A theoretical model for the velocity field and the surface profile of bores and hydraulic jumps is developed. The turbulence is assumed to be concentrated in a wedge that originates at the toe of the front and spreads towards the bottom, and the turbulent closure used is a simplified k-ε model allowing for non-equilibrium in the turbulent kinetic energy. The flow equations are satisfied in depth-integrated form (method of weighted residuals), and measured deviations from static pressure are analysed and shown to have a negligible effect on the results. Comparison with measurements shows good agreement, but there is a clear need for further experimental results in the highly turbulent region near the free surface. Some basic mechanisms of the flow are discussed and explained from the theory.

Employing a high-speed video system and hydrogen bubble-wire flow visualization, the characteristics of the low-speed streaks which occur in the near-wall region of turbulent boundary layers have been examined for a Reynolds-number range of 740 [les ] Reθ < 5830. The results indicate that the statistics of non-dimensional spanwise streak spacing are essentially invariant with Reynolds number, exhibiting consistent values of $\overline{\lambda^{+}} \approx 100$ and remarkably similar probability distributions conforming to lognormal behaviour. Further studies show that streak spacing increases with distance from the wall owing to a merging and intermittency process which occurs for y+ [simg ] 5. An additional observation is that, although low-speed streaks are not fixed in time and space, they demonstrate a tremendous persistence, often maintaining their integrity and reinforcing themselves for time periods up to an order of magnitude longer than the observed bursting times associated with wall region turbulence production. A mechanism for the formation of low-speed streaks is suggested which may explain both the observed merging behaviour and the streak persistence.

Laminar natural convection from a horizontal plate is studied by a finite-difference analysis and by experiments for Rayleigh numbers from 10 to 104. The plate with uniform surface temperature or concentration on one side and insulated on the other is situated in an ‘infinite’ fluid medium. The buoyancy near the surface is directed either outward or inward normal to the active surface – equivalent to a heated plate facing upward or downward. The effect of insulated vertical and horizontal extensions to the plate are also examined.

Finite-difference solutions are obtained for a heated strip in a two-dimensional domain for a Prandtl number of 0·7. Mass-transfer experiments are performed with square naphthalene plates in air. Both numerical and experimental results justify a 1/5-power law in the present range of Rayleigh number – i.e. Nusselt number or Sherwood number proportional to the Rayleigh number raised to the 1/5 power. The horizontal extensions cause a limited reduction in the transfer rate for the plate generating ‘outward buoyancy’, and a larger reduction with ‘inward buoyancy’. The vertical walls block the fluid flow directly, and thus greatly lower the transfer rate with either outward or inward buoyancy.

The steady motion of an eccentrically positioned sphere in a circular cylindrical tube filled with viscous fluid is considered as a regular perturbation of the axisymmetric problem. A sequence of boundary-value problems is formulated involving Stokes equations and some linear boundary conditions. Solutions of the first- and second-order problems yield the leading terms in the perturbation series of the additional drag and the torque on the spheres. The results are found to be in good agreement with the previous off-axis solutions.

The dynamics of the formation of the axisymmetric meniscus around a cone contacting a free liquid surface are discussed. An approximate phenomenological model is set up. In the case considered, where Re [Lt ] 1 and viscosity dominates the retarding forces, this leads to a differential equation relating the height of the circle of contact to time. Solutions are derived, involving one or more unknown parameters, which describe the time dependence of the height of the circle of contact.

Experimental data, obtained from delayed flash photographs of the meniscus profiles of silicone fluid climbing over the surface of glass cones, provide general support for the model. The agreement between the predicted and observed height as a function of time is sufficiently close to justify the model as a useful description.

We present an asymptotic study of steady two-dimensional radial flow between converging plane walls (Jeffery–Hamel flow) when the viscosity μ and density ρ vary with the angular coordinate θ. Two representative situations are considered, the first, being a two-layer system (in which μ and ρ are uniform except for discontinuities at an interface θ = θI), and the other involving a fluid for which μ and ρ vary continuously with θ. The flow is analysed in the asymptotic limit when a parameter c related to the wall pressure gradient is large; this corresponds to converging flow at large Reynolds number. Solutions are derived for the boundary layers at the walls and for the shear layer at the interface; the results are shown to agree well with some exact (numerical) profiles.

The solutions obtained are not unique, though for given c they represent the ‘simplest’ type of profile, and the one that seems most likely to be stable. We demonstrate the non-uniqueness by deriving in §3 an alternative solution for the interfacial shear layer. This solution, however, can exist for only restricted ranges of values of the density and viscosity ratios, and involves an outgoing jet, suggesting that it is likely to be unstable.

In this paper we concern ourselves with the theoretical description of curved converging shock waves, where nonlinear interaction effects, between the shock fronts and the flow behind them, and refraction effects are equally important. In a non-viscous, isoenergetic and isentropic flow the problem can be described by a nonlinear wave equation for the pressure field. This equation then admits an analytical solution with the help of the method of strained coordinates provided that the nonlinear terms contain only derivatives with respect to two independent variables. This restrictive condition is approximately fulfilled if the incoming wave is only slightly curved.

Replacing in the solution the strained coordinates – which themselves depend on the solution – by physical coordinates, we get an accurate description of the transition from the shock pattern obtained by the geometric-acoustics approach (very weak shocks) to the pattern determined by Whitham's shock dynamics (strong shocks). Furthermore, the solution describes the complete flow field and agrees very favourably with experimental data by Sturtevant & Kulkarny.

Exact expressions are derived for the centroid and variance as functions across the flow when there has been an initially uniform contaminant release in an oscillatory flow. Two examples are given to demonstrate that there can be a substantial region of the flow (where the velocity shear is relatively large) in which the contaminant distribution exhibits contraction after flow reversal. This effect, and the sensitivity of the variance to the precise time of discharge, is most marked when the flow oscillations are rapid relative to the timescale for cross-sectional mixing.

The problem of finite-amplitude thermal convection in a porous layer with finite conducting boundaries is investigated. The nonlinear problem of three-dimensional convection is solved by expanding the dependent variables in terms of powers of the amplitude of convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. Square-flow-pattern convection is found to be preferred in a bounded region [Gcy ] in the (γb, γt)-space, where γb and γt are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid. Two-dimensional rolls are found to be the preferred pattern outside [Gcy ]. The qualitative features of the convection problem appear to be essentially symmetric with respect to γb and γt. The dependence of the heat transported by convection on γb and γt is computed for the various solutions analysed in the paper.

Experiments with fluids whose viscosity depends strongly on temperature are used to study the effect of viscosity variations in the range 10−105 on the heat transfer and horizontally averaged temperature of a convecting layer between horizontal isothermal boundaries. At large viscosity variations (3 × 103 and 105) and Rayleigh numbers less than the critical value given by linear theory, the system can be either conductive or convective depending on whether the Rayleigh number is increased from an earlier conductive state or decreased from a preexisting convective state. At higher Rayleigh numbers and for the entire range of viscosity variation studied the heat transfer differs little (< 20%) from that of a uniform-viscosity fluid when the Rayleigh number is defined in terms of the viscosity corresponding to a temperature equal to the average of the boundary temperatures. The relationship between Nusselt number and supercriticality (Ra/Rc) is even more remarkable being independent of viscosity variation and indistinguishable from that of a uniform-viscosity fluid with appropriate Prandtl number. The horizontally averaged temperature becomes increasingly asymmetrical with increasing viscosity variation due to the relatively large temperature change across the cold, more-viscous boundary layer, and results in an isothermal interior temperature significantly hotter than the average of the boundary temperatures. The measured temperature and convective heat transfer as a function of depth show that for viscosity variations greater than about 100 most of the viscosity change occurs within a stagnant conductive layer that develops above the actively convecting part of the system.

Numerical solutions of the two-dimensional shallow-water equations are found for motion on a sloping beach generated by a single bore and by a periodic succession of bores, both incident at small angles. For the latter, periodic longshore motion can always be found if bottom friction (described here by the Chezy law) in included. The timescale of the development of longshore motion is also considered.

Methods to investigate the existence of overhanging gravity waves of permanent form at the interface between two uniform fluids of different density are discussed. Numerical results which demonstrate their existence are presented.

The vertical displacements Z(t) of fluid elements passing through a source z = 0 at t = 0 in a horizontal mean flow with stably stratified statistically stationary turbulence (with buoyancy frequency N and velocity time-scale T), under the action of random pressure gradients and damping by internal wave motions, are investigated by a model Langevin-like equation, and by a general Lagrangian analysis of the displacements, of the density flux and of the energy of fluid elements. Solutions for the mean-square displacement $\overline{Z^2}(t)$, the mean-square velocity $\overline{w^2}$, and the autocor-relation of the velocity are calculated in terms of the spectrum [Fcy ](s) of the pressure gradient. We use model equations for the momentum of fluid elements and for the exchange of density fluctuations between fluid elements, taking the elements’ diffusion timescale to by γ−1 times the buoyancy timescale N−1, where γ is a measurable parameter.

In the case of moderate-to-strong stable stratification (i.e. NT [gsim ] 1), we find the following.

1. (i) When there is no change of the fluid elements’ density (γ = 0), the mean-square displacement $\overline{Z^2}$ of marked particles ceases to grow when t [gsim ] N−1, and its asymptotic value is proportional to $\overline{w^2}/N^2$, where the constant of proportionality, $\overline{\zeta^2}(\infty)$, is O(1) and a decreasing function of (NT)−1. This result is also shown to be a general consequence of the finite potential and kinetic energy in the stationary turbulence.

2. (ii) If there is a small diffusive interchange of density between fluid elements (i.e. γ [Lt ] 1), the marked particles’ mean-square displacement has a slow linear growth (i.e. $\overline{Z^2}\sim\overline{w^2}/N^2(1+O(\gamma^2)tN) $).

3. (iii) Such molecular processes must also dilute the initial concentration of contaminants (e.g. dye or smoke) in those fluid elements that diffuse above the limit in (i).

4. (iv) The mean-square fluctuation of density is proportional to the product of the asymptotic mean-square displacement of marked particles and the square of the mean density gradient $(-{\cal G})\;({\rm i.e.}\overline{\rho^{\prime 2}}={\cal G}^2\overline{Z^2}=\frac{1}{2}\overline{\zeta^2}(\infty ,\gamma = 0){\cal G}^2\overline{w^2}/N^2)$.

5. (v) The flux Fρ of density in a turbulent flow can be expressed exactly as the sum of two terms, the first $\frac{1}{2}(d\overline{Z^2}/dt)\,{\cal G}$, being caused by the growth of the displacements of fluid elements, and the second $\overline{\dot{Z}(\rho + \rho^{\prime})}$ being caused by the mixing between fluid elements. In stable flows, it is shown that the second element is dominant, and $F_{\rho}\sim\gamma\overline{w^2}N\rho_og $ while the first is smaller by O(γ).

Previous laboratory and field measurements of $\overline{w^2}, \overline{Z^2}, \overline{\rho^{\prime 2}} $ and Rw(t) are discussed in some detail and shown to be consistent with this model.

Equipment is described for performing measurements of the Rayleigh–Bénard instability in a sample of rotating liquid 4He. Data are presented on the dependence of the critical Rayleigh number for the onset of convection on Prandtl number in the range 0·49 < Pr < 0·76 and over a range of dimensionless angular velocities 0 < Ω < 200. Evidence for the existence of subcritical convection is presented.

An exact solution for the squeeze-film motion in an upper convected Maxwell fluid is given for both the plane and axisymmetric cases. Inertia and viseoelastic effects are included, and it is shown that the solution depends only on the product of the Weissenberg and Reynolds numbers. Solutions are generated for values of this product up to 500 without numerical problems. The solutions show wave propagation and show a reduced load capacity relative to the Newtonian case.

A new system has been developed for estimating experimentally some of the principal physical variables of fluid flows, through flow-visualization and image-processing techniques. Distributions of stream function, vorticity and pressure are calculated by this system with reasonable accuracy for two examples of two-dimensional flow: namely unsteady twin-vortex flow behind a circular cylinder accelerated impulsively to constant speed, and Kármán vortices behind a circular cylinder moving at constant speed. A detailed explanation of the image-processing technique and the numerical calculation process is given first, and then some consideration is given to calculated results in these two types of flow. Comparison shows that some results of the unsteady twin-vortex experiment coincide well with those of previously published experimental investigations and theoretical calculations. Errors introduced at each stage of this system are estimated in some detail.

The pattern of dispersion and uptake of an inhaled slug of tissue-soluble gas is examined within a branching model of the bronchial airways of the human lung, considered as an assembly of segments from infinitely long, straight rigid tubes with absorbing walls of finite thickness. The model is based on the first three (time-dependent) spatial moments of the solute distribution in such tubes, determined by the Aris method of moments. Poiseuille flow in each airway is assumed, and the solute distribution is taken to be initially zero in the tissue and radially uniform in the gas. First, the time dependence of axial velocity and mixing coefficient of the advancing solute in infinitely long tubes is shown and the mechanisms responsible are discussed. Transit times, uptake, uptake efficiency and mixing coefficient predicted from the model are then shown for different flow rates and solubilities, as functions of the generation of branching. As is expected, greater penetration is found for lower-solubility gases. However, of greater interest is the model prediction that uptake decreases with increasing flow rate whereas uptake efficiency increases, a result consistent with experimental indications. Finally, the mixing coefficient is shown to fall, with distance into the lung, to a value which may be much smaller than the molecular diffusivity, depending on the solubility.

We use the method of Phythian & Curtis (1978) to obtain a self-consistent calculation, in lowest order of perturbation theory, for the α-coefficient and effective diffusivity of a magnetic field in a plasma with Gaussian turbulence.

If different contaminant species are subject to different transverse drift rates (e.g. gravitational settling), then there is a tendency for the species to separate out. The efficiency of this separation depends upon the relative shapes of the longitudinal concentration distributions. Jayaraj & Subramanian (1978) have drawn attention to the disparity between their computed skew concentration distributions and the symmetric Gaussian distributions predicted by one-dimensional diffusion models. Here it is shown that a one-dimensional delay-diffusion model yields suitably skew predictions. The model equation is used to investigate the extent to which the separation of different contaminant species can be improved by pre-treating the sample (i.e. allowing differential drift) in a stationary fluid before being eluted into the shear flow. Pretreatment is found to be very effective for plane Poiseuille flow but not for the thermogravitational columns.

Elementary calculations indicate that the effect of the Earth's rotation is likely to be important in the dynamics of most internal waves in oceans, lakes and the atmosphere. Here we present measurements of the structure and properties of one class of such waves, namely solitary internal Kelvin waves, in which the Coriolis force generated by wave motion in a stratified fluid is opposed by a pressure gradient and hence change in wave amplitude along its crest. We confirm that the wave speed is independent of the rate at which the system rotates and depends only on the stratification and maximum wave amplitude. However, rotation is shown to have a large effect on both the rate at which the amplitude varies with time and the cross-stream’ structure of the wave. In accordance with well-established theory, the amplitude transverse to the direction of propagation varies exponentially. This results in a decreasing wave speed with increasing distance from the wall, which in turn requires the wave front be curved backwards in order for the wave as a whole to propagate at a speed given by its maximum amplitude. Such a front curvature is not contained within the available theories. The rapid decay of wave amplitude is found to be due to the generation of inertial waves in the homogeneous fluid above and below the internal wave, and a reasonably successful scaling of this effect has been found. We also discuss the adjustment of the waves to geostropic balance and comment on applications of our results to natural systems.