A study is made of the entrainment interface in a two-layer flow with one layer turbulent owing to an imposed surface stress. Turbulent and wave parameters were measured from films of an experiment carried out by Kantha, Phillips & Azad (1977) and are analysed to determine relationships between wave and turbulent length and velocity scales and bulk scales.

A measure of interface distortion is calculated and it is found to be related to the mixed-layer depth only at small overall Richardson number. The local velocity of advance of the interface is determined and is found to vary with overall Richardson number tending to the Kolmogorov scale at small Ri and to Phillips’ (1966) quasilaminar scale at large Ri.

Kelvin-Helmholtz type instabilities are observed and their wavelength, amplitude and lifetime are measured. The wavelength is not found to be related to the mixed-layer depth but some dependence on overall Richardson number is observed. An interfacial mean flow Richardson number is estimated and the variation of wave slope at maximum amplitude with this Richardson number is found to be comparable with observations of Thorpe (1973a) in a laminar two-layer flow. A relationship between the wavelength of an instability and the distortion length scale is found and is discussed.

The problem of fluid transport by cilia is investigated using the Green's function for a Stokeslet between two parallel plates. The discrete-cilia approach is used in building the model, and a readily usable expression for the velocities is obtained. Dependence on the direction of the metachronal wave and on time is not averaged out. Velocity fields, pressure fields and fluxes due to a single Stokeslet and to an infinite line of Stokeslets are discussed. It is found that the flux associated with Stokeslets in between two parallel plates is always zero, in contrast to a Stokeslet parallel to, and above, one plate. In the model one also has to add a plane Poiseuille flow, which incorporates non-zero flux. The flow due to the Stokeslet solution imposes a positive pressure gradient downstream, and the Poiseuille flow a negative pressure gradient. Calculated velocity profiles, in the pumping range, are seen to be time-independent in the centre of the channel and vary between a negative parabolic profile and a plug flow. The reason for these profiles and some possible biological applications are discussed.

A photographic technique has been used to make observations of the motion of heavy suspended particles in an open channel with a smooth bottom. The observations showed that the path of an individual heavy particle consists of an alternation between upward and downward paths traced by the particle as it travels close to the bottom. The path records appear to reveal most of the flow patterns near the wall shown up by earlier visualization observations; the measured kinematical quantities concerning the particle motion are in accord with those of earlier observations (e.g. Nychas, Hershey & Brodkey 1973). Making use of the present observations and following Offen & Kline's (1975) model of the bursting process in turbulent boundary layers, an attempt was made to explain the mechanism of particle suspension close to the wall in turbulent flows.

Proposals are made for modelling the pressure-containing correlations which appear in the transport equations for Reynolds stress and heat flux in a simple way which accounts for gravitational effects and the modification of the fluctuating pressure field by the presence of a wall. The predicted changes in structure are shown to agree with Young's (1975) measurements in a free stratified shear flow and with the Kansas data on the atmospheric surface layer.

Turbulence produced by a grid which simultaneously imparts a mean temperature profile varying linearly with height was investigated in a specially constructed wind tunnel. While the mean temperature profile is preserved downstream of the grid in accordance with the theory of Corrsin (1952), the downstream evolution of the r.m.s. temperature fluctuation is at variance with his prediction. The reason for this discrepancy is shown to lie in the neglect of molecular diffusivity, which leads to unbounded growth of the fluctuations. Along with conventional correlations and spectra, the filtered heat-transfer correlation is presented. About 60% of the heat transport is accomplished by the low wavenumber components having length scales equal to or larger than the integral scale. An intriguing feature of the present experiments is the presence of an inertial-convective subrange for the temperature field notwithstanding the low Reynolds number and the consequent absence of an inertial subrange for the velocity field. Experimental results show that the temperature has a positive skewness everywhere in contrast to the velocity components, which are symmetrically distributed. Measurements of the joint probability density function of the vertical component of the velocity and the temperature indicate that, while the assumption of joint normality is not uniformly valid, the conditional expectations nearly follow the normal law. Marginal and joint moments of up to fourth order are presented. Odd-order joint moments are clearly sensitive to the skewness of the temperature.

Dean's series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms. Analysis suggests that convergence is limited by a square-root singularity on the negative axis of the square of the Dean number. An Euler transformation and extraction of the leading and secondary singularities at infinity render the series accurate for all Dean numbers. For curvature ratios no greater than $\frac{1}{250} $, experimental measurements of the laminar friction factor agree with the theory over a wide range of Dean numbers. In particular, they confirm our conclusion that the friction in a loosely coiled pipe grows asymptotically as the one-quarter power of the Dean number based on mean flow speed. This contradicts a number of incomplete boundary-layer analyses in the literature, which predict a square-root variation.

Velocity and pressure fields for Stokes flow due to a force singularity (Stokeslet) of arbitrary orientation and at arbitrary location inside an infinite circular pipe are obtained. Two alternative expressions for the solution, one in terms of a Fourier-Bessel type expansion, and the other as a doubly infinite series, are given. The latter is especially suitable for computational purposes as it is shown to be an exponentially decaying series. From the series it is found that all velocity components decay exponentially to zero up- or downstream away from the Stokeslet. This is also true for pressure fields of Stokeslets perpendicular to the axis of the pipe. A Stokeslet parallel to the axis of the pipe raises the pressure difference between − ∞ to + ∞ by a finite non-zero amount. Some numerical results for a Stokeslet parallel to the axis are given. Comparison of the results with flow in a two-dimensional channel is also discussed.

The flow field of an ‘axisymmetric’ vortex breakdown has been mapped using a laser-Doppler anemometer. The interior of the recirculation zone is dominated by energetic, non-axisymmetric, low frequency periodic fluctuations. Spectra for a number of points inside this zone, as well as time-averaged swirl and axial velocity profiles both inside and outside the recirculation zone, have been obtained. The time-averaged streamlines in the interior show an unexpected two-celled structure attributed to the action of the fluctuations. Although the present experiment deals with one particular breakdown, flow-visualization studies indicate that the case examined is typical of the ‘axisymmetric’ form of breakdown over a range of flow conditions.

Heat-transfer measurements have been carried out in a right circular cylinder of fluid which is heated from above and rotated steadily about its vertical axis. Convection is produced relative to solid-body rotation through the coupling of the centrifugal acceleration and density variations in the fluid. Two silicone oils having kinematic viscosities of 350 cS and 0·65 cS were used in the experiments. In the former case viscous forces are important throughout the cylinder whereas in the latter case Ekman layers form and the Coriolis acceleration controls the interior flow.

With the 350 cS oil the Nusselt number for heat transfer from the top to the bottom of the cylinder is a function of Grω and r0, where Grω is a Grashof number defined by employing the centrifugal acceleration evaluated at the outer edge of the cylinder in place of the gravitational acceleration, and r0 is the cylinder aspect ratio.

The behaviour is quite different for the 0·65 cS oil. Ekman layers form on the horizontal surfaces and heat is convected by Ekman suction. The Nusselt number is given by \[ Nu = 4.16\beta^{0.822}\epsilon^{-0.499}r_0^{0.173},\quad Ac\leqslant 0.025,\quad\sigma\beta\epsilon^{-\frac{1}{2}} > 0.7, \] where β is the thermal Rossby number, ε is the Ekman number, σ is the Prandtl number, and Ac is the ratio of gravitational to centrifugal accelerations. This is consistent with previous theories which indicate that the system should depend on the parameters σβε−½ and r0 in the limit as ε and β approach zero.

Kelvin and Poincaré waves are generated when an ocean wave arrives at a nominally rectilinear coastline and interacts with coastal irregularities. The discussion of this problem given by Howe & Mysak (1973) is extended in this paper in order to examine the role of multiple scattering of the Kelvin and Poincaré waves. An integro-differential kinetic equation is derived to describe these processes in the limit in which the irregularities are small compared with the characteristic wavelength. In the absence of dissipative mechanisms it is verified that this description of interactions with the coast conserves total wave energy. The theory is applied to a variety of idealized problems which model tidal and storm surge events, including the generation and decay of Kelvin waves by extensive and localized Poincaré-wave forcing, and the influence of multiple scattering on the radiation of Poincaré-wave noise into the open ocean.

Measurements have been made in a water channel of the flow in and around the separation region due to a rearward-facing step. Detailed profiles of mean velocities, turbulence intensities and Reynolds shear stress are presented. The turbulence measurements reveal the development of a new shear layer, which splits at reattachment with about one-sixth of the mass flow deflected upstream. The new shear layer is associated with a region of roughly constant values of both the non-dimensional mixing length l/x and the shear correlation coefficient K. The mixing-length ratio is larger than that found in plane mixing layers, whereas the shear coefficient is roughly the same. There is strong evidence that near the wall the length scales increase more rapidly with distance from the wall than in an attached boundary layer, and that a local maximum occurs.

The concept of diffusion by bulk convection formulated by Bradshaw is applied to the transport equations for the turbulent kinetic energy, turbulent shear stress and an integral length scale. The resulting set of hyperbolic partial differential equations is solved by an explicit finite-difference scheme for the cases of incompressible axisymmetric wakes and jets in a coflowing air stream. It is found that the profiles of mean velocity and shear stress are almost insensitive to the empirical input whereas the profiles of kinetic energy are very sensitive.

Symmetrical flow past a double wedge is studied for a high subsonic Mach number. The main features of the flow field are discussed in the physical and the hodograph plane. It is shown how a regular solution to the hodograph equation exhibits limit lines in the physical plane, which are eliminated through a ‘cut’ corresponding to the shock wave. In the case of the wedge it is assumed that the shock which is likely to be the most stable is the weakest possible.

The boundary-value problem is solved in the hodograph plane using Telenin's method, which has proved to be successful when dealing with equations of mixed type. The bounded analytic solution which is thus constructed is regular in the hodograph plane but presents three folds in the physical plane.

The computation is carried out for flow past a 4·5° half-angle wedge at a Mach number M∞ = 0·89. These figures are chosen so that the problem may be justifiably treated by potential theory, the entropy gradient behind the shock being negligible. In this case the mapping of the solution into the physical plane gives the pressure distribution along the double wedge, the sonic line and the shock wave. Of particular interest is the point where the sonic line meets the shock. This configuration is in agreement with the hypothesis of Nocilla, according to which the shock terminates in the supersonic domain. Experimental evidence cannot be obtained, however, because of the lack of resolution in this region.

A Navier-Stokes direct spectral simulation code was modified to produce stationary and nearly isotropic turbulence in three dimensions. An approximate $-\frac{5}{3}$ energy spectrum was maintained over the entire range of wavenumbers by simultaneously driving the fluid and supplementing the ordinary viscosity with a subgrid-like energy sink in the last 15% of the spectrum. Half-tone and contour plots of the fluctuations in the vorticity, rate-of-strain tensor and helicity show increasing ‘spottiness’ as the system evolves in time. Probability distributions and cross-correlations among these three quantities were also obtained. The flatness factor of the longitudinal velocity derivative, the longitudinal structure functions and the fluctuations in the locally averaged dissipation rate are consistent with some degree of intermittency, but do not unambiguously demonstrate its presence in the simulated flows.

A two-dimensional mathematical model is described for the calculation of the depth-averaged velocity and temperature or concentration distribution in open-channel flows, an essential feature of the model being its ability to handle recirculation zones. The model employs the depth-averaged continuity, momentum and temperature/concentration equations, which are solved by an efficient finite-difference procedure. The ‘rigid lid’ approximation is used to treat the free surface. The turbulent stresses and heat or concentration fluxes are determined from a depth-averaged version of the so-called k, ε turbulence model which characterizes the local state of turbulence by the turbulence kinetic energy k and the rate of its dissipation ε. Differential transport equations are solved for k and ε to determine these two quantities. The bottom shear stress and turbulence production are accounted for by source/sink terms in the relevant equations. The model is applied to the problem of a side discharge into open-channel flow, where a recirculation zone develops downstream of the discharge. Predicted size of the recirculation zone, jet trajectories, dilution, and isotherms are compared with experiments for a wide range of discharge to channel velocity ratios; the agreement is generally good. An assessment of the numerical accuracy shows that the predictions are not influenced significantly by numerical diffusion.

Analyses of fluid flow over a wavy wall are of interest because of their applications to the physical problems mentioned in § 1. The authors have therefore devoted their attention to the effect of waviness of one of the walls on the flow and heat-transfer characteristics of an incompressible viscous fluid confined between two long vertical walls and set in motion by a difference in the wall temperatures. The equations governing the fluid flow and heat transfer have been solved subject to the relevant boundary conditions by assuming that the solution consists of two parts: a mean part and a disturbance or perturbed part. To obtain the perturbed part of the solution use has been made of the long-wave approximation. The mean (zeroth-order) part of the solution has been found to be in good agreement with that of Ostrach (1952) after certain modifications resulting from the different non-dimensionalizations employed by Ostrach and the present authors respectively. The perturbed part of the solution is the contribution from the waviness of the wall. The zeroth-order, the first-order and the total solution of the problem have been evaluated numerically for several sets of values of the various parameters entering the problem. Certain qualitatively interesting properties of the flow and heat transfer, along with the changes in the fluid pressure on the wavy and flat wall, are recorded in §§ 5 and 6.

Hot-wire measurements are presented of the onset of instability in developed axial and tangential flow due to inner-cylinder rotation in annuli of radius ratios 0·9, 0·81 and 0·58 for axial-flow Reynolds numbers between 86 and 2000. Measurements of the minimum critical Taylor number are reported along three radii at azimuthal locations 90° apart. At the largest radius ratio these azimuthal measurements show considerable variation but the sensitivity of marginal stability to angular orientation becomes negligible at a radius ratio of 0·58, when measurements and the Galerkin predictions of Hasoon & Martin (1977) are in close accord. The greater sensitivity to orientation as the radius ratio increases appears to correlate with measured percentage circumferential variations in annulus width arising from manufacturing tolerances and non-uniform curvature of the surfaces.

The following problem is treated: given a plane acoustic wave propagating through an unbounded field of turbulence, calculate the amount of acoustic energy converted into turbulent kinetic energy. The fluid velocities due to the acoustic waves and the turbulence are assumed to be small compared with the speed of sound. Thus the sound-turbulence interaction is weak and the turbulent field may be considered to be incompressible. The analysis is based on the interaction of two opposite effects: the acoustic distortion of the turbulence (producing anisotropic Reynolds stresses) and the redistribution of the kinetic energy among components (tendency towards isotropy) and among wavenumbers (energy cascade and dissipation). These phenomena are described using semi-empirical turbulence arguments. It is seen that the simplest model for the redistribution among components is not sufficient for unsteady flows. A more complete model is used which is modified to agree with the exact instantaneous distortion analysis of Ribner [map ] Tucker to first order. Owing to the two redistribution effects, the Reynolds stress behaves inelastically and is out of phase with the acoustic field. Thus there is an average production of turbulent energy corresponding to the absorption of acoustic energy and attenuation of the incident wave. For nearly isotropic turbulence, the attenuation coefficient is found to be proportional to the rate of viscous dissipation and independent of the frequency.

In order to compare the theory with experiment several constants involved in the semi-empirical model of the turbulence must be found. Owing to the lack of better information these constants are estimated here by order-of-magnitude considerations. No existing experiments correspond to the homogeneous turbulence assumed by the theory. Comparison with the few reasonably applicable experiments shows qualitative agreement though the importance of the turbulent absorption is generally of nearly the same order as the measurement error. Several discrepancies between jet noise experiments and aerodynamic noise predictions may be roughly explained using the above analysis.

Periodic permanent waves in shallow water are stable to periodic disturbances in the same direction, but are unstable to certain oblique periodic disturbances. A computer-assisted stability analysis is made of such waves for oblique disturbances with wavelengths comparable to and long compared with the fundamental wavelength. Regions of instability are calculated, and an explanation is given for the occurrence of instability. It is shown that disturbances in the same direction with a small margin of stability may cause a greater modification to the permanent wave in practice than do oblique unstable disturbances.

In flows around three-dimensional surface obstacles in laminar or turbulent streamsthere are a number of points where the shear stress or where two or more component,s of the mean velocity are zero. In the first part of this paper we summarize and extend the kinematical theory for the flow near these points, particularly by emphasizing the topological classification of these points as nodes or saddles. We show that the zero-shear-stress points on the surface and on the obstacle must be such that the sum of the nodes ΣN and the sum of the saddles Σs satisfy \[ \Sigma_N -\Sigma_S = 0. \] If the obstacle has a hole through it, such as a passageway under a building, \[ \Sigma_N -\Sigma_S =-2. \] If the surface is a junction between two pipes, \[ \Sigma_N -\Sigma_S =-1. \] We also consider, in two-dimensional plane sections of the flow, the points where the components of the mean velocity parallel to the planes are zero, both in the flow and near surfaces cutting the sections. The latter points are half-nodes N′ or half-saddles S′. We find that \[ (\Sigma_N +{\textstyle\frac{1}{2}}\Sigma_{N^{\prime}}-(\Sigma_{S^{\prime}}+{\textstyle\frac{1}{2}}\Sigma_{S^{\prime}}) = 1-n, \] where n is the connectivity of the section of the flow considered.

In the second part new flow-visualization studies of laminar and turbulent flows around cuboids and axisymmetric humps (i.e. model hills) are reported. A new method of obtaining a high resolution of the surface shear-stress lines was used. These studies show how enumerating the nodes and saddle points acts as a check on the inferred flow pattern.

Two specific conclusions drawn from these studies are that:

1. for all the flows we observed, there are no closed surfaces of mean streamlines around the separated flows behind three-dimensional surface obstacles, which con-tradicts most of the previous suggestions for such flows (e.g. Halitsky 1968);

2. the separation streamline on the centre-line of a three-dimensional bluff obstacle does not, in general, reattach to the surface.