A numerical method is described which is suitable for solving the equations governing the steady motion of a viscous fluid through a slightly curved tube of circular cross-section but which is also applicable to the solution of any problem governed by the steady two-dimensional Navier–Stokes equations in the plane polar co-ordinate system. The governing equations are approximated by a scheme which yields finite-difference equations which are of second-order accuracy with respect to the grid sizes but which have associated matrices which are diagonally dominant. This makes them generally more amenable to solution by iterative techniques than the approximations obtained using standard central differences, while preserving the same order of accuracy.

The main object of the investigation is to obtain numerical results for the problem of steady flow through a curved tube which corroborate previous numerical work on this problem in view of a recent paper (Van Dyke 1978) which tends to cast doubt on the accuracy of previous calculations at moderately high values of the Dean number; this is the appropriate Reynolds-number parameter in this problem. The present calculations tend to verify the accuracy of previous results for Dean numbers up to 5000, beyond which it is difficult to obtain accurate results. Calculated properties of the flow are compared with those obtained in previous numerical work, with the predictions of boundary-layer theory for large Dean numbers and with the predictions of Van Dyke (1978).

Calculations of the flow field under laminar conditions in a helical semicircular duct have been made by numerically solving the Navier–Stokes equations. With the flat wall of the duct being the outer wall, the solution of the momentum equations for Dean numbers below 105 gave, for the secondary flow, twin counter-rotating vortices of Taylor–Goertler type. However, above a Dean number of Dn = 105, two solutions were possible. One solution was similar to that obtained for Dn < 105. The other solution revealed four vortices for the secondary flow. For Dn > 105, convergence to either flow pattern depended on the initial guess used in the numerical solution. Flow visualization confirmed the possibility of the presence of both types of secondary flow patterns.

The subject is the small-scale structure of a magnetic field in a turbulent conducting fluid, ‘small scale’ meaning lengths much smaller than the characteristic dissipative length of the turbulence. Philip Saffman developed an approximation to describe this structure and its evolution in time. Its usefulness invites a closer examination of the approximation itself and an attempt to place sharper limits on the numerical parameters that appear in the approximate correlation functions, topics to which the present paper is addressed.

A Lagrangian approach is taken, wherein one makes a Fourier decomposition of the magnetic field in a neighbourhood that follows a fluid element. If one construes the viscous-convective range narrowly, by ignoring magnetic dissipation entirely, then results for a magnetic field in two dimensions are consistent with Saffman's approximation, but in three dimensions no steady state could be found. Thus, in three dimensions, turbulent amplification seems to be more effective than Saffman's approximation implies. The cause seems to be a matter of geometry, not of correlation times or relative time scales.

Strictly-outward spectral transfer is a characteristic of Saffman's approximation, and this may be an accurate description only when dissipation suppresses the contributions from inwardly directed spectral transfer. In the spectral region where dominance passes from convection to dissipation, one can generate expressions for the parameters that arise in Saffman's approximation. Their numerical evaluation by computer simulation may enable one to sharpen the limits that Saffman had already set for those parameters.

A linear non-homogeneous analysis is presented for the standing waves produced on the hollow core of an irrotational vortex by an arbitrary obstacle on the wall of the tube containing the vortex. The group-velocity criterion based upon Kelvin's corresponding dispersion relation predicts whether a certain asymptotic wave pattern appears upstream or downstream of the obstacle. The analysis leads to amplitude singularities for the standing waves at certain critical radii of the core. The particularly interesting case of a counter-helix for which the wave energy is propagating upstream appears for a first-mode angular disturbance. For this situation it seems to be possible that the helix ends in a hydraulic jump and is continued by a counter-helix downstream, as the core size gradually diminishes due to the deceleration of the flow caused by viscous effects (not included in the analysis). The capillary-wave pattern produced by surface tension is also considered. A brief outline for the analogous wave problem is given for the case where the fluid rotates like a rigid body.

Photographic observations of hollow-core vortices in water flow are presented which confirm the qualitative predictions of the analysis, both for the response to an axisymmetric area contraction and also to a 90° bend at the downstream end of the vortex tube.

The extensional viscosity of a dilute suspension of spherical particles (rigid spheres, viscous drops or gas bubbles) is computed for the case when the Reynolds number of the microscale disturbance motion R is not restricted to be small, as in the classical analysis of Einstein and Taylor. However, the present theory is restricted to steady axisymmetric pure straining flow (uniaxial extension). The rate of energy dissipation is expressed using the Bobyleff-Forsythe formula and then conditionally convergent integrals are removed explicitly. The problem is thereby reduced to a determination of the flow around a particle, subject to pure straining at infinity, followed (for rigid particles) by an evaluation of the volume integral of the vorticity squared. In the case of fluid particles, further integrals over the volume and surface of the particle are required. In the present paper, results are obtained numerically for 1 [les ] R [les ] 1000 for a rigid sphere, for a drop whose viscosity is equal to the viscosity of the ambient fluid, and for an inviscid drop (gas bubble). For the last case, limiting results are also obtained for R → ∞ using Levich's approach.

All of these results show a strain-thickening behaviour which increases with the viscosity of the particle. The possibility of experimental verification of the results, which is complicated by the inapplicability of the approximation of material frame-indifference in this case, is discussed.

The motion of the head of a gravity current travelling down a slope of angle θ to the horizontal is investigated in the laboratory. The head is produced by suddenly initiating a buoyancy flux from a line source at the top of the slope. It is found that for very small slopes (θ [les ] 0.5°) the head decelerates with distance from the source, but at greater slopes the buoyancy force is large enough to overcome frictional effects and a steady head velocity results. Over a wide range of slope angles the front velocity Uf, non-dimensionalized by the cube root of the buoyancy flux (g′0Q)1/3, is almost independent of the slope angle and Uf/(g’0Q)1/3 = 1.5 ± 0.2 for 5° [les ] θ [les ] 90°. This result is shown to follow from some simple analysis which relates the velocity of the front to the following flow. For a Boussinesq plume the front velocity is found to be approximately 60% of the mean velocity of the following flow. This means that the head increases in size as it travels down the slope, both by direct entrainment into the head itself and by addition of fluid from the following flow. We find that direct entrainment increases with increasing slope and accounts for one-tenth of the growth of the head at 10° and about two-thirds at 90°.

The effect of homogeneous strain on passive scalar fluctuations, and the resultant evolution of the scalar field when the strain is removed, is experimentally studied by passing thermal fluctuations in decaying grid turbulence through a four-to-one axisymmetric contraction. Using a mandoline (Warhaft & Lumley 1978a) to vary the scale size of the initial thermal fluctuations and hence the pre-contraction mechanical/thermal time-scale ratio, r, it is shown, for values of r greater than unity, that as r is increased so is the post-contraction thermal decay rate, i.e. the contraction does not cause the thermal-fluctuation decay rate to equilibrate to a constant value. In these experiments the post-contraction thermal decay rate is always greater than the pre-contraction decay rate, i.e. the contraction accelerates the thermal-fluctuation decay. Moreover, the mechanical/thermal time-scale ratio in the post-contraction region is driven further from unity. In terms of scale size the uniform strain has the effect of increasing the thermal length scale by an amount equal in value to the contraction ratio if the pre-contraction thermal length scale is comparable to that of the pre-contraction velocity scale. However, if the pre-contraction thermal length scale is smaller than the pre-contraction velocity scale then the effect of the contraction on the thermal scale is less marked. The contraction induces significant negative cross-correlation ρuθ between the longitudinal velocity u and thermal fluctuations θ even if the pre-contraction cross-correlation is close to zero. The magnitude of ρuθ and hence the post-contraction heat flux is varied and the coherence structure is studied. It is shown that the thermal-fluctuation decay rate is insensitive to the magnitude of the heat flux, the latter of which decays rapidly compared to the relatively slow decay of turbulence energy in the post-contraction region. It is also shown that ρuθ tends towards zero in this axisymmetric homogeneous flow at a faster rate than in isotropic turbulence. In accord with previous investigations, the return toward isotropy of the velocity field is very slow.

A simplified model problem has recently been suggested by Schaeffer (1980) in order to explain the results obtained by Benjamin (1978) in his investigation of Taylor vortices in short cylinders. In particular Schaeffer reproduces the results obtained by Benjamin for cylinders so short that only two-cell or four-cell flows are possible. The model given by Schaeffer has artificial conditions imposed on the fluid velocity field at the end walls. These conditions depend on a parameter α and reduce to the no-slip condition when α = 1. If α = 0 the conditions require that the normal component of the velocity and the normal derivative of the tangential velocity vanish at the ends. In this case the onset of Taylor vortex-like motion occurs as a bifurcation from purely circumferential flow. If α is now taken to be small and positive, there is no bifurcation and the circulatory flow develops smoothly. We shall use perturbations method for the case of small α. The imperfect bifurcation problem which we obtain predicts some results consistent with those of Benjamin.

A two-dimensional stability analysis is presented of flow of low Froude number over an erodible bed. Particular regard is given to the modelling of the turbulent flow close to the bed. In contrast to previous theories that use a constant eddy-viscosity approach the present theory predicts the occurrence of two separate modes of instability, with wavelengths related to the roughness of the bed and the depth of the flow. It is postulated that these two modes correspond to the formation of ripples and dunes respectively. The results are strongly dependent on the two parameters z0, the roughness length of the bed, and β, the effect of the local bed slope on the bed-load transport. Using physically plausible estimates for these parameters the results of the analysis are in good agreement with observations for both ripples and dunes.

Under some conditions of temperature and flow an ice-water interface in the presence of a turbulent stream has been observed to be unstable. In this paper the source and the conditions for the instability were investigated for a well-defined turbulent boundary-layer flow. It was found that the instability resulted from the interaction that occurs between a wavy surface and a turbulent flow over it. Such an interaction results in a heat transfer variation which is 90 to 180 degrees out of phase with the surface wave shape – a result which is consistent with the calculations of Thorsness & Hanratty (1979a, b).

The main factor controlling damping of the instability at an ice-water interface was found to be the rate at which heat is conducted away from the interface into the ice.

In the past it has been found that when an ice layer is melting, that is when the heat conduction in the ice is small, the ice surface is highly unstable. In the present study it was found that for a sufficiently large temperature ratio (Tf−Tw)/(T∞−Tf), a steady-state ice layer is also unstable. Furthermore it is predicted, from the present observations, that a growing ice layer with a ratio of ice-side to water-side heat fluxes of up to 2.3 could be unstable.

Under sufficiently unstable conditions waves on the ice surface grow to an amplitude at which flow separations occur near the wave crests. This results in a ‘rippled’ ice surface pattern very similar to the patterns observed on mobile bed surfaces (Kennedy 1969) or surfaces which are being dissolved into a flowing stream (Allen 1971). The development of a ‘rippled’ ice surface results in a very substantial increase in the mean heat-transfer rate which would have an important influence on predictions of ice formation in the presence of a turbulent stream.

The 120th Euromech Colloquium on experimental techniques pertaining to sizing and tracking of submicron- to millimetre-sized particles suspended in fluids was held at Lyngby on 20–22 August 1979 with forty-one participants from ten countries. Twenty-seven papers were presented during the two-and-a-half day meeting. Techniques included optical and acoustical sizing and velocity measurements, tracking, sedimentation and deposition of particles in various gas and liquid flows.

General relations are derived for the forcing of a mean zonal flow in a damped rotating barotropic system through the action of weak eddies. In particular it is found that if the eddies are forced at localized latitudes the induced motion away from these latitudes is likely to be counter to the rotation (i.e. ‘easterly’). Over the forcing latitude the mean motion is always westerly except when the forcing provides a sink of relative momentum. In the case when the background field is deformed topographically to generate eddies the divergence of the Reynolds stress is balanced at lowest order by a dynamic pressure drag, and the mean motion takes the direction of propagation of the forcing.

The relations are applied to a linearized Rossby wave field in a viscous fluid driven by a moving system of boundary sources and sinks or hills and hollows. The results are compared with laboratory experiments. All major predictions are confirmed qualitatively, but the discrepancies in detail indicate the influence of nonlinear effects other than those incorporated in the theory.