A further refinement of the Howard-Kochar-Jain theorem is given which allows the estimation of the range of complex wave velocity for growing perturbations in a stratified shear flow. According to the results obtained, the boundary of this region depends both on the minimum Richardson number and on the wavenumber of the perturbations. The effect of external boundaries on the stability of parallel flows is defined. An estimate of the maximum rate of growth versus dimensionless wave-number is found. The theoretical results are compared with numerical computations and laboratory experiments of other authors.

A theoretical calculation is made of the decay of turbulence energy in the presence of coherent internal gravity waves of various intensities. The wave-turbulence interaction considered is energy production by wave shear. The production term (stress) is calculated by a second-order closure, with temperature fluctuations accounted for by buoyancy subrange theory. This formalism applies to large or small turbulence Froude number, both extremes of which are often encountered in experimental turbulence decay. The theoretical turbulence decay is shown to be a universal function of the wave shear (strain rate) and wave frequency provided that the energy is expressed in terms of the buoyancy wavenumber kB, and time is expressed in terms of N, the Brunt-Väisälä frequency. With the amplitude of wave shear characterized by a gradient Richardson number Ri0, the turbulence decay is found to undergo a sudden transition from rapid decay to a much slower oscillating decay when Ri0 is less than about 0.4. The transition time occurs at about t ≈ 2πN−1. If Ri0 exceeds 0.8 the rate of decay exceeds that of a neutral fluid. A transition in turbulence decay was observed in experiments by Dickey & Mellor (1980). It might explain the continued presence of turbulence in dynamically stable regions of oceans or atmosphere. The theory is compared in much detail with the Dickey & Mellor experiment. A briefer comparison is also made with other experiments, and with previous calculations of turbulence maintenance by steady mean shear. A simple explanation is proposed of why a transition is observed in a vertical grid experiment but not in horizontal grid experiments.

The laser-Doppler velocimeter is used to measure the mean and the fluctuating velocity for turbulent flow over a solid sinusoidal wave surface having a wavelength λ of 50.8 mm and a wave amplitude of 5.08 mm. For this flow, a large separated region exists, extending from x/λ = 0.14 to 0.69. From the mean velocity measurements, the time-averaged streamlines and therefore the extent of the separated region are calculated. Three flow elements are identified: the separated region, an attached boundary layer, and a free shear layer formed by the detachment of the boundary layer from the wave surface. The characteristics of these flow elements are discussed in terms of the properties of the mean and fluctuating velocity fields.

A simple two-point closure scheme for homogeneous axisymmetric turbulence is developed. For the isotropic case it is essentially an eddy-viscosity assumption in real space for the Kármán-Howarth equation. The eddy-viscosity function for large internal Reynolds numbers is derived from Kolmogoroff's 1941 theory. For moderate Reynold's numbers of order 102, approximately the same expression for the eddy-viscosity function is determined from experimental data. The resulting closed equation for the double-correlation function is solved numerically for both large and moderate Reynolds numbers, and the results are compared with experimental data. Self-similar solutions of the basic equation predict turbulent energy decay inversely proportional to time. It is shown that the departure from this ‘initial-period decay law’ observed in laboratory data is due to the behaviour of grid-produced correlation functions for large separation distances.

Measurements are presented of the velocity structure function on the axis of a turbulent jet at Reynolds numbers Rλ ≤ 852 and in a turbulent duct flow at Rλ = 515. Moments of the structure function up to the eighteenth order were calculated, primarily with a view to establish accurately the dependence on the order of the inertial range power-law exponent and to draw conclusions about the distribution of energy transfer in the inertial range. Adequate definition of the probability density of the structure function was achieved only for moments of order n ≤ 10. It is shown, however, that, although the values of moments of n > 10 diverges from their true values, the dependence of the moment of the structure function on the separation r is still given to a fair accuracy for moments up to n ≈ 18. The results demonstrate that the inertial-range power-law exponent is closely approximated by a quadratic dependence on the power which for lower-order moments (n [lsim ] 12) would be consistent with a lognormal distribution. Higher-order moments diverge, however, from a lognormal distribution, which gives weight to Mandelbrot's (1971) conjecture that ‘Kolmogorov's third hypothesis’ is untenable in the strict sense. The intermittency parameter μ, appearing in the power-law exponent, has been determined from sixth-order moments 〈(δμ)6〉 ∼ r2−μ to be μ = 0.2 ± 0.05. This value coincides with that determined from non-centred dissipation correlations measured in identical conditions.

The dynamics of propagation and disintegration of laminar liquid jets moving in air has been investigated theoretically. It is assumed that the jet is thin, i.e. the ratio of the characteristic transverse size to the longitudinal one is small. It is assumed also that the lateral surface of the jet is free of shearing forces and is ‘almost free’ of normal ones in the sense that the normal tractions other than isotropic pressure are small in comparison with the internal stresses acting in the jet cross-section.

Asymptotic quasi-one-dimensional equations of the continuity, momentum and moment of momentum of liquid motion in the jet have been derived. These equations were used as a basis for studying the process of growth of long-wave bending (transverse) disturbances of high-velocity jets of circular cross-section during their motion through air. The instability condition has been obtained and the growth rate of small bending disturbances of the jet has been found; the evolution of the jet shape at the stage of finite disturbances is investigated.

The plane steady flow of a grounded ice sheet is analysed under the assumption that the ice behaves as a nonlinearly viscous fluid with a strongly temperature-dependent rate factor. It is supposed that the accumulation/ablation distribution on the (unknown) free surface is prescribed, and that there is a given basal sliding condition connecting the tangential velocity, tangential traction and normal pressure. The basal boundary is defined as the smooth contour which describes the mean topography viewed on the ice-sheet lengthscale, and is assumed to have small slope. The perturbation analysis which reduces the isothermal or constant rate factor equations to an ordinary differential equation for the leading-order profile is now extended with similar success to the non-isothermal problem when the temperature distribution is prescribed. That is, the thermomechanically coupled energy balance is not solved, but families of temperature distributions qualitatively compatible with observed patterns are adopted to exhibit the effects of significant creep-rate variation with temperature.

The problem of transient natural convection in a cavity of aspect ratio A < 1 driven by internal buoyancy sources and sinks distributed linearly in the horizontal and uniformly in the vertical is considered. Scaling analysis is used to show that a number of possible transient flow regions are possible, collapsing ultimately onto one of conductive, transitional, or convective steady-state flow regimes. A number of numerical solutions are obtained, and their relationships to the scaling analysis are discussed.

This paper concerns the effects of topographic changes upon the dispersion of contaminants in rivers. Exact analytical results are derived for the time-of-arrival, temporal variance and skewness far downstream of a sudden contaminant release. It is demonstrated that there can exist optimal discharge sites, such that at all positions far downstream the maximum concentrations are minimized.

Around a horizontal cylinder submerged beneath waves with its axis parallel to the wave crests there exists, owing to streaming flow in the boundary layer, circulation with the same sense of rotation as that of the orbital motion in the waves. Experimental results, for conditions in which the effects of separation are not important, confirm theoretical predictions of the mass-transport velocity at the outer edge of the oscillatory boundary layer. Pressure measurements on the cylinder reveal a nonlinear component which is interpreted as a consequence of circulation induced by steady streaming.

Experiments on and calculation methods for flow in straight non-circular ducts involving turbulence-driven secondary motion are reviewed. The origin of the secondary motion and the shortcomings of existing calculation methods are discussed. A more refined model is introduced, in which algebraic expressions are derived for the Reynolds stresses in the momentum equations for the secondary motion by simplifying the modelled Reynolds-stress equations of Launder, Reece & Rodi (1975), while a simple eddy-viscosity model is used for the shear stresses in the axial momentum equation. The kinetic energy k and the dissipation rate ε of the turbulent motion which appear in the algebraic and the eddy-viscosity expressions are determined from transport equations. The resulting set of equations is solved with a forward-marching numerical procedure for three-dimensional shear layers. The model, as well as a version proposed by Naot & Rodi (1982), is tested by application to developing flow in a square duct and to developed flow in a partially roughened rectangular duct investigated experimentally by Hinze (1973). In both cases, the main features of the mean-flow and the turbulence quantities are simulated realistically by both models, but the present model underpredicts the secondary velocity while the Naot-Rodi model tends to overpredict it.

The flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles is studied using statistical methods analogous to those used in the kinetic theory of gases. Two theories are developed: one for the Couette flow of particles having arbitrary coefficients of restitution (inelastic particles) and a second for the general flow of particles with coefficients of restitution near 1 (slightly inelastic particles). The study of inelastic particles in Couette flow follows the method of Savage & Jeffrey (1981) and uses an ad hoc distribution function to describe the collisions between particles. The results of this first analysis are compared with other theories of granular flow, with the Chapman-Enskog dense-gas theory, and with experiments. The theory agrees moderately well with experimental data and it is found that the asymptotic analysis of Jenkins & Savage (1983), which was developed for slightly inelastic particles, surprisingly gives results similar to the first theory even for highly inelastic particles. Therefore the ‘nearly elastic’ approximation is pursued as a second theory using an approach that is closer to the established methods of Chapman-Enskog gas theory. The new approach which determines the collisional distribution functions by a rational approximation scheme, is applicable to general flowfields, not just simple shear. It incorporates kinetic as well as collisional contributions to the constitutive equations for stress and energy flux and is thus appropriate for dilute as well as dense concentrations of solids. When the collisional contributions are dominant, it predicts stresses similar to the first analysis for the simple shear case.

The interactions of compliant coatings with laminar, transitional and turbulent boundary layers are investigated. A 2 m long flat plate is towed in the range of speeds of 20–140 cm/s in an 18 m water channel using a carriage riding on an oil film. Isotropic and anistropic compliant coatings are used to cover about 20% of the working Plexiglas surface. The compliant material used is a viscoelastic plastisol gel produced by heating a mixture of polyvinyl chloride resin, a plasticizer and a stabilizer, and allowing them to gel. The shear modulus of rigidity of the plastisol was varied by changing the percentage of PVC in the mix. Anisotropy is introduced by placing the plastisol on a rubber surface having longitudinal grooves scaled with the low-speed streaks in the turbulent boundary layer. The most pronounced effect of the surface compliance in a turbulent boundary layer is a hydroelastic instability in the form of a spanwise wave structure on the compliant surface. The compliant-surface deformation was measured using a novel remote optical technique. The onset speed of the hydroelastic instability waves depends on the thickness and the modulus of rigidity of the plastisol. Their wavelength, wave speed and amplitude are found to depend on these plastisol parameters as well as on the towing speed. In a laminar boundary layer with freestream speeds of over twice the corresponding onset velocity for the turbulent case, no similar instability is observed.

Using the method of matched asymptotic expansions, the interaction between axisymmetric laminar boundary layers and inviscid supersonic external flows is investigated in the limit of large Reynolds numbers. The resulting triple-deck equations are solved numerically for two different cases of body shapes: a cylinder-cone configuration and a configuration consisting of two concentric cylinders which are connected by a smooth curve. Solutions to the linearized as well as the fully nonlinear equations are presented.

The establishment of an upstream intrusion of a buoyant fluid discharged into an open-channel flow of uniform density and finite depth is studied numerically using the full Navier-Stokes and diffusion equations. The problem is posed as an initial-boundary-value problem for the laminar motions of a Boussinesq fluid. The equations are integrated numerically by finite-difference methods. The flow patterns produced are controlled by the influx of buoyancy; therefore they are characterized by an inflow densimetric Froude number. A comparison with available experimental data provides favourable support to the theoretical predictions. The critical value of densimetric Froude number of the source of a vertically downward inflow at the free surface of a channel is determined. For densimetric Froude number less than critical, an intrusion is established on the upstream side of the source. Because dissipative mechanisms associated with viscosity take a finite time to intervene, the intrusion starts as an inviscid gravity current with a propagation speed greater than the surface velocity of the stream. The front speed is proportional to the phase velocity of long internal waves. Subsequently, the front decelerates as the interfacial friction, and, if applicable, the boundary frictional forces increase simultaneously with mass entrainment across the interface. The current slows down towards a two-zone equilibrium: (1) the zone encompassing the current behind the frontal zone, where a steady state is approached with respect to the inertial frame of reference; (2) the frontal zone, where the upstream speed approaches a steady speed of frontal advance, albeit small. The upstream intrusion alters the flow pattern of the ambient stream dramatically. A significant feature of both the upstream and downstream currents is the presence of surface convergence with concomitant downwelling near the fronts. As the upstream front decelerates, wavelike disturbances are excited just behind the front at frequencies characteristic of internal waves. As the front approaches steady state, these disturbances appear to be damped. This problem has practical implications in the design of once-through cooling-water systems for power plants taking their cooling water from rivers.

Experiments with hydraulic jumps in a layered flow with a small density difference at the interface are described. Two different configurations are examined, with shear between the fluids either upstream or downstream of the jump. It is shown that when the shear stress on all interfaces is small enough for there to be no mixing either the theory which assumes hydrostatic pressure on the face of the jump or that which assumes energy conservation in one of the layers describes the results. In the experiments for a jump in the lee of a towed obstacle this condition is always satisfied, but for a jump advancing into stationary layers it is only satisfied when the ratio of the height behind the jump to that in front is less than about 2. Beyond this limit there is mixing behind the first wave of the undular jump and the flow behaves like the head of a gravity current. The theory with energy conservation in one layer is extended to the case of a stationary jump, and for this case it is shown that for a given downstream control an approximate value of the fluid entrained can be computed.

Rotationally symmetric vortex flows between concentric cylinders with the inner one rotating and the outer one at rest have been investigated by numerical simulation and by laser-Doppler velocimetry for an annulus of aspect ratio Γ = 1.05 with a radius ratio η = 0.5066. Stationary states and relaxation towards them were explored close to the transition from the primary flow, which is mirror symmetric with respect to the midplane of the annulus, to a flow which gradually loses the symmetry. Detailed comparison of numerically simulated and measured velocity fields is made.

Measurements were taken of a developing complex turbulent flow formed by the merging of two far turbulent wakes of equal cylinders. The lower wake only was slightly heated, permitting the use of the conditional-sampling technique to study the interaction of the two wakes in detail. A special four-wire probe (Fabris 1978) enabled simulataneous uncontaminated measurement of all three instantaneous velocity components and temperature. The single-wake data of Fabris (1979) served as a reference. The interaction of the two single turbulent wakes resulted in a striking enhancement of the lateral transfer of heat. Undulations of the thermal interface are significantly wider laterally, but shorter streamwise than in the basic single wake. Lateral velocities of the heated and the unheated intermittent lumps are amplified two- to threefold by the interaction process. Levels of all three components of the kinetic energy of turbulence reach their highest relative maxima in the heated zones as they cross the upper cold wake. These maxima are higher than appropriate conventional or conditional maxima in the single wake. In addition to physical insights into the phenomena of turbulent flow, the data should be useful in refining and validating predictive methods, possibly spurring their further development.

The effects of curvature on the fluid dynamics of rapidly rotating gas centrifuges are studied. A governing system of a linear partial differential equation and boundary conditions is derived based on a linearization of the equations for viscous compressible flow. This system reduces to the Onsager pancake model if the effects of curvature are neglected. Approximations to the solutions of the governing equations with and without curvature terms are obtained via a finite-element method. Two examples are considered: first where the flow is driven by a thermal gradient at the wall of the centrifuge, and then for the flow being driven by the introduction and removal of mass through the ends of the centrifuge. Comparisons of the results obtained show that, especially for the second example, the inclusion of the terms due to curvature in the model can have an appreciable effect on the solution.

The phenomenon of cavitation inception was studied on four axisymmetric bodies whose boundary layers underwent a laminar separation and subsequent turbulent reattachment. The non-cavitating flow was studied by holographic and schlieren flow-visualization techniques. Surface distributions on the mean and the fluctuating pressures were also measured. The conditions for cavitation inception and desinence were determined and holograms were recorded just prior to and at the onset of cavitation. The population of microbubbles and the subsequent development of visible cavitation was determined from the reconstructed image. In every case the appearance of visible cavitation was preceded by a cluster of microscopic bubbles in a small portion of the flow field providing clear evidence that cavitation is initiated from small nuclei. The inception zone was located within the turbulent shear layer downstream of transition and upstream of the reattachment region of the bodies with large separation regions. The location and the shape of this cavitation suggested a relation to the mixing-layer eddy structure. The inception region on the body with the smallest separation zone, a hemisphere-cylinder body, was located in the reattachment region, but the cavities were still detached from the surface. Instantaneous minimum-surface-pressure measurements do not account for observed cavitation-inception indices except for the hemisphere body, where the correlation is satisfactory. The rate of cavitation events was estimated from measurements of nuclei population, and fluctuating-pressure statistics in the portion of the flow susceptable to cavitation. It was demonstrated for the hemisphere body that at least one such cavitation event could occur every second. These findings are consistent with what is observed visually at the onset of cavitation and support the location of inception determined holographically.