Conventional boundary-layer theory cannot be applied when the fluid velocity outside the layer changes direction, and the leading edge of a finite body changes ends. In this paper an approximate method for examining the details of the boundary layer during a single flow reversal (occurring at t = 0) is described. It is based on the expectation that (a) long before reversal (t < -t1), there will be a quasi-steady boundary layer appropriate to flow in one direction; (b) long after reversal (t > t2) there will be a quasi-steady boundary layer appropriate to flow in the opposite direction, and (c) in between there will be a period of pure diffusion. The method is applied to a simple heat-transfer problem, in which a fluid of thermal diffusivity D flows with uniform velocity U = At over the plane y = 0; the strip 0 < x < L of the plane is maintained at temperature T1, while the restof the plane and the fluid far away have temperature T0. The approximate solution is compared with an exact solution of the boundary-layer equation, and is shown to give an accurate prediction of the heat transfer as a function of time. The boundary-layer approximation itself breaks down in regions of length O(D2/3A−1/3) near the ends of the heated strip, as usual; it also breaks down in the neighbourhood of the point x = 1/2At2, t > 0, at which the influence of the new leading edge is first felt after flow reversal. In a solution of the full equation, this region is examined in detail, and boundary-layer theory is shown to be sufficiently accurate for the calculation of heat transfer.

Past study of the large-scale consequences of forced small-scale motions in electrically conducting fluids has led to the ‘α-effect’ dynamos. Various linear kinematic aspects of these dynamos have been explored, suggesting their value in the interpretation of observed planetary and stellar magnetic fields. However, large-scale magnetic fields with global boundary conditions can not be force free and in general will cause large-scale motions as they grow. I n this paper the finite amplitude behaviour of global magnetic fields and the large-scale flows induced by them in rotating systems is investigated. In general, viscous and ohmic dissipative mechanisms both play a role in determining the amplitude and structure of the flows and magnetic fields which evolve. In circumstances where ohmic loss is the principal dissipation, it is found that determination of a geo- strophic flow is an essential part of the solution of the basic stability problem. Nonlinear aspects of the theory include flow amplitudes which are independent of the rotation and a total magnetic energy which is directly proportional to the rotation. Constant a is the simplest example exhibiting the various dynamic balances of this stabilizing mechanism for planetary dynamos. A detailed analysis is made for this case to determine the initial equilibrium of fields and flows in a rotating sphere.

The 51st Euromech Colloquium, on mixing in stratified fluids, was held at the University of Cambridge from 24-27 June 1974, with one of the present authors (J.S.T.) acting as Chairman of the Organizing Committee. There were fifty-two participants from twelve countries, and thirty-two contributions were presented and discussed during the four days of the meeting.

The magnetic induction equation is solved numerically in a sphere for a variety of prescribed fluid flows. The models considered are the so-called ‘αω dynamos’, in which both small-scale turbulence and large-scale shearing play a significant role. Solutions are obtained by marching the finite–difference equations forward in time from some initial field. For a critical value of the magnetic Reynolds number Rm solutions which neither grow nor decay are found.

Further calculations are performed with a ‘cut-off effect’ in which an attempt is made to simulate the effect of the Lorentz forces on the turbulence. For supercritical values of R, the magnetic field now stabilizes a t a finite value instead of increasing indefinitely. The form of the final field is compared with that produced at critical Rm in the absence of the cut-off effect.

A sharp wedge expansion flap was tested in the von Kámán Institute Long-shot tunnel at Mach 16 and data on the wall pressure and heat transfer were obtained. Pitot pressure measurements in the boundary layer just ahead of the expansion flap were also made. The surface data are compared with predictions from a characteristics solution for the boundary-layer expansion and from a simple heat-transfer theory.

Free convection has been studied in gaseous helium at low temperatures in a cylindrical vessel of aspect ratio (diameterlheight) 2·5. Compared with measurements in fluids at room temperature the present arrangement has the advantages of small size, a short time constant and improved accuracy. As the Rayleigh number was varied from 60 to 2 × 109, the Nusselt number rose from 1 to 69, obeying the relation Nu = 0·173Ra0·2800±0·0005 over the upper four decades of Ra. The critical Rayleigh number was 1630, but the conditions of the experiment did not allow reliable measurements at such low values of Ra. The very high sensitivity within a given experiment showed the presence of several ‘discrete transitions’, which were often step like and not merely a change of gradient as reported by other workers. The largest of these, at Ra = 3 · 105, involved a drop in heat flux of some 6% and was somewhat hysteretic. The temperature fluctuations increased markedly as the step was crossed.

The resonant response of a harbour H of depth scale d and area A to excitation of frequency ω through a mouth M of width a is calculated in the joint limit a2/Aω2A/gd↓0. The results are relevant to the tsunami response of narrow-mouthed harbours. 16 is assumed that an adequate approximation to the radiation impedance of the external domain is available (Miles 1972). The boundary-value problem for H is reduced to the solution of ∇. (h∇ϕ) = −1/A, where h is the relative depth, the normal derivative of ϕ is prescribed in M and vanishes elsewhere on the boundary of H, and the spatial mean of ϕ must vanish. The kinetic energy in H is proportional to an inertial parameter M that is a quadratic functional of ϕ. It is demonstrated that decreasing/increasing h increases/decreases M. Explicit lower bounds to M are developed for both uniform and variable depth. The results are extended to coupled basins (inner and outer harbours). Several examples are considered, including a model of Long Beach Harbor, for which the calculated resonant frequency of the dominant mode is within 1% of the measured value. The effects of entry-separation and bottom-friction losses are considered; the latter are typically negligible, whereas the former may be comparable with, or dominate, radiation losses.

The stalling characteristics of an airfoil in laminar viscous incompressible fluid are investigated. The governing equations in terms of the vorticity and stream function are solved using an implicit finite-difference scheme and point successive relaxation procedure. The development of the impulsively started flow, the initial generation of circulation, and the behaviour of the forces at large times are studied.

Following the impulsive start, the lift is at first very large and then it rapidly drops. The subsequent growth of circulation and lift is associated with the starting vortex. After incipient separation, the lift increases owing to enlargement of the separation bubble and intensification of the flow rotation in it. The extension of this bubble beyond the trailing edge causes it to rupture and brings about the stalling characteristics of the airfoil. Subsequently, new bubbles are formed near the upper surface of the airfoil and are swept away. The behaviour of the lift acting on the airfoil is explained in terms of the strength and sense of these bubbles. The lift increases when attached clockwise bubbles grow and when counterclockwise bubbles are swept away and vice versa.

It is shown how a two-dimensional surface wave can be either totally reflected or totally transmitted in the presence of two parallel vertical barriers each containing a small gap. Total transmission of a surface wave past obstacles has been known to occur in many situations in water-wave theory, but total reflexion is a comparatively new phenomenon which could be of practical use in the design of breakwaters.

The similarity equations for rotationally symmetric flow above an infinite counter–rotating disk are investigated both numerically and analytically. Numerical solutions are found when α, the ratio of the disk's angular speed to that of the rigidly rotating fluid far from it, is greater than −0.68795. It is deduced that there exists a critical value αcr, of α above which finite solutions are possible. The value of α and the limiting structure as α → αcr are found using the method of matched asymptotic expansions. The flow structure is found to consist of a thin viscous wall region above which lies a thick inviscid layer and yet another viscous transition layer. Furthermore, this structure is not unique: there can be any number of thick inviscid layers, each separated from the next by a viscous transition layer, before the outer boundary conditions on the solution are satisfied. However, comparison with the numerical solutions indicates that a single inviscid layer is preferred.

Using hot-wire-anemometer dynamic-calibration methods, fully developed pipe-flow turbulence measurements have been taken in the Reynolds-number range 80 × 103 to 260 × 103. Comparisons are made with the results of previous workers, obtained using static-calibration methods. From the dynamic-calibration results, a consistent and systematic correlation for the distribution of turbulence quantities becomes evident, the resulting correlation scheme being similar to that which has previously been established for the mean flow. The correlations reported have been partly conjectured in the past by many workers but convincing experimental evidence has always been masked by the scatter in the results, no doubt caused by the difficulties associated with static-calibration methods, particularly the earlier ones. As for the mean flow, the turbulence intensity measurements appear to collapse to an inner and outer law with a region of overlap, from which deductions can be made using dimensional arguments. The long-suspected similarity of the turbulence structure and its consistency with the established mean-flow similarity appears to be confirmed by the measurements reported here.

First, the linear stability of the flow between two concentric cylinders when the outer one is a t rest and the inner has angular velocity Ω{1+ εcosωt} is considered. In the limit in which ε and ω tend to zero it is found that the critical Taylor number a t which instability first occurs is decreased by an amount of order ε2 from its unmodulated value, the stabilizing effect a t order ε2ω2 being slight. The limit in which ω tends to infinity with ε arbitrary is then studied. In this case it is found that the critical Taylor number is decreased by an amount of order ε2ω−3 from its unmodulated value.

Second, the effect of taking nonlinear terms into account is investigated. It is found that equilibrium perturbations of small but finite amplitude can exist under slightly supercritical conditions in both the high and low frequency limits. Some comparisons with experimental results are made, but these indicate that further theoretical work is needed for a broad band of values of ω. In appendix B it is shown how this can be done by an alternative formulation of the problem.

It was shown theoretically in part 1 (Shercliff 1969) that, when a horizontal magnetic field and electric current field are imposed upon a conducting liquid with a free surface, waves excited at the surface have an anisotropic dispersion relation. Here the possibility of demonstrating this phenomenon in the laboratory is considered. The analysis is completed for waves a t the interface between a conducting and non-conducting fluid of similar density, taking full account of surface tension, and then viscosity. Surface tension is found to have a considerable influence in reducing the degree of anisotropy to be expected. The problem of choosing suitable experimental parameters is discussed, and an apparatus is described in which it was possible to demonstrate the existence of anisotropic surface waves, and to compare the phase velocity with the theoretical results. Also, an experiment is described which verifies the relationship between the orientation of the anisotropy and the relative orientation of the imposed magnetic and current fields.

Stability characteristics of a single-phase free convection loop are presented. In the experiments, water was placed inside a toroidal glass loop oriented in a vertical plane. The lower half of the loop was heated and the upper half was cooled. At low heat-transfer rates and also a t high heat-transfer rates the free convection flow was observed to be steady. For the intermediate range, however, the flow was found to be highly oscillatory. Stability predictions are also developed. The comparison between theory and experiment yields favourable agreement.

Observations of unstable behaviour have been reported previously for single-phase fluids in the vicinity of the thermodynamic critical point. In these situations it has been assumed that the unusual behaviour of the fluid properties in the near-critical region necessarily constitutes the underlying cause of such instabilities. In contrast t o this view, analyses by Keller (1966) and Welander (1967) indicate that instabilities can occur for ordinary fluids as well. Results of the present study confirm this contention, since instabilities were clearly observed for water at atmospheric pressure and moderate temperatures.

A few statistical properties of fine-scale velocity and temperature fluctuations have been measured on the axis of symmetry of a heated turbulent round jet. The probability density of ∂θ/∂x, the streamwise derivative of the temperature fluctuation, is strongly negatively skewed, indicating a lack of isotropy for the fine-scale temperature structure. An estimate of the correlation between the velocity and temperature dissipation fields has been obtained by assuming that the dissipation of velocity and dissipation of temperature can be approximated by (∂θ/∂x)2, where u is the streamwise velocity fluctuation, and (∂θ/∂x)2r respectively. The correlation between the quantities (∂θ/∂x)2r and (∂θ/∂x)2r averages over a volume of linear dimension r, is fairly high and depends on the choice of r. An analysis shows that this correlation plays a vital role in the prediction of high-order structure functions of u and θ. The assumed lognormality of the probability density of (∂θ/∂x)2r and (∂θ/∂x)2r and of their joint density is found to be reasonable over a range of r corresponding to the inertial subrange.

A continuously stratified fluid, when subjected to a weak periodic horizontal acceleration, is shown to be susceptible to a form of parametric instability whose time dependence is described, in its simplest form, by the Mathieu equation. Such an acceleration could be imposed by a large-scale internal wave field. The growth rates of small-scale unstable modes may readily be determined as functions of the forcing-acceleration amplitude and frequency. If any such mode has a natural frequency near to half the forcing frequency, the forcing amplitude required for instability may be limited in smallness only by internal viscous dissipation. Greater amplitudes are required when boundaries constrain the form of the modes, but for a given bounding geometry the most unstable mode and its critical forcing amplitude can be defined.

An experiment designed to isolate the instability precisely confirms theoretical predictions, and evidence is given from previous experiments which suggest that its appearance can be the penultimate stage before the traumatic distortion of continuous stratifications under internal wave action.

A preliminary calculation, using the Garrett & Munk (197%) oceanic internal wave spectrum, indicates that parametric instability could occur in the ocean at scales down to that of the finest observed microstructure, and may therefore have a significant role to play in its formation.

The surface elevations in a strip 0 < y < b and in a rectangle -d < y < 0, |x| < a are expressed as a Fourier integral and Fourier series respectively. Using a Galerkin technique to match boundary conditions, the solution to the general problem of the response of a rectangular resonator to inviscid shallow-water waves in infinite and semi-infinite channels is obtained.

Theoretical results are compared with laboratory experiments of James (1970, 1971a, b). Agreement is generally very good, except that the inviscid and linear theory does not predict the observed large energy losses in the resonator at resonance.

The theory is also applied to a geometry corresponding to the Gulf of Carpentaria, and the calculated response of the Gulf to semi-diurnal tides gives a zero response at Kuramba, in agreement with observations. The full response of the Gulf is calculated in subsequent work (Williams 1974) which takes the effects of the earth's rotation into account.

It is known that head seas cannot travel without deformation along a horizontal cylinder of full constant cross-section. Calculations are given which indicate that the waves are refracted away from the axis of the cylinder. Similar refraction effects are found for waves generated by a pulsating source on the cylinder, and also for the Kelvin wave pattern generated by a long ship of nearly constant cross-sectiop moving with constant speed in the axial direction.

The theory of boundary-layer growth at a rear stagnation point, first presented by Proudman & Johnson, is here extended to cover fully three-dimensional rear stagnation points. Supporting numerical solutions of the full initial-value problem establish the relevance of the in viscid similarity solutions obtained.

This paper is concerned with the nonlinear stability of the flow between two long eccentric rotating cylinders. The problem, which is of interest in lubrication technology, is an extension both of the authors’ earlier work on the linear eccentric case and of still earlier work by Davey and others on the nonlinear concentric analysis of Taylor-vortex development. There are four parameters which are assumed small in the analysis; they are the mean clearance ratio, the eccentricity, the amount by which the Taylor number exceeds its critical value; and the Taylor-vortex amplitude. Following the earlier work mentioned above, relation-ships are specified between these parameters in order to develop a satisfactory perturbation scheme. Thus a non-local solution is obtained to the nonlinear stability problem, in which the whole flow field is taken into account.

Of some importance in the analysis is the fact that it is necessary to allow for the development of a pressure field substantially bigger than that associated with Taylor vortices in the concentric case, owing to the Reynolds lubrication effect in a viscous fluid moving through a converging passage. I n order to achieve this mathematically, it is necessary to solve the continuity equation to a higher order than is necessary for the momentum equations.

It is found that the angular position for maximum vortex activity, which is 90° downstream of the maximum gap in the linear case, can taken on any value between 0 and 90°, depending on the value of the supercritical Taylor number. For a particular experiment of Vohr (1968) acceptable agreement is obtained for this angle (50°), though the ‘small’ parameters are somewhat outside the expected range of perturbation theory. Formulae are obtained for the torque and forces acting on the inner cylinder.