Experiments on transition to turbulence in a purely oscillatory pipe flow were performed for values of the Reynolds number Rδ, defined using the Stokes-layer thickness δ = (2ν/ω)½ and the cross-sectional mean velocity amplitude Û, from 19 to 1530 (or for values of the Reynolds number Re, defined using the pipe diameter d and Û, from 105 to 5830) and for values of the Stokes parameter λ = ½d(ω/2ν)½ (ν = kinematic viscosity and ω = angular frequency) from 1·35 to 6·19. Three types of turbulent flow regime have been detected: weakly turbulent flow, conditionally turbulent flow and fully turbulent flow. Demarcation of the flow regimes is possible on Rλ, λ or Re, λ diagrams. The critical Reynolds number of the first transition decreases as the Stokes parameter increases. In the conditionally turbulent flow, turbulence is generated suddenly in the decelerating phase and the profile of the velocity distribution changes drastically. In the accelerating phase, the flow recovers to laminar. This type of partially turbulent flow persists even at Reynolds numbers as high as Re = 5830 if the value of the Stokes parameter is high.

The ideas of Lagrange, Poisson, Kelvin and Truesdell are reviewed. It is shown that in order for a bounding surface not to be a material surface either u. n = c must fail or more than one deformation can be associated with the velocity field. Examples are given.

Critical speeds for the onset of Taylor vortices and for the later development of wavy vortices have been determined from torque measurements and visual observations on concentric cylinders of radius ratios R1/R2 = 0·894–0·954 for a range of values of the clearance c and length L: c/R1 = 0·0478–0·119 and L/c = 1–107. Effectively zero variation of the Taylor critical speed with annulus length was observed. The speed at the onset of wavy vortices was found to increase considerably as the annulus length was reduced and theoretical predictions are realistic only for L/c values exceeding say 40. The results were similar for all four clearance ratios examined. Preliminary measurements on eccentrically positioned cylinders with c/R1 = 0·119 showed corresponding effects.

A hot-film anemometer has been used to investigate the mean velocity and turbulence intensity distributions in turbulent radial jets. A geometric parameter termed the constraint ratio, defined as the ratio of nozzle diameter to separation distance, is shown to characterize radial-jet behaviour. Large values of the constraint ratio typify ‘constrained’ radial jets, for which the nozzle walls constrain the flow leaving the orifice to be parallel; a small constraint ratio is representative of two opposing free axisymmetric jets, the collision of which produces an ‘impinged’ radial jet. It is found that the well-behaved constrained radial jet spreads at the same rate as does the familiar plane jet, whereas the impinged radial jet spreads at a rate more than three times as fast. Neither type of radial jet is amenable to a self-similar analytic solution; however, while the impinged jet is shown to require numerical solution techniques, an empirical solution for the constrained jet is demonstrated.

A conditional averaging technique to extract the underlying vortex pattern from a turbulent bluff body wake is described. Ensemble averages of wake velocities are developed on the basis of a reference phase position, determined from the outer flow irrotational fluctuations. The method is applied to the wakes of a stationary and oscillating D-shape cylinder, where, in the latter case, the vortex shedding is locked to the frequency of body movement. Direct comparisons of average circulation and vortex street spacings are obtained and these demonstrate the significant change in wake structure that accompanies and sustains vortex-induced vibrations. It is observed in both conditions that only 25% of the estimated shed vorticity is found in the fully developed wake. In addition the analysis produces profiles of vorticity and velocity in an ‘average vortex cycle’. A model, developed to help interpret these results, suggests that a good representation of an average wake situation is obtained by the addition of considerable mean shear to a street of finite area axisymmetric vortices.

A previous analysis of the acoustic radiation from multipole sources is extended to include additional components of the dipole and quadrupole sources. It is found that, unlike the components of the sources considered in the previous paper, the exponent of the Doppler factor now depends on the location of the sources within the jet.

The linear stability of modulated circular Couette flow to axisymmetric disturbances is examined in the narrow-gap limit. The outer cylinder is assumed stationary, while the inner is modulated both with and without a mean rotation. The equations governing the disturbance motion are solved by a Galerkin expansion with time-dependent coefficients, and the stability of the motion determined by Floquet theory. Modulation is found, in general, to destabilize the flow due to steady rotation, although weak stabilization is found for some modulation amplitudes at intermediate frequencies.

The Lagrangian and Hamiltonian for nonlinear gravity waves in a cylindrical basin are constructed in terms of the generalized co-ordinates of the free-surface displacement, {qn(t)} ≡ q, thereby reducing the continuum-mechanics problem to one in classical mechanics. This requires a preliminary description, in terms of q, of the fluid motion beneath the free surface, which kinematical boundary-value problem is solved through a variational formulation and the truncation and inversion of an infinite matrix. The results are applied to weakly coupled oscillations, using the time-averaged Lagrangian, and to resonantly coupled oscillations, using Poincaré's action—angle formulation. The general formulation provides for excitation through either horizontal or vertical translation of the basin and for dissipation. Detailed results are given for free and forced oscillations of two, resonantly coupled degrees of freedom.

The viscous damping of cnoidal waves progressing over a smooth horizontal bed is investigated. First approximations are derived for the attenuation of wave height with distance and for the friction coefficient at the bed. Attenuation coefficients are larger than those predicted on the basis of shallow-water sinusoidal wave theory and, unlike the case of sinusoidal waves, they are not independent of wave height. The limiting case of the solitary wave, considered previously by Keulegan (1948), is also discussed.

This paper considers the general problem of laminar, steady, horizontal, Oseen flow at large distances upstream and downstream of a two-dimensional body which is represented as a line source of horizontal or vertical momentum, or as a line heat source or heat dipole. The fluid is assumed to be incompressible, diffusive, viscous and stably stratified. The analysis is focused on the general properties of the horizontal velocity component, as well as on explicit calculation of the horizontal velocity profiles and disturbance stream-function fields for varying degrees of stratification. For stable stratifications, the flow fields for all four types of singularities exhibit the common feature of multiple recirculating rotors of finite thicknesses, which leads to an alternating jet structure both upstream and downstream for the horizontal velocity component and to leewaves downstream in the overall flow. The self-similar formulae for the velocity, temperature and pressure at very large distances upstream and downstream are also derived and compared with the Oseen solutions.

A body is started impulsively from rest and moves on an arbitrary path in an incompressible, stably stratified, rotating fluid. The phase configuration of the waves which are generated is studied using small amplitude wave theory. The theory is compared with experiment for a few special cases which include a horizontal cylinder (a) oscillating about a position fixed in the fluid, (b) moving with constant velocity and (c) moving with a constant angular velocity in a circular path relative to the fluid. Theory and experiment show reasonable agreement except where wakes interfere with the wave pattern.

Hot-wire anemometer measurements have been made in a dilute sea-water/claymineral suspension. For fully developed turbulent flows in an open channel with a smooth mud (from the North Sea) bottom, mean streamwise velocity profiles were measured for Reynolds numbers between 5400 and 27 800 (i.e. non-eroding and eroding flow rates) and compared with Newtonian flows under the same experimental conditions. For the clay-mineral suspensions, measurements of the kinematic viscosity v, Kármán's constant k and the mean streamwise velocity $\overline{u}$ of the logarithmic layer seemed to verify a Newtonian flow structure. Although the distributions of concentration showed no substantial increase towards the wall, it was found that beneath this Newtonian core there existed a viscous sublayer whose thickness was enhanced by a factor of 2–5. The friction velocity u* determined by the gradient method in the viscous sublayer was reduced by as much as 40 %. The mean flow structure exhibited an additional new layer in the region 10 < y+ < 30.

The measurements indicate that turbulent-drag reduction occurs for the experimental clay-mineral suspension at non-eroding and also at eroding velocities. Agglomeration of suspended clay-mineral particles is suggested as possible explanation of this phenomenon.

A boundary-layer model, based on computational results, describes a number of features of two-dimensional convection in a porous medium: the heat flux, velocity, length and temperature scales and the pattern of flow. The cell structure is different from that for convection in a viscous fluid. The model is valid for a limited range of the Rayleigh number.

The diffusion of slowly varying, weak magnetic fields by a statistically isotropic and stationary velocity field in a perfectly conducting fluid is studied by Eulerian analysis. The characteristic wavenumber and variance of the velocity field are k0 and 3v20, thus defining the eddy-circulation time τ0 = 1/v0k0. The velocity field is assumed constant on intervals of duration 2τ1 and statistically independent for distinct intervals. Thus the correlation time is τ1. The α-effect dynamo mechanism in the quasi-linear approximation is corroborated. Both the quasilinear and the direct-interaction approximations give identical diffusion of magnetic and passive scalar fields in reflexionally invariant turbulence. This result is found to be exact for τ1/τ0 → 0 but is demonstrated to be incorrect in general for finite τ1/τ0 because of effects of helicity fluctuations. The nature of the failure of the direct-interaction approximation is exhibited by an exactly soluble model system. Analysis based on a double-averaging device shows that longrange, persistent helicity fluctuations in reflexionally invariant turbulence give an anomalous negative contribution to the magnetic diffusivity which depends on the helicity covariance function. We term this the α2 effect. The magnitude of the effect depends sensitively on the turbulence statistics. If the characteristic scales of the helicity fluctuations are sufficiently larger than τ0 and 1/k0, the magnetic diffusivity is negative, implying unstable growth, while a passive scalar field diffuses normally. On the other hand, a crude estimate suggests that the α2 effect is small in normally distributed turbulence.

This paper examines large-scale nonlinear thermal convection in a rotating selfgravitating sphere of Boussinesq fluid containing a uniform distribution of heat sources. Conservative finite-difference forms of the equations of axisymmetric laminar motion are marched forward in time. The surface is assumed to be stress free and at constant temperature. Numerical solutions are obtained for Taylor numbers in the range 0 ≤ Λ ≤ 104 and Rayleigh numbers with \[ R_c \leqslant R\lesssim 10R_c. \] For high Prandtl number (P > 5) the solutions are steady and most of them resemble the solutions of the linear stability equations, though other steady solutions are also found. For P [lsim ] 1, the steady solutions have horizontal wavenumber l = 1 and nearly uniform angular momentum per unit mass, rather than nearly uniform angular velocity. This rotation law seems to be independent of many details of the model and may hold in the convective core of a rotating star.

A study has been made of the wake of a cylinder vibrating in line with an incident steady flow. The Reynolds number for the experiments was 190, and the vortex shedding was at all times synchronized with the vibrations of the cylinder, which were in a range of frequencies near twice the Strouhal shedding frequency for the stationary cylinder. Two distinct vortex wake patterns were encountered. The first is a complex regime in which two vortices are shed during each cycle of the vibration and form an alternating pattern of vortex pairs downstream. The second pattern is an alternating street which results from the shedding of a single vortex during each cycle of the cylinder's motion. The street geometry in the latter case shares many basic characteristics with the wake of a cylinder vibrating in cross-flow. These include the effects of vibration amplitude and frequency on the longitudinal and transverse spacing of the vortices. The results obtained from these experiments in air are in agreement with previous findings from free- and forced-vibration experiments in water at both higher and lower Reynolds numbers.

The problem of a uniform transverse flow past a prolate spheroid of arbitrary aspect ratio at low Reynolds numbers has been analysed by the method of matched asymptotic expansions. The solution is found to depend on two Reynolds numbers, one based on the semi-minor axis b, Rb = Ub/v, and the other on the semi-major axis a, Ra = Ua/v (U being the free-stream velocity at infinity, which is perpendicular to the major axis of the spheroid, and v the kinematic viscosity of the fluid). A drag formula is obtained for small values of Rb and arbitrary values of Ra. When Ra is also small, the present drag formula reduces to the Oberbeck (1876) result for Stokes flow past a spheroid, and it gives the Oseen (1910) drag for an infinitely long cylinder when Ra tends to infinity. This result thus provides a clear physical picture and explanation of the ‘Stokes paradox’ known in viscous flow theory.

When evaporation takes place in a surrounding vacuum, the expanding flow from a cylindrical surface is expected to start subsonic and to become supersonic in a short distance. A detailed treatment of this transition is given based on moment equations derived from the BGK model equation using an ellipsoidal approximant to the distribution function. Asymptotic solutions are developed for large source Reynolds numbers and compared with previous treatments. For moderate source Reynolds numbers a numerical procedure is used. In the latter case the treatment predicts that the flow never approaches a state of translational equilibrium.

This paper presents the results of experimental studies of the behaviour of different fluids (mercury, Pr = 0·0246; water, Pr = 7·16; and 5 cS silicone oil, Pr = 63) when each is contained in a rotating cylindrical annulus and subjected to an imposed radial temperature difference across the annulus. The results are summarized in the form of two-parameter regime diagrams (thermal Rossby number RoTvs. Taylor number Ta) at different Prandtl numbers Pr. The fluids are in contact above and below with rigid insulating boundaries. The range of thermal Rossby and Taylor numbers surveyed is 10−3 < RoT < 10, 105 < Ta < 109.

The regime diagram for water with a rigid upper lid in contact with the fluid resembles in certain respects the more familiar regime diagram for water with a free upper surface obtained by Fultz (1956) and by Fowlis & Hide (1965). Significant differences do, however, occur and are discussed in a separate paper by Fein (1973). The regime diagram for silicone oil possesses no lower symmetric regime within the range of thermal Rossby and Taylor numbers surveyed; that for mercury possesses no upper symmetric regime within the range surveyed. In mercury, turbulence is observed at high Rossby numbers. An experimental traverse across the regime diagram in which the imposed temperature difference is held constant at ΔT = 5 °C and the rotation rate Ω is changed monotonically reveals a highly conductive temperature structure in mercury and a highly convective temperature structure in both water and silicone oil. The regular wave regime, which appears at all three Prandtl numbers, is found to shift towards higher Taylor and Rossby numbers with decreasing Prandtl number. As a result, a single point in two-dimensionless-parameter space (RoT = 5 × 10−1, Ta = 1 × 106) lies in the upper symmetric regime for 5 cS silicone oil (Pr = 63), in the regular wave regime for water (Pr = 7·16) and in the lower symmetric regime for mercury (Pr = 0·0246).

The axisymmetric streaming Stokes flow past a body which contains a surface concave to the fluid is considered for the simplest geometry, namely, a spherical cap. It is found that a vortex ring is attached to the concave surface of the cap regardless of whether the oncoming flow is positive or negative. A stream surface ψ = 0 divides the vortex from the mainstream flow, and a detailed description of the flow is given for the hemispherical cup. The local velocity and stress in the vicinity of the rim are expressed in terms of local co-ordinates.