It is shown that two-dimensional MHD turbulence is in certain respects closer to three-dimensional than to two-dimensional hydrodynamic turbulence. A second-order closure indicates that:

1. at zero viscosity and magnetic diffusivity, a singularity appears at a finite time;

2. there is an energy cascade to small scales and an inverse cascade of squared magnetic potential, in agreement with a conjecture of Fyfe & Montgomery (1976);

3. small-scale magnetic energy acts like a negative eddy viscosity on large-scale magnetic fields;a

4. (iv) upon injection of magnetic energy, a stationary state is obtained which has zero magnetic energy for a positive magnetic diffusivity λ (anti-dynamo theorem); however, this stationary state is preceded by a very long non-zero magnetic energy plateau which probably extends to infinite times as λ → 0.

It is suggested that direct numerical simulation of the two-dimensional MHD equations with high resolution (a 5122 or 10242 grid) could lead to a better understanding of the small-scale structure of fully developed turbulence, especially questions of intermittency and geometry.

The focusing of very weak and slightly concave symmetrical shock waves is examined. The equation that describes this focusing is derived and the resulting similitude discussed. The initial conditions come from a formal matching of this nonlinear description with the linear solution. The maximum value of the pressure coefficient is shown to be proportional to the two-thirds power of both the initial strength of the wave front and a parameter characterizing its rate of convergence.

Two problems involving the unsteady motion of two-dimensional vortex sheets are considered. The first is the roll-up of an initially plane semi-infinite vortex sheet while the second is the power-law starting flow past an infinite wedge with separation at the wedge apex modelled by a growing vortex sheet. In both cases well-known similarity solutions are used to transform the time-dependent problem for the sheet motion into an integro-differential equation. Finite-difference numerical solutions to these equations are obtained which give details of the large-scale structure of the rolled-up portion of the sheet. For the semi-infinite sheet good agreement with Kaden's asymptotic spiral solution is obtained. However, for the starting-flow problem distortions in the sheet shape and strength not predicted by the leading-order asymptotic solutions were found to be significant.

The results of an experimental investigation of the thermal transpiration effect (the thermomolecular pressure difference or t.p.d. effect) in a single glass capillary with a length-to-radius ratio of 250 are presented. The temperatures of the gas in the ‘cold’ and ‘hot’ chambers were 273·2°K and 293°K, respectively. A modified relative method has been used. To measure the t.p.d. effect, a capacitance differential digital micromanometer with sensitivity 4·5 × 10−5N/(m2Hz) was used. The gases investigated were He, Ne, Ar, Xe, H2, D2, N2, CO2, CH4 and SF6. It was discovered that in the intermediate flow regime the thermal-creep flow rate does not depend on the (non-isothermal) tangential momentum accommodation coefficient. From the experimental data on the viscous slip flow regime, the Eucken factors and the accommodation coefficients are calculated. For inert gases the Eucken factor is found to be equal to 2·5 within the experimental error, while the accommodation coefficients differ significantly from unity.

The authors present the matched asymptotic expansion type of solution for the unsteady viscous incompressible flow past a sphere. Most of the analysis is developed under the assumption that a constant rectilinear velocity is suddenly imparted to a sphere in an otherwise quiescent infinite body of fluid. The Reynolds number based on that velocity is taken to be small, and the analysis is then extended to other transient flows that satisfy this requirement. Evidently in the unsteady cases under discussion one can recognize inner and outer regimes. The leading terms in the expansions representing the flow in these are governed by the unsteady Stokes and the hitherto unreported unsteady Oseen equations. The streamline patterns calculated show the ‘birth’ of a ring vortex close to the equator and its gradual migration downstream and outwards. This result is also verified qualitatively by a crude experiment.

Some of the dense fluid at the front of an advancing gravity current is observed to be mixed with the ambient fluid. This process continues when the cross-stream non-uniformities at the head of the current are suppressed by advancing the floor beneath the head. In the resulting two-dimensional flow regular billows are visible. This paper considers experimentally and analytically the inviscid gravity current head and specifically includes the observed mixing at the head. Experimental results were obtained with an apparatus in which the head of the gravity current was brought to rest by an opposing uniform flow. The mixing appears to occur through Kelvin-Helmholtz billows generated on the front of the head and controls the dynamics of the head. A momentum balance is used to analyse the flow and the problem is closed by quantitatively introducing the billow structure.

Detailed measurements of turbulent concentration parameters in a round free methane jet up to 70 diameters downstream from the jet source are described. The concentration changes were obtained using a Raman spectrometer designed for the measurement of a methane vibrational transition function, and processed using a photon correlator. Mean and fluctuating concentration levels are given, together with the concentration probability density distribution.

A theory for interpreting the statistical aspects of the signal is presented and results are compared with other data from the literature on jets of uniform and non-uniform density. It is shown that the intensity of concentration fluctuations has an asymptotic value of 28·5% in the far field region of the jet. One interesting feature of the data is the deviation from Gaussian statistics along the jet centre-line.

The investigation of the effects which a changing mean flow has on a uniform train of internal gravity waves (Thorpe 1978a) is continued by considering waves in a uniformly accelerating stratified plane Couette flow with constant density gradient. Experiments reveal a change in the mode structure and phase distribution of the waves, and their eventual breaking near the boundary where the mean flow is greatest, the phase speed of the waves being positive. A linear numerical model is devised which accurately describes the waves up to the onset of their breaking, and this is used to investigate their energetics. The working of the Reynolds stress against the mean velocity gradient results in a very rapid transfer of energy from the waves to the mean flow, so that by the time breaking occurs only a small fraction of their initial energy remains for possible transfer into potential energy of the fluid.

The consequences have important applications in oceanography and meteorology, to flow stability and flow generation, and explain some earlier laboratory observations.

Solutions have been obtained for a family of unsteady three-dimensional boundary-layer flows which approach separation as a result of the imposed pressure gradient. These solutions have been obtained in a co-ordinate system which is moving with a constant velocity relative to the body-fixed co-ordinate system. The flows studied are those which are steady in the moving co-ordinate system. The boundary-layer solutions have been obtained in the moving co-ordinate system using the technique of semi-similar solutions. The behaviour of the solutions as separation is approached has been used to infer the physical characteristics of unsteady three-dimensional separation.

In the numerical solutions of the three-dimensional unsteady laminar boundary-layer equations, subject to an imposed pressure distribution, the approach to separation is characterized by a rapid increase in the number of iterations required to obtain converged solutions at each station and a corresponding rapid increase in the component of velocity normal to the body surface. The solutions obtained indicate that separation is best observed in a co-ordinate system moving with separation where streamlines turn to form an envelope which is the separation line, as in steady three-dimensional flow, and that this process occurs within the boundary layer (away from the wall) as in the unsteady two-dimensional case. This description of three-dimensional unsteady separation is a generalization of the two-dimensional (Moore-Rott-Sears) model for unsteady separation.

By applying small lateral oscillations to a glass tube from which smoke was issuing, perfectly periodic coflowing jets and wake structures were produced at Reynolds numbers of order 300-1000. These structures remained coherent over long streamwise distances and appeared to be perfectly frozen when viewed under stroboscopic light which was synchronized with the disturbing oscillation. By the use of strobing laser beams, longitudinal sections of the structures were photographed and an account of the geometry of these structures is reported.

When the tube was unforced, similar structures occurred but they modulated in scale and frequency, and their orientation was random.

A classification of structures is presented and examples are demonstrated in naturally occurring situations such as smoke from a cigarette, the wake behind a three-dimensional blunt body, and the high Reynolds number flow in a plume from a chimney. It is suggested that an examination of these structures may give some insight into the large-scale motion in fully turbulent flow.

A new interpretation of a nonlinear wind-wave system is proposed. It is proposed that, for steady wind blowing in one direction, a nonlinear wind-wave system can be completely characterized, to a good first approximation, by a single nonlinear wave train having a carrier frequency equal to that of the dominant frequency in the wind-wave spectrum. In this model, the spectral components of the wind-wave system are not considered a random collection of free waves, each obeying the usual dispersion relation, but are effectively non-dispersive bound-wave components of a single dominant wave, travelling at the speed of the dominant wave. To first order, the nonlinear wind-wave system is considered to be a coherent bound-wave system which propagates energy only at the group velocity of the dominant wave and is governed by nonlinear self-interactions of the type found in amplitude-modulated wave trains. The role of short free waves in the system is discussed. Results of laboratory experiments performed by the authors and by Ramamonjiarisoa & Coantic (1976) are found to provide evidence supporting the applicability of such a model to wind waves under virtually all laboratory conditions. Preliminary consideration is given to possible application of the model to oceanic wind waves and conditions are identified for which the model would be most likely to apply.

A first-order analysis is presented for the propagation of a blast wave through a dilute spray of non-reactive liquid droplets that are suspended in a non-reactive gas-phase carrier. The analysis permits straightforward computation of decay rates and internal wave structure for wave strengths in the approximate Mach number range 4 ≤ Ms ≤ 15, and loading factors (mass of spray per unit mass of carrier) less than about 0·4. The droplets must be sufficiently small to completely change phase in a distance behind the shock that is at all times negligible compared with the wave radius. Representative calculations are presented and discussed. These show more rapid decay rates and higher pressures, densities, and particle velocities in two-phase blast waves when compared against equivalent gas-phase blast waves. A simplification of the analysis for the regime of higher wave Mach numbers (strong waves) is also given, which for that case allows direct algebraic calculation of early wave characteristics.

The developing flow in the entry region of a heated horizontal pipe is analysed. The asymptotic solution of the developing flow near the entrance of the heated straight pipe, distance O(a), is obtained by perturbing the solution of the developing flow in an unheated straight pipe. Two vortices result from the combination of the radial-directional and the downward motions of the fluid particles which are induced by the displacement of the boundary layer and develop along the pipe. The axial velocity has a concave profile in the inviscid core with its maximum off the centre-line near the entrance and it grows toward a uniformly distributed profile downstream. The downward stream caused by the displacement of the secondary boundary layer forces the axial velocity profile to turn anticlockwise continuously along the pipe if the flow is from left to right. The core flow induced by the axial boundary-layer displacement generates a favourable pressure gradient. Simultaneously, the secondary boundary-layer displacement affects the core flow to induce a favourable pressure gradient on the bottom of the pipe and an unfavourable pressure gradient on the top wall. The effect of the axial boundary-layer displacement is stronger than that of the secondary boundary layer near to the entrance. Downstream the growth of the boundary-layer thickness is suppressed by the inviscid secondary flow. It is expected that the displacement effect of the secondary boundary layer becomes dominant downstream from the region of O(d) when Gr is large.

Previous measurements in the moderate to small Reynolds number range of isotropic turbulence have all shown the skewness factor $S\equiv -{\overline{(\partial u/\partial x)^3}}/[\overline{(\partial u/\partial x)^2}]^{\frac{3}{2}}$ of the streamwise velocity derivative to increase with decreasing Reynolds number. This ‘paradoxical’ trend was found for 150 ≥ Rλ ≥ 4. New data covering the range 4 ≥ Rλ ≥ 1 show a maximum S for Rλ between 4 and 3 and a rapid decrease for Rλ < 2.

A series of initially planar shock waves was allowed to expand from a shock tube into a near half-space. The strength of the wall shock was measured at two positions on the slightly concave front wall. These measurements are compared with shock strengths predicted by a shock-dynamic model based on the cylindrical expansion of a critical shock. Chisnell's (1957) theory is used to account for the effect of the increasing surface area of the expanding wall shock, and Whitham's (1957) treatment to correct for the curvature of the wall. The critical shock strength is obtained from Skews’ (1967) experimental measurements of the Mach number of the self-similar wall shock following two-dimensional diffraction at a 90° edge.

The model predicts the relatively small degree of attenuation observed between the measuring stations, but overestimates the absolute shock strength. The most likely cause is that, in the early stages of expansion, the wall shock experiences further attenuation owing to its interaction with the expanding flow. These effects are shown to be short range and of negligible importance at the first measuring station, 1·86 tube diameters from the axis. Thus, using the experimental results at this station as the starting point, the model predicts accurately the shock strength at 3·76 diameters. It is concluded that Chisnell's theory can be applied to the weakening of the wall shock in ducts with large abrupt changes in cross-section only when the wall shock is some distance from the entrance to the area change.

Previous measurements of the decay rate of the fluctuation intensity of passive scalars in grid-generated turbulence show large variation. New results presented here show that the decay rate of passive temperature fluctuations produced by heating the grid is a function of the initial temperature fluctuation intensity. Although a full reason for this is wanting, spectra of the temperature fluctuations show that, by varying the heat applied to the grid, the wavenumber of the maximum in the temperature spectrum changes, indicating that the geometry of the thermal fluctuations is being altered in some way. In these experiments the one-dimensional temperature spectrum shows an anomalous $-\frac{5}{3}$ slope. In order to eliminate the dependence of the decay rate of the temperature fluctuations on their intensity, we describe a new way of generating temperature fluctuations by means of placing a heated parallel array of fine wires (a mandoline) downstream from the unheated grid. Results of this experiment show that the decay rate of passive thermal fluctuations is uniquely determined by the wave-number of the initial temperature fluctuations. In this type of flow there appears to be no equilibrium value for the thermal fluctuation decay rate and hence for the mechanical/thermal time-scale ratio since the thermal fluctuation decay rate does not change within the tunnel length, which is the equivalent of nearly one turbulence decay time.

The hydrodynamic forces due to the motion of a flexible foil in a large amplitude curved path in an inviscid incompressible flow are analysed. A parametric study of large amplitude oscillatory propulsion, with special emphasis on the effect of chordwise flexibility of the fin, is presented. This flexibility was found to increase the propulsive efficiency by up to 2% while causing small decreases in the overall thrust, compared with similar motion with rigid foils.

The spectral method of Orszag & Patterson has been extended to calculate the static pressure fluctuations in incompressible homogeneous decaying turbulence at Reynolds numbers Reλ [lsim ] 35. In real space 323 points are treated. Several cases starting from different isotropic initial conditions have been studied. Some departure from isotropy exists owing to the small number of modes at small wavenumbers. Root-mean-square pressure fluctuations, pressure gradients and integral length scales have been evaluated. The results agree rather well with predictions based on velocity statistics and on the assumption of normality. The normality assumption has been tested extensively for the simulated fields and found to be approximately valid as far as fourth-order velocity correlations are concerned. In addition, a model for the dissipation tensor has been proposed. The application of the present method to the study of the return of axisymmetric turbulence to isotropy is described in the companion paper.

The growth of small disturbances in the incompressible axisymmetric wake of a slender body of revolution has been studied experimentally. The experiments were performed in a running water channel at a Reynolds number of 3600 based on the body diameter and free-stream velocity, without any artificial stimulation of the wake. Digital techniques have been used to analyse the output signals of a multichannel hot-film anemometer system.

Analysis of the mean velocity profiles and the velocity fluctuations shows that the wake can be divided into three regimes: the near wake or adjustment region, a region of locally parallel flow, and the far wake. The near wake is characterized by the adjustment of the mean velocity profile from that in the boundary layer of the body to the wake profile. At the same time, the structure of the disturbances in the boundary layer seems to adjust to a different structure in the wake. Large velocity gradients in both the streamwise and the radial direction are present in this region. In the region of locally parallel flow, a helical mode disturbance is superseded by a mode which appears to be mostly a planar oscillation of higher frequency and amplification rate. Higher-order harmonics of dominant frequency components have been identified in these experiments. The far wake is the region in which nonlinear interactions appear to predominate. The experimental measurements and hydrogen bubble photographs indicate the presence of large scale transverse disturbances in this region, associated with roll-up of the fluid.

In this paper calculations are made of the two-dimensional flow field of an incompressible viscous fluid in a long parallel-sided channel whose walls pulsate in a prescribed way. The study covers all values of the unsteadiness parameter α and the steady-streaming Reynolds number. The wall motion is, in general, assumed to be of small amplitude and sinusoidal. Particular attention is given to the steady component of the flow at second order in the amplitude parameter ε. The results for the corresponding problem in axisymmetric geometry are given in an appendix.

Next the following problem is considered: the calculation of the wall motion which will result, in response to prescribed unsteady pressures imposed at the ends of the channel and outside its walls, if the walls are assumed to respond elastically to variations in transmural pressure. It is found that the system has a natural frequency of oscillation, and that resonance will occur if this frequency is close to a multiple of the frequency of the external pressure fluctuations. Finally the preceding work is applied in a discussion of blood flow in the coronary arteries of large mammals.