An investigation of the structure of jets in rotating systems is presented. The fluid is assumed to be homogeneous and the flow laminar and quasi-geostrophic. When the rate of rotation is small, the dynamics is shown to be identical to that of jets in non-rotating systems, being a balance between inertial terms and lateral dissipation. As the rate of rotation increases the Ekman layers become important and in the strongly rotating case the friction in the Ekman layers dominates lateral dissipation. The jet in a non-rotating system entrains fluid at its edges and the downstream momentum flux is independent of the distance downstream. In the strongly rotating case, however, the jet ejects fluid at its edges and the downstream momentum flux decreases with downstream distance due to dissipation in the Ekman layers. A similarity solution for the general case with both types of friction is obtained and the transition from a jet in which lateral dissipation dominates to one in which Ekman friction is more important is discussed. General features of jets in strongly rotating systems are studied and implications for the Gulf Stream are mentioned.

A unified linear viscous stability theory is developed for a certain class of stratified parallel channel and boundary-layer flows with Prandtl number equal to unity. Results are presented for plane Poiseuille flow and the asymptotic suction boundary-layer profile, which show that the asymptotic behaviour of both branches of the curve of neutral stability has a universal character. For velocity profiles without inflexion points it is found that a mode of instability disappears as η, the local Richardson number evaluated at the critical point, approaches 0.0554 from below. Calculations for Grohne's inflexion-point profile show both major and minor curves of neutral stability for 0 < η [les ] 0.0554; for \[ 0.0554 < \eta < 0.0773 \] there is only a single curve of neutral stability; and, for η > 0.0773, the curves of neutral stability become closed, with complete stabilization being achieved for a value of η of about 0·107.

Remarkable differences in dispersion of a tracer material injected into turbulent pipe flows of water and water containing as little as 2·5 parts per million by weight of a soluble high-molecular-weight drag-reducing polyoxyethylene additive have been measured. Analysis of the tracer response curves in terms of a simple one-parameter model shows that the observed results are compatible with a drag-reduction mechanism based on thickening of the viscous sublayer adjoining the wall. Other experiments, reported briefly, suggest that polymer adsorption on to the wall is responsible for this thickening.

The field radiated by an acoustic monopole in the presence of an infinite membrane, or plate, is studied, with emphasis on the case when fluid loading effects are small and when a free wave in the surface has supersonic phase speed relative to the fluid. Coupling between fluid and surface is then specified by a Mach angle θM and by a fluid loading parameter ε, with ε [Lt ] 1. Asymptotic expressions for the field are derived which are uniform in the observation angle θ, measured from the surface. Previous descriptions have suggested the formation of a strong two-dimensional beaming effect along the surface of the Mach cone θ = θM. Here it is shown that this effect is a spurious consequence of nonuniform asymptotics. A beam is indeed formed, and persists without attenuation or distortion to large distances k0R ∼ ε−2. However, the beam amplitude is small compared with that of the three-dimensional reflected field, while at distances k0R [Gt ] ε−2 only the reflected wave survives. Some interesting features of the reflexion coefficient and of the field near to the membrane are also discussed. In particular, it is shown that the pressure field generated by a subsonic surface wave is also confined to a conical zone, the transition across the generators of the cone being described by Fresnel functions of a familiar kind.

Robins showed in 1742 that a transverse aerodynamic force on a rotating sphere could be detected by suspending it as a pendulum. Differences of periodic time in conical pendulum motion with spin and orbit parallel and opposed have been found to give a reasonably accurate measure of the lift coefficient, and the results shown extend knowledge of the effect down to a Reynolds number of 2 × 103 and up to a ratio of 12 between the peripheral and translational velocities.

Two-dimensional frequency-wave-number spectra [Fcy ](kx, ω) and [Fcy ](kz, ω) of the longitudinal velocity component are presented for the sublayer in fully developed turbulent pipe flow, at Reynolds numbers between 10600 and 46400. All of these sublayer spectra apparently scale by introducing dimensionless quantities based on a chara cteristic length scale ν/UT and a characteristic time scale ν/UT2.

Representative convection velocities have been obtained from the [Fcy ](kx ω) spectra. The characteristic convection velocity in the sublayer is independent of wave-number and is the same at all positions in the layer cx ≃ 8·0UT. This result has led to the conclusion that sublayer turbulence is wave-like.

Existing visualization data seem to indicate that the sublayer waves are also relatively periodic at least at low values of Reynolds number. Characteristic dimensions of the sublayer waves are λ+x ≃ 630, and λz+ = 135. Results of the visualization studies of Fage & Townend (1932) and of Runstadler, Kline & Reynolds (1963) and Kline et al. (1967) do not appear to conflict with a wave model for the sublayer.

All of the existing measurements of the sublayer have been for relatively low Reynolds numbers. Some of the present results for positions just outside the sublayer suggest that at Reynolds numbers greater than 30000, the structure and properties will change substantially from those observed to date. In particular the streaky structure which is commonly regarded as being characteristic of the sublayer will probably not be detected at sufficiently high Reynolds numbers.

The results of an experimental investigation of separation in oscillating laminar boundary layers is reported. Instantaneous velocity profiles obtained with multiple hot-wire anemometer arrays reveal that the onset of wake formation is preceded by the initial vanishing of shear at the wall, or reverse flow, throughout the entire cycle of oscillation. Correlation of the experimental data indicates that the frequency, Reynolds number and dynamic history of the boundary layer are the dominant parameters and oscillation amplitude has a negligible effect on separation-point displacement.

The thermal boundary layer near a heated vertical plate in a poorly conducting liquid is subject to a horizontal d.c. electric field. If the electric field is strong enough, the boundary layer becomes unstable. In this paper a theory is developed to predict the onset of this instability. Experiments measuring the threshold voltage for instability are compared with the theoretical predictions. Other experiments are reported which determine the effect of this instability on the heat transferred from the heated plate.

This paper examines a region of rapidly varied flow in a two-layered density stratified system with one layer flowing, and the other stationary. The analogous phenomenon in open channel hydraulics is the hydraulic jump. In density stratified flows the phenomenon is referred to as a density jump because it is generally accompanied by a change in density of the flowing layer.

It is shown there is a fundamental difference between the hydraulic jump and the density jump in that flow conditions on either side of a density jump are not uniquely related. A density jump with given flow conditions upstream has a range of possible states which may be attained downstream. The rate of entrainment of ambient fluid into a density jump, and the conditions downstream of the jump, are determined by the downstream control and the upsteam conditions. The particular case of a density jump controlled by a broad crested weir downstream is examined in detail.

For a viscous incompressible fluid the low Reynolds number description of the flow generated by a three-dimensional point source of oscillating strength situated on a wall is approximately quasi-steady in the neighbourhood of the singularity. This quasi-steady solution contains a number of unacceptable features, the principal one being that it is not a uniformly valid approximation within a small region surrounding the source point. In addition to this the vorticity in this region is predicted to be zero everywhere on the wall except at the singular point where it is infinite, which does not seem to be a physically reasonable distribution. When account is taken of the finite radius of the hole through which the fluid is driven and the finite width of the wall, the above difficulties are resolved yielding results that are quite realistic and informative.

The response characteristics of a hot wire operated at constant temperature and exposed to a mean-velocity gradient along its length are examined both analytically and experimentally. The shear sensitivity of local wire temperature distributions, as measured with an infrared microscope, are compared with predicted temperature distributions in order to select a convective heat transfer correlation which can be applied locally along a wire in shear flow. On the basis of this correlation, the steady-state and dynamic response behaviour of platinum and tungsten wires in shear flow are examined by means of computer-generated data. Response curves of general applicability are presented which can be used to correct local mean-velocity and turbulence intensity measurements whenever a mean-velocity gradient exists along a wire.

The buoyant rise of chimney plumes is discussed for relatively large distances from the source, where atmospheric turbulence is the dominant cause of mixing (rather than turbulence due to the plume's own upward motion). A simple theory is developed which shows a number of different shapes plumes can have under different atmospheric conditions (particularly in an unstable environment).

Experimental data are then presented, which were collected in the field near Toronto, with reasonably detailed supporting information on the atmospheric temperature structure. These (and earlier results) often show quite complex plume behaviour at large distances from the source, which can, however, be readily understood in terms of the simple theory presented.

The suggestion made by the authors in a previous paper (Hurle & Jakeman 1969) that the Soret effect could give rise to overstable solutions of the thermosolutal Rayleigh–Jeffreys problem is investigated theoretically and experimentally.

Oscillatory instability is shown to occur in initially homogeneous layers of water-methanol mixtures when they are heated from below. This instability triggers a finite-amplitude steady mode. The magnitude and sign of the Soret coefficient was changed by varying the composition of the mixture; as predicted, overstable modes were observed when the sign of the coefficient was such as to produce a stabilizing contribution to the density gradient. The observed critical Rayleigh numbers and temporal frequencies are consistent with theory.

When a container of fluid of arbitrary shape is heated from below and the temperature gradient exceeds a critical value ([Rscr ]c2) the conduction solution with no motion becomes unstable and is replaced by convection. The convection may have two forms: one with ‘upflow’ at the centre of the container and one with ‘downflow’ there. Here we study the stability of the two forms of convection. Both forms are here shown to be stable to infinitesimal disturbances. When the viscosity varies with the temperature or the conduction profile is not linear, etc., the steady convection can be driven with finite amplitudes |ε| at subcritical values of the temperature contrast ([Rscr ]2 < [Rscr ]c2). This subcritical convection is stable when the convection is strong (|ε| > |ε*| > 0) but is unstable when the convection is feeble (|ε| > |ε*|). Hence, when |ε| > |ε*| and [Rscr ]2 < [Rscr ]c2 either ‘upflow’ or ‘downflow’, but not both, is stable. When [Rscr ]2 > [Rscr ]c2, however, both the ‘upflow’ and the ‘downflow’ can be stable. This contrasts with the corresponding situation which is known to hold when the container is an unbounded layer. In the layer there is only one stable form of convection. The difference between the bounded domain with two forms of convection and the layer with just one stable form is traced to the mathematical property of simplicity of [Rscr ]c2 when viewed as an eigenvalue of the linear stability problem for the conduction solution. It is argued that [Rscr ]c2 is a simple eigenvalue in most domains, but in the layer [Rscr ]c2 can have infinite multiplicity. The explanation of the transition from the bounded domain to the unbounded layer is sought (1) in the chaotic conditions which frequently prevail at the edges of a ‘bounded’ layer and (2) in the fact that in the layer of large horizontal extent, the higher eigenvalues crowd [Rscr ]c2. In the course of the explanation, a new exact solution of the linear Bénard problem in a cylinder with a rigid side wall and a stress-free top and bottom is derived.

A thermally conducting body floating in a fluid that is stably stratified by heat may develop unstable oscillations provided the body temperature shows a large enough time-lag relative to the fluid temperature. A gravitationally stable, non-Boussinesq fluid may itself become unstable in an oscillatory way through a similar diffusive time-lag. One case, that is investigated in the present work, occurs when the basic vertical temperature gradient β = dT/dz and the thermal expansion coefficient α vary in an opposite sense: β is large when α is small and vice versa. The variation in β is here assumed to be caused by internal heat sources and sinks.

If vertical oscillations are started in such a fluid temperature variations will be produced in the regions of large β but the buoyancy forces do not develop until these perturbations have diffused to regions of large α. With an appropriate lag, the buoyancy forces may give a positive work and the oscillations can grow.

Two models are investigated. The first one is a non-viscous two-layer model with β = 0, α = α0 in one layer and β = β0, α = 0 in the other layer. For this model analytical results are derived. The second model is more realistic, having continuous profiles β(z) and α(z), viscosity and horizontal boundaries. The case is studied by a numerical technique, solving the equations directly in time.

A discussion of the numerical method is given in an appendix by K. Holmaker.

Forced axial flow in an annular gap of a cylindrical rotor is investigated analytically and experimentally. At small rotation rates and narrow gap widths, the axial flow is a simple Poiseuille flow over most of the rotor. The distance required for this Poiseuille flow to get established is estimated. An instability is observed at large rotation rates with certain input geometries.

The hydrodynamic stability of an ideal mixture of two viscous, dissimilar liquids contained between two concentric rotating cylinders is analyzed. The basic flow of the mixture is determined by coupling the mass and momentum equations with an equation for the equilibrium concentration distribution. Infinitesimal, axisymmetric disturbances are assumed, and the disturbance equations are written for the limiting case of large Schmidt numbers (no diffusion).

The presence of density and viscosity variations leads to a twelfth-order eigenvalue problem with two-point boundary conditions that has the appearance of a combined Taylor and density-stratified shear flow problem. A numerical technique is devised to determine the stability boundary and to calculate Taylor numbers and oscillation frequencies for different growth rates.

It is found that very small mean density gradients alter the critical Taylor number and that oscillations occurring in both the growing and neutral solutions are the dominant mode.

The transient process by which an incompressible dissipative rotating stratified fluid adjusts to a small change in the rotation rate of its container is examined theoretically. The aim is to clarify the effects of the imposed density stratification and of the boundary condition specified for the density perturbation on the behaviour of the fluid, particularly during the time span when the adjustment is performed in a homogeneous fluid. For a weakly stratified fluid in a cylinder, it is shown how these two factors govern the nature and intensity of boundary layers on the vertical wall which close the secondary meridional circulation generated by Ekman layers along the horizontal boundaries. For a more strongly stratified fluid, the usefulness and importance of potential vorticity conservation in determining the quasi-steady motion is verified, and a calculation for a spherical container demonstrates some new features that arise only when the container boundaries are not normal or parallel to the rotation axis. It is shown that experimental results of Holton (1965) are in less good agreement with predictions of the linear theory than had been previously indicated.

A primarily experimental investigation was undertaken to determine the internal structure of steady, quasi-uniform, non-separated, axisymmetric flows in circular pipes with sinusoidal wall profiles. The quantities measured include radial and longitudinal distributions of mean velocity, pressure, and total head; the Reynolds shear stress and all three components of turbulence velocity; and boundary shear stress and pressure. Two different wall-wave steepnesses were investigated, and a constant Reynolds number of 1·13 × 105 (based on the average pipe diameter) was maintained in most experiments. The boundary shear stress was found to be shifted upstream relative to the boundary wave, whereas the wall pressure is shifted slightly downstream. The turbulence measurements revealed that there is a central core extending over some 60% of the pipe radius in which the turbulence quantities are constant along the pipe. Near the boundary, however, the turbulence velocities and stress vary periodically along the boundary waves. The longitudinal component of mean velocity was found to be distributed radially according to the power law, but with an exponent that varies along each wave; a simple analytical model is constructed to predict the variation of the exponent. It was not found possible to relate the local boundary shear stress to just the local flow characteristics, since the convective or ‘history’ effects play a significant role in its determination. An empirical formula is derived relating the local boundary shear stress to the local velocity distribution and the first two derivatives of the boundary profile.

Vapour bubble collapse problems lacking spherical symmetry are solved here using a numerical method designed especially for these problems. Viscosity and compressibility in the liquid are neglected. Two specific cases of initially spherical bubbles collapsing near a plane solid wall were simulated: a bubble initially in contact with the wall, and a bubble initially half its radius from the wall at the closest point. It is shown that the bubble develops a jet directed towards the wall rather early in the collapse history. Free surface shapes and velocities are presented at various stages in the collapse. Velocities are scaled like (Δp/ρ)½ where ρ is the density of the liquid and Δp is the constant difference between the ambient liquid pressure and the pressure in the cavity. For \[ \Delta p/\rho = 10^6 {\rm cm}^2/\sec^2 \approx 1\, \hbox{atm/density of water} \] the jet had a speed of about 130m/sec in the first case and 170m/sec in the second when it struck the opposite side of the bubble. Such jet velocities are of a magnitude which can explain cavitation damage. The jet develops so early in the bubble collapse history that compressibility effects in the liquid and the vapour are not important.