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.

The objective of this study is to investigate for turbulent flow the fluid motions very near a solid boundary, and to create a physical picture which relates these motions to turbulence generation and transport processes. An experimental technique was developed which permitted detailed observations of the regions very near a pipe wall, including the viscous sublayer, without requiring the introduction of any injection or measuring device into the flow. This technique involved suspending solid particles of colloidal size in a liquid, and photographing their motions with a high-speed motion picture camera moving with the flow. To provide greater detail, the field of view was magnified.

Fluid motions were observed to change in character with distance from the wall. The sublayer was continuously disturbed by small-scale velocity fluctuations of low magnitude and periodically disturbed by fluid elements which penetrated into the region from positions further removed from the wall. From a thin region adjacent to the sublayer, fluid elements were periodically ejected outward toward the centreline. Often there was associated with these events a zone of high shear at the interface between the mean flow and the decelerated region that gave rise to the ejected element. When the ejected element entered this shear zone, it interacted with the mean flow and created intense, chaotic velocity fluctuations. These ejections and resulting fluctuations were the most important feature of the wall region, and are believed to be a factor in the generation and maintenance of turbulence.

Flow of a conducting fluid along the annular channel between two non-conducting circular cylinders is examined by a numerical method for concentric and eccentric cases. Solutions have been obtained for Hartmann numbers ranging from 0·1 to 40 and, for some of these, details of velocity distribution and of induced current are given. The results obtained enable the development of the patterns of velocity and of current flow to be traced as the Hartmann number increases. The details of the development of the current flow patterns for eccentric cylinders are particularly interesting and are discussed in detail. At the higher values of Hartmann number studied the solutions are in excellent agreement with the results of Todd's (1967) high Hartmann number analysis and it is possible to determine at what value of Hartmann number Todd's analysis becomes applicable within a specified accuracy. The effect of eccentricity of the cylinders on the flow rate at a fixed pressure gradient is shown to diminish rapidly with increasing Hartmann number. The net flow of current around the annulus, which occurs when the cylinders are eccentric, has a maximum value for each case studied at a Hartmann number of 3, approximately.

This paper considers the linear response of a homogeneous uniformly rotating ocean of infinite horizontal extent with a discontinuity in depth to a variable horizontal wind stress. It is shown that, for either a transient or time-periodic wind stress which is suddenly applied to an initially calm sea surface, the asymptotic response far from the forcing region is dominated by an outgoing dispersive wave which is trapped along the depth discontinuity, i.e. a double Kelvin wave. Plots of the forced wave patterns in the neighbourhood of the depth discontinuity itself are also presented.

An outline of the classical far-and near-field solutions for small-amplitude one-dimensional unsteady flows in a general inviscid relaxing gas is given. The structure of the complete flow field, including a non-linear near-frozen (high frequency) region at the front, is obtained by matching techniques when the relaxation time is ‘large’.

If the energy in the relaxing mode is small compared with the total internal energy, the solution in the far field is, in general, more complex than that predicted by classical theory. In this case the rate process is not necessarily able to diffuse all convective steepening. An equation valid in this limit is derived and discussed. In particular, a sufficient condition for the flow to be shock-free is established. For an impulsively withdrawn piston it is shown that the solution is single-valued both within and downstream of the fan. Some useful similarity rules are pointed out.

The corresponding formulation for two-dimensional steady flows is also noted in the small energy limit.

The effect of a sudden change in vertical velocity of a vessel partially filled with fluid is considered. It is shown that very small free surface disturbances can be amplified so strongly by a velocity change that instability can occur. This instability frequently causes a jet to emanate from the free surface. Conditions causing free surface disturbances in a vessel in free fall are considered and it is shown that a contact angle between the fluid and the wall of the vessel different from ½π radians can distort the surface in such a way that the amplification and instability upon vertical impact results in a central jet. The results of experiments of this effect are shown. The generation of waves due to transient side wall motions, such as those that might result from fluid pressure at impact, are considered in the appendix. It is shown that such waves would have a different form than those observed in the experiments.

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.

The energy of internal waves tends to be propagated along certain characteristic paths inclined at fixed angles to the vertical direction; the angle of inclination depending only on the wave frequency and the density stratification (not on the wavelength). The reflexion of such waves by smooth plane surfaces has been discussed recently by Sandstrom (1966).

In the present paper the role of surface roughness is examined. Surprisingly, it appears that quite small-scale irregularities can completely alter the reflecting properties of a surface; the tangential scale of the roughness elements may be much smaller than the wavelength of the incident or reflected waves. All scales of roughness are relevant, down to those comparable in magnitude to the thickness of the oscillatory boundary layer. For tidal waves in the ocean this thickness is of the order of 1 m.

The behaviour of the coefficient of transmission as a function of the angle of incidence appears at first sight to be extremely complicated. Some simple examples of periodic surface roughnesses are discussed and elucidated: a sawtooth, a square-topped wave and a simple sine-wave. The transmission coefficient T for a sine-wave, for example, is shown in figure 9. An approximate expression for T is also derived in the case of a slowly modulated sine-wave (figure 10). These results are for a non-viscous fluid. The effects of viscosity are also considered qualitatively.

A considerable body of experimental data now exists concerning turbulent boundary layers with air injection at the wall, both at subsonic and at supersonic speeds. In the present report these data for Mach numbers up to 6·5 have been analyzed to find the parameters which occur in the law of the wall as deduced from mixing-length theory. Although the absolute values of the parameters are subject to error because of the lack of accurate skin-friction measurements, the trends of these parameters with Mach number and injection mass flow are clearly defined.

Experiments are performed to demonstrate the occurrence and explore the characteristics of a secondary flow superposed upon the natural convection main flow on an inclined plate. A flow visualization technique is employed whereby the flow pattern is made visible by local changes of colour of the fluid itself, the colour change being brought about by a change in pH. The secondary flow consists of longitudinal vortices or rolls distributed more or less periodically across the width of the plate. The number of such vortices increases with the temperature difference between the surface and the ambient fluid, but appears to be relatively insensitive to the inclination angle of the plate. The secondary flow results from the destabilizing effect of the buoyancy force component, which acts normal to the plate surface. The longitudinal vortices are the first stage of the laminarturbulent transition process. This is in contrast to the case of natural convection on a vertical plate, where the first stage of transition is Tollmien-Schlichting waves.

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.

This paper considers the effect of continuous convection from small sources of buoyancy on the properties of the environment when the region of interest is bounded. The main assumptions are that the entrainment into the turbulent buoyant region is at a rate proportional to the local mean upward velocity, and that the buoyant elements spread out at the top of the region and become part of the non-turbulent environment at that level. Asymptotic solutions, valid at large times, are obtained for the cases of plumes from point and line sources and also periodically released thermals. These all have the properties that the environment is stably stratified, with the density profile fixed in shape, changing at a uniform rate in time at all levels, and everywhere descending (with ascending buoyant elements).

The analysis is carried out in detail for the point source in an environment of constant cross-section. Laboratory experiments have been conducted for this case, and these verify the major predictions of the theory. It is then shown how the method can be extended to include more realistic starting conditions for the convection, and a general shape of bounded environment. Finally, the model is applied quantitatively to a variety of problems in engineering, the atmosphere and the ocean, and the limitations on its use are discussed.

The variational problem introduced by Howard (1963) for the derivation of an upper bound on heat transport by convection in a layer heated from below is analyzed for the case in which the equation of continuity is added as constraint for the velocity field. Howard's conjecture that the maximizing solution of the Euler equations is characterized by a single horizontal wave-number is shown to be true only for a limited range of the Rayleigh number, Ra. A new class of solutions with a multiple boundary-layer structure is derived. The upper bound for the Nusselt number, Nu, given by these solutions is Nu ≤ (Ra/1035)½ in the limit when the Rayleigh number tends to infinity. The comparison of the maximizing solution with experimental observations by Malkus (1954a) and Deardorff & Willis (1967) emphasizes the similarity pointed out by Howard.

The steady rise of a given amount of gas in an infinite liquid under the action of buoyancy forces is examined. The shape of the bubble is that of a spherical cap. Behind the bubble there is a region of closed streamlines where the flow is assumed to be steady and rotational. The predicted velocity and geometry of the cap agree well with observation for a limited range of values of the Reynolds number.

A theoretical analysis of two-dimensional, finite-amplitude, thermal convection is made for a fluid which has an infinite Prandtl number. The vertical velocity disturbance is expanded in a double Fourier series which satisfies the horizontal and lateral boundary conditions. The resulting coupled sets of non-linear differential equations are solved numerically. It is found that for a particular Rayleigh number the number and size of the convection cells that form depend upon the ratio of the distance between the lateral boundaries to the depth of the fluid layer and on the initial conditions. The steady-state solutions are not unique and the solution for which the heat transport is a maximum is not necessarily the solution that results. Where there are no lateral boundaries, the lateral edges of the cells tend to tilt and the Nusselt number increases slightly.

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 unified approach to the problem of jet efflux from symmetrical channels of finite width and possessing a general curvilinear nozzle shape is presented. The nozzle may be composed of polygonal and/or curved-arc segments. Precise nozzle shapes cannot be initially prescribed, however. The solution is based on the combined use of conformal mapping and the Riemann-Hilbert solution to a mixed boundary-value problem. The selection of an appropriate curvature function is described; examples show possible applications.

Measurements of flow and fluctuating heat transfer were made for a circular cylinder in cross-flow with a transverse standing sound field imposed simultaneously. Reynolds numbers were of the order of 104, known to be in the disturbance-sensitive range, and sound intensities were as large as 140 db. The frequencies of the sound field were of the order of the disturbance frequency in the separated shear layers, reported first by Bloor.

With a sound field having its frequency matched sufficiently closely to that occurring naturally in the shear layer, the growth of the instability is enhanced with the processes of vortex fusion and possibly vortex breakdown being detectable. At the same time, the vortex street frequency is only very weakly affected, although the vortex formation region length is reduced when the instability in the shear layer is enhanced. It is suggested that the discretization of vorticity in the shear layers is one factor significant in reducing the formation length. Heat transfer at the rear of the cylinder fluctuates at frequencies centred on the shedding frequency. The fluctuation level increases as the formation region shortens.

The steady, two-dimensional, incompressible flow past a circular cylinder is calculated for Reynolds numbers up to ten. An accurate description of the flow field is found by employing the semi-analytical method of series truncation to reduce the governing partial differential equations of motion to a system of ordinary differential equations which can be integrated numerically. Results are given for Reynolds numbers between 0.4 and 10.0 (based on diameter). The Reynolds number at which separation first occurs behind the cylinder is found to be 5.75. Over the entire Reynolds number range investigated, characteristic flow parameters such as the drag coefficient, pressure coefficient, standing eddy length, and streamline pattern compare favourably with available experimental data and numerical solution results.

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.