This paper examines the energy conversion mechanisms which govern the emission of low frequency sound from an axisymmetric jet pipe of arbitrary nozzle contraction ratio in the case of low Mach number nozzle flow. The incident acoustic energy which escapes from the nozzle is partitioned between two distinct disturbances in the exterior fluid. The first of these is the free-space radiation, whose directivity is equivalent to that produced by monopole and dipole sources. Second, essentially incompressible vortex waves are excited by the shedding of vorticity from the nozzle lip, and may be associated with the large-scale instabilities of the jet. Two linearized theoretical models are discussed. One of these is an exact linear theory in which the boundary of the jet is treated as an unstable vortex sheet. The second assumes that the finite width of the mean shear layer of the real jet cannot be neglected. The analytical results are shown to compare favourably with recent attenuation measurements.

The recent analysis by Acrivos & Lo (1978) concerning the breakup of a long slender droplet in an axisymmetric straining motion is extended to the case of a two-dimensional hyperbolic flow. It is found that, although the cross-section of the droplet becomes significantly non-circular, the theoretical criterion for breakup is effectively the same as in the axisymmetric case. The theoretical predictions are in good agreement with the available experimental results.

We use a calculation of periodic homogeneous isotropic turbulence to simulate the experimental decay of grid turbulence. The calculation is found to match the experiment in a number of important aspects and the computed flow field is then treated as a realization of a physical turbulent flow. From this flow, we compute the large eddy field and the various averages of the subgrid-scale turbulence that occur in the large eddy simulation equations. These quantities are compared with the predictions of the models that are usually applied in large eddy simulation. The results show that the terms which involve the large-scale field are accurately modelled but the subgridscale Reynolds stresses are only moderately well modelled. It is also possible to use the method to predict the constants of the models without reference to experiment. Attempts to find improved models have not met with success.

It is shown that the pressure and velocity fluctuations of the unsteady motion on a transversely sheared mean flow can be expressed entirely in terms of the derivatives of two potential functions. One of these is a convected quantity (i.e. it is frozen in the flow) that can be specified as a boundary condition and is related to a transverse component of the upstream velocity field. The other can be determined by solving an inhomogeneous wave equation whose source term is also a convected quantity that can be specified as a boundary condition in any given problem. The latter is related to the curl of the upstream vorticity field. The results are used to obtain an explicit representation of the three-dimensional gust-like or hydrodynamic motion on a transversely sheared mean flow. It is thereby shown that this motion is ‘driven’ entirely by the two convected quantities alluded to above.

The general theory is used to study the interaction of an unsteady flow with a scmi-infinite plate embedded in a shear layer. The acoustic field produced by this interaction is calculated in the limits of low and high frequency. The results are compared with experimental one-third octave sound pressure level radiation patterns. The agreement is found to be excellent, especially in the low frequency range, where the mean-flow and convective effects are shown to have a strong influence on the directivity of the sound.

Thermoelectric currents in the presence of magnetic fields can cause pumping or stirring of liquid-metal coolants in nuclear reactors or stirring of molten metal in industrial metallurgy. The interaction between the thermal and magnetohydrodynamic fields is a mutual one owing to alterations in the thermal convection and to the Peltier and Thomson effects (although these are usually small). This paper sets up the equations of magnetohydrodynamics and thermal convection when coupled by thermoelectricity and solves some illustrative problems in which the thermal field is known ab initio. Examples where the effects are due to either continuous or discontinuous variation of material composition are included. Practical magnitudes are discussed for the case of a fusion-reactor blanket, where the effects are potentially important owing to the unusual thermoelectric power of lithium.

The inviscid flow above an obstacle in slow transverse motion inside a rotating vessel is analysed to study the influence of the container depth on the basic steady flow structure. An asymptotic theory is presented for an arbitrarily small Rossby number Ro = U0/2Ωl under a fixed H = hRo/l (where Ω is the angular velocity of the container, U0 the obstacle velocity relative to the vessel, h the depth of the container, and l a body length measured transversely to the rotation axis). The equations when linearized for a thin obstacle or shallow topography take on the form of the inertial-wave equation; their solutions for non-vanishing H are obtained for obstacles of three-dimensional as well as ridge-like two-dimensional shapes. In all cases analysed, the solution possesses a bimodal structure, of which one part is column-like with a vorticity proportional to the body elevation (or ground topography). The other part is confined mainly to a region enclosing the body, extending a distance O(H½) upstream of the obstacle and behind a wedge-shaped caustic front at large distances; its contribution consists of lee waves similar to that discussed by Cheng (1977) for an infinite depth. The field associated with the lee waves is then biased on the downstream side, but there is little indication of any tendency to tilting in the sense of Hide, Ibbetson & Lighthill (1968).

The early attempts at calculating effective transport properties of suspensions of interacting spherical particles resulted in non-absolutely convergent expressions. In this paper we provide a physical interpretation for these convergence difficulties and we present a new method of determining the effective transport properties which clarifies difficulties in existing methods.

This method, which is described for simplicity in the context of the thermal conduction problem, is based on an expression that gives the temperature gradient ∇T at a point x in the matrix in terms of integrals over the surrounding particles and an integral over a large surface Γ which encloses x and which we term the ‘macroscopic boundary’. Without the integral over Γ, this expression for ∇T would be non-absolutely convergent, for the contribution to ∇T(x) from a distant particle is proportional to 1/r3, where r is the distance of the particle from x. On comparing the expression for ∇T with the formula used by Rayleigh (1892) in his investigation of the effective conductivity of a cubic array of spheres, we find that Rayleigh's convergence difficulties arose simply from an incorrect assessment of the macroscopic boundary integral.

By combining the expression for ∇T(x) with a formula for the dipole strength of a sphere placed in an ambient temperature field, we obtain a convergent expression relating the dipole strength of a sphere to integrals over the surrounding particles. An expression for the effective conductivity of a random suspension of spheres correct to O(ϕ2) is obtained simply by averaging this expression for the thermal dipole strength. By a similar procedure we obtain expressions for the effective viscosity and effective elastic moduli correct to O(ϕ2). Most of these results have been obtained by earlier workers using a ‘renormalization’ procedure due to Batchelor; the method presented here has the advantage that the renormalization quantity arises naturally from the macroscopic boundary integral referred to earlier, so there is no uncertainty about its choice.

Mean velocity profiles, turbulence intensity distributions and streamwise energy spectra are presented for turbulent air flow in a smooth-walled, high aspect ratio rectangular duct with small streamwise curvature, and are compared with measurements taken in a similar straight duct.

The results for the present curved flow are found to differ significantly from those for the more highly curved flows reported previously, and suggest the need to distinguish between ‘shear-dominated’ flows with small curvature and ‘inertia-dominated’ flows with high curvature. Velocity defect and angular-momentum defect hypotheses fail to correlate the central-region mean flow data, but the wall-region data are consistent with the conventional straight-wall similarity hypothesis. A secondary flow of Taylor–Goertler vortex pattern is found to occur in the central flow region.

An examination of the flow equations yields a model for the mechanisms by which streamline curvature affects turbulent flow, in which a major effect is a direct change in the turbulent shear stress through a conservative reorientation of the turbulence intensity components. Data for the streamwise and transverse turbulence intensities show behaviour consistent with that expected from the equations, and the distribution of total turbulence energy in the central flow region is found to be nearly invariant with Reynolds number and wall curvature, in agreement with the model.

Energy spectra for the streamwise component are examined in terms of a Townsend-type two-component turbulence model. They indicate that a universal, ‘active’ component exists in all flow regions, with an ‘inactive’ component which affects only the low wavenumber spectra intensities. This is taken to imply that the effects of streamline curvature are determined by the central-region flow structure alone.

Measurements are presented of the fluctuating pressure recorded at a point 90° from the mean position of the forward stagnation point on a circular cylinder oscillating in a water flow. The aspect ratio of the cylinder was 9·5 and the turbulence level in the free-stream was 5·5%. The cylinder Reynolds number was 2·4 × 104 and the cylinder was forced to oscillate transverse to the main flow at amplitudes up to 1·33 cylinder diameters. The reduced velocity was varied over the range 3–18 and the experiments spanned the vortex-shedding lock-in range. Measurements of phase difference between pressure and displacement show that the maximum out-of-phase lift force occurs at an amplitude of about half a diameter. Good agreement is found between measurements on forced and freely oscillating cylinders. A simple potential-flow model gives reasonable predictions of the pressure fluctuations at the body frequency and at twice the body frequency at reduced velocities away from lock-in.

Submerged resonant ducts offer an approach to the design of wave-energy extraction devices consistent with the need for maximum seaworthiness. This paper gives a full account of one type of analysis of these systems, based upon two-dimensional wave hydrodynamics and linearized duct dynamics. The mathematical analyses are given in detail in § 2 while § 1 describes as concisely as possible (i) the assumptions underlying each analysis, (ii) its results and their implications for design, and (iii) any available experimental comparisons.

One theoretical prediction, unexpected when it was first made but since confirmed by experiment (Knott & Flower 1979), is that the effective pressure fluctuations to which a resonant duct responds can be substantially greater than those that would be present at the level of the duct mouth if the duct were absent. Other important predictions are concerned with added mass, radiation damping and the conditions for optimum energy extraction, calculated below for a wide variety of mouth design configurations and internal duct geometries. Broad tentative conclusions from the analyses are given at the end of § 1.

Numerical integrations of the Navier-Stokes equations for flow past a smooth, three-dimensional, surface-mounted obstacle are presented. The variation of the flow with Reynolds number, and with geometric ratios such as the maximum slope of the obstacle, are investigated. The separated flow is investigated using visualizations of the surface-stress patterns, and also particle trajectories through the flow.

The fluid mechanics of heart-valve motion is investigated experimentally and theoretically. From the experiments, it is found that the principal mechanism ensuring optimal mitral-valve closure is the pressure field induced by a strong flow deceleration through the valve. A complete theory, evolved from the Bellhouse & Talbot analytical model, is developed for both the mitral and the aortic valve to provide a relationship between valve motion and valve flow. Results predicted by this theory agree consistently well with those obtained from experiments.

The problem of subgrid modelling, that is, of representing energy transfers from large to small eddies in terms of the large eddies only, must arise in any large eddy simulation, whether the equations of motion are open or direct (unaveraged) or closed (averaged). Models for closed calculations are derived from classical closures, and these are used to determine the effect of filter shape, grid-scale spectrum and grid-scale anisotropy on the effective eddy viscosity: the Leonard or resolvable-scale stress is calculated separately and is found to account for 14% of the total drain in a typical high Reynolds number case.

The validity of using these eddy viscosities in an open calculation is considered. It is concluded that this is not unreasonable, but that the simulation would be much improved if the gross drain could be separated into net drain and backscatter.

This work describes experiments in which a stably stratified salt solution is subjected to steady lateral temperature gradients. The criteria for the onset of layered convection are observed to agree with previously published stability analysis. Convective layers formed in this way are found to be statically unstable; they always merge two into one to form a new system. This process continues until the density jump at the solute interface between layers is larger than the density deficit produced by the side wall heating/cooling. The eventual stable system develops into well-mixed convective layers separated by sharp solute interfaces. The conditions for any subsequent layer intermixing and its effect on the lateral heat transfer across the slot, are also described.

A combined experimental and theoretical investigation was conducted on the shock-tube side-wall ionizing boundary-layer induced by a shock wave moving into argon. The dual-wavelength interferometric boundary-layer data were obtained by using a 23 cm diameter Mach—Zehnder interferometer with the 10 × 18 cm Hypervelocity Shock Tube at initial shock Mach numbers of 13 and 16, an initial pressure of 5 torr and a temperature of 300° K. The plasma density and electron number density in the boundary layer were measured and compared with numerical profiles obtained by using an implicit finite-difference scheme for a two-temperature, chemical non-equilibrium, laminar boundary-layer flow in ionizing argon. The analysis included the variations of transport properties based on elastic-scattering cross-sections, effects of chemical reactions, radiation-energy losses and electron-sheath wall boundary conditions. Considering the difficulties involved in such complex plasma flows, satisfactory agreement was obtained between the analyses and experiments. A comparison was made with the flat-plate case and despite the very different velocity boundary conditions at the wall for the two flows the experimental data appear to be quite similar. The experimental bump in the profile of electron number density which was found in the flat-plate case was not found in the side-wall case. Comparisons and discussions of the results for the different types of boundary layer are presented, including a comparison between experimentally derived and analytical plasma-temperature profiles.

The instabilities of two-dimensional convection rolls in a horizontal fluid layer heated from below are investigated in the case when the Prandtl number is seven or lower. Two new mechanisms of instability are described theoretically as well as experimentally. The knot instability causes the transition to spoke-pattern convection at higher Rayleigh numbers while the skewed varicose instability accomplishes a change to larger horizontal wavelengths of the convection rolls. Both instabilities disappear in the limits of small and large Prandtl number. Although the experimental methods fail in realizing closely the infinitely conducting boundaries assumed in the theory, the observations agree in all qualitative aspects with the theoretical predictions.

The flow of an incompressible fluid contained between two infinitely long circular cylinders is considered when the inner cylinder performs small harmonic oscillations about the centre of the larger (fixed) cylinder. Numerical solutions are presented for the Navier–Stokes equations governing the steady-streaming component of the motion. Special attention is then paid to this flow when the Reynolds number of the steady streaming is large, and when the radius of the outer cylinder is much larger than that of the inner. Results obtained show an improved correlation with experimental results and indicate strongly that the finiteness of the domain is the major cause of the discrepancies between experiment and previous theoretical studies.

A new theory is given for calculating the wave pattern and wave resistance of a ship moving at low Froude number F. It applies to ships of any width, either full-bodied or slender. In this theory, the waves travel along rays which start at source points, such as the bow and stern, on the water-line. They propagate with the speed of waves in deep water, but are also advected by the double body flow. This is the flow about the ship and its image in the undisturbed free surface. The phase of a wave at any point on a ray is the optical length of the ray from the source to that point. The amplitude is determined by an excitation coefficient, which determines its initial value, and by an integral along the ray. The total wave height at any point is the sum of the heights on all the rays through the point. The theory is incomplete because the excitation coefficients are known only for thin ships. As an illustration, the theory is applied to the thin ship case, and the results then agree with Michell's thin ship solution evaluated for F small.

A new class of ships, which we call streamlined ships, is introduced next. The usual linear free surface condition applies to the waves they produce. The ray theory is developed for these waves at low F, and it involves straight rays produced at all points on the rear half of the water-line. In addition, as an alternative to the ray theory, another method is presented for obtaining the waves at low F. It involves a Schrödinger-like equation in which distance along the ship's centre-line is the time-like co-ordinate.

A problem of major practical interest is the variation with x and t of the statistical properties of Γ(x, t), the distribution of concentration of a contaminant in a cloud containing a finite quantity Q of contaminant, released in a specified way at t = 0 over a volume of order L30. Of particular relevance is the case of relative diffusion (when x is measured throughout each realization relative to the centre of mass of the cloud), when important properties are L(t), the linear dimension of the cloud, C(x, t), the ensemble mean concentration, $\overline{c^2}({\bf x}, t)$, the variance of the concentration, and p(y, t), the distance-neighbour function. Much fundamental work has led to a knowledge of the way L varies with t, but not of the way the other properties vary. Hitherto therefore, prediction of such variation has normally used unjustifiable empirical concepts such as eddy diffusivities, but this is ultimately unsatisfactory, practically as well as theoretically. Hence the exact equations have been used to obtain a quite new description of the structure of a dispersing cloud, which it is hoped will serve as a basis for future practical work.

When κ = 0 (where κ is the molecular diffusivity) the magnitude of p(y, t) is of order Q/L3 for most y, but of order Q/L30 when |y| is very small. By a variety of arguments it is shown that these facts can be explained (for many, if not all, flows) only if the distributions of C and $\overline{c^2}$, as well as that of p, have a core-bulk structure. In the bulk of the cloud C and $\overline{c^2}$ have magnitudes of order Q/L3 and Q2/L30L3 respectively, but there is a core region of thickness decreasing to zero surrounding the centre of mass within which they have much greater magnitudes. In one case, examined in some detail, the magnitudes in the core are of order Q/L30 and Q2/L60.

It is then shown that the core and bulk exist even in the real case when κ ≠ 0. In the real case the core thickness no longer tends to zero but to a constant of order λc, the conduction cut-off length. As a consequence almost entirely of molecular diffusion acting in the core region, the magnitudes of C and $\overline{c^2}$ in both the core and the bulk decay to zero in a way which depends on the details of the fine-scale structure of the velocity field. Several examples of the decay are discussed.

The distribution of vorticity in the wake of a hovering bird or insect is considered. The wake is modelled by a chain of coaxial small-cored circular vortex rings stacked one upon another; each member of the chain is generated by a single wing-stroke. Circulation is determined by the animal's weight and the time for which a single ring must provide lift; ring size is calculated from the circulation distribution on the animal's wing. The theory is equally applicable to birds and insects, although the mechanism of ring formation differs. This approach avoids the use of lift and drag coefficients and is not bound by the constraints of steady-state aerodynamics; it gives a wake configuration in agreement with experimental observations. The classical momentum jet approach has steady momentum flux in the wake, and is difficult to relate to the wing motions of a hovering bird or insect; the vortex wake can be related to the momentum jet, but adjacent vortex elements are disjoint and momentum flux is periodic.

The evolution of the wake starting from rest is considered by releasing vortex rings at appropriate time intervals and allowing them to interact in their own velocity fields. The resulting configuration depends on the feathering parameter f (which depends on the animal's morphology); f increases with body size. At the lower end of the wake rings coalesce to form a single large vortex, which breaks away from the rest of the wake at intervals. Wake contraction depends on f; the minimum areal contraction of one-half (as in momentum-jet theory) occurs only in the limit f → 0, but values calculated for smaller insects of just over one-half suggest that the momentum jet may be a good approximation to the wake when f is small.

Induced power in hovering is calculated as the limit of the mean rate of increase of wake kinetic energy as time progresses. It can be related to the classical momentum-jet induced power by a simple conversion factor. For an insect or hummingbird the usual momentum-jet estimate may be between 10 and 15% too low, but for a bird it may be as much as 50% too low. This suggests that few, if any, birds are able to sustain aerobic hovering, and that as small a value of f as possible would be necessary if the bird were to hover.

Tip losses (energy cost of the vortex-ring wake compared with the equivalent momentum jet) are negligible for insects, but can be in the range 15–20% for birds.