Numerical methods are used to investigate the motion of a horizontal vortex pair through a stably stratified atmosphere. The vortices carry with them a mass of fluid whose density differs from that of the air through which it descends, and the surface of this accompanying fluid becomes a vortex sheet, which is modelled by a set of discrete line vortices.

It is shown that, at first, the vortex pair slows down with the shape of the envelope of the accompanying fluid remaining constant. Later, vorticity concentrates at the rear, initiating detrainment and causing a downward acceleration of the vortex pair. Throughout the motion, the vortices approach each other.

Some new measurements and a reassessment of previous data on statistical properties of the breakdown coefficients qr,l in high Reynolds number turbulence show the existence of a range of scale similarity for scales larger than those in the viscous range (l [ges ] 36η). The rate of variation of the probability density p(qr,l) with changing outer scale l/η decreases as l/η increases, becoming fairly insignificant for the largest values of l/η. Measurements of characteristic functions of the probability densities show a substantial degree of statistical independence for sequential adjoint values of qr,l, consistent with the small values of the correlation coefficients for these variables. The data for the moments of qr,l exhibit a behaviour very close to that predicted by the scale-similarity theory when only data for r [ges ] 36η are considered, i.e. data for smaller inner length scales are excluded. The moments and corresponding values of the parameters μp are in good agreement with our previous results and with some earlier data of Kholmyansky, but some rather large unresolved differences in the probability densities of qr,l are found on comparing the present data with those of Kholmyansky. The present measurements of breakdown coefficients for ζ1 = [uscr ]− 1∂u/∂t = ∂(In [uscr ])/∂tand ζ2 = U−1∂u/∂t the time derivatives of the streamwise velocity and its logarithm measured in the atmospheric boundary layer, resolve some previous questions concerning the sensitivity of the results obtained to the choice of positive variable, varying sampling rates and the values of external parameters.

For low sampling rates, a systematic change in the shape of the probability densities p(qr,l) with varying digital sampling rate is found using either ζ1 or ζ2. For sufficiently high sampling rates, the probability densities are independent of the sampling rate; and invariant results are obtained when the sampling rate is at least one-quarter of the Kolmogorov frequency associated with the viscous length scale based on the turbulent dissipation rate. The probability densities p(qr,l) measured using either ζ1 or ζ2 are very similar to the corresponding spectra of ζ1 or ζ2 respectively. Comparison of the mean-square values of ζ1 and ζ2 with an extended form of Taylor's hypothesis shows that the variable ζ1 is not a good approximation to the true spatial derivative ∂u/∂x, and the use of such an approximation can lead to results that are both qualitatively and quantitatively incorrect.

Finite amplitude axisymmetric thermoconvective flows in a bounded cylindrical layer of fluid heated from beneath were calculated using Galerkin's method. Both systems with insulated and with conducting lateral boundaries were investigated for radius-to-depth ratios of 1, 2·25, 2·55 and 2·66. Flows bounded by a conducting lateral boundary were found to be unstable for Rayleigh numbers greater than 1·1 times the critical value. The cell size was found to be an increasing function of the Rayleigh number.

This paper describes a reformulation of the Lighthill (1952) theory of aerodynamic sound. A revised approach to the subject is necessary in order to unify the various ad hoc procedures which have been developed for dealing with aerodynamic noise problems since the original appearance of Lighthill's work. First, Powell's (1961 a) concept of vortex sound is difficult to justify convincingly on the basis of Lighthill's acoustic analogy, although it is consistent with model problems which have been treated by the method of matched asymptotic expansions. Second, Candel (1972), Marble (1973) and Morfey (1973) have demonstrated the importance of entropy inhomogeneities, which generate sound when accelerated in a mean flow pressure gradient. This is arguably a more significant source of acoustic radiation in hot subsonic jets than pure jet noise. Third, the analysis of Ffowcs Williams & Howe (1975) of model problems involving the convection of an entropy ‘slug’ in an engine nozzle indicates that the whole question of excess jet noise may be intimately related to the convection of flow inhomogeneities through mean flow pressure gradients. Such problems are difficult to formulate precisely in terms of Lighthill's theory because of the presence of an extensive, non-acoustic, non-uniform mean flow. The convected-entropy source mechanism is actually absent from the alternative Phillips (1960) formulation of the aerodynamic sound problem.

In this paper the form of the acoustic propagation operator is established for a non-uniform mean flow in the absence of vortical or entropy-gradient source terms. The natural thermodynamic variable for dealing with such problems is the stagnation enthalpy. This provides a basis for a new acoustic analogy, and it is deduced that the corresponding acoustic source terms are associated solely with regions of the flow where the vorticity vector and entropy-gradient vector are non-vanishing. The theory is illustrated by detailed applications to problems which, in the appropriate limit, justify Powell's theory of vortex sound, and to the important question of noise generation during the unsteady convection of flow inhomogeneities in ducts and past rigid bodies in free space. At low Mach numbers wave propagation is described by a convected wave equation, for which powerful analytical techniques, discussed in the appendix, are available and are exploited.

Fluctuating heat sources are examined: a model problem is considered and provides a positive comparison with an alternative analysis undertaken elsewhere. The difficult question of the scattering of a plane sound wave by a cylindrical vortex filament is also discussed, the effect of dissipation at the vortex core being taken into account.

Finally an approximate aerodynamic theory of the operation of musical instruments characterized by the flute is described. This involves an investigation of the properties of a vortex shedding mechanism which is coupled in a nonlinear manner to the acoustic oscillations within the instrument. The theory furnishes results which are consistent with the playing technique of the flautist and with simple acoustic measurements undertaken by the author.

A convective instability is produced by salt water diffusing onto the surface of a fresh-water layer in a Hele Shaw cell. Although the horizontal wavelength of the initial instability is small, an increase in the horizontal wavelength of the convective flow with time and depth is observed as the resulting two-dimensional convection develops. The phenomenon of wavelength variation is confirmed numerically, but quantitative observational and theoretical comparison is limited to small Rayleigh numbers. It is shown that perturbations in the density field cause horizontal pressure gradients, which in turn cause convective elements to combine.

The stability of a horizontal fluid layer when the thermal (or concentration) gradient is not uniform is examined by means of linear stability analysis. Both buoyancy and surface-tension effects are considered, and the analogous problem for a porous medium is also treated. Attention is focused on the situation where the critical Rayleigh number (or Marangoni number) is less than that for a linear thermal gradient, and the convection is not (in general) maintained. The case of constant-flux boundary conditions is examined because then a simple application of the Galerkin method gives useful results and general basic temperature profiles are readily treated. Numerical results are obtained for special cases, and some general conclusions about the destabilizing effects, with respect to disturbances of infinitely long wavelength, of various basic temperature profiles are presented. If the basic temperature gradient (considered positive, for a fluid which expands on heating, if the temperature decreases upwards) is nowhere negative, then the profile which leads to the smallest critical Rayleigh (or Marangoni) number is one in which the temperature changes stepwise (at the level at which the velocity, if motion were to occur, would be vertical) but is otherwise uniform. If, as well as being non-negative, the temperature gradient is a monotonic function of the depth, then the most unstable temperature profile is one for which the temperature gradient is a step function of the depth.

Convective flow of the cellular type has been investigated in vertical cylinders of axisymmetric and annular shape. The critical Rayleigh numbers were determined as functions of the box diameter and the gap width respectively. A distinct influence of side walls with different thermal conductivities was observed.

The characteristics of some flows that occur when fluid is driven through a curved tube are disclosed for an imposed pressure gradient of pulsatile nature, varying sinusoidally with time about a non-zero mean. The fully developed motion depends on three parameters, a traditional Dean number D, a frequencyrelated parameter β and a secondary Reynolds number Rs, it being assumed that the pipe's radius of curvature is much greater than its cross-sectional dimensions. The theoretical description of the flow field is extended from the steady and purely oscillatory limits hitherto studied to all the key situations arising when Rs is of order unity and one of the other parameters β or D takes a large or small value. During this analysis, which in certain cases involves the interactions between steady boundary layers and Stokes layers, a number of pulsatile motions are revealed and the manner in which at high frequencies the secondary motion can change its direction, from inward ‘centrifuging’ to outward, is also explained. Two further illustrations of pulsating motions, stemming from the steady limit, produce an alternative mode of transition from steady boundary-layer flow to the boundary-layer flows occurring when Rs ∼ 1. The study, which mainly deals with the flow in an arbitrary cross-section, lays down a formal basis for deriving the fundamental attributes of many physical situations, some of which are expressible in terms of crucial modifications to, or combinations of, flow problems whose properties are already appreciated.

A European Mechanics Colloquium (Euromech 56) on ‘Nonlinear processes in rotating fluids’ was held at University College, London from 14–18 April 1975. It was attended by 77 mathematicians, engineers and physicists from 14 countries, and 33 papers were presented and discussed. The topics included motions in homogeneous and stratified rotating fluids, with applications to the atmosphere, the oceans, the earth's core and centrifuges. Brief summaries of the papers presented are contained in this report.

Experiments are described in which a body was towed at large Richardson numbers through a uniformly stratified fluid, with particular attention being given to the parameter range in which diffusive effects are likely to play an important role. When the Richardson number is only moderately large (i.e. less than about 106) it appears that a linear viscous diffusive theory fails to model the experimental observations, but at larger Richardson numbers it seems that such a theory is a strong candidate for modelling the physical situation. At Richardson numbers of about 103 a non-diffusive viscous model (see the theories of Graebel 1969; Janowitz 1971) appears to give a fairly good description of the experimental results, except for some discrepancies between the theories and the observed flows in the wake downstream from the body. However, the conditions under which the experiments were carried out were not completely favourable for the application of these theories.

A brief survey and new interpretations of some of the work on the rotatingfluid counterparts of the present experiments are also given. In view of the theoretical similarities between the two situations we have tried to assess and compare (where possible) the results of experimental investigations in each field and we feel that the observations in the rotating-fluid experiments are in good qualitative agreement with the present results. No measurements of the drag acting on the body were made in the present experiments but this has been done carefully in the rotating-fluid experiments, and the results of those measurements are in poor agreement with the theoretical predictions. Because of the close similarities between the axisymmetric model for viscous rotating fluids (cf. Moore & Saffman 1969) and the two-dimensional model for diffusive stratified fluids (Freund & Meyer 1972), an explanation for the discrepancy should be sought before the linear viscous diffusive model can be applied with any confidence to the flow of stratified fluids at very large Richardson numbers.

This paper describes a model problem where compact surface vibration radiates sound into a subsonically flowing fluid. There are two distinct acoustic effects. First, the radiation is increased by flow by an amount proportional to 5M2 and that increase is shown by a general argument to arise from an enhanced surface damping and work done by the flow to overcome drag in the ratio 2:1. Second, the acoustic source strength is affected and resonance frequencies are significantly modified by flow. The main effect is that flow induces on the surface a force proportional to the displacement which opposes the action of natural surface elasticity. A critical velocity exists beyond which the surface is unstable; the stability limit is determined. The surface motion might be regarded as an acoustic monopole, but since aerodynamic fields are determined by the rate of change of the rate of mass outflow, the frequency dependence is more that of a quadrupole. Convective amplification of the sound is also shown to be that characteristic of quadrupole sources. This result indicates that real simple fields may be more sensitive to convection than might be expected from past studies of simple inhomogeneities satisfying a convected wave equation.

Supersonic boundary-layer flow on a flat plate near a point of injection cut-off is considered. The goal is to develop a non-singular solution in the limit of large Reynolds number in the vicinity of the boundary-condition change. An application of Goldstein-type singular solutions shows that an interaction-type theory is required even for transverse velocity jumps on the boundary O(R− ½). The interaction analysis is developed in terms of the linearized triple-deck theory described by Smith & Stewartson (1973a). The analytically derived solution provides a continuous pressure and wall-shear distribution.

The necessity for developing a boundary-layer theory in the case of blast waves stems from the fact that inviscid flow solutions often yield physically unrealistic results. For example, in the classical problem of the so-called non-zero counterpressure explosion, one obtains infinite temperature and zero density in the centre at all times even after the shock front deteriorates into a sound wave. In reality, this does not occur, as a consequence, primarily, of heat transfer that modifies the structure of the flow field around the centre without drastically affecting the outer region. It is profitable, therefore, to consider the blast wave as a flow field consisting of two regions: the outer, which retains the properties of the inviscid solution, and the inner, which is governed by flow equations including terms expressing the effects of heat transfer and, concomitantly, viscosity. The latter region thus plays the role of a boundary layer. Reported here is an analytical method developed for the study of such layers, based on the matched asymptotic expansion technique combined with patched solutions.

Similarity solutions are determined for the turbulent wake of a self-propelled body (thrust = drag). The momentumless wake is shown to behave in a manner intermediate to homogeneous grid turbulence and more familiar free-shear flows such as the drag wake or jet. In essence the decay of momentumless-wake turbulence is similar to that of grid turbulence, but proceeds at a somewhat greater rate owing to lateral diffusion. The mean velocity difference is coupled to the difference $\overline{u^2}-\overline{v^2}$ between the axial and radial components of the mean-square fluctuating velocity. It is necessary to consider governing relations for various second-order turbulence quantities. Previously developed closure approximations yield far-wake decay rates that agree well with available measurements. Production of turbulent energy is negligible asymptotically; thus there is no balance between production and dissipation, and the far-wake behaviour does not become independent of the initial (near-wake) conditions. Even the radial profiles depend on the initial conditions, and there is no natural length scale with which to characterize the far wake.

Mean velocities and temperatures were measured in a turbulent plume above a line heat source at the base of a vertical adiabatic wall. Various values of the buoyancy flux and distances above the source in the weakly buoyant regime of the flow were considered. Correlating expressions for the data were determined by assuming streamwise similarity of mean flow quantities, and a fixed entrainment constant and local skin-friction coefficient. The results yielded an entrainment constant based on the average velocity of 0·095±0·005. This value of the entrainment constant is in fair agreement with earlier work on wall plumes, but is less than half the value reported for free line plumes. Local flow properties are only weakly dependent upon the value of the skin-friction coefficient, with variations of 0-0·01 causing only a 3% change in the mean velocity and weight density defect. The results suggest that the presence of a wall, as opposed to a plane of symmetry, has a strong influence on entrainment, although the specific drag characteristics of the wall are of less importance. The reduced entrainment causes higher velocities and higher values of the weight density defect in a wall plume than in a comparable free plume.

The paper describes an investigation of a subjectively distinguishable element of high speed jet noise known as ‘crackle’. ‘Crackle’ cannot be characterized by the normal spectral description of noise. It is shown to be due to intense spasmodic short-duration compressive elements of the wave form. These elements have low energy spread over a wide frequency range. The crackling of a large jet engine is caused by groups of sharp compressions in association with gradual expansions. The groups occur at random and persist for some 10−1s, each group containing about 10 compressions, typically of strength 5 × 10−3 atmos at a distance of 50 m. The skewness of the amplitude probability distribution of the recorded sound quantifies crackle, though the recording process probably changes the skewness level. Skewness values in excess of unity have been measured; noises with skewness less than 0·3 seem to be crackle free. Crackle is uninfluenced by the jet scale, but varies strongly with jet velocity and angular position. The jet temperature does not affect crackle, neither does combustion. Supersonic jets crackle strongly whether or not they are ideally expanded through convergent-divergent nozzles. Crackle is formed (we think) because of local shock formation due to nonlinear wave steepening at the source and not from long-term nonlinear propagation. Such long-term effects are important in flight, where they are additive. Some jet noise suppressors inhibit crackle.

The effect of large mass injection on the following three-dimensional laminar compressible boundary-layer flows is investigated by employing the method of matched asymptotic expansions: (i) swirling flow in a laminar compressible boundary layer over an axisymmetric surface with variable cross-section and (ii) laminar compressible boundary-layer flow over a yawed infinite wing in a hypersonic flow. The resulting equations are solved numerically by combining the finite-difference technique with quasi-linearization. An increase in the swirl parameter, the yaw angle or the wall temperature is found to be capable of bringing the viscous layer nearer the surface and reducing the effects of massive blowing.

A relation connecting the reflexion and transmission coefficients for scattering of water waves by a fixed body with the far-field radiated waves due to forced motions of the same body is derived. Two alternative derivations are given, including a simple argument based on the analysis of an appropriate linear superposition of the two problems, and a more formal application of Green's theorem to the two potentials. For bodies with horizontal symmetry, the transmission and reflexion coefficients are related to the phase angles of the far-field radiated waves associated with symmetric and antisymmetric forced motions of the body. Some general conclusions follow for arbitrary symmetric bodies, and these are verified in specific cases by comparison with existing solutions. The applicability of these relations to other types of wave problem is noted.

The linearized initial-value problem for a two-dimensional, unbounded, exponentially stratified, plane Couette flow is considered. The solution is used to evaluate the evolution of wave-packet perturbations to the mean flow for all Richardson numbers J > ¼, demonstrating that a consistent wave-packet approach to wave propagation in these flows is possible for all J > ¼. It is found that the vertical influence of a wave-packet perturbation is limited to a distance of order (J − ¼)½/k0, where k0 is the magnitude of the initial central wave vector. These results are used to clarify the J [gsim ] ¼ conclusions of an earlier treatment by Booker & Bretherton.

Neither the phenomenological nor the structural approach to the determination of constitutive equations has yet shown itself to be capable of producing useful and predictive descriptions of the majority of technologically important complex fluids. In the present paper we explore the suggestion that significant progress can be made when these two complementary approaches to rheology are combined. For this initial study we restrict our attention to materials which can be modelled as a suspension of particles in a Newtonian fluid, thereby including most polymer solutions while excluding polymer melts. By applying phenomenological techniques to the basic formulation of suspension mechanics we are able to deduce a common simplified constitutive model for all suspension-like materials and to reveal its physical origin. The present analysis demonstrates that the constitutive model of Hand (1962), involving a single second-order tensor, is not sufficiently general for a rigorous description of the majority of suspension-like materials. Consideration of the constitutive forms for the limiting cases of near-equilibrium and strongly non-equilibrium microstructure suggests, however, that Hand's model may provide a reasonable approximation to the exact constitutive behaviour which is useful over the whole range of flow strengths.