Although the form and dimensions of steep vortex ripples are well studied in relation to the oscillating flow which generates them, nevertheless the accompanying fluid motion is not yet understood quantitatively. In this paper we present a method of calculation based on the assumption that the sand-water interface is fixed and that the effect of sand in suspension is, to a first approximation, negligible.

The method employs a simple conformal transformation of the fluid flow onto the exterior of a polygon, and thence onto the interior of a unit circle. The initial, irrotational flow is represented by a logarithmic vortex at the centre of the circle. Other vortices within the fluid are each represented by a symmetric system of P vortices and their images in the unit circle, P being the number of sides of the original polygon. Typically P is equal to 5. However, P is not limited to integer values but may be any rational number greater than 2 (see § 15).

To proceed with the calculation it is assumed that separation of the boundary layer takes place at the sharp crests of the ripples, and that the shed vorticity can be represented by discrete vortices, with strengths given by Prandtl's rule. (For a typical time sequence see figures 7 and 8.) After a complete cycle, a vortex pair is formed, which can escape upwards from the neighbourhood of the boundary.

The total momentum per ripple wavelength and the horizontal force on the bottom are expressible very simply in terms of the shed vortices at any instant. The force consists of two parts: an added-mass term which dissipates no energy, and a ‘vortex drag’, which extracts energy from the oscillating flow.

The calculation is at first carried out with point vortices, in a virtually inviscid theory. However, it is found appropriate to assume that each vortex has a solid core whose radius expands with time like [ε(t − tn)]½, where tn denotes the time of birth, and ε is a small parameter analogous to a viscosity. The expansion of the vortex tends to reduce the total energy (which otherwise would increase without limit) at a rate independent of ε. If the cores of two neighbouring vortices overlap they are assumed to merge, by certain simple rules.

Calculation of the effective vortex drag in an oscillating flow yields drag coefficients $\overline{C}_D$ of the order of 10−1, in good agreement with the measurements of Bagnold (1946) and of Carstens, Nielson & Altinbilek (1969). The tendency for the highest drag coefficients to occur when the ratio 2a/L of the total horizontal excursion of the particles to the ripple length is about 1·5 is confirmed. When 2a/L = 4, the drag falls to about half its value at ‘resonance’.

Time-exposure photographs of the buoyancy-driven flow adjacent to a vertical ice surface melting in pure water are presented for ambient water temperatures between 3·9 and 8·4 °C. These conditions are of special interest since between about 4 and 8 °C the buoyancy force distribution is locally bi-directional across the thermal transport region, owing to the density extremum at about 4 °C. Although knowledge of the transport for such circumstances is important in both environmental and technological applications, previous experimental studies have provided only limited information, concerning only the gross aspects of the resulting fluid motions. It is now possible, from the extensive experimental results presented here, to understand some of the more subtle mechanisms which arise in such flows. Photographs of the entire flow field document the many complicated flow configurations which occur for these circumstances. As the ambient water temperature is increased from 3·9 to 8·4 °C, regimes of upward, locally bi-directional, and downward flow are observed. Bi-directional flow is seen to result from the reversal of part or all of the upward wake above the top of the ice surface. Local velocities and surface heat-transfer rates, measured from the photographs, are compared, where possible, with the analytical results of Carey, Gebhart & Mollendorf (1980). It was found that the flow velocities for ambient temperatures from 4·05 to 4·70 °C, depart only slightly from the analytical predictions, in spite of an interaction of the outer portion of the upward flow with downward-moving remnants of the upward wake. However, near the surface, the velocity profiles, and consequently the heat transfer and melting rates, agree well with the analytical results. Comparisons at ambient temperatures below 4·05 °C and above 5·9 °C show that the velocity profile and surface heat transfer are in excellent agreement with the results of Carey et al. (1980). At ambient water temperatures between 4·7 and 5·0 °C the flow was found to be bi-directional and weakly time dependent. This weak time dependence is attributed to the instability of the upward wake, which in turn is the source of the outside downward flow in such circumstances.

An investigation is carried out into the structure of the laminar boundary layer originating from the forward stagnation point of a prolate spheroid at incidence in a uniform stream, assuming that the external velocity distribution is given by attached potential theory. The principal new results of the study are:

1. A new transformation of the body co-ordinates is devised which facilitates the computation of the solution near the nose.

2. Two variations of the standard box method of solving the equations are devised to enable solutions to be computed in regions of cross-flow reversal. They are referred to as the zigzag box and the characteristic box.

3. Whereas in two-dimensional flows the effect of the boundary layer approaching separation on the external flow may be represented by a blowing velocity, in the present study we find that this is only true near the windward line of symmetry. Near the leeward line of symmetry the blowing velocity must be replaced by a suction velocity even though the boundary layer is being significantly thickened.

4. The boundary layer over the whole of the spheroid cannot be computed in an integration from the forward stagnation point. The accessible region is largely bounded by the separation line, if α ≤ 6°, and develops a wedge-like shape whose apex is named the accessibility ok, pointing towards the nose of the spheroid. On the windward side of this line the solution develops a singularity; on the leeward side the situation is less clear but it is also believed to occur there.

5. For α ≥ 15° the accessible region on the leeward side of the ok is largely determined by the external streamline through the ok.

Numerical results are presented for the flow field in a rapidly rotating gas. These results are compared with the predictions of asymptotic theories, particularly those of Brouwers concerning viscous effects. The comparison shows that the effect of the viscous core is important and the extent of the different flow regimes is well predicted by the theory of Brouwers.

Kinematic and dynamic conditions for the existence, or otherwise, of viscous eddies due to point, ring or a line distribution of stokeslets near no-slip boundaries are investigated. Boundaries considered are (i) a single plane boundary, (ii) two parallel plane boundaries, (iii) an infinite cylinder, and (iv) a finite cylinder. It is found that the following constraints on the fluid lead to the existence of eddies (i) a zero flux condition, (ii) confinement due to boundaries, (iii) streamline convergence near the singularity, and (iv) the interaction of flow fields due to adjacent stokeslets. The existence or non-existence of various viscous eddies is illustrated and discussed in detail for the case of infinite line distributions of stokeslets (i.e. a two-dimensional stokeslet). The paper suggests that flow fields produced by sessile micro-organisms are determined primarily by the container geometry in which they are located.

A method of renormalized expansions in theory of turbulence is developed with the use of the Lagrangian position function. The introduction of this function makes it easy to express the Lagrangian development of the velocity field. A simple truncation of a set of renormalized expansions is shown to lead to an approximation which is compatible with Kolmogorov's inertial range energy spectrum.

A method of time-dependent conformal mapping is introduced to simplify the power-series solution procedure for time- and space-periodic standing waves in deep water. A solution has been found to 25th order in the wave amplitude. The values of certain coefficients are determined by the requirement that secular terms must be suppressed. Because the series for the wave profile is not always uniformly convergent, Padé approximants are used for summation. For very high waves, the slope of the surface has at least two relative maxima. The singularity structure of the solution is also discussed.

An optical method for the direct measurement of vorticity in liquid flows is described. At the present state of development it is capable of responding to vorticity fluctuations with a response time of about 1 msec and a spatial resolution of better than 50 μm. Small spherical particles suspended in the flow rotate with angular velocity accurately equal to half the local vorticity; thus measurements of the rotation rates of such particles indicate the vorticity. Transparent spherical particles of less than 50 μm diameter, each containing embedded planar crystal mirrors, have been developed for this purpose and are suspended in a refractive-index-matched liquid. Measurements of the times required for laser reflections from the mirrors to rotate through the small angle defined by a pair of slits yields the rotation rate, and thus the vorticity. Production and physical properties of the probe particles are reported. Theoretical capabilities and limitations of the method, including accuracy, spatial and temporal resolution, data rate, and background noise are calculated and found to be coupled to the optical geometry and flow field. Analysis yields procedures for selective optimization of each parameter as dictated by the particular application. Measurements of steady-state, laminar, two-dimensional Poiseuille flows demonstrate the effectiveness of the technique and confirm theoretical predictions.

Results of hot-wire measurements in an incompressible rectangular jet issuing into a quiet surrounding at ambient conditions are presented. The quantities measured include distributions of mean velocity and the turbulence shear stresses in the two central planes of the jet at stations up to 115 widths (small dimension of the nozzle) downstream of the nozzle exit. The flow field of the jet was found to be characterized by the presence of three distinct regions, defined by the axial mean velocity decay, which are referred to as: a potential core region, a two-dimensional-type region, and an axisymmetric-type region. The onset of the axisymmetric region occurs at a downstream location where the two shear layers from the short edges of the nozzle meet. In the central plane which contains the small dimension of the nozzle, similarity was found both in the mean velocity and shear stress profiles beyond 30 widths downstream of the nozzle exit; however, profiles of r.m.s. velocity show similarity in the second but not the third region. The mean velocity, shear stress and r.m.s. velocity profiles in the central plane containing the long dimension of the nozzle do not show geometrical similarity.

Exact, positive definite expressions both in power series and in closed form are derived for the potential energy density in a continuously stratified incompressible fluid. They are useful for determining the energy of large displacements or mixing processes in regions of rapidly varying buoyancy frequency, such as a sharp pycnocline.

An exact, local, positive definite expression is obtained for the potential energy density in a wide class of stratified compressible fluids. This expression is an extension of that derived for incompressible stratified fluids in the preceding paper by Holliday & McIntyre (1981), and also represents a finite-amplitude analogue of the disturbance potential energy density that is familiar in small-amplitude theory. Its volume integral reduces to Lorenz’ (1955) available potential energy under suitable choice of a hydrostatic reference state, provided that the fluid is contained within a fixed volume enclosed by rigid impermeable boundaries.

The nominally self-preserving mixing layers that originate at the upper and lower lips of a ‘two-dimensional’ jet nozzle meet at a downstream distance x equal to about 6 times the nozzle height h, but self-preservation, this time as a fully-developed jet, is not regained until x = 20h or more. In the intervening region, and possibly in the fully-developed jet, the flow can be thought of as two interacting mixing layers. Dimensional quantities like the mean velocity and Reynolds stresses are of course altered by the interaction, but it may be expected that the dimensionless structure parameters of the turbulence will be less affected.

In interacting turbulent boundary layers in plane duct flow (Dean & Bradshaw 1976) the turbulence structure of each boundary layer is not significantly altered by the interaction, which means that even a fully-developed duct flow can be predicted by a calculation method using empirical data derived solely from isolated boundary layers (Bradshaw, Dean & McEligot 1973). Similar calculations for jets and wakes (Morel & Torda 1973) were less satisfactory, and implied significant structural changes due to the interaction.

The present experiments were designed to investigate the structural changes in a jet in still air. Fluid originating from one mixing layer was permanently marked by heating, and contributions to turbulence properties were measured separately for the instantaneously ‘hot’ and ‘cold’ zones, as done by Dean & Bradshaw and by other workers. Differences between hot-zone structure parameters in the isolated mixing layers and in the interacting region could then be deduced. The results show that near the centre-line the behaviour of the triple velocity products that effect turbulent transport of Reynolds stress is greatly altered by the interaction, the implication being that the large eddies from either shear layer do not ‘time share’ near the plane of symmetry in the simple way they appear to do in the duct. However these changes near the centre line seem not to extend to the maximum-shear region, and calculations using the superposition procedure, with a more refined basic turbulence model than that used by Morel & Torda, are in quite good agreement with experiment. Further improvement to this or any other calculation method would probably require the introduction of a transport equation for the triple product $\overline{uv^2}$ appearing in the transport equation for $\overline{uv}$.

A rivulet is a narrow stream of liquid located on a solid surface and sharing a curved interface with the surrounding gas. Long-wave instabilities are investigated for flat rivulets on a vertical wall. Three cases of contact-line conditions are investigated: (i) fixed contact lines, (ii) moving contact lines having fixed contact angles and (iii) moving contact lines whose contact angles vary smoothly with contact-line speeds. For case (i), the straight, unidirectional rivulet is stable below a critical Reynolds number. For case (ii), the rivulet is unconditionally unstable. For case (iii), the variation, G1 ≡ [dα/dUUCL = 0, in contact angle α with contact-line speed UCL stabilizes the rivulet and if G1 is large enough can completely stabilize the flow. The analysis lends support to the idea that contact-angle steepening with contact-line motion is a purely dissipative process.

A semipermeable membrane forms part of a plane boundary which separates pure solvent from a solution of concentration Cb. Stirring motions in the solution cause it to flow parallel to the plane with uniform wall shear rate α. The velocity of osmotic solvent flow across the membrane into the solution at any point is PΔC, where P is the osmotic permeability of the membrane and ΔC is the local concentration difference across it. ΔC is reduced below Cb by a position-dependent factor γ because of the concentration boundary layer over the membrane, which is thicker than in the absence of osmosis as a result of advection by the osmotic flow itself. The concentration boundary layer is analysed, on the assumption that it is two-dimensional, for both small and large values of the dimensionless longitudinal co-ordinate \[ \xi = PC_b (9x/\alpha D^2)^{\frac{1}{3}}, \] where x is the dimensional co-ordinate and D is the solute diffusivity. These expansions are used to compute γι, the average value of the flux-reduction factor γ over the whole membrane, as a function of β′, which is the value of ξ at the downstream end of the layer, x = l. It is shown that the standard physiological model, in which the layer has a given thickness δ and the stirring motions are not explicitly considered, gives accurate results for γl as a function of β = PCb δ/D as long as δ is given by \[ \delta = 1.22(Dl/\alpha)^{\frac{1}{3}}, \] so that β is proportional to β′.

Flow visualization studies of the zero-pressure-gradient turbulent boundary layer over the Reynolds-number range 500 < Reθ < 17500 have shown large Reynolds-number effects on boundary-layer structure.

At high Reynolds numbers (Reθ > 2000, say) the layer appears to consist very largely of elongated hairpin vortices or vortex pairs, originating in the wall region and extending through a large part of the boundary-layer thickness or beyond it; they are for the most part inclined to the wall at a characteristic angle in the region of 40–50°. Large-scale features, which exhibit a slow overturning motion, appear to consist mainly of random arrays of such hairpin vortices, although there is some evidence of more systematic structures.

At low Reynolds numbers (Reθ < 800, say) the hairpin vortices are very much less elongated and are better described as horseshoe vortices or vortex loops; large-scale features now consist simply of isolated vortex loops (at the very lowest Reynolds numbers), or of several such loops interacting strongly, and show a relatively brisk rate of rotation.

A theoretical and experimental study is carried out for the problem of the wall effect experienced by a fluid body moving with a constant speed along the axis of a vertical circular tube filled with a highly viscous liquid. In the theoretical study the body is limited to being either spherical or cylindrical and an optimization process with least squares is used to write the no-slip condition on the tube wall. Comparisons between the hydrodynamic and kinematic behaviour of a rigid, liquid and gaseous body are established. Furthermore, from an experimental investigation, based upon a fine visualization technique and rising-speed measurements, the respective limits of validity of the calculations have been found in the case of an air bubble. Information concerned especially with the shape of this bubble, and the hydrodynamic field that it generates, is given for the whole domain of the bubble and tube diameter ratio ranging from no wall influence to maximum wall influence.

Intermittency effects, in magnitude comparable to the early Batchelor & Townsend (1949) experiments, are studied for stationary, homogeneous, isotropic turbulence by means of a direct spectral simulation on a 643 lattice. The turbulence is kept stationary by a coupling to modes external to the spectral code that model the straining effects of large scales on smaller ones. The rate of energy input and viscosity are free parameters. The interrelations of intermittency and parametrizations of the large scales are discussed. Small-scale universality and a local cascade are necessary if comprehensive models of the large scales are to prove tractable. An iterative method to determine the otherwise arbitrary parameters in such a scheme is proposed but not implemented.

The equations for energy and vorticity balance are checked as a function of wave-number. The nonlinear (e.g. vortex stretching) terms in the spectral simulation account for nearly 95% of the vorticity production with the external forcing supplying the rest. The non-dimensionalized one-dimensional energy spectrum agrees well with experiments in the dissipation range at Rλ ∼ 100. The locality of the energy cascade in wavenumber is also examined.

First- and second-derivative flatness factors of order 4·5-5·0 and 9·0 respectively are found under stationary conditions with bursts to higher values. Resolution and other systematic errors are explored by extensive runs with a 323 code; de-aliasing all higher-order derivative statistics; and recomputing selected averages after zeroing in succession the highest- and lowest-wavenumber bands. The latter analysis is of some relevance to the experimental problem of gauging how a finite-length hot wire biases a flatness measurement. A host of other higher-order derivative statistics are computed, including the vorticity/rate of strain correlations. Three-dimensional plots of the vorticity reveal persistent and extended tubes, sheets and blobs.

Two-dimensional buoyant clouds moving along inclined boundaries under a gravitational force are investigated theoretically and experimentally. It is found that the ‘thermal theory’ gives a good description of the flow in the slope angle range 5° [lsim ] θ ≤ 90°. In this range the spatial growth rates of the cloud height and length are constant for a given slope angle and show a linear dependence on θ. For a cloud released with zero initial velocity the front velocity Uf first increases and then decreases, with the characteristic time of acceleration predicted by theory. In the decelerating state Uf/(g0′Q0/xf)½ is 2·6 ± 0·2 at θ ≃ 15°, and then reduces uniformly with increasing θ to a value of 1·5 ≃ 0·2 at 90° (where g0′Q0 is the released buoyancy and xf is the front position measured from a virtual origin). The shape of the cloud is well approximated by a half-ellipse. The variation of the ratio of the principal axes of the half-ellipse with slope angle is identical with that of the head of an inclined starting plume (Britter & Linden 1980). However, the cloud has a greater growth rate than the head of a starting plume.

The main velocity redistribution in steady flow through curved conduits of shallow rectangular cross-section is considered. Its mechanism is analysed using a mathematical model of steady incompressible laminar flow in coiled rectangular pipes. The transverse transport of main-flow momentum by the secondary circulation is shown to be the principal cause of this velocity redistribution. The importance of the side-wall regions, even in shallow channels, is assessed and the neglect of the influence of the side walls in the commonly applied simplified models of flow through shallow curved channels is shown to be strongly limiting in case of long bends with a rectangular cross-section.

The method of multiple scales is used to determine a first-order uniform expansion for the effect of counter-rotating steady streamwise vortices in growing boundary layers on oblique Tollmien–Schlichting waves. The results show that such vortices have a strong tendency to amplify oblique Tollmien-Schlichting waves having a spanwise wavelength that is twice the wavelength of the vortices. An analytical expression is derived for the growth rates of these waves. These exponential growth rates increase linearly with increasing amplitudes of the vortices. Numerical results are presented. They suggest that this mechanism may dominate the instability.