A model is proposed which attempts to explain the complete ‘burst cycle’. This model views the wall streak as a sub-boundary layer, within the conventionally defined boundary layer, and the lift-up stage of bursting either as an upwelling motion of this sub-boundary layer which is similar to a local, convected separation or, equivalently, as the consequence of a vortex roll-up. ‘Sweeps’ are thought to represent the passage of a previous burst from further upstream. They appear either to impress on the wall streak the temporary adverse pressure gradient required to bring about its lifting or, alternatively, to provide the outer vortex which rolls up with the vortex associated with the wall streak. The model is also used to explain how the interactions between a burst and a sweep bring about (i) breakup, as well as (ii) new wall streaks further downstream.

Arguments are presented to demonstrate that the three kinds of oscillatory growth reported by Kim, Kline & Reynolds (1971) may be associated with just one type of flow structure: the stretched and lifted vortex described by Kline et al. (1967).

Following the ideas suggested by Landahl (1967, 1975), some model calculations of the fluctuating velocity field in the wall region of a turbulent boundary layer have been carried out. It was assumed that the turbulent stresses are generated intermittently on small scales in time and space owing to bursting-type motions. The Reynolds-stress distribution during bursting periods and the mean velocity profile were assumed to be known, and the linear large-scale response to a random system of bursts was computed using an idealized model for the joint probability distribution in time and space of the occurrence of bursts. Computed energy spectra of the streamwise velocity fluctuations display scales in the spanwise and streamwise directions and time which are in good agreement with measurements by Morrison, Bullock & Kronauer (1971). However, the wavenumber band-widths of the computed spectra are narrower than those of the measured ones. This discrepancy is probably due to the crudeness of the model employed for the Reynolds stress during bursting.

The probability density function and the first three statistical moments of the velocity, acceleration and dynamic pressure are obtained for a Gaussian, stationary, homogeneous, random gravity-wave field in deep water, using infinitesimal wave solutions. It is shown that the velocity, acceleration and pressure are non-Gaussian. While the horizontal accelerations and vertical velocity component are of zero mean and unskewed, the dynamic pressure, vertical acceleration and horizontal velocity components are skewed and have non-zero mean.

The time-averaged vorticity field within the free-surface boundary layer associated with a general class of propagating gravity waves is considered. The principal results are applied in a calculation of the mass transport velocity field for edge waves.

The purpose of this paper is to consider the effect of one-dimensional random depth variation on the propagation of planetary waves in a homogeneous layer of fluid having a free upper surface. We begin by determining the dispersion relation for the coherent part of the wave using the vorticity equation for the transport stream function and a previously described perturbation method. Then, from the resulting first-order expressions for the wavenumber, we obtain the phase speeds for the two possible planetary-wave solutions. These are compared with the corresponding phase speeds of planetary waves over a smoothly varying topography; the validity limits of the approximations are discussed. For the most physically realizable situation, of random depth correlation lengths much shorter than a typical wavelength, we find that the phase speed of the shorter (longer) wave component is less (greater) over a randomly varying topography than over a smoothly varying topography. In the case of the shorter waves, greatest relative changes in phase speed occur when the associated fluid motions are at right angles to the ‘strike’ of the roughness elements, while for both long and short waves there is no relative change in phase speed if fluid motions are parallel to the roughness contours. Moreover, both types of waves are shown to lose energy in the direction of energy propagation as a result of scattering. Numerical values are then obtained using hydrographic charts of the western North Pacific, and show that the randomness may significantly decrease the phase speed of the shorter planetary-wave component. Finally, we give a brief descriptive explanation of the results based on the effect of the topography on the wave restoring mechanism.

The propagation of internal gravity waves in a shear flow in a rotating fluid is examined for the case when the rotation vector is inclined to the vertical. It is shown that internal gravity waves approaching a critical level, where ω*, the Doppler-shifted frequency, equals 2ΩV, the vertical component of the Coriolis parameter, will be either transmitted or absorbed according as \[ W_g\omega^{*}2\Omega_V\{m(U_z + 2\Omega_H)-lV_z\}\lessgtr 0; \] here Wg is the vertical group velocity, 2ΩH is the horizontal component of the Coriolis parameter, l and m are the easterly and northerly wavenumber components, and Uz and Vz are the shear rates of the easterly and northerly components of the mean flow. Between critical levels, wave action flux is conserved. However, for a wave absorbed at a critical level, the wave action flux is attenuated by a factor exp { − 2π|m(Uz + 2ΩH) − lVz]/(lUz + mVz)|}. The phenomenon is also analysed using a WKBJ approximation.

This paper discusses critically observations of mixing processes across density interfaces in laboratory experiments and inferences that have been made and can be made from these observations. Fluxes of heat or salt and entrainment velocities have been found to depend on negative powers of an overall Richardson number Ri* based on the buoyancy jump across the interface, the depth of the homogeneous layer and the intensity of the turbulence near the source. When the Reynolds and Péclet numbers are large, the fluxes or entrainment velocities appear to be proportional to the minus one and minus three-halves powers of Ri* for flows with and without mean shear respectively, and this difference has caused speculation about the accuracy of the experimental data and about the reasons for the two laws if the difference is real. In the present discussion, we accept the accuracy of the two laws and attribute the higher entrainment rate for shear flows to the decrease of r.m.s. velocities near the interface with increasing Ri* in the case of zero shear. A plausible argument yields the unifying result that the entrainment rates in both cases are proportional to Ri−1, where Ri is a Richardson number based on the buoyancy jump and velocities and lengths characteristic of the turbulence near the interface. It is suggested that the $Ri^{-\frac{3}{2}}$ behaviour inferred by Turner is based on an erroneous interpretation of experimental data.

In the course of the discussion, it is shown that the drag coefficient in flow of a stratified fluid over a rough surface is independent of the Richardson number (or density jump across the interface or inversion) and depends only on the ratio of the roughness length to the depth of the homogeneous layer. This has obvious implications for problems of parameterizing the momentum flux near the ground in the atmosphere.

On the basis of a small Froude number theory it is shown that previous horizontalduct models of the withdrawal of fluid from a stratified tank with a free surface and a fixed bottom are inappropriate at large times and that the flow should be modelled by that from an infinite vertical duct. The falling horizontal free surface in the tank is replaced by a vertically moving column of stratified fluid and the tank bottom is modelled by a stagnant pool of fluid below. This model is analysed and a solution uniformly valid in time is presented. The properties of the solution are then compared with existing theories and experiments.

The effect of a small surface-tension coefficient on the classical theory of free-streamline separation from a sharp trailing edge is studied. The classical solution fails in a small region surrounding the edge, where it predicts singular behaviour, and an inner solution, satisfying linear boundary conditions, is required to obtain a uniformly valid first approximation. The solution valid near the edge removes the curvature and pressure-gradient singularities of the classical solution and predicts a standing capillary wave along the free streamline.

In this paper we study the development of large-scale wavelike eddies in a two-dimensional turbulent jet, extending earlier work on the mixing region (Liu 1974). The basic mean flow develops from one of mixing-region type with an initially specified boundary-layer thickness into a fully developed jet. This study brings out the role of the varicose and sinuous modes as they develop in a growing mean flow. In general, it is found that, for a given frequency parameter, the varicose mode has a shorter streamwise lifetime than the sinuous mode. For lower frequencies, the latter persists past the end of the potential core only to become subject to dissipation by the enhanced fine-scale turbulent activity in that region.

The objective of this study was to make velocity measurements in drag-reducing flows which would be sufficient in scope and accuracy to test proposed models of drag-reducing flows and to yield new information about the mechanisms of drag reduction. Consequently, measurements of the mean and turbulence intensity of the streamwise velocity component were made in fully developed, turbulent, drag-reducing flow in a two-dimensional channel with a laser-Doppler anemometer. The anemometer was operated in the individual-realization mode and corrections were made to eliminate statistical biasing of the data. Two polyacrylamides and a polyethylene oxide were used to produce seven flows which had drag reductions ranging from 24 to 41 %. Measurements were also made in water to establish the standard characteristics of the flow channel.

The data show that the drag-reducing mean velocity profile can be divided into three zones: a viscous sublayer, a buffer or interactive region and a logarithmic region. There is no evidence that the viscous sublayers of the drag-reducing channel flows are thicker than those in the solvent flows. In addition the normalized streamwise fluctuations are essentially the same in both the solvent and drag-reducing sublayers. The changes caused by the polymer addition occur in the buffer region. The drag-reducing buffer region is thicker and the velocity profile in the outer flow region adjusts in order to accommodate this buffer-region thickening. The measurements of the streamwise velocity fluctuations also show that the polymer additives redistribute the primary turbulent activity over a broadened buffer region. The normalized magnitude of these fluctuations is, however, considerably lower in these two-dimensional drag-reducing channel flows than in those previously reported by Rudd (1972), Logan (1972) and Kumor & Sylvester (1973). Moreover, the mean velocity profiles in the buffer region do not confirm the hypothesis of Virk, Mickley & Smith (1970) that the data will follow their proposed ‘ultimate profile’ when the drag reduction is less than that given by the maximum asymptote. The mean velocity measurements also show that the proposed methods for predicting the upward shift in the outer portion of the mean velocity profile are inconsistent and lack universality. However, these results do confirm the previous suggestions of Virk (1971), Tomita (1970) and Lumley (1973) that the buffer region is the area of importance and change in drag-reducing flows.

Extensive boundary-layer measurements have been made on a cone-ogive-cylinder model at a free-stream Mach number of 7·0 and momentum-thickness Reynolds number of 8500. Mean flow transformations and calculated turbulence correlations are presented which are in good agreement with previous incompressible results. New quantitative turbulence measurements including measurements of the first higher moment and probability density of fluctuations in mass flow and total temperature in hypersonic flow are also presented. The higher moment and probability density data show that the characters of the fluctuation modes of the mass flow and total temperature are significantly different in the wall region and in the outer part of the boundary layer. These differences together with data on the turbulence scale and lifetime obtained from autocorrelation and space-time correlation measurements are discussed.