To avoid high pollution levels along the banks of a river it is desirable to site steady discharges somewhere near the centre of the stream. Here it is shown that the precise location of the optimal discharge site is at the single zero crossing of the first advection–diffusion eigenmode. Simple examples reveal that this position tends to be weighted towards the deeper parts of the channel, and towards the inside bank at the beginning of bends.

We treat the problem of slow flow through a periodic array of spheres. Our interest is in the drag force exerted on the array, and hence the permeability of such arrays. It is shown to be convenient to formulate the problem as a set of two-dimensional integral equations for the unknown surface stress vector, thus lowering the dimension of the problem. This set is solved numerically to obtain the drag as a function of particle concentration and packing characteristics. Results are given over the full concentration range for simple cubic, body-centred cubic and face-centred cubic arrays and these agree well with previous limited experimental, asymptotic and numerical results.

The initial development of the viscous flow in the vicinity of the sharp trailing edge of a symmetrical body, impulsively set into motion is studied within the framework of boundary-layer theory. For a blunt trailing edge there exists a similarity solution for the inviscid outer flow as has been pointed out by Proudman & Johnson (1961). The present work shows that for sharp trailing edges, however, no such solution exists.

For small or moderate trailing-edge angles, a moving singularity occurs in the solution fairly early in the flow development. The flow in the vicinity of this singularity exhibits the characteristics of unsteady separation. For large trailing-edge angles, the boundary layer becomes so thick that solutions must be terminated before there is any evidence of a singularity. For a cusped trailing edge, the solution is arbitrarily terminated to avoid large computational times and with the realization that the trailing edge flow must eventually be influenced by the leading edge at which time the present solutions cease to have meaning.

Even mild longitudinal wall curvature is known to produce significant effects on the time-averaged turbulent transport in a boundary layer. The present study was undertaken to study the manner in which the instantaneous structure of turbulence in the boundary layer responds to mild streamline curvature. Both convex and concave boundary layers with a boundary-layer thickness to wall radius ratio of about 0·01 were studied. Attention was directed mainly to two events characterizing the instantaneous turbulence structure. These were the so-called ‘bursting’ and ‘zero-crossing’. Quantitative data on the statistics of these events were obtained using a combination of analog instrumentation and visual counting (from continuous film records). These data were compared with data from flat-wall boundary layers obtained from similar signal-processing techniques. The results indicate that neither the individual nor the joint statistics of these events are significantly affected by curvature in the vicinity of the wall. On the other hand, curvature seems to affect appreciably at least some properties of these events at large distances from the wall. Careful examination of these results shows, however, that neither the process of turbulent production near the wall nor the turbulent dissipative process anywhere in the boundary layer is significantly affected by mild curvature. Apparent curvature effects on the instantaneous structure in the outer part of the boundary layer can be explained as being due to the strong effect of streamline curvature on the turbulent diffusion process. This explanation is consistent with previous observations of the time-averaged structure of the flow. The results of the present study indicate the need to re-examine some of the recent turbulence models for curved flows that involve modification of the production and dissipation terms rather than the diffusion term in the transport equations.

The characteristics of a self-excited axisymmetric air jet driven by a whistler (i.e. pipe-collar) nozzle have been explored experimentally for various choices of the controlling parameters: namely, pipe length, jet diameter, collar length, step height, and jet speed. By appropriate choices of these parameters as well as the stage and mode (half- and full-wave), the self-sustained excitation has been induced at specific values of the excitation amplitude and the Reynolds number RD. The jet characteristics up to RD ≃ 3·1 × 105 and x/D ≃ 60 have been documented for both laminar and turbulent flows at the pipe exit. Comparison with corresponding unexcited-jet data reveals that self-excitation produces a large increase in the fluctuation intensity in the near field of the jet, while it increases the jet spread and decay rate for the entire x-range of measurement. The dependence of the jet structure on the initial condition is stronger when self-excited than when unexcited. The first stage of excitation always produces the highest turbulence augmentation and the spectral evolution is significantly modified by self-excitation up to x/D ≃ 6. The excitation produces a significant increase in the broad-band turbulence level over that of the unexcited jet. The broad-band amplification is maximized at x/D ≃ 4 and is the highest at the largest RD studied.

These data suggest interesting possibilities for the self-excited jet in the augmentation or control of entrainment, mixing and aerodynamic noise production.

The behaviour of a periodic wavetrain propagating obliquely over water of slowly varying depth is studied. The depth contours are taken to be straight and parallel. The wave properties used are those of ‘numerically exact’ solutions for waves on water of uniform depth. Comparison is made with linear theory which proves to be quite accurate for predicting wave direction unless the waves are propagating in a direction within about 25° of the contours. The results give a direct indication of where waves may break, but do not include dissipation.

Examples are given which correspond to waves ‘trapped’ within a region of limited depth. They are related to edge waves and to caustics of the linear theory. The behaviour of solutions is consistent with earlier work on deep-water waves. This includes behaviour we term ‘anomalous refraction’, which is to be discussed in another paper.

Analysis of a considerable body of new data, based upon flow-visualization experiments, reveals a simple criterion for the occurrence of vortex breakdown at a fixed location in a tube: ReB ∼ Ω-3R−1, where Ω is the circulation number, R the ratio of the radial to tangential velocities in the inflow region, and ReB the pipe Reynolds number at which vortex breakdown occurs. The constant of proportionality is found to be practically independent both of the pipe flare angle α and the type of breakdown observed (bubble, spiral, etc.) although the latter is shown to depend on ReB. Theoretical support for the experimental results is derived from the analysis of Benjamin (1962) combined with similarity arguments.

From rotating-screen annulus experiments the entrainment rate, we, normalized by the friction velocity, u*, has been found to be a function of both the overall Richardson number, RT, and the inverse Froude number, Rv. The RT−½ dependence deduced by Price (1979) and Thompson (1979) satisfactorily explains the present data if multiplied by an approximate Rv−1·4 dependence. The measurements indicate that Rv is a variable that is influenced by side-wall friction, time after onset of the surface stress, or other factors. The greater we/u* values of experiments of the type of Kantha, Phillips & Azad (1977) over that of the Kato & Phillips (1969) experiment can be explained by somewhat greater Rv values in the latter case.

A close connection is now apparent between entrainment experiments in two-layer systems designed to have only one velocity scale (the interfacial velocity jump, Δv), and the rotating-screen annulus experiments having two velocity scales (u* and Δv). The former also have (at least) two velocity scales, the second one being associated with the presence of turbulence throughout one or both of the fluid layers.

The turbulent layer is found to be quite well mixed in density only if we/u* does not exceed about 0·03, or we/|Δv| does not exceed about 0·003. The present data suggest more rapid entrainment when temperature rather than salt provides the density jump, as first noted by Turner (1968) in oscillating grid experiments. If this is a Péclet-number effect, the trend did not continue for still greater Pe values, the data for kaolin (clay) being very compatible with that for salt.

The effect of the free-stream turbulence on the flow past a circular cylinder was studied experimentally in the subcritical and critical regimes. Several grids were used to produce approximately homogeneous turbulent fields with longitudinal integral scales ranging from 0·30 to 3·65 cylinder diameters and with the longitudinal intensities ranging from 1·4 to 18·5%.

The critical Reynolds number Rc at which the time-mean drag coefficient obtains the value of 0·8 was found to satisfy the relation Rc1·34T = 1·98 × 105, where T is the Taylor number defined in terms of the longitudinal integral scale. The time-mean drag coefficient, the base-pressure coefficient and the spanwise correlation length of the surface-pressure fluctuations in the vicinity of the separation point were fairly well correlated with the parameter R1·34T, R being the Reynolds number. It was argued that the parameter R1·34T will control some aspects of the flow past a circular cylinder immersed in turbulent streams.

An experimental study of the evolution to breaking of a nonlinear deep-water wave train is reported. Two distinct regimes are found. For ak [les ] 0·29 the evolution is sensibly two-dimensional with the Benjamin-Feir instability leading directly to breaking as found by Longuet-Higgins & Cokelet (1978). The measured side-band frequencies agree very well with those predicted by Longuet-Higgins (1978b). It is found that the evolution of the spectrum is not restricted to a few discrete frequencies but also involves a growing continuous spectrum, and the description of the evolution as a recurrence phenomenon is incomplete. It is found that the onset of breaking corresponds to the onset of the asymmetric development of the side bands about the fundamental frequency and its higher harmonics. This asymmetric evolution, which ultimately leads to the shift to lower frequency first reported by Lake et al. (1977), is interpreted in terms of Longuet-Higgins’ (1978b) breaking instability. For ak [ges ] 0·31 a full three-dimensional instability dominates the Benjamin-Feir instability and leads rapidly to breaking. Preliminary measurements of this instability agree very well with the recent results of McLean et al. (1981).

Results of experiments on the low-Reynolds-number flow of non-neutrally buoyant drops through a straight circular tube are reported. The undeformed radii of the drops are comparable to the size of the tube, and the drops adopt an eccentric lateral position owing to a density difference between the drop and the suspending fluid. Measured values for the extra pressure difference caused by the presence of the drop, the relative velocity of the drop, and the shape of the drop are correlated with the minimum gap width between the eccentrically located drop and the tube wall using simple lubrication approximations. The viscosity ratio, density difference, volumetric flow rate and drop size are varied in the experiment. Comparisons with previous results for concentric, neutrally buoyant drops show that the effects of eccentric position can be substantial for surprisingly small values of the density difference. Both Newtonian and viscoelastic suspending fluids are considered, and the results suggest that both viscometric and time-dependent non-Newtonian effects are present. For the Newtonian case, the data are compared with the predictions of available theories which account explicitly for the eccentric drop position.

In part 1 of this work (Brown & Stewartson 1980b) we examined the nonlinear interaction of a forced internal gravity wave in a stratified fluid with its critical level. Although the Richardson number J was taken to be large, the method described there was, in principle, applicable to all Richardson numbers and as such we did not take advantage of the asymptotic properties of the solution of the linearized equations. Here in part 2 we re-develop the linearized solution for a general basic shear and temperature profile when J [Gt ] 1 as the large-time limit of an initial-value problem for a wave incident from above the shear layer. On this time scale it is known that the reflection and transmission coefficients are 0(e−νπ), ν = (J −¼)½. It is shown that, when J [Gt ] 1, the solution in the neighbourhood of the critical layer consists only of algebraically decaying elements with a direction of propagation parallel to the layer (critical-level noise) below a certain level, but of critical-level noise and a wavelike term, corresponding to the imposed incident wave, above this level. On a longer time scale, specifically $t = 0(\epsilon^{-\frac{2}{3}})$, where ε is the amplitude of the forced wave, the nonlinear terms are no longer negligible; the development of the reflection and transmission coefficients on this time scale is the subject of part 3 (Brown & Stewartson 1982).

In part 2 (Brown & Stewartson 1982) of this paper we set out the linearized solution of the critical layer that is expected to hold some time after the forced internal gravity wave of small amplitude e is initiated at an infinite distance above the shear layer. This differed from that presented in part 1 (Brown & Stewartson 1980) in that in part 2 we exploited to a much greater extent the fact that the Richardson number J was large and obtained the solution in a form consisting of explicit functions rather than infinite integrals. Also it was demonstrated that at times t = O(1) the wave did not penetrate beyond a certain level in the critical layer, and that critical-layer noise only was created above this line and transmitted below it. In this paper we examine the development of this solution on a longer time scale $\tau (\propto \epsilon^{\frac{2}{3}}t)$ and show how the reflection and transmission coefficients which are exponentially small, \[ O\{\exp(-(J-\frac{1}{4})^{\frac{1}{2}}\pi)\}, \] when τ = 0 increase with time. As in part 1 we obtain a reflection coefficient for the first harmonic that is O(τ3), and because of the simpler formulation of the linearized solution are able to obtain a reflected second harmonic. These harmonics appear as complementary functions that are induced by singularities in the particular integrals of the equations. It is shown that the interaction between the initial-noise term in the lower part of the critical layer and an induced-noise term at the sixth stage of the expansion will eventually lead to a transmitted wave. This appears at the ninth stage of the expansion and its transmission coefficient is O(τ12) though it is not explicitly calculated here.

An experimental investigation of the near wake of a thin airfoil at various incidence angles is reported in this paper. The airfoil (NACA 0012 basic thickness form) was located in a wind tunnel, and the wake structure was measured using hot-wire sensors. The measurements of mean-velocity, turbulence intensity and Reynolds-stress components across the wake at several distances downstream show the complex nature of the near wake and its asymmetrical behaviour. The asymmetry in the wake property, which is maintained up to a length of 1·5 chords downstream of the trailing edge of the blade, is dependent on the incidence angle of the inlet flow. The streamwise velocity defect in an asymmetric wake decays more slowly compared to that of a symmetric wake. The streamline curvature due to the blade loading has a substantial effect on the mean velocity profile as well as the turbulence structure. The numerical study of the same wake indicates that the existing turbulence closure models need some modification to account for the asymmetric characteristics of the wake.

In the first part of this paper we introduce a path-integral formalism for the internal-wave field of the ocean. The intent is to show that this type of formalism may be useful in suggesting improvements to current calculations, as it provides a framework for applying a wide variety of approximations that have been and are currently being developed in other areas of physics. We demonstrate the method by deriving equations for a self-consistent field approach (also known as the direct-interaction approximation). The experience in other areas of physics is that the self-consistent field approximation is more reliable than lowest-order perturbation theory. The end result of the DIA is the determination of an effective linear model for the description of internal waves in the deep-ocean environment. In the second part of the paper we obtain Hasselmann's source function by a prescribed limiting process and are able to indicate possible improvements in related calculations by comparing the limiting assumptions with numerically computed values.

Meander bends typically show certain systematic deviations from simple Cartesian sinusoidal forms. Bends tend to be round and full, or ‘fat’, often to the point of possessing double-valued plan-forms, as Langbein & Leopold (1966) have noted. Bends also tend to be characteristically skewed in such a fashion that their direction of migration can be determined directly from an aerial photograph of the planform; the water margin of the downstream accreting half of a point bar describes a convex planform, whereas the upstream eroding side has a concave shape.

In the present paper a generalized nonlinear equation of bend migration is treated based on the analysis of Part 1 (Ikeda, Parker & Sawai 1981). An expansion technique reminiscent of the Stokes expansion for water waves is developed to perform a nonlinear stability analysis. This analysis provides an explanation of skewing and fattening, and also indicates that lateral and downstream migration rates should increase as bend amplitude develops. These results agree qualitatively with field observations.

The two-dimensional diffraction of a long surface wave by a deformation of the bottom is calculated through a conformal-mapping algorithm (Kreisel 1949). The result is applied to obtain the complex reflection coefficient for a rectangular trench. The corresponding reflection coefficient for oblique incidence is obtained through a variational formulation.

A linear model of a wave packet in a laminar Ekman boundary layer is proposed for tracing the development of an initially localized pulsed perturbation at the boundary. The model disturbance was built up from a linear combination of growing modes summed numerically over all wavenumbers and frequencies. The input spectrum was assumed to be flat so that there was no biasing at any wavenumber or frequency and the evolution was calculated on the basis of linear-stability theory. The wave packet generated by the summation of modes developed uniformly downstream of the disturbance source location and, depending upon the choice of the Reynolds number, vividly displayed results representative of the different kinds of instabilities that are present in the Ekman layer. Outputs in the form of perspective plots are given in order to explain the evolution of the packet into either a single wave patch or a sum of individual wave patches.

Grimshaw (1979) discussed the mean flow induced by an internal gravity-wave packet propagating in a shear flow. The present paper analyses the effect of dissipative processes on this problem. In a manner similar to that described by Longuet-Higgins (1953) for water waves, frictional effects in the Stokes boundary layers modify the mean-flow field just outside the boundary layers. Just outside the bottom boundary layer there is a wave-induced mean Lagrangian velocity, whose magnitude is proportional to the square of the wave amplitude, while just below the free-surface boundary layer there is a wave-induced mean-velocity gradient. In the interior of the fluid the presence of dissipation in the wave field will induce a significant mean-flow field whenever the group velocity of the wave packet exceeds the phase speed of a longwave mode. Ultimately, this interior mean flow will be modified by diffusion from the boundaries of effects induced in the aforementioned Stokes layers.

Optimal co-ordinates were introduced by Kaplun (1954) as a result of his study on the role of co-ordinate systems in boundary-layer theory. In this paper the basic ideas of optimal co-ordinates are examined and many restrictions of Kaplun's optimal co-ordinates are removed. His rule for constructing optimal co-ordinates is found to apply unaltered to axisymmetric flows, to flows with oncoming streams containing vorticity, to free boundary layers such as jets, to free-convection flows and to compressible flows. It is also extended to unsteady boundary layers, and to three-dimensional boundary layers using a pair of stream functions.