In a laboratory study, the class of instabilities of the laminar Ekman layer called type II (or class A) are found to have the spectral characteristics of narrow-band noise (Q ∼ 5). The unperturbed laminar profile resembles very closely the ideal Ekman solution. The frequency of the spectral peak varies with the Reynolds number as predicted theoretically by Lilly (1966), but the measured frequencies are only 60% of the predicted value. The critical Reynolds number for this instability is found to be 56·7, in good agreement with Lilly's analysis. The measured boundary layer profile of the magnitude of the spectral peak has the behaviour predicted by Lilly. A sudden onset of turbulence is found at a Reynolds number of 148.

It is shown that a random superposition of inertial waves in a rotating conducting fluid can act as a dynamo, i.e. can systematically transfer energy to a magnetic field which has no source other than electric currents within the fluid. Dynamo action occurs provided the statistical properties of the velocity field lack reflexional symmetry, and this occurs when conditions are such that there is a net energy flux (positive or negative) in the direction of the rotation vector Ω.

If the magnetic field grows from an infinitesimal level, then the mode of maximum growth rate dominates before the back-reaction associated with the Lorentz force becomes significant. This mode is first determined, and then the back-reaction associated with it alone is analysed. It is shown that the magnetic energy grows exponentially during the stage when the Lorentz forces are negligible, then reaches a maximum depending on the values of the parameters \[ R_m = u_0 l/\lambda,\quad Q = \Omega l^2/\lambda, \] (u0 = initial r.m.s. velocity, l = length scale characteristic of the velocity field, λ = magnetic diffusivity) and ultimately decays as t−1 (equation (5.15)). This decay is coupled with a decay of the velocity field due to ohmic dissipation, and it occurs because there is no external source of energy for the fluid motion.

This paper attempts to emulate the great study by Goldstein (1929) ‘On the vortex wake of a screw propeller’, by looking for a dynamical theory of how another type of propulsion system has evolved towards ever higher performance. An ‘undulatory’ mode of animal propulsion in water is rather common among invertebrates, and this paper offers a preliminary quantitative analysis of how a series of modifications of that basic undulatory mode, found in the vertebrates (and especially in the fishes), tends to improve speed and hydromechanical efficiency.

Posterior lateral compression is the most important of these. It is studied first in ‘pure anguilliform’ (eel-like) motion of fishes whose posterior cross-sections are laterally compressed, although maintaining their depth (while the body tapers) by means of long continuous dorsal and ventral fins all the way to a vertical ‘trailing edge’. Lateral motion of such a cross-section produces a large and immediate exchange of momentum with a considerable ‘virtual mass’ of water near it.

In § 2, ‘elongated-body theory’ (an extended version of inviscid slender-body theory) is developed in detail for pure anguilliform motion and subjected to several careful checks and critical studies. Provided that longitudinal variation of cross-sectional properties is slow on a scale of the cross-sectional depth s (say, if the wavelength of significant harmonic components of that variation exceeds 5s), the basic approach is applicable and lateral water momentum per unit length is closely proportional to the square of the local cross-section depth.

The vertical trailing edge can be thought of as acting with a lateral force on the wake through lateral water momentum shed as the fish moves on. The fish's mean rate of working is the mean product of this lateral force with the lateral component of trailing-edge movement, and is enhanced by the virtual-mass effect, which makes for good correlation between lateral movement and local water momentum. The mean rate of shedding of energy of lateral water motions into the vortex wake represents the wasted element in this mean rate of working, and it is from the difference of these two rates that thrust and efficiency can best be calculated.

Section 3, still from the standpoint of inviscid theory, studies the effect of any development of discrete dorsal and ventral fins, through calculations on vortex sheets shed by fins. A multiplicity of discrete dorsal (or ventral) fins might be thought to destroy the slow variation of cross-sectional properties on which elongated-body theory depends, but the vortex sheets filling the gaps between them are shown to maintain continuity rather effectively, avoiding thrust reduction and permitting a slight decrease in drag.

Further advantage may accrue from a modification of such a system in which (while essentially anguilliform movement is retained) the anterior dorsal and ventral fins become the only prominent ones. Vortex sheets in the gaps between them and the caudal fin may largely be reabsorbed into the caudal-fin boundary layer, without any significant increase in wasted wake energy. The mean rate of working can be improved, however, because the trailing edges of the dorsal and ventral fins do work that is not cancelled at the caudal fin's leading edge, as phase shifts destroy the correlation of that edge's lateral movement with the vortex-sheet momentum reabsorbed there.

Tentative improvements to elongated-body theory through taking into account lateral forces of viscous origin are made in §4. These add to both the momentumandenergyof the water's lateral motions, but mayreduce the efficiencyof anguilliform motion because the extra momentum at the trailing edge, resulting from forces exerted by anterior sections, is badly correlated with that edge's lateral movements. Adoption of the ‘carangiform’ mode, in which the amplitude of the basic undulation grows steeply from almost zero over the first half or even two-thirds of a fish's length to a large value at the caudal fin, avoids this difficulty.

Any movement which a fish attempts to make, however, is liable to be accompanied by ‘recoil’, that is, by extra movements of pure translation and rotation required for overall conservation of momentum and angular momentum. These recoil movements, a potentially serious source of thrust and efficiency loss in carangiform motion, are calculated in § 4, which shows how they are minimized with the right distribution of total inertia (the sum of fish mass and the water's virtual mass). It seems to be no coincidence that carangiform motion goes always with a long anterior region of high depth (possessing a substantial moment of total inertia) and a region of greatly reduced depth just before the caudal fin.

The theory suggests (§5) that reduction of caudal-fin area in relation to depth by development of a caudal fin into a herring-like ‘pair of highly sweptback wings’ should reduce drag without significant loss of thrust. The same effect can be expected (although elongated-body theory ceases to be applicable) from widening of the wing pair (sweepback reduction). That line of development of the carangiform mode in many of the Percomorphi leads towards the lunate tail, a culminating point in the enhancement of speed and propulsive efficiency which has been reached also along some quite different lines of evolution.

A beginning in the analysis of its advantages is made here using a ‘twodimensional’ linearized theory. Movements of any horizontal section of caudal fin, with yaw angle fluctuating in phase with its velocity of lateral translation, are studied for different positions of the yawing axis. The wasted energy in the wake has a sharp minimum when that axis is at the ‘three-quarter-chord point’, but rate of working increases somewhat for axis positions distal to that. Something like an optimum regarding efficiency, thrust and the proportion of thrust derived from suction at the section's rounded leading edge is found when the yawing axis is along the trailing edge.

This leads on the present over-simplified theory to the suggestion that a hydromechanically advantageous configuration has the leading edge bowed forward but the trailing edge straight. Finally, there is a brief discussion of possible future work, taking three-dimensional and non-linear effects into account, that might throw light on the commonness of a trailing edge that is itself slightly bowed forward among the fastest marine animals.

Numerical methods have been used to investigate the steady incompressible flow past oblate and prolate spheroids for Reynolds numbers up to 100. The ratio of minor to major axis of the spheroids investigated were 0·9, 0·5 and 0·2, together with 1·0, which represents the limiting case of a sphere. The pressure distribution and the skin and form drag coefficients were numerically evaluated for the various Reynolds numbers. Streamlines, equi-vorticity lines and equivelocity lines are presented and show in detail the flow characteristics.

An analytical method of calculating the body motion was given in an earlier paper. Viscosity and surface tension were neglected, and the equations of motion were linearized. It was found that, for a half-immersed horizontal circular cylinder of radius a, the vertical motion at time τ(a/g)½ is described by the functions h1(τ) (for an initial velocity) and h2(τ) (for an initial displacement) where \begin{eqnarray*} h_1(\tau) &=& \frac{1}{2\pi}\int_{-{\infty}}^{\infty}\frac{e^{-iu\tau}du}{1-\frac{1}{4} \pi u^2(1+\Lambda(u))}\\ {\rm and}\qquad\qquad\qquad h_2(\tau) &=& -\frac{1}{8}i\int_{-\infty}^{\infty}\frac{u(1+\Lambda(u))e^{-iu\tau}du}{1-\frac{1}{4}\pi u^2(1+\Lambda (u))}. \end{eqnarray*} The function ∧(u) in these integrals is the force coefficient which describes the action of the fluid on the body in a forced periodic motion of angular frequency u(g/a)½. To determine ∧(u) for any one value of u an infinite system of linear equations must be solved.

In the present paper a numerical study is made of the functions h1(τ) and h2(τ). The integrals defining h1(τ) and h2(τ) are not immediately suitable for numerical integration, for small τ because the integrands decrease slowly as u increases, for large τ because of the oscillatory factor e−iur. It is shown how these difficulties can be overcome by using the properties of ∧(u) in the complex u-plane. It is found that after an initial stage the motion of the body is closely approximated by a damped harmonic oscillatory motion, except during a final stage of decay when the motion is non-oscillatory and the amplitude is very small. It is noteworthy that the motion of the body can be found accurately, although little can be said about the wave motion in the fluid.

The idea of distributing static probe cross-sectional areas so as to render the probe insensitive to Mach number is combined here with that of using non-circular cross-sections to render probes insensitive to yaw and angle of attack. Appropriate non-circular cross-sections are described in detail, and a general means of designing blunt or slender probes to have zero sensitivity to yaw and angle of attack in potential flow is described. Four experimental probes have been tested, and test results are presented. These results show that the probes are quite insensitive to yaw and angle of attack within certain limiting angles, which are assumed to correspond to the onset of flow separation.

Mean streamwise vorticity in turbulent flow is shown to arise both from mean flow skewing and from the inhomogeneity of anisotropic wall turbulence. The structure of the Reynolds stress tensor is examined in several flows where the latter mechanism predominates. On the basis of a simple model for the anisotropy, the direction of the secondary currents is deduced for the corner boundary layer, the salient edge flow, and in the non-uniform nominally two-dimensional boundary layer.

Observations are reported on the growth of vortices in the vortex sheets bounding the jet emerging from a sharp-edged two-dimensional slit and from a sharp-edged circular orifice. A regular periodic flow is observed near the orifice for both configurations when the Reynolds number of the jet lies between about 500 and 3000. The two-dimensional jet produces a symmetric pattern of vortex pairs with a Strouhal number of 0·43. Vortex rings are formed in the circular jet with a Strouhal number of 0·63. Computer experiments show that a growing pair of vortices in two parallel vortex sheets produces a symmetric pattern of vortices upstream from the original disturbance.

The boundary-layer flow over a semi-infinite horizontal circular cylinder heated to a constant temperature and immersed in a uniform axial free stream is discussed in five situations corresponding to successively greater displacements from the leading edge. In the first three cases the drift velocity due to buoyancy is assumed small compared to the axial velocity component. Close to the leading edge of the cylinder the techniques of Seban & Bond are extended to include the drift velocity; far from the leading edge the asymptotic series methods of Stewartson, of Glauert & Lighthill, and of Eshghy & Hornbeck are employed to obtain a solution for the drift velocity. In the intermediate zone where the series solutions do not apply the appropriate partial differential equations are solved numerically. Still further downstream than the region where the ‘asymptotic’ solutions hold it is assumed that the boundary-layer flow is primarily convective and that the boundary layer is thin compared with the radius of the cylinder. A series solution is obtained which is valid near the lowest generator of the cylinder. Numerical methods are used to advance this solution upwards around the cylinder by solving the full boundary-layer equations step-by-step.

Kennett (1963) calculated the hypersonic flow fields past the lower (compression) surface of a delta wing, using the one-strip approximation of the method of integral relations. He obtained solutions only for wings with detached shocks. In this paper, his solutions are extended to wings with attached shocks. Here, the sonic point is inboard of the leading edge which makes the problem mixed. The solutions compare very well with the numerical solutions of the full equations by Babaev (1963a) both in the shock shapes and pressure distributions for various Mach numbers.

As an extension of our previous work (Sato & Kuriki 1961), an experimental study was made on the laminar-turbulent transition of a two-dimensional wake, with emphasis on the non-linear interaction of velocity fluctuations. In the natural transition, a sinusoidal velocity fluctuation first appears as a result of the selective amplification of irregular disturbances. When the amplitude of the fluctuation becomes large, a non-linear interaction takes place: higher harmonics of the sinusoidal fluctuation and a slow, irregular fluctuation are generated. The transition into turbulence is gradual, and there are no abrupt ‘breakdowns’.

When a sound with two frequencies (f1 and f2) is introduced into the wake, two velocity fluctuations are induced. They are amplified independently when amplitudes are small. If amplitudes exceed certain values, a mutual interaction takes place, and the growth of each component is suppressed by the presence of the other. At the same time, fluctuations with frequencies of f1 − f2, 2(f1 − f2) … and f1 + f2, 2(f1 + f2) … are produced. The behaviour of the slow, irregular fluctuation in the natural transition closely resembles that of the f1 − f2 component. They seem to be produced by the same mechanism.

In the transition region of the wake, a small variation in the amplitude of velocity fluctuation is enhanced by the suppression effect, and a small irregularity in the frequency is amplified by the generation of the f1 − f2 component. These are two elementary processes of randomization of regular fluctuations, which lead to the development of turbulence.

Many situations of practical importance both in fluid mechanics and elsewhere are governed by perturbed forms of conservation laws. Generally the perturbations are in the nature of positive dissipation terms in the sense that any initial disturbance from a uniform state ultimately decays to that state. Diverse examples of these are discussed briefly.

A situation in which the perturbation results naturally in a negative dissipation term, in the sense that initial disturbances grow, although not necessarily indefinitely, arises in what has been accepted for a model for the Gunn (1963) effect and other so-called bulk negative resistance effects in semiconductors. The Gunn effect, which is of immense importance in electron-device technology (comparable with transistors), is the appearance of coherent microwave current oscillations in the crystals of a suitable semiconductor, in particular Gallium Arsenide, when they are subjected to a large electric field generally of the order of several kilovolts per centimetre. It now seems to be accepted that the effect is a consequence of the negative resistance (that is the electron drift speed decreases with increasing electric field) properties of the semiconductor crystal.

A typical model with negative resistance properties is described in detail, the resulting perturbed (both singularly and otherwise) non-linear conservation equations ((2.19) and (2.21)) are studied for practical situations of interest and the physical implications discussed in the light of experimental facts. Particular care is given to the shocks or discontinuities that must appear in the solutions when the diffusion is zero. As a result of these comparisons with experiment a simpler model is suggested which should suffice for a large number of practical situations and various quantitative features of this model are given.

This work deals with an experimental investigation of natural convection inside a horizontal cylinder. The fluid, geometry, and thermal boundary condition were chosen so as to have a high Prandtl number and unit-order Grashof number.

The thermal boundary condition was established by imposing temperatures at two points, 180° apart, on the circumference of the cylinder. The resulting boundary condition for the full 360° was found experimentally and is presented. The apparatus was constructed so that the entire cylinder could be rotated in order to introduce an arbitrary heating angle into the boundary condition.

Temperature profiles and streamline patterns were observed at steady state for various values of this heating angle and for various initial conditions. For some cases, velocity profiles were also plotted. It was found that the interior ‘core’ of fluid was thermally stratified when the diameter containing the two imposed temperatures was horizontal. The flow occurred primarily in the region close to the cylinder wall. The cylinder was rotated so that the diameter containing the two imposed temperatures made an angle with the horizontal. The hotter of the imposed temperatures was always below the horizontal diameter of the cylinder. As this angle of rotation increased, it was found that the velocities encountered in the fluid increased and the degree of thermal stratification in the core region decreased. It was also found that the steady-state results were identical for the different initial conditions imposed. The results of this study are compared with previous work, both analytic and experimental.

Experiments to explore the effect of initial disturbance length-scale on turbulence developed in the presence of a uniform mean shear are the subject of this paper.

Flows with nearly the same mean shear (8·6 sec−1) and initially different turbulent scales are generated in a wind tunnel test-section by placing grids just downstream of a honeycomb of uniform cell diameter (¼ in.) and non-uniform cell length. Both round-rod grids of uniform square mesh and parallel-rod construction with roughly equal solidity (0.34) are used. Grid mesh sizes range from $\frac{1}{24}$ in. to 2 in.

From the results it is concluded that for a given value of mean shear the imposed length scale fixes the energy level of the resulting turbulence, provided the scale is sufficiently large. When it is reduced below some minimum value the turbulence decays. Also, it is found that two-dimensional flow-generator geometries are more effective than three-dimensional geometries in producing a roughly homogeneous turbulent field with a higher fluctuation level in a shorter distance.

Many numerical integrations support Goldstein's theory of the structure of the solution of a laminar boundary layer near the point of separation O when the mainstream is prescribed, and in particular confirm that the solution is singular there. The existence of the singularity, however, implies that the hypotheses of the boundary layer break down in the neighbourhood of O, and it has been suggested that the disturbance to the mainstream near O is sufficient to smooth out the singularity and enable the solution to pass over into another conventional boundary layer downstream of O containing a region of reversed flow. The aim of this paper is to explore this possibility in detail using the methods of the triple-deck, developed by the author and others, which have proved successful in somewhat related problems.

Granted the hypothesis that the interaction between the boundary layer and the mainstream is significant near separation and manifests itself through a triple deck, it is found that its streamwise extent is O(ε2l) where ε−8 is a characteristic Reynolds number, ε [Lt ] 1, and l a characteristic length of the problem. The upper deck is of width O(ε2l), lies entirely outside the boundary layer, and in it the flow is inviscid. The main deck is of width O(ε4l) and constitutes the majority of the boundary layer near O, and the perturbations in the velocity are largely inviscid. Finally, the lower deck is of lateral extent $O(\epsilon^{\frac{9}{2}}l)$ and is controlled by a linear equation of boundary-layer type. The whole structure is found to be consistent provided a certain integro-differential equation can be solved, which takes different forms according as the mainstream is supersonic or subsonic. When the mainstream is subsonic it is found that there is no solution to this equation that is sufficiently smooth on the downstream side of the triple deck. When the mainstream is supersonic it is found that the triple deck can at best postpone the breakdown of the assumed structure which still must occur within a distance O(ε2l) of O.

It is concluded that the singularity is not removable by the methods proposed and it is inferred that the singularity is a real phenomenon terminating the flow which, at high Reynolds number, exists upstream of O.

Significant experimental results from a study of pressure in the water-entry cavity are presented. Projectiles were fired into water at velocities up to 250 ft./sec and entry angles of 90°, 60°, and 45°. Pressure data obtained using underwater probes were correlated with high-speed motion pictures taken of the entries. Results indicate that the cavity pressure drop prior to surface closure is an order of magnitude greater than previously assumed. As the entry angle is decreased from 90°, the pressure drop decreases. The minimum cavity pressure decreases linearly with increasing entry velocity over the test range. As the entry angle is increased, the minimum entry velocity required to produce a measurable pressure drop becomes greater. An improved pressure-volume correlation is obtained if the volume enclosed by the cavity walls is corrected to account for re-entrant jet volume and air volume enclosed by the splash walls. Cavity pressure during the closed cavity phase behaves approximately according to the isentropic pressure–volume relation. Pressure drop and history are strongly dependent on projectile nose geometry. No appreciable cavity pressure gradient, axial or transverse, was found to exist. Deep closure or cavity collapse is accompanied by relatively high-pressure pulses.

Goldstein's expansion of the Oseen drag of a sphere in powers of Reynolds number is extended to 24 terms by computer. The convergence is found to be limited by a simple pole at R = − 4·18172. The series is recast using an Euler transformation and other devices to yield accurate results for large R.

Structure functions of turbulent velocity fluctuations up to fourth order have been measured at several heights in the atmospheric boundary layer over the open ocean, and the results are compared with theoretical predictions for separations in the inertial subrange. The behaviour of second- and third-order quantities shows substantial agreement with the predictions of Kolmogorov's original theory over a wide range of separations, but the results of a recent modification of the theory, attempting to account for intermittency in the local dissipation rate, are also consistent with the data over somewhat shorter separation intervals. The behaviour of the measured fourth-order structure function disagrees with that predicted from Kolmogorov's original work, but good agreement is found with the results of the modified theory.

The paper is concerned with a class of non-Newtonian fluids, ν-fluids, all of whose properties are determined by a single dimensional constant of the same dimensions as a viscosity. A regular nth-order ν-fluid is then defined to be one whose nth order time derivative of stress is a regular function of the local stress and flow fields and any of their space and time derivatives. The regularity condition determines the constitutive relation of such a fluid completely in terms of a finite set of non-dimensional constants which define the fluid.

An obvious property of these fluids is that their motions obey the same principles of Reynolds number similarity as those of a Newtonian fluid, and the primary aim of the paper is to examine the extent to which their flow properties are the same as those of the mean turbulent flow of a Newtonian fluid.

It is shown that a third-order fluid is the simplest ν-fluid which shares enough properties with turbulent motion to be worth further consideration in this context. At infinite Reynolds number, the constitutive relation for such a fluid reduces to the form \[ AS\ddot{S} + B\dot{S}^2 + CS^2 S^{\prime\prime} + DSS^{\prime 2} + Eu^{\prime 2}S^2 = 0, \] where A,B, …,E, are isotropic tensor constants of the fluid, S is the stress tensor, u’ is the total rate of strain tensor, dots denote total time derivatives, and primes denote space derivatives. A number of illustrative examples of the properties of such a constitutive relation are then considered, representing the decay of a homogenous stress field, the effect of rigid-body rotation on such a decay, the structure of the equilibrium stress field in the presence of homogeneous rate of strain, both with and without vorticity, and the nature of flow near a plane boundary. In all cases, the results appear to be consistent with known properties of turbulent motion, to the extent that the analysis is taken.

Finally, the effect of finite Reynolds number on the decay of an isotropic and homogeneous stress field is shown to be consistent with observations on the decay of isotropic turbulence.

The results of a numerical evaluation of the Navier-Stokes equations of motion for the case of a viscous fluid streaming past a sphere are presented in terms of the length of the standing eddy behind the sphere and in terms of the angle of flow separation at the sphere. Emphasis was placed on calculating these quantities at Reynolds numbers between 20 and 40 where no reliable theoretical or experimental values are available. In support of these calculations, it is shown that the values for the drag on a sphere previously calculated by us from the Navier-Stokes equations of motion by the same numerical technique as that used for calculating the eddy length and angle of flow separation agree well with our recent, extensive drag measurements for a wide Reynolds number interval. Our results are used to make a comparison between drag and flow field as predicted by analytical solutions and numerical solutions to the Navier-Stokes equations of motion. Some limitations of the analytical solutions to predict correct values for the drag, and to describe the correct nature of the flow field, are pointed out. It is shown further that a plot of [(D/Ds) – 1] versus log NRe, where D is the actual drag on a sphere, Ds is the Stokes drag, and NRe is the Reynolds number, reveals that the variation of the drag on a sphere with Reynolds number follows well defined régimes, which correlate well with the régimes of the flow field around a sphere. A similar relationship between ‘drag-régime’ and flow field pattern is discussed for the case of viscous flow past a cylinder.