The development of the boundary layer accompanying the formation of a free surface at y′ = 0, from the two-dimensional uniform shear flow u′ = ωyω, is discussed. The analysis shows that the surface velocity and surface position vary as the cube root of the distance downstream, while the mass-transfer coefficient varies inversely as the cube root of this distance. It is shown how these may be applied to the formation of capillary jets.

The stability with respect to ‘supersonic’ disturbances of the laminar mixing of two parallel streams of a compressible fluid is studied. For locally supersonic disturbances there will be waves propagating outward from the mixing layer. The numerical results show that the flow is generally unstable with respect to supersonic disturbances, although the amplification rate is smaller than that for subsonic disturbances. The flow is more unstable at lower Mach number, and the increase of the angle between the disturbance wave-number vector and the principal flow direction tends to increase the instability.

The present experiments lay a foundation for a detailed study of interface propagation in a mixed laminar–turbulent flow between counter-rotating concentric cylinders. Such mixed flows, including one particularly well-organized pattern called spiral turbulence, are found to be a dominant feature of transition in Couette flow. In spiral turbulence, the laminar and turbulent regions of the flow form an alternating pattern of helical stripes, rotating with approximately the mean angular velocity of the two cylinders. Stable right-handed or left-handed spirals occur with equal probability when the flow is established from rest. Hot-wire measurements have been made of the mean cross-sectional shape of the interfaces in the axial mid-plane for a spiral-turbulent flow having low dispersion in the interface position. A nose of turbulence associated with the leading interface projects into the laminar region near the outer cylinder, while a corresponding tail is associated with the trailing interface near the inner cylinder. Fluctuations in interface position and in apparent length of the turbulent region are closely Gaussian, and the fluctuations are rapid compared with the period of rotation of either cylinder.

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.

An approximately homogeneous turbulent shear flow is generated in an open-return wind tunnel test-section by a plane parallel-rod grid of uniform rod diameter and non-uniform rod spacing. The grid design is based upon an analysis by Owen & Zienkiewicz (1957). Hot-wire measurements taken in this flow include mean velocities, component turbulence intensities, shear and two-point space correlations, and energy spectra. In addition, microscales, obtained both from instantaneous time derivatives of the hot-wire signal and from two-point space correlations, and integral scales, calculated both from correlations and energy spectra, are reported.

Based upon these results, it is concluded that, far enough away from the grid and the test-section wall boundary layers:

1. The turbulence intensities are maintained at uniform values by the nearly constant mean shear.

2. The turbulent shear stress approaches an asymptotic value.

3. Measured two-point space correlation coefficients and one-dimensional energy spectra attain self-preserving forms.

4. When distance downstream of the grid is measured in terms of the number of ‘local’ grid rod spacings, (see discussion of microscales obtained from time derivatives), the Taylor microscale defined by the correlation coefficient Ruu(rX, 0, 0) grows linearly with this ‘effective’ distance over most of the region measured.

5. The limited number of integral scale determinations and experimental uncertainty allow only the statements that the magnitude of the longitudinal scale is roughly one-eighth the lateral dimension of the square test-section and tends to increase slightly with ‘effective’ distance from the grid.

6. The lateral integral scales are approximately one-half the longitudinal scales and also increases with distance from the grid.

7. The integral scale which characterizes the size of the eddy primarily responsible for momentum transfer is roughly one-tenth the test-section lateral dimension (measured at one point only).

The collapse time of a closed cavity that is initially at rest in an incompressible, inviscid fluid of density ρ and ambient pressure p∞ has the form \[ t_1 = \{\rho/(p_{\infty} - p_c)\}^{\frac{1}{2}}\ell, \] where pc is the internal pressure, which is assumed to remain constant during collapse, and [ell ] is a length that depends only on the geometry of the cavity. A variational formulation of the dynamical problem is constructed from Jacobi's statement of the principle of least action. A single-degree-of-freedom approximation is developed from the similarity hypothesis that the cavity collapses through a family of similar surfaces with volume as the generalized co-ordinate. Two-degree-of-freedom approximations are given for both prolate and oblate spheroidal cavities and are used to obtain error estimates for the similarity approximation (approximately 2% for the limiting case of a needle-like, prolate spheroid and approximately ½ % for a disk-like, oblate spheroid). A perturbation analysis is developed for an approximately spherical cavity, which is found to have the same collapse time as a spherical cavity of equal volume within a factor 1 + O(e4), where e is a representative eccentricity. A first-order correction for surface tension is obtained.

Measurements at moderately large Reynolds numbers in a finite laminar circular Couette flow show that the tangential motion near the axial plane of symmetry is two-dimensional, within experimental accuracy, but is nevertheless strongly modified by end conditions.

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 cavity in which the magnetic field of two arbitrary line currents is confined by a streaming plasma which is assumed cold and perfectly conducting is investigated by using conformal transformations. When the magnetic field at the boundary is always directed in the same sense the finite breadth of the cavity at infinity depends only on the algebraic sum of the inducing currents and not their position. If the two line currents are of opposite sign the boundary magnetic field may change sign at two ‘pseudo-singularities’.

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.

The study of free oscillations of a homogeneous liquid under gravity in rotating rectangular basins of uniform depth is undertaken from both theoretical and experimental considerations.

The theoretical study is in the framework of the quasistatic equations. It is also assumed that the curvature of the free surface can be ignored. Numerical computations for the frequencies and modal structures were carried out for several of the slowest antisymmetric and symmetric modes in a square basin and in a rectangular basin of two-to-one dimension ratio, without any restriction on the angular speed of rotation of the basin. These computations are in agreement with a numerical value obtained many years ago by Taylor, and also with several values found by Corkan & Doodson. They exhibit the typical frequency-splitting associated with certain multiplets in the zero-rotation spectrum. Further, the theoretical calculations indicate that the slopes of the curves of frequency versus speed of rotation change sign for some of the modes in rectangular geometry. Such behaviour is not present in circular basins. Negative modes are found to be ‘unstable’ in the sense of Corkan & Doodson; that is, they are transformed into positive modes for sufficiently high rotation. Calculations were also made for the slowest longitudinal oscillations in highly elongated basins to demonstrate the decreasing importance of rotation on the frequencies of these modes.

Experimental work was carried out in a flat-bottomed square tank for the slowest positively and negatively propagating antisymmetric modes and the slowest positively propagating symmetric mode. Good agreement was found between theory and experiment.

The problem studied here is that of the attenuation of internal waves through turbulent mixing in a weakly and exponentially stratified fluid. The equations are linearized and it is assumed that the action of turbulence can be parametrically represented by eddy mixing coefficients and that the influence of bottom friction is restricted to a thin bottom boundary layer. The simple case where there is no rotation and only one component to the stratification is first examined in detail, and the modifications caused by introducing rotation and a second component are subsequently investigated. Subject quantitatively to the choice made for the eddy coefficients, but qualitatively not strongly dependent on that choice, the following conclusions are drawn: (i) very short internal waves (length < 100 m) are strongly damped in basins of all depths; (ii) long internal waves or seiches in shallow seas (depth ≃ 100 m) will not last more than a few cycles as free oscillations; (iii) the attenuation rate for long internal tides is small enough that these should be observable very far from the coasts, but large enough to exclude the possibility of oceanic standing wave systems; (iv) for very long internal waves the damping is predominantly due to the effect of bottom friction, and the attenuation rate becomes almost independent of the actual form of the stratification present in the fluid.

An inviscid transonic theory appears to be inadequate to describe the flow near the throat of a converging–diverging nozzle during the transition from the symmetrical Taylor (1930) type of flow to the subsonic–supersonic Meyer (1908) flow. A viscous transonic equation taking account of heat conduction and longitudinal viscosity has been developed previously (Cole 1949; Sichel 1963; Szaniawski 1963). An exact, nozzle-type of similarity solution of the viscous transonic equation, similar to the inviscid solution of Tomotika & Tamada (1950), has been found. This solution does provide a description of the gradual transition from the Taylor to the Meyer flow and shows the initial stages in the development of a shock wave downstream of the nozzle throat. The solution provides a viscous, shock-like transition from an inviscid, supersonic, accelerating flow to an inviscid, subsonic, decelerating flow.

A theory is developed for an incompressible fluid in a steady three-dimensional rotational flow. Solutions are obtained subject to the restriction of small perturbations and are determinant provided that the vorticity distribution along the axis of rotation is known. Effects of viscosity are included. Closed-form expressions for the zeroth-order circulation and stream function and first-order circulation are given, with other higher-order expressions requiring high-speed computers. Experimental results of radial variation of axial velocity in the core show a distribution more elaborate than Gaussian.

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.

A critical scrutiny of the nature of the three-dimensional characteristics of the vortex wake of a circular cylinder serves to suggest lines for further investigation and furnishes some ideas on the nature of the growth and development of these non-uniformities. It is suggested that the basic occurrence in the growth of three-dimensionality is the continuation of vortex lines, oriented more or less parallel to the body, into the direction of the free stream. The causes of this vary, as do the details of the development with the particular situation considered.

Experiments were performed in a wind tunnel at Reynolds numbers based on cylinder diameter of 85, 235 and 2 × 104, at which stable, transitional and turbulent vortices were investigated.

The problem considered is the entry of a thin symmetric wedge impinging normally on the free surface of a compressible inviscid liquid, gravitational effects being neglected. Since the body is thin, the problem is a linear one, and its solution is possible for the whole range of Mach numbers of the body's motion in the liquid. It is shown that the free-surface elevations are lowered considerably as the Mach number increases, and that the presence of the free surface acts so as to lower the pressure differences arising from compressibility.

The solution for the flow of a fully ionized gas over a wedge of finite angle is known for the case when the applied magnetic field is aligned with the incident stream. In this flow there are current sheets on the surfaces of the wedge. When the magnetic field is allowed to deviate slightly from the stream, the current sheets may move into the gas and become shock waves. The magnetic fields adjacent to the wedge above and below it have to be matched. A perturbation method is introduced by means of which expressions for the unknown quantities in the different regions may be determined when there are four shocks attached to the wedge. The results give insight into the manner in which the shock-wave pattern develops as the obliquity of the magnetic field to the stream increases. The question of the stability of the shock waves is also examined.

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 \[ a/d = 9.97 \times 10^7 Re_i^{-2.46}(D/d)^{-0.48} (L/d)^{0.74}. \]

Taylor's formula for the velocity of an oscillating sheet in a viscous fluid is extended to larger amplitudes of motion and his formula for the velocity of a rotating or oscillating filament is proved to be valid for amplitudes of motion small but greater than the filament radius. A simple formula is derived for the velocity of a sheet or filament in any general oscillatory motion of small amplitude. The forces and energy involved in oscillating tubes are calculated.