An infinitesimal centre disturbance is imposed on a fully Ldveloped plane Poiseuille flow at a Reynolds number R slightly greater than the critical value Rc for instability. After a long time, t, the disturbance consists of a modulated wave whose amplitude A is a slowly varying function of position and time. In an earlier paper (Stewartson & Stuart 1971) the parabolic differential equation satisfied by A for two-dimensional disturbances was found; the theory is here extended to three dimensions. Although the coefficients of the equation are coinples, a start is made on elucidating the properties of its solutions by assuming that these coefficients are real. It is then found numerically and confirmed analytically that, for a finite value of (R-Rc)t, the amplitude A develops an infinite peak at the wave centre. The possible relevance of this work to the phenomenon of transition is discussed.

The semi-empirical heat-transfer laws of Collis & Williams (1959) and Davies & Fisher (1964) give values of the heat-transfer rates for the flow past fine wires which are generally very different from one another. This paper describes some measurements of heat-transfer and convective time constants which show that the relationship of Collis & Williams is the more representative expression.

For the uniform flow past a semi-infinite flat plate subject to a blowing velocity profile equal to C(Uv/x),½ the conventional boundary-layer approximations break down as C approaches 0middot;6192. Here, we consider the structure of the flow for large Reynolds numbers R when C exceeds this critical value. It is shown that, for C > 0·6192, a region containing injected fluid O(R-1/3)) in thickness forms directly above the plate. To a first approximation the flow in this region is inviscid and the pressure a function of x only. This blowing region is separated from the free stream by a free shear boundary layer of thickness O(R-½). Thus the flow domain consists of three distinct regions which interact to yield a similarity solution valid for large values of Rx. This solution is then extended to higher order by expanding the stream function in each region in powers of (Rx)-1/3 and evaluating the first four terms in the resulting series using standard matching techniques. Finally, more general blowing profiles which also lead to boundary-layer ‘blow off’ are considered and an expression, valid far downstream of boundary-layer detachment, is derived for the position of the streamline separating the injected fluid from that of the free stream. For the case of uniform blowing the blowing region takes on the shape of a wedge, indicating that no solution can exist for the corresponding external flow if the plate is truly semi-infinite.

A noise generation mechanism for a nearly ideally expanded supersonic jet is proposed. It is suggested that the dominant part of the noise of a supersonic jet is generated at two rather localized regions of the jet. These regions are located at distances quite far downstream of the nozzle exit. Large-scale instabilities of the jet flow are believed to be responsible for transferring the kinetic energy of the jet into noise radiation. An analysis based on a simple mathematical model reveals that two large-scale unstable waves are preferentially amplified in a supersonic jet. The rapid growth of these waves causes the oscillations of the jet to penetrate the mixing layer at two locations and to interact strongly with the ambient fluid there. This gives rise to intense noise radiation. Theoretical results based on the proposed noise generation mechanism are found to compare favourably with experimental measurements. A simple scaling formula is also derived.

The physiology of the cochlea (part of the inner ear) is briefly examined in conjunction with a description of the ‘place’ theory of hearing. The role played fluid motions is seen to be of importance, and some attempts to bring fluid mechanics into a theory of hearing are reviewed. Following some general fluid-mechanical considerations a potential flow model of the cochlea is examined in some detail. A basic difference between this and previous investigations is that here we treat an enclosed two-dimensional cavity as opposed to one-dimensional and open two-dimensional models studied earlier. Also the two time-scale aspect of the problem, as a possible explanation for nonlinear effects in hearing, has not previously been considered. Thus observations on mechanical models indicate that potential flow models are applicable for times of the same scale as the frequency of the driving acoustic inputs. For larger time scales mechanical models show streaming motions which dominate the qualitative flow picture. The analytical study of these effects is left for a future paper.

Acoustic liners are often perforated screens backed by sound-absorbent material. Turbulence can interact with these screens to generate additional sound. The dynamics of the generation process is examined in this paper, where the liner is modelled as an infinite rigid plane boundary with a homogeneous array of circular orifices or rigid pistons. The acoustic properties of these boundaries are derived in the long wavelength limit. Small-scale turbulence is scattered by individual apertures into sound. Acoustically transparent surfaces support dipole scattering centres while more ‘opaque’ surfaces have monopoles at the apertures which convert turbulence into sound more effectively. It is shown that the process can be described once the response of an individual aperture in an infinite baffle is known. At low Mach numbers the screen can increase the sound radiated by adjacent turbulence by a factor equal to the inverse fourth power of the Mach number. Mean-flow effects are ignored but they are thought to increase the effects deduced in this preliminary study.

A theory is developed to describe the evolution of the entrainment interface in turbulent flow, in which the surface is convoluted by the large-scale eddies of the motion and at the same time advances relative to the fluid as a result of the micro-scale entrainment process. A pseudo-Lagrangian description of the process indicates that the interface is characterized by the appearance of ‘billows’ of negative curvature, over which surface area is, on average, being generated, separated by re-entrant wedges (lines of very large positive curvature) where surface area is consumed. An alternative Eulerian description allows calculation of the development of the interfacial configuration when the velocity field is prescribed. Several examples are considered in which the prescribed velocity field in the z direction is of the general form w = Wf(x – Ut), where the maximum value of the function f is unity. These indicate the importance of leading points on the surface which are such that small disturbances in the vicinity will move away from the point in all directions. The necessary and sufficient condition for the existence of one or more leading points on the surface is that U [les ] V, the speed of advance of an element of the surface relative to the fluid element at the same point. The existence of leading points is accompanied by the appearance of line discontinuities in the surface slope re-entrant wedges, In these circumstances, the overall speed of advance of the convoluted surface is found to be W + (V2 – U2)½, where W is the maximum outwards velocity in the region; this result is independent of the distribution f.

When the speed U with which an ‘eddy’ moves relative to the outside fluid is greater than the speed of advance V of an element of the front, the interface develops neither leading points nor discontinuities in slope; the amplitude of the surface convolutions and the overall entrainment speed are both reduced greatly. In a turbulent flow, therefore, the large-scale motions influencing entrainment are primarily those that move slowly relative to the outside fluid (with relative speed less than V). The experimental results of Kovasznay, Kibens & Blackwelder (1970) are reviewed in the light of these conclusions. It appears that in their experiments the entrainment speed V is of the order fifteen times the Kolmogorov velocity, the large constant of proportionality being apparently the result of augmentation by micro-convolutions of the interface associated with small and meso-scale eddies of the turbulence.

The two-dimensional low Mach number sound field from the passage of a line vortex around the edge of a rigid half plane is calculated by singular perturbation methods. Simple exact expressions are given for the space-time variation of the acoustic field, and for the total radiated energy and its frequency spectrum. These expressions show in particular that the effect of the half plane is to increase the energy radiated by a nearby vortical flow by M−3, where M is a characteristic Mach number.

Dispersion in curved tubes and channels is treated analytically, using the velocity distribution of Topakoglu (1967) for tubes and that of Goldstein (1965) for curved channels. The result for curved tubes is compared with that obtained previously by Erdogan & Chatwin (1967) and it is found that the presentdispersion coefficient contains the Erdogan & Chatwin result as a limiting case.

The most striking difference between the results is that Erdogan & Chatwin predict that the dispersion coefficient is always decreased by curvature if the Schmidt number exceeds 0.124, which is the ease for essentially all systems of practical interest. In contrast, the present result, equation (76), predicts that the dispersion coefficient may be increased substantially by curvature in low Reynolds number flows, particularly in liquid systems which would be of interest in biological systems.

Two competing mechanisms of dispersion are present in curved systems. Curvature increases the variation in residence time across the flow in comparison with straight systems and this in turn increases the dispersion coefficient. The secondary flow which occurs in curved tubes creates a transverse mixing which decreases the dispersion coefficient. The results demonstrate that the relative importance of these two effects changes with the Reynolds number, since the dispersion coefficient first increases and then decreases as the Reynolds number increases. Since secondary flows are not present in curved channels the dispersion coefficient is increased over that in straight channels for all cases.

Source-sink flows in a rotating pie-shaped basin provide a laboratory analogue of wind-driven ocean circulation (Stommel, Arons & Faller 1958). Experiments and theory are presented here for flows which are mildly nonlinear. Theory and experiment show satisfactory agreement for the intense flow in the western boundary-layer region which contains the strongest nonlinear effects. The strengths of the sources and sinks were increased in the experiments in an attempt to induce an instability in the western boundary layer. However, the western boundary layer was always stable, even for relatively large Rossby numbers. Photographs from experiments with a basin of semicircular cross-section show the difference between eastern and western boundary layers in a striking manner.

Simultaneous measurements of the sea surface displacement and the longitudinal component of the wind velocity at several levels are reported. They were obtained at the Marine Tower under various conditions, with the air and the waves moving either in the same or in opposite directions. The spectral analysis was made. The cross-correlation coefficient between the sea surface displacement and the wind velocity is large at the layer adjacent to the surface and decreases with increasing mean wind velocity and height. Below a certain level which is several times or several tens of times higher than the height of the critical level where the wind velocity component in the direction of wave propagation equals the wave velocity, the phase lag of the Fourier component of the wind velocity compared with the surface elevation component is about 160 to 190°. Above this layer the wave-induced wind component is very weak and the phase reversal takes place at the height where the mean wind velocity equals 1·2 to 1·5C, C being the phase velocity of wave. When the wind blows in the opposite direction from that of the wave propagation, the wind fluctuation is in phase with respect to the wave motion and the amplitude of wave-induced wind component is relatively large. Some discrepancies are shown between the observations and the predictions from the theory of inviscid fluids.

The existence theory for steady vortex rings of small cross-section is used to derive asymptotic formulae that describe the shape and overall properties of such rings. A certain two-parameter family of rings is studied in detail, to a first approximation; for members of this family, the ratio ω/r (of vorticity to cylindrical radius) falls from a positive maximum at a central point of the core cross-section to a value at the core boundary that can be substantially smaller or even negative. The case of uniform ω/r is considered to a higher order of approximation, and the formulae given for this case appear to be useful for quite substantial cross-sections.

The interfacial hydrodynamics of pools of dilute binary solutions heated from below is examined using linear stability analysis. Results show that the stabilizing effect of the solute can vary greatly with its interfacial properties, its concentration, and the rate and mechanism of its transfer between the bulk phase and the surface.

Asymptotic and numerical solutions of the unsteady boundary-layer equations are obtained for a main stream velocity given by equation (1.1). Far downstream the flow develops into a double boundary layer. The inside layer is a Stokes shear-wave motion, which oscillates with zero mean flow, while the outer layer is a modified Blasius motion, which convects the mean flow downstream. The numerical results indicate that most flow quantities approach their asymptotic values far downstream through damped oscillations. This behaviour is attributed to exponentially small oscillatory eigenfunctions, which account for different initial conditions upstream.

The local entrainment rate in the initial region of axisymmetric turbulent air jets has been measured by a novel method, which is an adaptation of the ‘porous-wall’ technique used by Ricou & Spalding (1961). The local entrainment rate, which is independent of the nozzle Reynolds number for values greater than 6 × 104, is strongly dependent upon the axial distance. At an axial distance of one nozzle diameter the local entrainment rate is only about one-third of that in the fully developed jet; the entrainment rate increases with increasing axial distance to reach the fully developed value at an axial distance of about thirteen nozzle diameters.

Most of the wave resistance of blunt bow displacement ships is caused by the bow-breaking wave. A theoretical study of the phenomenon for the two-dimensional steady flow past a blunt body of semi-infinite length is presented. The exact equations of free-surface gravity flow are solved approximately by two perturbation expansions. The small Froude number solution, representing the flow beneath an unbroken free surface before the body, is carried out to second order. The breaking of the free surface is assumed to be related to a local Taylor instability, and the application of the stability criterion determines the value of the critical Froude number which characterizes breaking. The high Froude number solution is based on the model of a jet detaching from the bow and not returning to the flow field. The outer expansion of the equations yields the linearized gravity flow equations, which are solved by the Wiener-Hopf technique. The inner expansion gives a nonlinear gravity-free flow in the vicinity of the bow a t zero order. The matching of the inner and outer expansions provides the jet thickness as well as the associated drag.

This paper investigates the boundary-layer characteristics on a helical blade of large chord length, enclosed in an annulus and rotating in a fluid otherwise at rest. The three-dimensional form of momentum integral equations is derived, and used to predict the boundary-layer growth and limiting streamline angles on the blade surface. The measurements are in general agreement with the predictions. The wall shear stress correlation, which includes both Reynolds number and rotation parameters, valid for a rotating blade operating at zero pressure gradient, is derived. Radial and tangential velocity profiles, the tangential component of turbulence intensity and blade static pressures are measured at several locations on the blade surface. The nature of flow near the blade tip is discussed. An expression for the radial velocity profile, valid in the outer region of the boundary layer, is derived theoretically.

This paper examines the linear stability of antisymmetric disturbances in incompressible plane Poiseuille flow between identical flexible walls which undergo transverse displacements. Using a variational approach, an approximate solution of the problem is formulated in a form suitable for computational evaluation of the (complex) wave speeds of the system. A feature of this formulation is that the varying boundary conditions (and the Orr-Sommerfeld equation) are satisfied only in the mean; this reduces the labour involved in determining the approximate solution for a variety of wall conditions without increasing the difficulty of obtaining solutions to a given accuracy. In this paper the symmetric stream function distribution across the channel is represented by a series of cosines whose coefficients are determined by the variational solution. Comparisons with previous work, both for the flexible-wall and rigid-wall problems, show that the method gives results as accurate as those obtained previously by other methods while new results, for flexible walls, indicate the presence of a higher wave-number stability boundary which joins the distorted Tollmien-Schlichting stability boundary at lower wave-numbers. In some cases this upper unstable region, which is characterized by large amplification rates, may determine the critical Reynolds number of the system.

Previously reported experiments with a self-propelled body submerged in a fluid with a stable vertical density gradient have demonstrated that the turbulently mixed wake first expands more or less uniformly and then collapses vertically while continuing to expand horizontally (Schooley & Stewart 1963). It was also shown that the vertical collapse of the wake generates internal waves. Essentially two-dimensional experiments have also been used to explore some of the build-up and decay characteristics of vertical wake collapse induced by a sub-merged burst of turbulent mixing (Wu 1969; Schooley 1968). The present paper reports new experimental measurements and a linear theoretical analysis of the internal wave field created in stratified water by a burst of submerged turbulent mixing. The forcing function has been obtained in integral form for an initial-value model of wake collapse in terms of a general Brunt-Väisälä frequency profile, using normal mode theory. Numerical results have been determined for the specialized case of a completely mixed circular wake in a constant Brunt-Väisälä profile. These results are compared to the experimental measurements.