A fluid is contained between rigid horizontal planes which move relative to each other with constant horizontal velocity. A gravitationally unstable temperature difference is maintained at the boundaries, and the heat flux and momentum flux (stress) transmitted by the fluid are measured. The Nusselt number, Nu, and the dimensionless momentum flux, Mo, are obtained for small mean rates of shear. The Rayleigh number, R, and the Prandtl number, σ, are both large in these experiments. The data are consistent with the following relations: \[ Nu \propto Mo \sigma^{\frac{1}{2}} \propto R^{\frac{1}{3}}. \]

Kraichnan's mixing-length theory of turbulent thermal convection is extended to the present situation, and the above experimental dependence of Nu and Mo on R and σ is obtained. The agreement between mixing-length theory and experiment provides strong support for Kraichnan's concise treatment of turbulent convection. The importance of this result is that heat and momentum fluxes may be calculated in this way for a variety of flows in geophysics.

This study of the boundary layer of steady, incompressible, plane, crossed-fields m.h.d. flow at large Reynolds number Re and magnetic Reynolds number Rm begins with a review of Hartmann's case, where a boundary layer occurs whose thickness is proportional to (Re Rm)−½. Following this clue, it is shown that in general the boundary layer is a ‘local Hartmann boundary layer’. Its profiles are always exponential and it is determined completely by local quantities. The skin friction and the total electric current in the layer are proportional to the square root of the magnetic Prandtl number, i.e. to (Rm/Re)½. Thus the exterior-flow problem, the solution of which precedes a boundary-layer solution, generally involves a current sheet at the fluid-solid interface.

This inviscid-flow problem becomes tractable if (Rm/Re)½ is small enough to permit a linearized solution. The flow field about a flat plate at zero incidence is calculated in this approximation. It is pointed out that the thin-cylinder solutions of Sears & Resler (1959), which pertain to Rm/Re = 0, can immediately be extended to small, non-zero values of this parameter by linear combination with this flat-plate solution.

A theory is derived for the class of long two-dimensional waves, comprising solitary and periodic cnoidal waves, that can propagate with unchanging form in heterogeneous fluids. The treatment is generalized to the extent that the waves are supposed to arise on a horizontal stream of incompressible fluid whose density and velocity are arbitrary functions of height, and the upper surface of the fluid is allowed either to be free or to be fixed in a horizontal plane. Explicit formulae for the wave properties and a general interpretation of the physical conditions for the occurrence of the waves are achieved without need to specify particular physical models; but in a later part of the paper, §4, the results are applied to three examples that have been worked out by other means and so provide checks on the present theory. These general results are also shown to accord nicely with the principle of ‘conjugate-flow pairs’ which was explained by Benjamin (1962b) with reference to swirling flows along cylindrical ducts, but which is known to apply equally well to flow systems of the kind in question here.

The theory reveals certain physical peculiarities of a type of flow model often used in theoretical studies of internal-wave phenomena, being specified so as to make the equation for the stream-function linear. In an appendix, some observations are also made regarding the ‘Boussinesq approximation’, which too is often used as a simplifying assumption in this field. It is shown, adding to a recent discussion by Long (1965), that finite internal waves may depend crucially on small effects neglected in this approximation.

Because of the relative simplicity of the statistical model for molecular interactions, numerically exact solutions of the Krook kinetic equation can be obtained. Comparison of exact and approximate solutions of the model equation allows the evaluation of approximate procedures for solving the Boltzmann equation. Exact and approximate numerical solutions have been obtained for Couette flow with heat transfer and the structure of a plane shock wave. The present paper summarizes the work of the author in obtaining the numerically exact solutions.

This is a continuation of our previous work (1965) on the Sears–Resler–Stewartson controversy, in the context of axially symmetric flow. A new approach is presented using boundary-layer arguments, which remove much of the old complexity.

For resistive bodies of shape R(x) we uncover a similarity with plane airfoils of shape F(x) = R2(x). As before, Stewartson's slug flows develop fore and aft. For bodies of very high conductivity the Sears–Resler (steady-state) solution turns out to be one possibility. It pertains to bodies (of much higher conductivity than the liquid) into which the initial magnetic field has not diffused.

A correlation scheme for velocity and temperature profiles is derived for turbulent boundary layers in adverse pressure gradients. The resulting analytical expressions are obtained by what could be referred to as ‘regional similarity’ arguments. This avoids the need to make use of the Reynolds analogy (explicitly, at least) or the usual local gradient-type diffusion expressions for momentum and thermal transport (‘the local similarity’ and Boussinesq concept). The expressions agree well with experimental data for the velocity profiles and encouraging correlation is shown for the temperature profiles. The expressions cover a wider part of the profile than given by the logarithmic law of the wall. Surface roughness and Prandtl-number effects are included in the analysis.

If a long wave of elevation travels in shallow water it steepens and forms a bore. The bore is undular if the change in surface elevation of the wave is less than 0·28 of the original depth of water. This paper describes the growth of an undular bore from a long wave which forms a gentle transition between a uniform flow and still water. A physical account of its development is followed by the results of numerical calculations. These use finite-difference approximations to the partial differential equations of motion. The equations of motion are of the same order of approximation as is necessary to derive the solitary wave. The results are in general agreement with the available experimental measurements.

The pattern of motion on the surface of a model shaped in plaster of paris and immersed in a water stream, can be made visible by reason of the marks caused when small discontinuities in the surface excite local fluctuations of velocity which lead to differential solution of the plaster and to small features of relief. This technique, which has so far been applied in geological studies, is illustrated by reference to motions about a cylinder on a flat plate and over symmetrical skewed steps. Current crescents and sand waves are briefly discussed in the light of these motions.

Some of the proposals made by Spalding (1964a) in his ‘Unified theory of friction, heat transfer, and mass transfer’ are examined. The two-component velocity-profile family proposed by Spalding is compared with measured boundary-layer and wall-jet velocity profiles and is shown to be adequate for flows with moderate wake components. The drag law implicit in the velocity-profile family is shown to be in good agreement with experimental data.

Recommendations are provided for entrainment functions for both boundary layers and wall jets.

Predictions of boundary-layer and wall-jet development, based on the recommended entrainment functions, are presented and compared with experiment. The predictions are in good agreement with experiment except in the vicinity of separation.

Liquid-into-liquid jets in short cylindrical vessels have been investigated under conditions of uniform flow by using an aqueous blue tracer solution in conjunction with tránsparent cylindrical tanks. The vessels had diameters D of 3, 6, 12 and 24in., length-to-diameter ratios L/D of 1, 2 and 3 and inlet diameters d between ¼ and 1 in. Reynolds numbers in the inlet tube, Rei, ranged between 100 and 28,000. Four main types of jet were observed: \[ \begin{array}{@{}l@{\qquad}l@{\quad}l@{}} & \hbox{dissipated-laminar jets} & (Re_i < 300\;\hbox{approx}.);\\ & \hbox{fully laminar jets} & (300 < Re_i < 1000\;\hbox{approx}.);\\ & \hbox{semi-turbulent jets} & (1000 < Re_i < 3000\;\hbox{approx}.);\\ \hbox{and} & \hbox{fully turbulent jets} & (Re_i > 3000). \end{array} \]

The laminar length a of sub-turbulent jets was investigated and correlated with Rei and the geometric parameters by the equation \[ a/d = 9.97 \times 10^7 Re_i^{-2.46}(D/d)^{-0.48} (L/d)^{0.74}. \]

A laboratory model of a tornado vortex has been produced, incorporating two features which are believed to be important to the understanding of the atmospheric phenomenon, but which have been largely ignored in previous studies. First, it has been shown that a vortex can be driven from above by a mechanism analogous to convection in a cloud, and that density differences within the funnel itself are not essential. Associated with this mechanism of formation is a circulation in the vertical, with an upflow in the centre surrounded by a compensating annular downflow. Secondly, the bottom boundary is seen to have a strong influence on the vortex, since the down and up flows are linked there by a rapid radial inflow in a thin boundary layer.

In the present paper an approximate theoretical description of such a vortex is proposed. The interior and boundary layer flows are first examined separately, and then a condition is sought which makes the two solutions consistent. The starting-point of the theory is the assumption of a form of stream function which describes a circulation in the vertical having the essential features of that observed. The result of the matching procedure is to fix both the form of the tangential velocity profile, and the relative magnitudes of the three components of velocity. These deductions are not critically dependent on the assumed form of the motion in the vertical, and are in good agreement with the first measurements in the laboratory vortices, though the quantitative experimental results are not emphasized here.

The characteristic lengths of the oscillating wakes of bluff bodies is discussed; in particular, those used in the universal non-dimensional frequencies proposed by Roshko (1954b) and Goldburg, Washburn & Florsheim (1965). It is concluded that these are equivalent at high Reynolds number. A closer examination leads to the conclusion that there are two simultaneous characteristic lengths; the scale of the formation region, and the width to which the free shear layers diffuse. Discussion of the mechanics of the formation region results in a physical basis for the determination of the frequency by these two characteristic lengths. The ideas developed are applied to the effects of splitter plates in the wake. The possibility of a high-Reynolds-number symmetrical formation region is suggested as an explanation of the very small lift values observed in the absence of free-stream disturbances.