An integral method is used to investigate the interaction between a two-dimensional, single frequency finite amplitude disturbance in a laminar, incompressible wake behind a flat plate at zero incidence. The mean flow is assumed to be a non-parallel flow characterized by a few shape parameters. Distribution of the fluctuation across the wake is obtained as functions of those mean flow parameters by solving the inviscid Rayleigh equation using the local mean flow. The variations of the fluctuation amplitude and of the shape parameters for the mean flow are then obtained by solving a set of ordinary differential equations derived from the momentum and energy integral equations. The interaction between the mean flow and the fluctuation through Reynolds stresses plays an important role in the present formulation, and the theoretical results show good agreement with the measurements of Sato & Kuriki (1961).

Flow of an incompressible inviscid fluid past a sphere is considered, where the flow upstream consists of a slight shear flow superimposed on a uniform stream. Secondary vorticity, produced by deformation of vortex elements as they are carried past the sphere, is determined by a method due to Lighthill (1956b). Components of vorticity are calculated from a drift function for which expressions were previously available only in part of the flow field. For the region in which no expansion is valid an exact integral expression is obtained to replace the rough numerical approximation used by Lighthill (1956b). The velocity distribution over the upstream part of the sphere is determined numerically using a Biot–-Savart law. These results are required for the calibration of certain forms of Pitot tubes.

Any slight tilt or tide on the fluid in a rotating cylinder causes periodic motions, whose radiation pressures in the viscous boundary layers force mean differential rotations of the fluid, which are found numerically. At certain fluid depths, even very small tilts can cause shears large enough for perturbations to overcome Ekman friction, causing turbulence. An experiment confirms the theory.

In the present paper the stability of plane Couette flow is studied. The stream function and distribution of vorticity and the Reynolds stresses for the linearized solution are computed. The stability effect of the non-linear terms are also discussed and it is found that for small amplitudes, the non-linear terms are destabilizing. A neutral curve determining the necessary amplitude in order to get instability, is found. The convergence of the expansion in the latter case is, however, somewhat uncertain and the result should therefore only be considered as a first, rough approximation.

Results of linear stability calculations and post-stability experimental observations are reported for horizontal fluid layers with upward heat flux bounded below by a stably stratified fluid. Stability calculations were done for several families of continuous and discontinuous temperature distributions, and it was found that as a rule the flow originating in the unstable layer penetrates into the stably stratified region, resulting in increased critical cell size and correspondingly decreased critical Rayleigh number. A notable exception to this occurs for an unstable layer with a linear temperature distribution adjacent to a stable layer of very high stable density gradient. In this case energy pumped from the unstable to the stable region is sufficient to raise the critical Rayleigh number above that of a solid boundary. It is also found that, for density distributions with a more gradual transition between the stable and the unstable regions, the effect of increased cell size upon the critical Rayleigh number is sometimes masked by effects of curvature in the density profile of the unstable region, which tends to increase the critical Rayleigh number. The inadequacy of the usual definition of Rayleigh number to characterize the stability of such complex systems is discussed. Experimentally, such a temperature distribution was produced by radiant energy from above as it was absorbed by the top few centimetres of the fluid. Within an uncertainty of ± 20%, it was found that the critical experimental Rayleigh number agreed with neutral stability calculations. The supercritical convective motion consisted of vertical jets of cool surface fluid which plunged downward into the interior of the fluid. The jets were not arranged in an orderly lattice but were in a constant state of change, each jet having a tendency to merge with a close neighbour. The net loss of jets due to merging was balanced by new jets spontaneously appearing. As Rayleigh number was increased, the mean number of jets and the intermittancy increased proportionally. Temperature scans taken with a movable probe showed that cool surface fluid plunging downward in the jets was confined to a fairly restricted region, the surrounding fluid being quite isothermal.

The method of matched asymptotic expansions is employed to calculate the mass transport velocity due to small amplitude oscillatory waves propagating in conditions of density and viscosity discontinuities. For progressive waves in a two-layer system, it is found that the velocity at the interface is in the direction of wave propagation; when the uppermost surface is free, the velocity there is in the direction opposite to that at the interface. If the difference in the densities is small, the calculated transport velocity associated with an internal wave can be of more importance than that associated with the surface wave as obtained from the work of Longuet-Higgins (1953).

Solutions are obtained for the self-similar form of the incompressible boundary-layer equations for all four second-order contributors, i.e. vorticity interaction, displacement speed, longitudinal and transverse curvature. These results are found to contain all previous self-similar solutions as members of the much larger family of solutions presented here. Numerical solutions are presented for a large number of cases, and several closed form solutions, which may have special significance for the separation problem, are also discussed.

Convective instability of a ferromagnetic fluid is predicted for a fluid layer heated from below in the presence of a uniform vertical magnetic field. Convection is caused by a spatial variation in magnetization which is induced when the magnetization of the fluid is a function of temperature and a temperature gradient is established across the layer. A linearized convective instability analysis predicts the critical temperature gradient when only the magnetic mechanism is important, as well as when both the magnetic and buoyancy mechanisms are operative. The magnetic mechanism predominates over the buoyancy mechanism in fluid layers about 1 mm thick. For a fluid layer contained between two free boundaries which are constrained flat, the exact solution is derived for some parameter values and oscillatory instability cannot occur. For rigid boundaries, approximate solutions for stationary instability are derived by the Galerkin method for a wide range of parameter values. It is shown that in this case the Galerkin method yields an eigenvalue which is stationary to small changes in the trial functions, because the Galerkin method is equivalent to an adjoint variational principle.

The dynamical effects of a sudden increase of as much as 100°C in boundary temperature upon fully turbulent boundary layers at low Reynolds numbers in air have been investigated in a wind tunnel. A section of the floor or the roof of the tunnel could be heated, so that the rate of working of gravitational forces on the turbulence could be made to represent either a gain or a loss of turbulent mechanical energy. Techniques of hot-wire anemometry were employed which enabled the instantaneous temperature and the instantaneous velocity to be measured simultaneously at a point in the non-homogeneous turbulent flow field.

In the case of a strong discontinuity in the floor temperature, a fine-scale convective structure developed from the highly unstable interface between the heated air just above the surface and the turbulent boundary layer; and the motion in this region was sufficiently vigorous that the mean pressure in the vicinity of the floor was reduced and a local wall jet was generated. The deduced pressure distribution is regarded as evidence of coupling between the free and forced convection modes which may lead to a series of local wall jets downstream of the discontinuity.

In the case of a strong discontinuity in the roof temperature, the interface between the heated air and the turbulent boundary layer was stable; and the boundary-layer turbulence, acting to spread this stable gradient over the vertical extent of the boundary layer, was required to do work against the gravitational field. A rate of working against gravity which was an order of magnitude less than the rate of supply of turbulent energy from the mean shear proved sufficient to suppress the turbulence in a very short time.

The stability of parallel shear flow of an inviscid compressible fluid is investigated by a linear analysis. The extension of the Rayleigh stability criterion and Howard's semi-circle theorem to compressible flows, obtained by Lees & Lin (1946) and Eckart (1963) respectively, are each rederived by a different approach. It is then shown that a subsonic neutral solution of the stability equation may be found when the basic flow is represented by the hyperbolic-tangent velocity profile. With the aid of this solution, the unstable eigenvalues, eigenfunctions and Reynolds stress are determined by numerical methods. A brief discussion of the results follows.

The decay of perturbations in a radiating gas is analyzed by first carrying out complete solutions for the decay of initial sinusoidal perturbations in the temperature, gas velocity, and pressure. These sinusoidal perturbations are superposed to yield solutions for the decay of initial ‘step’ temperature profiles consisting of constant initial temperature perturbations inside finite planar, cylindrical and spherical regions, with zero initial temperature perturbations outside. In contrast to the sinusoidal case, which may be described by a single radiation parameter, the decay of the step profile is determined by both the optical depth of the initial profile and the Boltzmann number, which is inversely proportional to the blackbody radiative flux. As the limits of zero and infinite Boltzmann numbers are approached, constant-density and constant-pressure cooling expressions are recovered. For a broad range of intermediate and small Boltzmann numbers the cooling proceeds in time from a constant-density process to a constant-pressure process. This transition is produced by gasdynamic waves generated near the profile edges by the radiative cooling. The temperature near the profile centre may increase during the transition period.

A theory is presented for two-dimensional incompressible potential flow external to a symmetrical bluff body and its wake. The desired flow-separation points are made the critical points of a conformal transformation to a complex plane in which surface sources in the wake create stagnation conditions at the critical points. The stagnation streamlines then transform to tangential separation streamlines in the physical plane, with separation at the desired pressure. The position and strength of the sources are determined by the requirements of separation position and pressure coefficient. The flow inside the separation streamlines is ignored and base pressure is assumed constant at the separation value. Features of the theoretical model include a finite wake width, a pressure distribution on the separation streamlines decreasing asymptotically towards the free stream value at infinity and a simple analytic expression for the pressure distribution on the body. Comparisons of the theory with experimental data and with other theories are presented for the normal plate, the circular cylinder, the 90° wedge, and the elliptical cylinder. Although simpler to apply than the other theories, the present theory produces at least as good agreement with the experimental data.

This paper pertains to the steady condensation on to, or evaporation from, a liquid droplet suspended in a mixture of its vapour and an inert gas. The treatment is from a kinetic theory viewpoint. The Maxwell moment method is used with Lees’ two stream Maxwellian representations for the distribution functions, to obtain a closed form result valid through the whole range from free molecule to continuum. In the diffusion control limit the formula reduces to approximate results obtained by Maxwell, Fuchs and others. In the limit where no inert gas is present the formula reduces to a result obtained earlier by the author. The formulae presented here for the mass and energy flux can now be used to calculate the growth rate of very small droplets, under a wide range of conditions.

The dynamical behaviour of a system of parallel line vortices in an inviscid fluid is studied numerically. The initial configuration of the system is assumed to be such that the points of intersection of the line vortices with a plane normal to the vorticity form a regular polygon. The numerical experiments show that the vortex polygon is rearranged due to non-linear interactions among the line vortices in such a way as to produce a more or less uniform distribution of vortices inside the fluid with an approximately constant mean separation. The average angular velocity of the rotation of the vortex lines about the instantaneous centroid of the vortex system remains approximately constant. These results agree with the conjecture of Raja Gopal (1964). The results may prove to be of some value in a macroscopic model of liquid helium based on hydrodynamical principles.

The train of capillary waves which appears on the forward face of a steep gravity wave is discussed by considering the capillary waves as stationary waves on a slowly varying running stream, using the non-linear capillary wave solution of Crapper (1957) and the recent method due to Whitham (1965a, b). Then energy input and damping are introduced into an energy equation which becomes a non-linear ordinary differential equation for the capillary wave steepness. Numerical solutions for various gravity wavelengths and steepnesses are discussed, and some light is thrown on the problem of the breaking of gravity waves.

The stability of a horizontal layer of fluid heated from above or below is examined for the case of a time-dependent buoyancy force which is generated by shaking the fluid layer, thus causing a sinusoidal modulation of the gravitational field. A linearized stability analysis is performed to show that gravity modulation can significantly affect the stability limits of the system. In this analysis, much emphasis is placed on qualitative results obtained by an approximate solution, which permits a rather complete stability analysis. A useful mechanical analogy is developed by considering the effects of gravity modulation on a simple pendulum. Finally, some effects of finite amplitude flows are considered and discussed.

The trailing edge region of a finite flat plate in laminar, incompressible flow is examined for the limit of high Reynolds numbers.

It is shown that the trailing edge region is an elliptic region of O(R−¾) and therefore a correct mathematical description must be based upon the full Navier–Stokes equations.

The ‘method of series truncation’ is used to reduce the full Navier–Stokes equations, written in parabolic co-ordinates, to an infinite set of non-linear, coupled, ordinary differential equations. Two sets of asymptotic boundary conditions, called simplified and exact boundary conditions, are determined by matching the Navier–Stokes region downstream with Goldstein's near wake solution.

By numerical integration the solutions for the first and second truncations are obtained for both sets of asymptotic boundary conditions. The results confirm that the size of the trailing edge region is of O(R−¾).

Is it possible to express the problem of calculating turbulent flame speeds as an eigenvalue problem that is analogous to the laminar flame speed problem? It is argued for grid turbulence that the answer is affirmative, and some benefits of pursuing such a calculation are exploited for the limiting case of a first-order reaction with vanishingly small heat release. The streamwise turbulent transport of reactant occupies a central role in the analysis. The equation governing the ensemble average of this quantity assumes different simplified forms in the limits of small-scale and large-scale turbulence. The criterion which is obtained for separating the small-scale and large-scale régimes differs from that of Damköhler and also from that of Kovasznay and Klimov. In the small-scale régime, turbulence produces a spatially varying diffusivity, the form of which can be ascertained only through an investigation of non-linear equations describing the statistical dynamics of production and decay of the velocity–concentration correlation. In the large-scale régime, which is of greater practical importance, the ensemble average of the streamwise turbulent reactant flux satisfies a linear ordinary differential equation whose solution for the growth and decay of the flux contains effects resembling wrinkling of the laminar flame, increasing of the effective diffusivity and augmentation of the effective reaction rate. An exact solution to the linear eigenvalue problem which arises in the large-scale limit reveals that turbulence enhances mean reactant consumption in the upstream portion of the flame and retards reactant consumption downstream. Formulas are given for the increase in flame speed and the increase in flame thickness that are produced by turbulence in the large-scale limit. Since the equations are relatively tractable in the large-scale limit, it is suggested that further study of these equations may yield improved descriptions of realistic turbulent flames.

A study is described of the forced inertial oscillations appearing in an axially rotating completely filled circular cylinder with plane ends. Excitation is provided by causing the top end to rotate about an axis inclined slightly to the rotation axis. Experiments demonstrate the presence of numerous low mode resonances in a densely spaced range of ratios of net cylinder height to radius in close conformance with linear inviscid theory. Where geometry permits simple corner reflexion, characteristic surfaces are revealed which confirm in part the theoretical predictions concerning their scale and form.

Detailed measurements are given of the amplitude at one point within the cylinder for the condition in which the disturbance frequency equals the rotation frequency. Amplitude column height spectra are compared with theoretical estimates, and the evolution of amplitude for the simplest mode of resonant oscillation is studied. A non-linear theory based on the integral energy of large amplitude oscillation is derived whose broad features are in fair quantitative and qualitative agreement with these observations.

Some investigation is made of the phenomenon of resonant collapse, in which larger amplitude resonant oscillations, after persisting in an apparently laminar form, degenerate abruptly into a state of agitation and disorder from which they do not recover. It is found that the time for emergence of this collapse after the introduction of the forcing disturbance has a close correspondence with the theoretical period of one ‘evolutionary’ cycle of momentum exchange between the main motion and the secondary oscillation.

Recent theoretical results for the turbulent flow of polymer solutions in round tubes have been extended to deduce the similarity laws for the boundary-layer flow of drag-reducing polymer solutions. The analysis shows directly how the drag reduction depends on the elastic properties of the fluid and thereby defines the levels of elasticity necessary to achieve significant reductions in drag.

Calculations employing available physical property measurements of highly elastic (0·1%) and moderately elastic (0·01%) polymer solutions indicate that, for boundary layers on large objects, drag reduction may not occur at polymer concentrations that are economically attractive. For example, at a Reynolds number of 109 the reduction in drag is predicted to be 60% and 10% for the concentrated and dilute polymer solutions respectively. Some savings in polymer however, may be realized by special injection techniques or fluid systems with specially tailored properties.