This work was motivated by recent experimental results on the spectra of fluctuating temperature gradients in a heated turbulent boundary layer obtained by Sreenivasan, Danh & Antonia. Standard techniques of turbulence theory are used herein to derive expressions relating the individual one-dimensional spectra of each of the three components of the spatial gradient ∂θ/∂xi in a locally isotropic turbulent scalar field. The results of the isotropic theory explain all of the new observed features of the temperature-gradient spectra. The spectra of ∂θ/∂y and ∂θ/∂z decrease monotonically with increasing wavenumber, in contrast to the well-known behaviour of the spectrum of ∂θ/∂x, which reaches a maximum value at roughly one-tenth the Kolmogorov wavenumber. The spectra of ∂θ/∂y and ∂θ/∂z are relatively rich in low frequency energy and relatively poor in high frequency energy compared with the spectrum of ∂θ/∂x. The absolute magnitudes of the spectra of ∂θ/∂y and ∂θ/∂z calculated from the spectrum of ∂θ/∂x using the isotropic relations are in generally good agreement with the corresponding measured spectra for a large range of wavenumbers, indicating second-order spectral local isotropy of the fine-scale scalar structure for sufficiently large wavenumbers. The form of the spectra of ∂θ/∂y and ∂θ/∂z in the inertial subrange is derived analytically.

An integral solution is described for flow of an electrically conducting fluid in a plane channel in a magnetic field which is aligned with the direction of the mean flow. It is shown that the presence of the magnetic field retards the development of the velocity profile by producing Lorentz forces which oppose the movement of fluid from the viscous wall region to the core. Solutions are presented for the entry length as a function of the magnetic interaction parameter. Solutions are also given for the dependence of the frictional component of the pressure drop on the magnetic field strength. The transverse pressure gradient produced by Lorentz forces is discussed for a typical case.

Disturbances in a two-dimensional jet of a viscous incompressible fluid are examined for the case where the jet has a parabolic velocity distribution at the nozzle mouth. Partial differential equations for a finite amplitude disturbance are solved by use of the finite-difference approximation, in which case the jet is analogous to that excited externally. Numerical calculations for various disturbance amplitudes clarify the nonlinearity of the solution. Moreover, the behaviour of the finite disturbance is compared with the behaviour of an infinitesimal disturbance determined from a linearized stability theory. Streaklines calculated from the finite amplitude solution indicate ‘rolling-up’. The computations are carried out over the range of jet Reynolds numbers 500-2000.

Irrotational flow of two-dimensional jets from a channel is treated without direct use of a logarithmic hodograph plane. An analytical approach is introduced for solving the general problem of two jets issuing from a channel with three end plates. Numerical values of the contraction coefficient and the angle of jet deflexion are obtained for the special case where the two jets are located symmetrically and all the end plates are in line. Limiting cases of the resulting single-jet problem are the symmetric and asymmetric configurations solved by von Mises. Results for the asymmetric case improve upon the theoretical values reported by von Mises, and compare favourably with existing experimental data.

The effects of substantial temperature and hence density changes on a low-speed ‘incompressible’ flow can be modelled by adopting Charles’ Law as one of the equations of state. It is found that planar radial inflows or outflows constitute a group of solutions which are self-similar for arbitrary temperature variations. When the temperature is written as a separable function of radius and polar angle, ordinary differential equations result. Permissible solutions include some with discontinuities in the temperature gradient across a radial line (streamline); this is a rough model of a diffusion flame and it is used to illustrate some features of a variable-density flow in a channel with radial walls in the presence of such a ‘flame’.

Exact analytical solutions are given for the situation in which temperature increases linearly with radius; no boundary layers appear for either outflow or inflow. Approximate analytical solutions are presented for the case of a relatively rapid inflow with temperature independent of the radius; a velocity boundary layer exists at the walls and in the neigh bourhood of the ‘flame’, although the latter is of small velocity amplitude.

This paper describes measurements of noise from two-phase flow over hydrofoils. The experiments were performed in a variable-pressure water tunnel which was acoustically calibrated so that sound power levels could be deduced from the sound measurements. It is partially reverberant in the frequency range of interest.

Cavitation was generated on a hydrofoil in the presence of either a separated laminar boundary layer or a fully turbulent attached boundary layer. The turbulent boundary layer was formed downstream of a trip which was positioned near the leading edge. High-speed photographs show the patterns of cavitation which were obtained in each case. The noise is shown to depend on the type of cavitation produced; and for each type, the dependence on speed and cavitation index has been determined. Dimensionless spectral densities of the sound are shown for each type of flow.

The propagation of internal Alfvén-ácoustic-gravity waves in a compressible, stratified, inviscid, perfectly conducting, isothermal atmosphere in the presence of a horizontal magnetic field is investigated by considering both the horizontal and the vertical component of the group velocity. The vertical component of the group velocity is important because it determines the speed at which energy travels upwards and becomes available for heating the upper regions. The regions of propagation and no propagation of waves are delineated for different magnetic Mach numbers, in a refractive-index domain. The horizontal and vertical group velocities are compared with the corresponding phase velocity of the wave motion. It is found that the horizontal group velocity of the internal waves is always less than the horizontal phase velocity for small magnetic fields and vice versa for large magnetic fields, whereas the vertical group velocity is always opposite in direction to the vertical phase velocity for small magnetic fields and vice versa for large magnetic fields. We have also drawn the reflexion condition in a wave-number-frequency domain for different Mach numbers.

The paper presents a new analytical and experimental study of transonic flow around spheres. The results of the analytical study, which employs the method of orthogonal collocation for simultaneous solution of the momentum equations, the equation of continuity and the energy equation, are compared with hitherto unpublished measurements obtained on spheres of various sizes (1·02, 2·54 and 3·81 cm in diameter) in air, in dry steam and in wet steam with free-stream Mach numbers in the transonic range (0·58 < M∞ < 0·97). The relationship θ h = 91·78 + 8·59 M∞ between the attached-shock angle and the free-stream Mach number was obtained by fitting the theoretical pressure distributions to the experimental ones.

The problem of small heave and pitch motions of a slender ship in shallow water including the effect of forward speed is analysed using the method of matched asymptotic expansions. Formulae valid to first order in slenderness are given for the added-mass and damping coefficients in terms of the frequency and subcritical Froude number.

When a contaminant molecule is released in a laminar flow in a straight tube its motion differs from that of the fluid particle with which it initially coincided because of its random motion, whose intensity is measured by the molecular diffusivity κ. For T = kt/a2 ≤ 0·25, where t is the time after release and a is a length characteristic of the cross-section, the statistics of its motion can be determined in the way described by Taylor (1953), Aris (1956) and Chatwin (1970). However, in many applications, including blood flow, the values of T which are attained are much smaller, and the purpose of this paper is to present first approximations for T [Gt ] 1 to some of the statistics measuring the deviations between the motion of the molecule and that of the fluid particle with which it initially coincided. To obtain these a technique due to Saffman (1960) is used for molecules released well away from the tube wall, and an extension of the same technique is used for molecules released near the tube wall. It is shown how the results can be used to describe the initial stages of dispersion of a cloud of contaminant molecules distributed arbitrarily over the cross-section in any laminar flow, steady or unsteady. Comparisons with some exact results for steady Poiseuille flow in a circular tube confirm that the approximations are formally correct, but show that in certain cases their practical use is limited to very small values of T, because of the high coefficients of the first two terms neglected in deriving the approximation.

Flow of a homogeneous inviscid fluid down a rotating channel of slowly varying crosssection is considered, with particular reference to conditions under which the flow is ‘hydraulically controlled’. This problem is a member of a general class of problems of which gas flow through a nozzle and flow over a broad-crested weir are examples (Binnie 1949). A general discussion of such problems gives the means for determining the position of the control section (which is generally flow dependent) and shows that at this position there always exist long-wave disturbances with zero phase speed (i.e. disturbances are always ‘critical’ at the control section). The general theory is applied to the rotating-channel problem for the case of uniform potential vorticity. For this problem, three parameters are needed to specify the upstream flow, and the control theory gives a relationship between these parameters which depends on the geometry of the channel.

Investigations have been carried out of some aspects of the fine-scale structure of turbulence in grid flows, in boundary layers in a zero pressure gradient and in a boundary layer in a strong favourable pressure gradient leading to relaminarization. Using a narrow-band filter with suitable mid-band frequencies, the properties of the fine-scale structure (appearing as high frequency pulses in the filtered signal) were analysed using the variable discriminator level technique employed earlier by Rao, Narasimha & Badri Narayanan (1971). It was found that, irrespective of the type of flow, the characteristic pulse frequency (say Np) defined by Rao et al. was about 0·6 times the frequency of the zero crossings.

It was also found that, over the small range of Reynolds numbers tested, the ratio of the width of the fine-scale regions to the Kolmogorov scale increased linearly with Reynolds number in grid turbulence as well as in flat-plate boundarylayer flow. Nearly lognormal distributions were exhibited by this ratio as well as by the interval between successive zero crossings.

The values of Np and of the zero-crossing rate were found to be nearly constant across the boundary layer, except towards its outer edge and very near the wall. In the zero-pressure-gradient boundary-layer flow, very near the wall the high frequency pulses were found to occur mostly when the longitudinal velocity fluctuation u was positive (i.e. above the mean), whereas in the outer part of the boundary layer the pulses more often occurred when u was negative. During acceleration this correlation between the fine-scale motion and the sign of u was less marked.

A fluid-mechanical model is developed for representing the mechanism of propulsion of a finite ciliated micro-organism having a prolate-spheroidal shape. The basic concept is the representation of the micro-organism by a prolate-spheroidal control surface upon which certain boundary conditions on the tangential and normal fluid velocities are prescribed. Expressions are obtained for the velocity of propulsion, the rate of energy dissipation in the fluid exterior to the cilia layer, and the stream function of the motion. The effect of the shape of the organism upon its locomotion is explored. Experimental streak photographs of the flow around both freely swimming and inert sedimenting Paramecia are presented and good agreement with the theoretical prediction of the streamlines is found.

The development of the flow field of a jet emanating from a point source of momentum in an infinite incompressible fluid of density σ is considered. The flow field is assumed to be due to the application of a constant force F0 at the origin. The problem is formulated in terms of the dimensionless variable λ = (vt)½/r, where v is the kinematic viscosity of the fluid, t the time from the application of the force and r the distance from the origin. At a station r the flow field is dipolar, with the dipole axis in the direction of F0, for all t satisfying the inequalities vt Lt; r2 and F0t2 [Lt ] 4πρr4. Also, at a given time t the streamlines of the developing flow field in a section through the axis of symmetry of the problem form closed loops about a stagnation point. If this occurs at λ = λm, the stagnation point propagates to infinity, along a straight line emanating from the origin, with speed v½/2λmt½, where λm = λm(F0) decreases as F0 increases. The larger F0 is the faster the steady state is established.

In the mixing of a jet with a cross-stream, it is found that in the near field, defined as the region of the flow from the jet exit to a distance of a few diameters downstream of this exit, a considerable amount of dynamical adjustment takes place. This near-field region characterizes the subsequent behaviour and development of the jet, its wake and the cross-stream in the vicinity of this mixing region. The rapid evolution of the flow gives rise to a pair of bound vortices attached to the lee side of the jet boundary, to fast development of the turbulent and mean vorticity, to a vortex-shedding system, and to the largest rates of entrainment of cross-stream flow into the jet. Furthermore, it is found that the geometrical configuration of the boundaries at the jet exit plays an important role in the mixing and development processes.

An intrinsic method is proposed for the delineation of the flow boundaries between the jet and the cross-stream. Calculations of mass, momentum and vorticity fluxes have been made. The vorticity flux gives evidence of the rapid stretching and tilting of the vorticity vector field in the near-field region.

In this paper we develop a separation of variables theory for solving problems of Stokes flow in wedge-shaped trenches bounded by radial lines and concentric circles centred at the vertex of the wedge. The theory leads to a set of Stokes flow eigenfunctions which in the full wedge reduce to the corner eigenfunctions studied by Dean & Montagnon (1949) and Moffatt (1964). Asymptotic formulae for the distribution of eigenvalues are derived, an adjoint system is defined and is used to develop an algorithm for the computation of the coefficients in an eigenfunction expansion of edge data prescribed on the circular boundaries. To illustrate the algorithm we find the motion and the shape of the free surface in a wedge-shaped cavity heated from its side.

Laboratory experiments on swirling flows through tubes often exhibit a phenomenon called vortex breakdown, in which a bubble of reversed flow forms on the axis of swirl. Mager has identified breakdown with a discontinuity in solutions of the quasicylindrical flow equations. In this study we define a tornado-like vortex as one for which the axial velocity falls to zero for sufficiently large radius, and seek to clarify the conditions under which the solution of the quasi-cylindrical flow equations can be continued indefinitely or breaks down at a finite height. Vortex breakdown occurs as a dynamical process. Hence latent-heat effects, though doubtless important to the overall structure and maintenance of the tornado, are neglected here on the scale of the breakdown process. The results show that breakdown occurs when the effective axial momentum flux (flow force) is less than a critical value; for higher values of the flow force, the solution continues indefinitely, with Long's (1962) similarity solution as the terminal state. When applied to the conditions of the 1957 Dallas tornado, the computed breakdown location is in agreement with Hoecker's analysis of the observations.

The effects of inertia on the upstream-growing boundary layer over a finite horizontal flat plate of length b moving uniformly with speed U0 in a linearly stratified \[ [(dp/dy)_{-\infty}- = - \rho_0\beta)], \] viscous, non-diffusive fluid under the Boussinesq approximation are studied. The nonlinear inertia terms are linearized by the Oseen approximation, but no boundary-layer approximation is required. The flow is governed by two parameters, namely the internal Froude number Fr[= U0/(βgb2)½] and a parameter L3[= βgb3/U0ν], where $L^{\frac{3}{4}}$ is proportional to the ratio of boundary-layer thickness to plate length for the case Fr = 0. Large values of L3 and Fr2 = 0 correspond to the case of an upstream boundary layer. By increasing the Froude number gradually, a transition occurs from an upstream boundary layer accompanied by an upstream wake to a downstream boundary layer with a downstream wake. The upstream boundary layer and wake are characterized by a balance of viscous and buoyancy forces, whereas the downstream boundary layer and wake are characterized by a balance of inertia and viscous forces. In the so-called critical-boundary-layer case, Fr4L3 = O(1), inertia, viscous and buoyancy forces are all important and this boundary layer is accompanied by both upstream and downstream wakes. Complete transition occurs when Fr4L3 increases from 10·0 to 1000·0. The drag on the plate is also calculated.

Mahony & Smith (1972) put forward a model to explain the phenomenon of energy transfer between nearly resonant oscillations at greatly differing frequencies. However, their model of ‘spatial resonance’ is restricted to situations where the geometry of the system is very simple. The present paper shows how to derive Mahony & Smith's equations in a general manner, and compares the theoretical predictions for a situation with circular symmetry with existing experimental results (Huntley 1972). In addition, it suggests a simple method for evaluating the resonance frequencies when a liquid-filled beaker is vibrated in one of its bell modes.

A numerical solution of the two-dimensional compressible laminar boundary-layer equations up to the point of separation is presented. For a particular mainstream velocity distribution it is necessary to specify the surface temperature (or the heat flux across the surface), the suction velocity, the free-stream Mach number and the viscosity-temperature relationship for a solution to be generated. The effect upon the position of separation of a hot or cold wall and of varying the free-stream Mach number is given special emphasis. The variations of the skin friction, heat transfer and various boundary-layer thicknesses for compressible flow past a circular cylinder and for flow with a linearly retarded mainstream were found. The behaviour of the solutions close to separation is investigated. Known functions which model the skin friction and heat transfer are introduced and are used to match the numerical solutions with the Buckmaster (1970) expansions.