A method is described for calculating the development of a three-dimensional turbulent boundary layer, over a flat or developable surface, in incompressible flow. The method involves the numerical integration of the equations of motion by an explicit finite-difference method. The shear stress is determined by a parallel integration of the turbulent energy equation modified by the inclusion of empirical functions of a form which has proved successful in two dimensions, and the additional assumption is made that the turbulent shear stress acts in the direction of the rate of strain of the mean motion. The treatment of the turbulent energy equation follows closely the work of Bradshaw, Ferriss & Atwell (1967) in two dimensions.

Comparison with experiment is found to be substantially more difficult than in two dimensions. Particular difficulty is encountered in translating the recorded details of the experiment into boundary conditions for the calculation. The comparisons submitted here give some indication that the method as a whole performs satisfactorily, but they do not provide a definitive assessment of the validity of the basic assumptions. A plea is made for an experiment to supply data in a suitable form for making a more careful assessment of methods of this type.

An experiment is described in which a constant stress is applied to the surface of an initially quiescent tank of fluid with a uniform density gradient. The development of the turbulent layer by entrainment of the underlying fluid is described and it is found that the entrainment coefficient E, the ratio of the entrainment velocity ue to the friction velocity u* is given in terms of the depth D of the mixed layer and the density jump δρ across the entrainment interface by the relation \[ E = \frac{u_e}{U_{*}} = 2.5\frac{\rho_0u^2_{*}}{g\delta\rho D}. \] The rate of increase of potential energy of the stratified fluid was found to be proportional to the rate of dissipation of kinetic energy per unit area in the turbulent layer. The form of these results is consistent with those found by Turner with an agitation tank, but the parameters used here allow direct application to entrainment in the ocean.

A slow viscous flow around a circular cylindrical post confined between two parallel flat plates is analyzed. The solution is given in the form of a series each term of which satisfies the equations of motion and continuity as well as the no-slip condition on the plates. A number of arbitrary constants left open in the series can then be determined according to the no-slip condition at the surface of the cylinder. Numerical results are presented. The series is found to be adequate when the ratio of the distance between the plates and the diameter of the post is not much larger than unity. The coefficient of resistance of the flow decreases rapidly as the sheet-thickness-to-post-diameter ratio increases. It is shown that the series converges rapidly when the thickness ratio is small, so that an approximate solution consisting of the two leading terms of the series, gives a reasonable representation of the velocity field and a good approximation to the flow resistance.

In the first part of the paper a method is developed for the relaxation or numerical solution of axisymmetric fully cavitating flows. Employing the technique suggested in a paper by Woods (1951 a) of working in a transformed (ϕ ϕ)-plane, solutions are obtained for cavities behind a disk and a sphere in different sizes of solid wall tunnel. Under certain conditions flow ‘choking’ occurs.

The results of a series of experiments carried out with such headforms are then reported. The apparent viscous effect on the position of separation from the sphere and thus on the drag proves to be of particular interest.

A response of viscous heat-conducting compressible fluid to an abrupt change of angular velocity of a containing thermally insulated circular cylinder under the existence of stable distribution of the temperature is investigated within the framework of the Boussinesq approximation for a time duration of the order of the homogeneous-fluid spin down time in order to resolve the Holton-Pedlosky controversy. The explicit expression of the solution is obtained by the standard method and Holton's conclusion is confirmed. The secondary meridional current induced by the Ekman layers spins the fluid down to a quasi-steady state within the present time scale. However, unlike the homogeneous case, the quasi-steady state is not one of solid body rotation. The final approach to the state of rigid rotation is achieved via the viscous diffusion in the time scale of the usual diffusion time.

This investigation demonstrates the measurement of low turbulence intensities in a flow of mercury using the constant-temperature hot-film technique. A simple equation for the determination of turbulence intensity is derived on the basis of observed calibration data for short, cylindrical, quartz-insulated, platinum hot-film sensors. This equation has the advantage of being independent of the changeable effects at the sensor-fluid interface and of the usually troublesome drift in anemometer output signal. The flow speeds considered are low, so that the Péclet number based on the outside diameter of the insulated sensor is less than unity. Various problems associated with the application of hot-film anemometry to mercury flows are discussed.

The oscillations of a drop moving in another fluid medium have been studied at low values of Reynolds number and Weber number by taking into consideration the shape of the drop and the viscosities of the two phases in addition to the interfacial tension. The deformation of the drop modifies the Lamb's expression for frequency by including a correction term while the viscous effects split the frequency into a pair of frequencies—one lower and the other higher than Lamb's. The lower frequency mode has ample experimental support while the higher frequency mode has also been observed. The two modes almost merge with Lamb's frequency for the asymptotic cases of a drop in free space or a bubble in a dense viscous fluid but the splitting becomes large when the two fluids have similar properties. Instead of oscillations, aperiodic damping modes are found to occur in drops with sizes smaller than a critical size ($\sim\hat{\rho}\hat{\nu}^2/T $). With the help of these calculations, many of the available experimental results are analyzed and discussed.

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.

Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.

As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

The flow of an incompressible, viscous fluid past a sphere is considered for small values of the Reynolds number. In particular the drag is found to be given by \[ D = D_s\{1+{\textstyle\frac{3}{8}}R+{\textstyle\frac{9}{40}}R^2(\log R+\gamma + {\textstyle\frac{5}{3}}\log 2 - {\textstyle\frac{323}{360}})+{\textstyle\frac{27}{80}}R^3\log R+O(R^3)\}, \] where Ds is the Stokes drag, R is the Reynolds number and γ is Euler's constant.

On the assumptions of incompressibility, and negligible thermal conduction, salinity diffusion and viscosity, simple expressions are derived for the conservation equations of mass, momentum and energy when internal waves encounter an unsteady non-uniform current. These expressions of conservation equations are valid for all kinds of internal waves without regard to their different characteristics. From the dynamical conservation equations, we find that a stress-like term, the ‘excess momentum flux tensor’, plays an important role in the interaction between internal waves and an unsteady non-uniform current. Furthermore, it is deduced from the energy balance equation that, in the encounter of interfacial waves with a steady non-uniform current in a two-liquid system, the waves are amplified in an adverse current but suppressed in an advancing current as a result of interaction of the waves with the current. This conclusion may explain the large amplitudes sometimes observed in internal waves near the confluence of currents and near fronts at the thermocline, the region in the ocean where the density gradient is a maximum.

The trapping of short-period gravity waves by islands and seamounts has been studied by Chambers (1965) and by Longuet-Higgins (1967). It was shown by the latter that in the absence of rotation, or when the wave frequency σ is large compared with the Coriolis parameter f, these waves cannot be perfectly trapped; some energy must always leak away to infinity. Very long-period oscillations in the presence of a sloping shelf surrounding an island, with σ [Lt ] f, have been studied by Mysak (1967) and Rhines (1967, 1969). Here perfect trapping is possible. However, as pointed out in Longuet-Higgins (1968), the rotation itself exerts a strong trapping effect not only when |σ| [Lt ] f, but also whenever a |σ| < f. It seems not to have been noticed that this effect is capable of trapping waves round an island in an ocean of uniform depth, in the absence of any shelf or sloping region offshore.

The existence of such waves is demonstrated for a circular island in § 1 of the present paper. It is shown that the waves exist only when the azimuthal wave-number n is at least 1. The waves always progress round the island in a clockwise sense in the northern hemisphere. At large distances r from the island, the wave amplitude decays exponentially, but this exponential trapping occurs only if the radius a of the island exceeds the critical value (n(n − 1)gh)½/f. When n = 1, this critical radius is zero, so that in theory the waves exist for any island of non-zero radius.

The application of these results to the ocean is discussed in § 2. Except possibly for baroclinic motions, it appears that only the waves corresponding to n = 1 could exist in fact, and that their frequency would be nearly equal to the inertial frequency f. It is unlikely that f could be regarded as constant over a sufficiently wide area for the model to apply without qualification. Nevertheless, the oscillations may be regarded as the local adjustment of the pressure field to inertial currents in the neighbourhood of the island. It is possible that the peak at about 0·73 c.p.d. in the spectrum of sea-level at Oahu, as found by Miyata & Groves (1968), can be interpreted in this way.

In this paper it is shown that the free convection boundary layer approaches a singular character if the Prandtl number tends to zero. The method of matched asymptotic expansions is used to integrate the equations for this extreme case. An expression is derived for the Nusselt—Grashof relation and the results are compared with those of previous investigations which attack the problem in a different way.

Pumping by means of an infinite train of peristaltic waves is investigated under conditions for which the relevant Reynolds number is small enough for inertial effects to be negligible and the wavelength to diameter ratio is large enough for the pressure to be considered uniform over the cross-section. Theoretical results are presented for both plane and axisymmetric geometries, and for amplitude ratios ranging from zero to full occlusion. For a given amplitude ratio, the theoretical pressure rise per wavelength decreases linearly with increasing time-mean flow. An experiment with a quasi-two-dimensional apparatus confirmed the theoretical values.

Calculations of the detailed fluid motions reveal that under many conditions of operation the net time-mean flow is the algebraic difference between a forward time-mean flow in the core of the tube and a backward (‘reflux’) time-mean flow near the periphery. The percentage of reflux flow can be very high. This reflux phenomenon is probably of physiologic significance in the functioning of the ureter and the gastro-intestinal system. A second fluid-mechanical peculiarity with physiological implications is that of ‘trapping’: under certain conditions an internally circulating bolus of fluid, lying about the axis, is transported with the wave speed as though it were trapped by the wave.