Some new measurements and a reassessment of previous data on statistical properties of the breakdown coefficients qr,l in high Reynolds number turbulence show the existence of a range of scale similarity for scales larger than those in the viscous range (l [ges ] 36η). The rate of variation of the probability density p(qr,l) with changing outer scale l/η decreases as l/η increases, becoming fairly insignificant for the largest values of l/η. Measurements of characteristic functions of the probability densities show a substantial degree of statistical independence for sequential adjoint values of qr,l, consistent with the small values of the correlation coefficients for these variables. The data for the moments of qr,l exhibit a behaviour very close to that predicted by the scale-similarity theory when only data for r [ges ] 36η are considered, i.e. data for smaller inner length scales are excluded. The moments and corresponding values of the parameters μp are in good agreement with our previous results and with some earlier data of Kholmyansky, but some rather large unresolved differences in the probability densities of qr,l are found on comparing the present data with those of Kholmyansky. The present measurements of breakdown coefficients for ζ1 = [uscr ]− 1∂u/∂t = ∂(In [uscr ])/∂tand ζ2 = U−1∂u/∂t the time derivatives of the streamwise velocity and its logarithm measured in the atmospheric boundary layer, resolve some previous questions concerning the sensitivity of the results obtained to the choice of positive variable, varying sampling rates and the values of external parameters.

For low sampling rates, a systematic change in the shape of the probability densities p(qr,l) with varying digital sampling rate is found using either ζ1 or ζ2. For sufficiently high sampling rates, the probability densities are independent of the sampling rate; and invariant results are obtained when the sampling rate is at least one-quarter of the Kolmogorov frequency associated with the viscous length scale based on the turbulent dissipation rate. The probability densities p(qr,l) measured using either ζ1 or ζ2 are very similar to the corresponding spectra of ζ1 or ζ2 respectively. Comparison of the mean-square values of ζ1 and ζ2 with an extended form of Taylor's hypothesis shows that the variable ζ1 is not a good approximation to the true spatial derivative ∂u/∂x, and the use of such an approximation can lead to results that are both qualitatively and quantitatively incorrect.

The stability of a horizontal fluid layer when the thermal (or concentration) gradient is not uniform is examined by means of linear stability analysis. Both buoyancy and surface-tension effects are considered, and the analogous problem for a porous medium is also treated. Attention is focused on the situation where the critical Rayleigh number (or Marangoni number) is less than that for a linear thermal gradient, and the convection is not (in general) maintained. The case of constant-flux boundary conditions is examined because then a simple application of the Galerkin method gives useful results and general basic temperature profiles are readily treated. Numerical results are obtained for special cases, and some general conclusions about the destabilizing effects, with respect to disturbances of infinitely long wavelength, of various basic temperature profiles are presented. If the basic temperature gradient (considered positive, for a fluid which expands on heating, if the temperature decreases upwards) is nowhere negative, then the profile which leads to the smallest critical Rayleigh (or Marangoni) number is one in which the temperature changes stepwise (at the level at which the velocity, if motion were to occur, would be vertical) but is otherwise uniform. If, as well as being non-negative, the temperature gradient is a monotonic function of the depth, then the most unstable temperature profile is one for which the temperature gradient is a step function of the depth.

Supersonic boundary-layer flow on a flat plate near a point of injection cut-off is considered. The goal is to develop a non-singular solution in the limit of large Reynolds number in the vicinity of the boundary-condition change. An application of Goldstein-type singular solutions shows that an interaction-type theory is required even for transverse velocity jumps on the boundary O(R− ½). The interaction analysis is developed in terms of the linearized triple-deck theory described by Smith & Stewartson (1973a). The analytically derived solution provides a continuous pressure and wall-shear distribution.

Similarity solutions are determined for the turbulent wake of a self-propelled body (thrust = drag). The momentumless wake is shown to behave in a manner intermediate to homogeneous grid turbulence and more familiar free-shear flows such as the drag wake or jet. In essence the decay of momentumless-wake turbulence is similar to that of grid turbulence, but proceeds at a somewhat greater rate owing to lateral diffusion. The mean velocity difference is coupled to the difference $\overline{u^2}-\overline{v^2}$ between the axial and radial components of the mean-square fluctuating velocity. It is necessary to consider governing relations for various second-order turbulence quantities. Previously developed closure approximations yield far-wake decay rates that agree well with available measurements. Production of turbulent energy is negligible asymptotically; thus there is no balance between production and dissipation, and the far-wake behaviour does not become independent of the initial (near-wake) conditions. Even the radial profiles depend on the initial conditions, and there is no natural length scale with which to characterize the far wake.

Neither the phenomenological nor the structural approach to the determination of constitutive equations has yet shown itself to be capable of producing useful and predictive descriptions of the majority of technologically important complex fluids. In the present paper we explore the suggestion that significant progress can be made when these two complementary approaches to rheology are combined. For this initial study we restrict our attention to materials which can be modelled as a suspension of particles in a Newtonian fluid, thereby including most polymer solutions while excluding polymer melts. By applying phenomenological techniques to the basic formulation of suspension mechanics we are able to deduce a common simplified constitutive model for all suspension-like materials and to reveal its physical origin. The present analysis demonstrates that the constitutive model of Hand (1962), involving a single second-order tensor, is not sufficiently general for a rigorous description of the majority of suspension-like materials. Consideration of the constitutive forms for the limiting cases of near-equilibrium and strongly non-equilibrium microstructure suggests, however, that Hand's model may provide a reasonable approximation to the exact constitutive behaviour which is useful over the whole range of flow strengths.

The interaction between internal gravity waves in a rotating frame and the mean flow is discussed for the case when the properties of the mean flow vary slowly on a scale determined by the local wave structure. The principle of conservation of wave action is established. It is shown that the main effect of the waves on the Lagrangian mean velocity is due to an appropriate ‘radiation stress’ tensor. A circulation theorem and a potential-vorticity equation are derived for the mean velocity.

The paper examines how a passive contaminant disperses along the axis of a tube in which the flow is driven by a longitudinal pressure gradient varying harmonically with time. This problem is of intrinsic interest and is relevant to some important practical problems. Two examples are dispersion in estuaries and in the blood stream. By means both of statistical arguments and an analysis like that used by Taylor (1953) in the case of a steady pressure gradient it is shown that eventually the mean distribution of concentration satisfies a diffusion equation (and is therefore a Gaussian function of distance along the axis) with an effective longitudinal diffusion coefficient K(t) which is a harmonic function of time with a period equal to one half of that of the imposed pressure gradient. Contrary to the supposition made in most previous work on this problem it is shown by examining some special cases that the harmonic terms in K(t) may have a noticeable effect on the dispersion of the contaminant and in particular on the rate at which it is spreading axially. The size of the effect depends on both the frequency and the Schmidt number and is particularly large at low frequencies. The paper concludes with an analysis of a model of dispersion in estuaries which has been used frequently and it is concluded that here too oscillatory effects may often be noticeable.

The paper discusses a theoretical model of statistically steady flow in a strongly stratified estuary. A halocline is assumed to be present and the lower layer is taken to be deep and non-turbulent. The outflowing upper fluid mixes with the salty lower fluid and the flux of the brackish water increases with distance from the head of the estuary. The mixing is assumed to be similar to that in laboratory models of mixing across density interfaces.

Two equations of mass conservation are used, one for the steady-state mass flux across a vertical section from top to bottom of the channel and one for the mass flux into a section of the upper fluid. A buoyancy conservation equation is used for the buoyancy flux across a vertical section. A final equation is obtained by integrating the horizontal equation of motion across a section of the upper fluid. The flow in this layer is assumed to be opposed by a frictional force proportional to the square of the velocity averaged over the layer. The pressure-gradient force arising from the slope of the free surface is solved for in terms of the thickness of the upper layer, the buoyancy difference across the interface, the slope of the interface and the horizontal density gradient in the upper layer. The derivation shows that the horizontal pressure-gradient force vanishes in the lower layer.

The mathematical problem reduces to two ordinary differential equations for the flux in the upper layer and its thickness. Attention is confined to the solution for subcritical flow, in which the interface falls with distance from the head, reaching a maximum depth at a certain section of the estuary. Beyond this the interface rises. At the mouth, where, by definition, the width of the estuary increases rapidly, it is shown that there must be a transition from subcritical to supercritical flow. This condition, applied to the solution for uniform width, determines a remaining unknown related to the depth of the halocline at the head of the estuary and the complete solution is obtained as a function of the freshwater influx per unit width, the r.m.s. turbulent velocity, the estuary length and the buoyancy of sea water.

The solution is complicated but has reasonable behaviour for variations of the given parameters of the problem. A basic feature for values of the constants appropriate to fjord-type estuaries is the dominance of friction, omitted in an earlier, incomplete investigation by Stommel. This is also revealed by the large drop in the free surface over the length of the estuary.

A comparison with two estuaries, Oslofjord and Knight Inlet, British Columbia, indicates that the former is very different from the model of this paper but that the latter may have a similar nature.

This paper reports experimental investigations on the characteristic decay and the radial-type decay regions of a three-dimensional isothermal turbulent wall jet in quiescent surroundings. The velocity and the length scale behaviour for both the longitudinal and the transverse directions are studied, and compared with the results of other workers. The estimated skin friction is discussed in relation to the available data from earlier investigations. Wall jet expansion rates and the behaviour of skin friction are also discussed. The rate of approach of turbulence components to a self-similar form is found to be influenced by the fact that the expansion rate of the wall jet in the longitudinal direction is different from that in the transverse.

A stably stratified free shear layer is created in a continuously circulating water channel in the laboratory. Two streams of salt water of different concentrations are brought together at the entrance to the open channel and a layered uniform flow field with a distinct sharp interface is produced in the test section. The maximum density difference between the two layers is Δρx = 0·0065ρw, where ρw is the density of water. The velocity of each layer can be adjusted at will to create free shear across the interface. At the end of the open channel, a mechanical device to separate the layers for recirculation is provided. The resulting flow field has a viscous region approximately 15 times larger than the scale of the salinity diffusion layer. Visual observations are made with hydrogen bubbles and dye traces. Interfacial waves are initiated by artificial excitation. The perturbation frequencies range from 0·476 to 10·40Hz. The measured wavelengths range from 0·46 to 3·02 cm. Damped waves as well as growing waves are observed at various exciting frequencies. Velocity profiles and instantaneous velocities are measured by a hot-film anemometer designed for use in salt water. Experimental values of the Richardson number, the dominant parameter characterizing the instability process, range from 1·23 to 14·45. The stability boundary is determined experimentally. Comparisons with Hazel's numerical results and the earlier results of Scotti & Corcos for low values of the Richardson number are also made.

Measurements have been made of the net force F acting on a bluff rigid body moving with velocity U (relative to a fluid rotating about a vertical axis with uniform angular velocity Ω) in a plane perpendicular to the axis of rotation. The force F is of magnitude 2ΩρVU, where ρ is the density of the fluid and V is a volume which depends on the size and shape of the body. The relative direction of F and U is found to depend on the quantity \[ {\cal S}\equiv \frac{2\Omega L}{U}\bigg(\frac{h}{D}\bigg), \] where L and h are horizontal and vertical lengths characterizing the object and D is the depth of the fluid in which the object is placed.

We consider the mixing between two miscible liquids of slightly different density (< 10%) when one of them (cargo) is injected into a tank partially filled with the other (inventory). The injection of the cargo is such that buoyancy and inertia act in concert on the plume produced by the cargo. The two basic processes that govern the mixing of the two liquids in the tank are the entrainment of tank liquid by the plume and the tank circulation set up by this entrainment and by the plume discharge. Unlike plumes in an environment of infinite extent, the plume in this case changes its environment continuously, which, in turn, has a continuously-varying effect on the plume. A mathematical model for the mixing of the two liquids is presented, from which one can compute the tank stratification that may result when given amounts of cargo and inventory are thus mixed. Plume entrainment theory is used for the plume dynamics and a ‘filling-box’ model is used for the tank circulation. The partial differential equations of the model are integrated by an original and unique numerical method. The problem was also treated experimentally. The tank stratification is expressed in terms of a normalized density-difference variable δ. Except for some very localized large discrepancies, due to certain local effects not included in the model, computed and experimental profiles of δ agree very well, their maximum and average deviations being within 4 and 2%, respectively. It is found that values of the empirical plume parameters α and λ that are used commonly for steady plumes in environments of infinite extent are approximately right for the time-dependent plumes under consideration too.