair on circulation and lift generation is investigated. Dimensionless circulations and lift coefficients are compared with experimental and theoretical results in the literature and the relevance of these results to insects and engineers is discussed.

A linear approximation for surface-wave radiation by two adjacent slender bodies is derived and compared with a three-dimensional numerical method. The approximation incorporates slender-body theory for a single body and accounts for wave interaction between the bodies. It is assumed that the distance between the bodies is on the order of their lengths. The far-field disturbance due to each body is obtained by distributing wave sources and dipoles on its centreline and solving a pair of coupled integral equations for their strengths and moments respectively. The hydrodynamic added-mass and damping coefficients are then calculated from simple expressions involving the source strengths and the hydrodynamic coefficients of each body separately. Wave exciting forces are also calculated from a far-field reciprocity relation. The approximation performs well even when the separation distance is comparable to the characteristic transverse dimension of each body.

The exact solution of the concentration field of jet fluid in a round laminar jet is presented. This analytical solution, which assumes constant kinematic viscosity and molecular diffusivity, establishes the dependence of the concentration field on the Schmidt number. This solution and a kinematic argument are used to calculate the shape of the lifted flame front in a round laminar jet. The observed shape of the lifted laminar propane flame front is compared with the prediction of this formulation. A spatial-stability criterion of the flame is developed and applied to examine the stability of the lifted flame in the flow field of the round laminar jet. The laminar flame blowout height and the corresponding Reynolds number are calculated from the stability criterion. The predictions agree well with the experimental values. The flame blowout Reynolds number of laminar fuel jets of pure fuels discharging from round pipes with fully developed laminar flow is shown to be directly proportional to the pipe diameter. At blowout the fuel concentration in the vicinity of the flame is found to attain a constant value which lies between the lean flammability limit and the fuel concentration at which the laminar flame speed is maximum. This stability criterion is generalized to laminar gas-jet flames of different fuels using three experimentally determined parameters describing their flame speed–concentration characteristics. The general form can account for dilution of fuel jets with inert gases. That flames can be lifted and blown out while they are still laminar is also demonstrated experimentally.

The linear field induced by the sudden starting of a wave on an infinite plane surface is found exactly. An acoustic wavefront, on which the pressure remains constant and finite, moves outwards from the surface at the sound speed c. Behind this front the pressure field consists of two distinct components. The first is recognizable as the field due to steady motion over a wavy wall, while the second is an acoustic transient which propagates through the fluid to accelerate it into its steady asymptotic state. We find that whenever the surface phase speed is subsonic, there is an equipartition of the energy between the steady evanescent field attached to the surface and the outward-travelling sound. If the surface has sonic phase speed then the pressure on the surface grows like t½, and becomes unbounded. For subsonic waves the final momentum of the fluid parallel to the surface is shown to be equal to the total energy radiated divided by the surface phase speed. In the asymptotic state the ratio of momentum in the far field to that in the near field is ½m2/(1 − ½m2), where m is the ratio of the surface phase speed to the sound speed. The transient field on the surface can be identified as consisting of two travelling sound waves. One travels in the same direction as the surface wave, and the other in the opposite direction. They have amplitudes which are proportional to (1 − m)−l and (1 + m)−1 respectively, and decay in time as t−½. Their wavelength parallel to the surface is, of course, the same as that of the surface wave that induced them. We show that when the surface wave is very subsonic, the evanescent field is established very slowly, settling down only after the surface wave has travelled about m−1 wavelengths.

Experimental results for the axial velocity distribution in the entry region of weakly curved circular ducts at high Dean numbers are presented. Estimates for the displacement thickness of the boundary layer and the wall shear stress obtained from these data are compared with theoretical predictions.

It is shown that viscous Poiseuille flow sustains wave propagation in its centre region for waves whose phase speed is less than the maximum flow speed. When, moreover, there are critical levels, there exists a range of phase speeds for which over-reflection occurs and this range corresponds exactly to the range for which unstable eigenmodes exist. Consistent with the conjecture of Lindzen & Barker (1985), it is found that viscous boundary layers around the critical level and at the wall replace the exponential regions and wave sinks required for inviscid over-reflection.

Over-reflection, we find is confined to phase speeds for which these two boundary layers are in close proximity rather than widely separated or substantially overlapping.

Over-reflection is inevitably associated with a wave phase tilt opposite in direction to the shear at the critical level. All other cases yield a phase tilt in the direction of the shear. The former is consistent with the condition for the Orr mechanism to produce amplification (Boyd 1983).

A Lagrangian stochastic model of two-point displacements which includes explicitly the effects of molecular diffusion and viscosity is developed from the marked-particle model of Durbin (1980) and used to study the influence of these molecular processes on scalar fluctuations in stationary homogeneous turbulence. It is shown that for the homogeneous scalar field resulting from a uniform-gradient source distribution or for a cloud produced by a source large compared with the Kolmogorov microscale, the variance of scalar fluctuations $\overline{\theta^{\prime 2}}$ is independent of the molecular diffusivity for large Reynolds number Re provided that the Prandtl number Pr is finite. In these circumstances $\overline{\theta^{\prime 2}}$ can be calculated from marked-particle-pair statistics.

For source sizes that are not large compared with the Kolmogorov microscale, $\overline{\theta^{\prime 2}}$ depends explicitly on the effect of molecular diffusion under the action of the straining motion on the Kolmogorov microscale. The model is consistent with Saffman's (1960) calculation for the mean concentration near such a small source and with Townsend's (1951) measurements on small heat spots. For the fluctuation field in stationary homogeneous turbulence these small-scale processes remain important far from the source.

It is shown that dissipation of $\overline{\theta^{\prime 2}}$ is intimately connected to the process of relative dispersion and that the non-dissipative model for $\overline{\theta^{\prime 2}}$ discussed by Corrsin (1952) and Chatwin & Sullivan (1979) corresponds to the point-sample, infinite-Pr limit of the two-point theory.

The two-point model is also used to examine the effect of instrumental averaging on $\overline{\theta^{\prime 2}}$. For finite Pr, the reduction in $\overline{\theta^{\prime 2}}$ due to a fixed sampling volume is eventually negligible because the lengthscale of $\overline{\theta^{\prime 2}}$ grows with time. For infinite Pr, the linear-strain field of the turbulence on the Kolmogorov microscale generates an increasingly fine-scale structure in the scalar field. Then a finite sampling volume reduces $\overline{\theta^{\prime 2}}$ but not as much as the effect of reducing Pr to a finite value. A finite sampling volume and infinite Pr is not necessarily equivalent to a finite value of Pr and a point sample.

Timescales for important stages in the development of the scalar field in a cloud or downwind of a continuous source (for example, the onset of dissipation or the stage at which fluctuations are dominated by internal structure or ‘streakiness’ within the cloud rather than bulk motion or ‘meandering’) have been estimated. For small sources and large Re these timescales are significantly less than the integral timescale tL. Many real flows evolve on the timescale tL, so that the present results for stationary homogeneous turbulence should apply to such flows for small sources and large Re, a situation typical of the atmosphere.

A Lagrangian stochastic model is used in conjunction with detailed wind-tunnel measurements to examine the structure and development of the temperature field behind a line source in grid turbulence. It is shown that on the scale of these experiments molecular diffusion and viscosity have an important influence on temperature fluctuations (particularly on the intensity of these fluctuations) and must be explicitly modelled. The model accounts for a wide range of the measured properties of the temperature field and provides a unified treatment of temperature fluctuations through all stages of the development of the temperature field. This development is discussed in terms of a simple physical picture in which the hot plume is initially smooth and is moved about bodily by the turbulence, but gradually develops increasing internal structure or patchiness as it grows with distance downstream until a self-similar state is reached in which this internal structure maintains the temperature fluctuations.

It is shown how the effects of the initial discharge profile, vertical drift, and boundary absorption (catalytic reaction) can be incorporated into a Gaussian approximation for the two-dimensional contaminant distribution in a parallel shear flow. Exact and asymptotic expressions are derived for the centroid displacement, shear-dispersion coefficient, and variance. Detailed results are presented for the effect of absorption at the bed and of vertical drift velocities upon contaminant dispersion in turbulent open-channel flow. For both cases the advantages of discharges close to the bed over surface discharges are made quantitative.

In the Boussinesq model, which is a standard frame for the analysis of internal-wave phenomena, the fluid has variable density but is incompressible, inviscid and non-diffusive. Without further approximations, which will not be made here, the dynamical equations are nonlinear and the evolutionary problem posed cannot be solved explicitly except by numerical means; but various interesting properties are accessible. In §1, where a previous summary account is recalled (Benjamin 1984), the model is reformulated as a system of integro-differential equations in which the dependent variables are density ρ and density-weighted vorticity σ. The aim subsequently is to survey the model's mathematical consequences in general rather than to examine particular solutions. Very few exact solutions are yet known although approximate solutions are on record describing solitary and periodic waves of permanent form.

In § 2 the Hamiltonian representation of the two-component system is noted, being the key to much of the analysis that follows. The complete symmetry group for this system is given in § 3. It is composed of nine one-parameter subgroups which are listed first in Theorem 1; then their collective significance in relation to Hamiltonian structure is discussed. In § 4 two theorems are given specifying necessary and sufficient conditions for a scalar function to be a conserved density for solutions of the Boussinesq system. There are found to be basically eight such conserved densities which are listed in Theorem 4; and the corresponding conservation laws in integral form for motions between horizontal planes are stated in Theorem 5.

The meaning of impulse according to the Boussinesq model is examined in § 5. The two linear components of impulse density and the density of impulsive couple are revealed by the preceding examination of symmetries and local conservation laws; but care is needed to identify physical interpretations of the integral conservation laws that involve impulse. Two laws relating impulse to kinematic properties of the density distribution are particularly strange. Separate treatments are needed for the cases where the fluid-filled domain D is the whole of ℝ2, where D is a half-space with rigid horizontal boundary (which case is in several respects the most delicate) and where D is a horizontal infinite strip. Finally, in § 6, a variational characterization of steady wave motions is explained as a concomitant of Hamiltonian structure, and its implications concerning the stability properties of such motions are reviewed.

Appendix A notes a semi-Lagrangian formulation which has a simpler Hamiltonian structure but a narrower range of application. Appendix B outlines an alternative confirmation of Hamiltonian properties by use of a lemma due to Olver (1980b).