Algebraic disturbances, a non-modal component of the linear perturbation fields, are shown to be an essential feature of stratified shear flows. We find that they must be included even in situations where the modes form a complete set, for such completeness does not extend to the space of these ill-behaved functions.

If the Richardson number Ri is less than ¼ anywhere in the flow, the algebraic disturbances are very generally instabilities of the system, growing without limit as time t → ∞.

Both of these results are in direct contradiction with the currently accepted viewpoint. We examine the previous research in this field to locate the source of this discrepancy.

The algebraic instabilities are not form preserving, and display extreme distortion as they evolve. In the asymptotic limit they appear as quasi-horizontal flow fields, with a vertical ‘wavelength’ that tends to zero. As such, they must be expected to induce secondary shear instabilities and cascade into motions of smaller (horizontal) scale.

Analyses based on a three-dimensional vortex-filament model are presented for the unsteady pressure field generated by a ducted propeller. An oscillating part is identified in the kernels and absolute terms of the governing equations for the harmonic components, allowing two methods to be developed for the solution of the higher harmonics. The first method is exact and is applicable to ducted propellers with practical configurations (small chord-to-diameter ratio) while the second is approximate but more suitable for ducted systems with large chord-to-diameter ratios. The second method was applied to a configuration for which experimental data were available and good agreement was obtained for pressure harmonic amplitudes downstream of the propeller and for phase angles upstream of the propeller.

Special consideration was given to the Kutta-Joukowski condition at the duct trailing edge and a general constraint developed for the doubly coupled governing integral equations.

A velocity-defect law for channel flow is shown to result from the single requirement of Reynolds-stress spectral ‘smoothness’ for any mean profile maintained free of inflexions by transient instabilities. The deduced velocity is a logarithmic function of position near boundaries and parabolic in mid-regions of the flow, independent of the detailed mechanisms of momentum transport by the fluid. However, if the ‘smoothness’ of the spectral tail is decreased, a second logarithmic layer of steeper slope emerges inside the first layer. Profile data from drag-reduction experiments gathered by Virk (1975) exhibit the deduced inner log layer and its transition region to the usual outer flow.

In this paper a theoretical investigation of convection currents in anisotropic porous media is performed. The porous layer is homogeneous and bounded by two infinite perfectly heat-conducting horizontal planes. The criterion for the onset of convection is derived. The supercritical steady two-dimensional motion, the heat transport and the stability of the motion are investigated. Some of the results may be applied in insulation techniques.

Melting from an electrically heated horizontal cylinder embedded in a paraffin (n-octadecane, fusion temperature 301·3 °K) has been studied experimentally. The shape of the solid-liquid interface has been determined photographically, and the local heat transfer coefficients have been measured using a shadowgraph technique. The experiments provide conclusive evidence of the important role played by natural convection in melting a solid due to an embedded cylindrical heat source. The four distinct pieces of quantitative evidence which contribute to this conclusion are the melt shape, surface temperature, local and average heat transfer coefficients and their variation with time.

The experimental findings prove the importance of natural convection in phase change problems involving melting and indicate that continued practice of neglecting the effects in the analysis of such problems does not appear reasonable. Natural convection should be considered in analysis and design of systems involving phase change.

The closing mechanism of the natural aortic valve is investigated experimentally in a two-dimensional analogue. The interaction between the sinus content and the aortic flow when the latter is decelerating is studied for different Strouhal numbers: for high Strouhal numbers the sinus content rotates virtually as a rigid mass around the centreline whereas for low Strouhal numbers, corresponding to the human situation, the cusp remains straight and slowly rotates around its point of attachment. Simple theoretical models based on the phenomena observed are proposed. Acceptable agreement between theory and experiment is found.

Swirling water bells are studied theoretically and experimentally. It is shown theoretically that, if the effects of gravity and the surrounding air are neglected, the shape of a swirling water bell will, under certain circumstances, be periodic along the axis of rotation. Under ideal conditions, a swirling water bell may thus be infinitely long. However, the experiments show that in reality the length of a swirling water bell will be limited owing to Kelvin-Helmholtz instabilities. Theoretically calculated shapes of swirling water bells are found to agree reasonably well with experimental results.

Temperature-sensitive constant-current wires operating at very low resistance ratios have been tested for temperature fluctuation response. A significant step in the response was found to occur with a centre-frequency of typically 1/6 Hz. The step size was observed to be as large as 30% and grew from zero to its maximum value in about a decade. Analysis shows that this phenomenon is associated with axial conduction of heat to and from the prongs. If it is recognized that prongs have finite thermal inertia then a modification of the boundary conditions to the equations of Betchov (1948) predicts this step, in agreement with the simple asymptotic analysis of Maye (1970).

Experiments indicate that a similar phenomenon occurs with velocity-sensitive wires. Axial conduction appears to be the most likely cause. Aeroelastic deflexions and non-uniform cooling caused by bowing of the wire make precise predictions impossible. Here the differences in step size between wires were observed to be as large as 10% (or 20% in mean-square energy), the centre-frequency was usually beyond 10 Hz for the wires tested and the step extended over a much broader frequency range than in the temperature-sensitive case. The effect occurred at all velocities, resistance ratios and wire geometries. An analysis based on non-uniform cooling of the wire filament predicts the correct frequency range and shows that steps of 10% in frequency response are quite plausible.

The stability characteristics of some simple steady magnetohydrodynamic flows within an axisymmetric container of arbitrary electrical conductivity are investigated. Attention is focused upon rapidly rotating fluids in which the unperturbed velocity and magnetic field are axially symmetric and purely zonal. Detailed solutions are obtained for the particularly simple basic state representing a rigidly rotating homogeneous fluid with a uniform axial current. The theory of dynamic (dissipationless) instabilities is reviewed and its shortcomings are elucidated. A stability criterion is derived for an inviscid fluid of small electrical conductivity within a perfectly conducting axisymmetric container and it is shown that a certain class of inertial modes is unstable for any non-zero magnetic field strength. When the effects of container conductivity are included it is found that a class of slow modes with westward phase speed may be unstable. These modes are shown to be unstable within a cylinder but appear to be stable within a sphere. The influence of density gradients within a spherical container is investigated and it is found that for a certain class of exceptional slow modes with westward phase speed, a bottom-heavy density gradient is destabilizing. This surprising behaviour is explained in terms of a new branch of the stability curve developed by Eltayeb & Kumar (1977).

The dynamics of an infinitely long one-dimensional vortex and a swirl are compared with the dynamics of a semi-infinitely long trailing vortex and trailing swirl. With increasing distance, the change in the axial velocity difference between the core of the trailing vortex and the surrounding region causes radial convection and some associated axial convection of angular momentum. In laminar or turbulent trailing vortices, we show that under most conditions of interest this is the dominant mechanism for the decrease in the velocities of swirl in the core and corresponding growth of the core. On the basis of theoretical considerations and experimental observations, we show that the axial velocity difference between the core of the trailing vortex and the surrounding region is necessary for the sustenance of turbulence in the vortex core. A theory of the turbulent trailing vortex is developed on the basis of these mechanisms and the results are compared with our experimental observations.

A detailed description of radiative interactions in laminar compressible boundary layers for moderate Mach numbers is presented by way of asymptotic analysis and supporting solutions. The radiation field is described by the differential approximation. While the asymptotic analysis is valid for large N (the ratio of photon mean free path to molecular mean free path) and arbitrary Boltzmann number, Bo (the ratio of convective heat flux to radiation heat flux), the solutions are obtained for Bo [Lt ] 1, the case of strong radiative interactions.

The asymptotic analysis shows the existence of an optically thin boundary layer for large N and all Bo. For Bo [Lt ] 1, two outer regions are observed — one optically thin (at short distances from the leading edge) and the other optically thick (at large distances from the leading edge). An interesting feature not pointed out in the previous literature is the existence of a wall layer at large distances from the leading edge where convective heat flux can be ignored to the leading order of approximation. The radiation field in all cases can be very well approximated by a one-dimensional description.

The solutions have been constructed using the ideas of matched asymptotic expansions by approximate analytical procedures and numerical methods. It is shown that, to the leading order of approximation, the radiation slip method yields exactly the same result as the more complicated matching procedure. Both the cases of linear and nonlinear radiation have been considered, the former being of interest in developing approximate methods which are subsequently generalized to handle the nonlinear problem. Detailed results are presented for both cases.

The entrance region has been divided into two parts, the inlet region and the filled region. At the end of the inlet region, the boundary layers meet at the pipe axis but the velocity profiles are not yet similar. In the filled region, adjustment of the completely viscous profile takes place until the Poiseuille similar profile is attained at the end of it. The boundary-layer equations in the inlet region and the Navier-Stokes equations with order-of-magnitude analysis in the filled region are solved using fourth-degree velocity profiles. The total length of the entrance region so obtained is ξ = x/R Re = 0·150, whereas the boundary layers are observed to meet at approximately one-quarter of the entrance length, i.e. at ξ = 0·036. Experiments reported in the paper corroborate the analytical results.

The effect of uniform wall suction on the structure of turbulence in a fully established turbulent pipe flow has been measured, with special attention to the critical layers close to the wall. Uniform suction was introduced into a pipe flow with a Reynolds number of 17250 by means of a porous-walled section 2·2 diameters in length with very fine perforations. The effect of suction on the turbulent energy balance was then measured over the entire cross-section at four axial locations. The results indicate the following.

1. The amplitudes of the three principal velocity fluctuation components are reduced by suction, but to differing degrees. Moreover, the effects of suction on the amplitudes of these fluctuations develop at differing rates such that the x-wise components are first affected, then the r-wise and lastly the ϕ-wise components.

2. The suction-induced perturbation in the turbulent structure propagates from the wall to the pipe centre-line with a velocity approximately equal to the friction velocity Uτ.

3. Even with very small rates of fluid extraction the maxima of the terms in the turbulent energy balance occurring close to the wall are drastically reduced. Nevertheless there is no tendency for the location of these maxima to move towards the wall.

4. The general reduction of the level of turbulent energy across the entire section is due to transport of this energy by the augmented mean radial velocity towards the wall, where it is dissipated since the boundary condition inhibits the passage of turbulent energy through the wall.

A study has been carried out of turbulent flow over a sinusoidally shaped solid boundary of large enough amplitude that the flow separates from the downstream side of the wave. Measurements are presented of the conditions under which separated flows exist, the extent of the separated region and the variation of the pressure and shear stress along and the velocity profile above the wavy surface. The results are consistent with the model D turbulent solution of the linear momentum equations discussed in Zilker, Cook & Hanratty (1977, hereafter referred to as I) in that reversed flows are possible for wave height to wavelength ratios 2a/λ > 0·033 and in that the range of flow rates for which reversed flows exist increases with increasing 2a/λ. For non-separated flows and for flows with thin separated regions the amplitude of the pressure variation changes linearly with wave height; however, for very large amplitude waves with thick separated regions it is much less sensitive to variations in wave height since the external flow that controls the pressure variation sees a wave profile consisting of a composite of the solid wave and the separated region. The variation of the velocity field along the wave surface in the non-separated region is similar to that predicted by linear theory for small amplitude waves. The flow separates in a region of increasing pressure and reattaches in the region where the pressure on the wave surface is a maximum. A large increase in the wall shear stress is noted immediately downstream of reattachment.

The effects of rotation on a salt-fingering interface between two mixed layers are studied experimentally. It is found that rotation causes an interface to thicken more rapidly than it does in the corresponding non-rotating experiment. In order to interpret this result, the collective-instability model of Stern (1969, 1975) is extended to include the Coriolis effect and the neutral-stability condition is derived. Rotation stabilizes the fingers, the degree of stabilization being dependent on the wavenumber of the perturbation. By assuming equal fluxes in the rotating and non-rotating experiments, the interface thickness data are found to be consistent with the extended collective-instability model.

The stability of an electrically conducting Boussinesq fluid which is confined between two horizontal planes a distance d apart is investigated. The fluid is heated from below, cooled from above and the whole system rotates rapidly with angular velocity Ωc about a vertical axis. A weak non-uniform horizontal magnetic field, whose strength is measured by the Alfvén angular velocity ΩM [[Lt ] Ωc, see (1.2)] permeates the fluid and corresponds to the flow of a uniform electric current parallel to the rotation axis. When the modified Rayleigh number R [see (2.1)] is greater than zero and q = κ/λ < 1, where κ and λ are the thermal and magnetic diffusivities respectively, instability sets in as a westward-propagating wave with a low frequency of order κ/d2.

When R = 0 and ΩM > 2(ν/λ)½ Ωc, where ν is the viscosity, Roberts & Loper (1979) have isolated an exceptional class of unstable fast inertial waves which grow on the magnetic diffusion time scale τλ = d2/λ. When R < 0 and Γ = τλΩ2M/τc, exceeds some value dependent upon q, a class of unstable slow waves also exists for a range of negative values of R. These waves propagate eastwards (westwards) when q is less (greater) than unity. In this case the fluid is stably stratified and the energy for the disturbance is taken from the magnetic field. The resulting description of the stability boundary for R < 0 in the Γ, R plane extends and clarifies the results of Roberts & Loper (1979), which are valid when both Γ and q are large.

Flow visualization, laser-Doppler anemometry and pressure measurements have been used to identify and delineate the regimes of vortex breakdown in a rotating cavity with a central axial flow of air. For the particular cavity tested (where the ratio of the outer to the inner radius was ten and the ratio of the axial width to the inner radius was approximately five), spiral breakdown and axisymmetric breakdown occur in both laminar and turbulent flow. Rossby numbers ε characterizing the boundaries between the breakdown modes were established from visual observations of flow behaviour, from discontinuities in velocity components and in the pressure drop across the cavity, and from changes in the velocity spectra. In laminar flow, spiral breakdown occurs for 1·6 [lsim ] ε [lsim ] 3·2 and axisymmetric breakdown occurs for 0·8 [lsim ] ε [lsim ] 1·5. In turbulent flow, spiral breakdown occurs for 21 [lsim ] ε [lsim ] 100 and 1·5 [lsim ] ε [lsim ] 2·6, and axisymmetric breakdown occurs for 2·6 [lsim ] ε [lsim ] 21 and 0·8 [lsim ] ε [lsim ] 1·5. At the higher Rossby numbers, the flow under laminar conditions is significantly different to that under turbulent conditions; at the lower Rossby numbers, it was found to be impossible to distinguish between laminar and turbulent flow.

Numerical results are presented for planar sinusoidal waves (amplitude α, wave-number k). The average swimming speed and power consumption are computed for a wide range of the parameters. The optimal sine wave for minimizing power consumption is found to be a single wave with amplitude αk ≈ 1. The power consumption is found to be relatively insensitive to changes in the flagellar radius. The optimal flagellar length is found to be in the range L/A = 20–40. The instantaneous force distribution and flow field for a typical organism are presented. The trajectory of the organism through one cycle shows that a wave of constant amplitude may have the appearance of increasing amplitude owing to the yawing motion of the organism.

The results are compared with those obtained using resistance coefficients. For organisms with small cell bodies (A/L = 0.05), the average swimming speed predicted by Gray-Hancock coefficients is accurate to within 10%. For large cell bodies (A/L = 0.2), the error in swimming speed is approximately 20%. The relative error in the predicted power consumption is 25–50%. For the coefficients suggested by Lighthill, the power is consistently underestimated. The Gray-Hancock coefficients underestimate the power for small cell bodies and overestimate it for large cell bodies.

The interaction between magnetic fields and convection in a fluid heated from below is investigated in an axisymmetric cylindrical geometry. When Rm, the magnetic Reynolds number, is large the field is concentrated into a thin rope on the axis of the cylinder. For weak magnetic fields a larger Rayleigh number is necessary to produce a flux rope than that needed for infinitesimal convection. For larger total fluxes, however, the opposite is true and the system is subcritically unstable to steady motions. The results are contrasted with those found by Busse (1975) for the corresponding two-dimensional roll problem.

A numerical analysis of two-dimensional wave scattering problems is performed. The treatment relies on the parabolic approximation and provides the forward scattered wave field. Two problems are considered in particular: (i) the scattering of plane waves by a cylindrical inhomogeneity of uniform refraction index, (ii) the scattering of plane waves by a viscous core vortex. The structure of the scattered field is examined in detail and the numerical solutions of the two problems are compared to analytical results obtained in the Born approximation and interpreted according to the method of smooth perturbation.