A numerical study is made of the temporal eigenvalue spectrum of the Orr-Sommerfeld equation for the Blasius boundary layer. Unlike channel flows, there is no mathematical proof that this flow has an infinite spectrum of discrete eigenvalues. The Orr-Sommerfeld equation is integrated numerically, and the eigenvalues located by tracing out the contour lines in the complex wave velocity (c = cr + ici) plane on which the real and imaginary parts of the secular determinant are zero. This method gives only a finite and small number of discrete eigenvalues for a wide range of Reynolds numbers and wavenumbers. The spectrum of plane Poiseuille flow is used as a guide to study the spectrum of an artificial two wall flow which consists of two Blasius boundary layers. As the upper boundary of this flow moves to infinity, it is found that the portion of the spectrum with an infinite number of eigenvalues moves towards cr = 1 and the spacing between eigenvalues goes to zero. It is concluded, on the basis of this result and the contour method, that the original few eigenvalues found are the only discrete eigenvalues that exist for Blasius flow over a wide portion of the c plane for cr < 1 and cr > 1. It is suggested that the discrete spectrum is supplemented by a continuous spectrum which lies along the cr = 1 axis for ci < −α/R.

Measurements are described of the fluid velocities close to rough beds oscillating in their own plane. The roughness with which most of the results were obtained consisted of smooth spheres closely packed in hexagonal formation. Some results are also given for beds of gravel. The beds were oscillated with simple harmonic motion in still air and the measurements were made with a hot-wire anemometer.

The measurements very close to the beds of smooth spheres show two maxima in the velocity profile during each half-cycle. One maximum corresponds to a component of velocity which varies nearly sinusoidally with time. The second forms quite a sharp peak and occurs close to ωt = 90°, 270°, where ω is the angular frequency of oscillation and t is time measured from the instant of maximum velocity of the plate. The phase at which this peak occurs shows little variation with distance from the bed. For values of βD > 3·0, where β = (ω/2ν)½, ν is the kinematic viscosity and D is the sphere diameter, the maximum velocity during each half-cycle is found at this peak over at least a certain range of distances from the bed. The variation with height of the nearly sinusoidal component of velocity is quite close to that given by Stokes’ (1851) solution for a flat plate. The peak at ωt = 90°, 270°, however, rises from zero at the bed to a maximum at a distance of about one-eighth of a sphere diameter above the crests and then falls off again.

The measurements with beds of gravel show a variation in velocity similar to that observed by Kalkanis (1957, 1964) and Sleath (1970). Because of the irregularity of the surface it is difficult to draw definite conclusions about the flow in the immediate vicinity of the bed.

A number of tests were carried out, with the beds of spheres, using a wire slanted at 45° to the bed in order to determine the velocity product uw.

The interface between two miscible fluids which have identical mechanical properties but disparate electrical conductivities and are stressed by an equilibrium tangential electric field is studied experimentally and theoretically. A bulk-coupled electrohydrodynamic instability associated with the diffusive distribution of fluid conductivity at the interface is experimentally observed.

The configuration is modelled using a layer of exponentially varying conductivity spliced on each surface to a constant-conductivity fluid half-space. Over-stable (propagating) modes are discovered and characterized in terms of the complex growth rate and fastest growing wavenumber, with the conductivity ratio and an inertia-viscosity time-constant ratio as parameters. In the low inertia limit, growth rates are governed by the electric-viscous time τ = η/εE2. Instability is found also with the layer of varying conductivity bounded by rigid equipotential walls. A physical mechanism leading to theoretically determined fluid streamlines in the form of propagating cells is described.

At relatively high electric fields, large-scale mixing of the fluid components is observed. Photocell measurements of distributions of average fluid properties demonstrate evolution in time on a scale determined by τ.

The stability of natural convection in fluid between two parallel vertical plates is investigated theoretically. The two plates are maintained at different temperatures and a uniform stable temperature gradient β is present in the vertical direction. The Prandtl number of the fluid is fixed at 7.5. An orthonormalization method is used in numerical integrations of the disturbance equations. It is shown how the critical Grashof number varies with β for both stationary and travelling disturbances. It is found that for β ≤ 7.1 × 10−3 the convection is unstable to stationary disturbances and for β ≥ 7.1 × 10−3 it is unstable to travelling disturbances. The critical Grashof number is given by \[ G_c = \left\{\begin{array}{@{}l@{\quad}c@{\quad}l@{}} 500 & {\rm for} & \beta < 1.0\times 10^{-3},\\ 1.3\times 10^6\beta^3 & {\rm for} &\beta > 4.1\times 10^{-2}, \end{array}\right. \] and even for intermediate values of β the variation of Gc is rather simple but not monotonic.

Flow control by a wide, deep weir in a rotating fluid is investigated theoretically and experimentally. A strong (vertical) vorticity constraint due to frame rotation is combined with conservation of the Bernoulli function along streamlines and a standard hydraulic control assumption to show that the volume flux over the barrier is \[ Q = g^{-1}\left[\frac{2}{3}g(H - b_0)-\frac{1}{3}f^2l^2 \right]^{\frac{3}{2}}, \] where H is the depth of the fluid column upstream, bo is the crest height, f is the Coriolis parameter, and l is a length-scale measure of the breadth of the weir. The component of the velocity parallel to the weir crest is computed from conservation of potential vorticity to be v = −fl; perpendicular to the crest, we recover the standard hydraulic relation u = (gh0)½.

Experimental investigations of upstream height and streamline deflexion as functions of rotation are described. It is found that agreement with theory is good up to a certain rate of rotation, above which the finite width of the experimental weir becomes important.

In three-dimensional Bénard convection regions of rising and sinking fluid are dissimilar. This geometrical effect is studied for axisymmetric convection in a Boussinesq fluid contained in a cylindrical cell with free boundaries. Near the critical Rayleigh number Rc the solution is obtained from a perturbation expansion, valid only if both the Reynolds number and the Péclet number are small. For values of the Nusselt number N ≤ 2 accurate solutions are provided by an expansion in a finite number of vertical modes. For Prandtl numbers p < 1 the form of the solution changes at large Reynolds number and becomes independent of p; in the limit p → 0 there is an effective critical Rayleigh number R* = 1.32Rc, which can also be derived by a perturbation procedure, and the Nusselt number is a function of the Rayleigh number only. Numerical experiments yield solutions for Rayleigh numbers R ≤ 100Rc and p ≥ 0.01. The results are similar to those for two-dimensional rolls and for R ≥ 5Rc the Nusselt number shows only a weak dependence on p. For p > 1 there is a viscous regime with N ≈ 2(R/Rc)1/3; when R/Rc [gsim ] p3/2, N increases more rapidly, approximately as R0.4. At high Rayleigh numbers a large isothermal region develops, in which the ratio of vorticity to distance from the axis is nearly constant.

Some insight into the mechanism of solid-particle transport by peristalsis is sought experimentally through a two-dimensional model study (§ 2). The peristaltic wave is characterized by a single bolus sweeping by the particle, resulting in oscillatory motion of the particle. Because of fluid-particle interaction and the significant curvature in the wall wave, the peristaltic flow is highly nonlinear and time dependent.

For a neutrally buoyant particle propelled along the axis of the channel by a single bolus, the net particle displacement can be either positive or negative. The instantaneous force acting upon the particle and the resultant particle trajectory are sensitive to the Reynolds number of the flow (§ 3 and 4). The net forward movement of the particle increases slightly with the particle size but decreases rapidly as the gap width of the bolus increases. The combined dynamic effects of the gap width and Reynolds number on the particle displacement are studied (§ 5). Changes in both the amplitude and the form of the wave have significant effects on particle motion. A decrease in wave amplitude along with an increase in wave speed may lead to a net retrograde particle motion (§ 6). For a non-neutrally buoyant particle, the gravitational effects on particle transport are modelled according to the ratio of the Froude number to the Reynolds number. The interaction of the particle with the wall for this case is also explored (§ 7).

When the centre of the particle is off the longitudinal axis, the particle will undergo rotation as well as translation. Lateral migration of the particle is found to occur in the curvilinear flow region of the bolus, leading to a reduction in the net longitudinal transport (§ 8). The interaction of the curvilinear flow field with the particle is further discussed through comparison of flow patterns around a particle with the corresponding cases without a particle (§ 9).

An experiment was performed in a 3·5 by 3·5 m variable-height, closed convection box, with conditions ranging from a Rayleigh number of 4 × 104 up to 7 × 109, using air as the working fluid. Heat-flux measurements made at Rayleigh numbers up to 7 × 109 yielded a Nusselt number Nu = 0·13Ra0·30. Velocities and temperatures were measured up to Ra = 1·7 × 107, and Fourier spectra calculated to find the predominant horizontal scales of the motion midway between the boundaries. The predominant scale at Ra ∼ 105 was approximately four times the distance between plates, changing to six as Ra increased to 106. With side walls introduced so that the transverse aspect ratio was equal to five, Fourier spectra indicated considerable smaller scale motions, approximately equal to the layer depth. These motions decreased in size as Ra was increased. The results are discussed in relation to previous experimental and theoretical work.

The stability of short-wave displacement perturbations on a vortex filament of constant vorticity in a weak externally imposed strain field is considered. The circular cross-section of the vortex filament in this straining flow field becomes elliptical. It is found that instability of short waves on this strained vortex can occur only for wavelengths and frequencies at the intersection points of the dispersion curves for an isolated vortex. Numerical results show that the vortex is stable at some of these points and unstable at others. The vortex is unstable at wavelengths for which ω = 0, thus giving some support to the instability mechanism for the vortex ring proposed recently by Widnall, Bliss & Tsai (1974). The growth rate is calculated by linear stability theory. The previous work of Crow (1970) and Moore & Saffman (1971) dealing with long-wave instabilities is discussed as is the very recent work of Moore & Saffman (1975).

A thermally driven steady axisymmetric flow of gas of small diffusivity in a vertical circular cylinder rotating rapidly about its axis of symmetry is studied. The side wall is a thermal insulator and the horizontal end plates are perfect conductors. The temperature of the top end plate is kept slightly higher than that of the bottom one.

The boundary-layer method is applied to solve the linearized basic equations and the following results are obtained.

1. The axial velocity in the inner core is fully controlled by the Ekman suction on the horizontal plates and is the same as that in the case of a perfectly conducting side wall.

2. The closed circulation in the side-wall Stewartson E½ layer is strongly suppressed compared with the case of a perfectly conducting side wall.

This situation is reflected in the inner temperature field, which deviates from that in the case of a perfectly conducting side wall. The critical parameter governing the solution is found to be (γ − 1) PrG0E−1/3/4γ, where Pr is the Prandtl number, γ the ratio of specific heats, E the Ekman number and G0 the square of the Mach number based on the peripheral speed of the cylinder.

The paper describes an experimental study of the structure of air-water mixtures flowing vertically. Resistivity probe techniques were applied to measurements of local void properties, including void fraction, gas-phase convection velocity, bubble size distributions and space-time correlation functions. The axial development of flows for six different air-water mixing conditions were examined. Measurements up to 108 diameters from inlet indicated that, while flow patterns for the different mixers may be significantly different initially, the flows tend to develop towards a common structure determined only by the flux rates of the two phases. This was evidenced by the convergence of the void and velocity profiles, and particularly the bubble size distributions, as the flow developed. Estimates of bubble sizes for these more developed conditions, based on a balance of energy between the interfacial structure and the turbulent structure, gave values of diameters which were on average 13% above experimental values. The void distributions obtained for bubbly flow conditions, after an adequate settling length, appear to be characterized by a local minimum at the centre of the pipe.

Slow, steady, mechanically driven, axisymmetric motion of a stably stratified, electrically conducting, rotating fluid is studied. Attention is focused upon the parameter values for which hydromagnetic effects first become important in a rotating stratified fluid and upon the nature of their influence on the interior flow of that fluid. It is found that hydromagnetic effects are able to alter the flow of a stratified rotating fluid at much weaker magnetic field strengths than the flow of an unstratified fluid. Specifically, the interior azimuthal flow is altered if E/α2 [Lt ] σS [Lt ] 1 or if E [Lt ] α2 and 1 [Lt ] σS, where E = ν/ΩL2, $E = \nu/\Omega L^2,\quad\alpha^2 = \overline{\sigma}B^2/\rho\Omega $ and $S = \overline{\alpha}\Delta Tg\nu/\Omega^2\kappa L$. The hydromagnetic effects act to decrease the vertical shear in the azimuthal flow from the levels which would occur in the absence of magnetic fields.

The steady laminar motion of fluid through pipes of circular cross-section, the curvature of whose centre-line varies locally, is analysed theoretically. The flow in three kinds of pipes whose centre-lines are specified by \[ \hat{y} = a(1+\kappa^2\hat{x}^2)^{\frac{1}{2}},\quad\hat{y} = a\tan h\kappa\hat{x}\quad{\rm and}\quad\hat{y} = a\sin\kappa\hat{x} \] are treated as the examples of once-, twice- and periodically-curved pipes, respectively. The analysis is valid for any other two-dimensionally curved pipes, when centre-line curvature is small. At very small Reynolds number, the position of maximum axial velocity shifts towards the inner side of the pipe section; at large Reynolds number, on the contrary, it tends to the outer side, owing to centrifugal force. Furthermore, in the latter case, adaptation of the flow follows the change of mean-flow direction, with a phase lag.

The method of Vickery for calculating the drag of plane lattice structures normal to a turbulent stream is extended to cases of increased solidity. The analysis incorporates an extended version of Taylor's theory for the flow through a porous plate, and a simplified version of Hunt's analysis of the distortion of a turbulent flow by the mean flow field of a body. Some comparisons are made with experimental data.

Internal Alfvén-acoustic-gravity waves propagating in an isothermal, perfectly electrically conducting, plane stratified, inviscid, compressible atmosphere permeated by a horizontal stratified magnetic field in which the mean horizontal velocity U(z) depends on the height z only exhibit singular properties at the Doppler-shifted frequencies \[ \Omega_{d} = 0,\quad\pm\Omega_A,\quad\pm\Omega_A/(1+M^2)^{\frac{1}{2}},\quad\pm (\Omega_c/2^{\frac{1}{2}})[1+M^2\pm \{(1+M^2)^2 - 4\Omega^2_A/\Omega^2_c\}^{\frac{1}{2}}]^{\frac{1}{2}} \] where ΩA is the Alfvén frequency, Ωc the sonic frequency and M the magnetic Mach number. The phenomenon of critical-layer absorption is studied using the momentum-transport approach of Booker & Bretherton (1967), the wave-packet approach (which is a consequence of the WKBJ approximation) of Bretherton (1966) and the technique involving wave normal curves of McKenzie (1973). The absorption effects are also illustrated, following Acheson (1972), by drawing ray trajectories. We find that the waves are absorbed at the critical levels Ωd = ± ΩA and ± ΩA/(1 + M2)½, and in particular we observe that these levels do not act like valves as observed by Acheson (1972). We also conclude that the combined effect of velocity shear and density and magnetic-field stratification is to increase the number of absorption levels.

The present paper and part 2 (adjacent) study the sound field produced by a convected point quadrupole embedded in and moving along the axis of a round plug-flow jet. Only subsonic eddy convection velocities are considered. We examine cold jets here and hot jets in part 2. A principal feature of the study is extensive comparison with jet-noise data. It appears that this simple model problem succeeds in explaining all the major interesting features of jet-noise data, on both hot and cold jets, for jet exit velocities in the low supersonic range. Particular success is achieved in explaining aspects of the data not explainable by the Lighthill acoustic-analogy approach. The picture of jet-noise generation that emerges (at least for jet velocities in the low supersonic regime) is in many respects a striking reaffirmation of the Lighthill point of view. It appears that there is an intrinsic or universal distribution of compact quadrupoles, whose strength and frequency distribution scale with the jet velocity and nozzle diameter as would be expected from simple dimensional reasoning, responsible for jet-noise generation. These quadrupoles are of course convected by the mean flow and satisfactory agreement with the data is obtained by assuming that they are devoid of any intrinsic directionality. There appears to be no significant jet Mach number (compressibility) or jet temperature effect on the scaling of this intrinsic distribution. The essential improvement over the Lighthill analysis is the incorporation of mean-flow shrouding effects on the radiation of the convected quadrupoles. It is perhaps no exaggeration to claim that, with the incorporation of such a shrouding effect, the problem of scaling jet noise with regard to the jet velocity, jet temperature, jet size and the angle from the jet axis appears to be completely resolved. (The ‘scaling’ principle cannot of course be very simply expressed and in fact needs calculations of the sort contained in the present paper to implement it.)

The linearized theory of axisymmetric hypersonic source flow past a slender pointed body is treated by using linearized potential theory, assuming that the hypersonic parameter is less than unity. The governing equation in the transformed co-ordinates with the ratio of specific heats γ = 1.5 shows a modified form relative to the equation for parallel flow, whereas that with γ = 2 shows essentially the same form. Some numerical calculations for a cone and a cone-cylinder are presented, and the surface pressure distributions obtained for both γ = 1.5 and 2 show that there is a considerable pressure-distribution difference between source flow and parallel flow. The pressure on a cone in hypersonic source flow approaches zero downstream, while, if the cone has a cylindrical afterbody, the pressure recovers to the free-stream value quickly. This suggests that the static pressure probe for supersonic parallel flow is useful also in source flow.

This paper continues the study of part 1 into the area of the noise of heated jets. First, this part of the study discusses how a convected wave equation approach based on Lilley's equation leads to additional dipole and simple source terms associated with the velocity fluctuations due to transverse gradients of the mean density of the flow. Once these source terms have been identified and roughly estimated, we revert to a plug-flow model of the jet flow (where now the jet temperature and jet density differ from the ambient values) to estimate the radiation of these singularities. Several novel physical aspects of hot-jet noise are uncovered by the analysis. Indeed the problem of hot-jet noise is the one where the greatest deviations from Lighthill's ideas on jet noise generation are evident. The results are applied to available data and a very satisfactory measure of agreement is obtained with respect to the various predictions of the theory. Mechanisms for ‘excess’ pure jet noise scaling on M6 and M4 are found to result from the density gradients of the mean flow. The satisfactory agreement with the data suggests a solution of the problem of scaling jet noise with regard to jet temperature effects. The ability to predict correctly the data also suggests that the jet temperature has very little effect on the turbulence source spectrum generating jet noise at least for jet exit velocities up to about 1·5 times the atmospheric speed of sound.

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.