Calculations are presented of the development of the turbulent boundary layer on a rotating cylinder with its axis along the stream, and the results are compared with measurements made by Parr. With the choice of a suitable co-ordinate system, the boundary layer downstream of the nose of the cylinder approximates closely to a condition of two-dimensionality, and a simple integral method of solution can be applied. The only evidence of three-dimensionality lies in the destabilizing effect of rotation on the turbulence structure of the layer and an analysis of this instability has been made which relates changes in mixing length and entrainment to an instability parameter in the form of a Richardson number.

In spite of the changes in shear stress distribution and entrainment brought about by rotation, mean velocity profiles and skin friction values are found to be related to H and Rθ in the same way as for the stationary flat plate.

This article describes experimental work on the mixed convection régime with flow normal to electrically heated cylinders. The forcing velocities used were in the range 0·0085–3 ft./sec (i.e. 10−2 < Ref < 45) and temperature differences in the range 30°C to 200°C (i.e. 10−3 < Ra < 10) were covered.

Correlations are proposed for the forced convection and natural convection conditions. A correlation is also developed for the combined forced and natural convection region by a vectorial addition of the flow parameters, which gives good agreement with the experiments except over a limited range in the contraflow régime.

Experiments are reported in which the error of a sharp-edged, circular static pressure hole normal to the boundary of a moving fluid is deduced from measurements made with pressure transducers, the diaphragms of which were set flush with the boundary surface. An error curve is presented which covers the range of hole Reynolds number up to a value of 2000.

The steady-state solution of the non-linear equation \[ h_t + hh_x + h_{xxx} = \delta h_{xx} \] with both damping and dispersion is examined in the phase plane. For small damping an averaging technique is used to obtain an oscillatory asymptotic solution. This solution becomes invalid as the period of the oscillation approaches infinity, and is matched to a straightforward expansion solution. The results obtained are compared with a numerical integration of the equation.

This paper describes some experiments in rotating flows in which solitary waves were observed.

In one set of experiments the waves were generated on a swirling flow whose circumferential velocity distribution resembled that of the Rankine combined vortex. This flow was established by stirring the liquid in a large cylindrical container, in much the same way as one stirs a cup of tea, and it was often found at the cessation of the stirring that a wave had been generated. This wave propagated along the vortex core and was reflected at the bottom of the container and at the free surface of the liquid and displayed the remarkable permanence characteristic of solitary waves. It appears that, to a first approximation, the speed of the waves may be calculated simply from the depression of the free surface of the liquid at the centre of the vortex. These waves are the rotating-fluid counterpart to the solitary waves in fluids of great depth recently discussed by Benjamin (1967b) and by Davis & Acrivos (1967).

In a second set of experiments, solitary waves were generated in a long cylindrical tube and are analogous to the familiar solitary wave of open-channel flows. The theory indicates that these waves are possible in any swirling flow in which the angular velocity is distributed non-uniformly. Thus, a long liquid-filled tube was started rotating about its axis with a uniform angular velocity, and waves were generated before the fluid had reached a state of uniform rotation. Using the known velocity distribution for a tube of infinite length, comparisons have been made between the observed wave forms and the theoretical calculations of Benjamin (1967a). There is good agreement between the observed wave forms and the theoretical predictions.

The existence of elementary, exponential solutions of the linear Boltzmann equation for gases is examined. Using the hard-sphere model of scattering, it is found that, in problems involving velocity perturbations, there are no discrete non-zero eigenvalues. Thus the relaxation to the asymptotic distribution is non-exponential and is described by the continuum eigenfunctions. For temperature perturbations, however, we find two non-zero discrete eigenvalues whose values are ±0·975 in units of the minimum scattering cross-section. Relaxation to the asymptotic distribution is therefore exponential, although still very rapid.

The conclusions stated above are based upon a truncation of the scattering kernel and a subsequent numerical solution of the resulting integral equations.

An energy principle is presented which gives necessary and sufficient conditions for exponential stability for a large class of dissipative systems. The maximal growth rate Ω of an unstable system is shown to be the least upper bound of a certain functional, giving a variational expression for Ω. These results are used to discuss the gravitational stability of incompressible viscous fluids and resistive magnetofluids in arbitrary geometries.

Measurements of the velocity distribution close to the bed have been made under laminar flow conditions in a wave tank. The classical solution for the velocity distribution was found to be valid when the bed was smooth, but considerable deviations between theory and experiment were observed with beds of sand. It is suggested that these deviations were caused by vortex formation around the grains of sand. The similarity between the velocity profiles obtained in these tests and those reported by other writers under supposedly turbulent conditions suggests that even at high Reynolds numbers vortex formation may continue to be the dominant effect in oscillatory boundary layers of this sort.

The study of natural convection in a rectangular cavity whose side walls are maintained at different fixed temperatures can be regarded as one of the classical problems of thermal convection (see Batchelor 1954; Elder 1965 and Gill 1966). One aspect of this problem is the question of stability of the laminar flow solution for given values of the three governing parameters, the Rayleigh number A = γgΔTL3/κν, the aspect ratio h = H/L and the Prandtl number σ = ν/κ. Here H is the height and L the width of the cavity, T is the temperature difference between the two walls and g the acceleration due to gravity. The fluid filling the container has coefficient of expansion γ, thermal diffusivity κ and kinematic viscosity ν. Instead of studying the stability of the exact laminar flow solution, which is not a parallel flow, the stability of the following solution of the governing eauations is examined instead:

The Stokes creeping flow, induced by the passage of a uniform current parallel to the axis of a stationary non-conducting ellipsoid of revolution in an incompressible viscous fluid occupying, apart from the ellipsoidal region, the whole space, is investigated. The magnetic field, which is due to the distortion of the uniform current by the ellipsoid, is zero all over the surface of the ellipsoid. The induced flow field is symmetric with respect to the axis, and also with respect to a plane through the centre perpendicular to the axis of the ellipsoid. The case of a non-conducting circular disk, with its plane perpendicular to the direction of the undisturbed current, is deduced from that of a planetary ellipsoid.

The boundary layers associated with gravity waves in a fluid with a linear variation of density are discussed in order to examine the steady Eulerian velocities induced by the Reynolds stress. For the case of a standing wave, the induced steady motion is shown to decay in an outer boundary layer which represents a balance between buoyancy and diffusion when the wave slope is sufficiently small but when viscous decay effects are even smaller. When the wave slope is larger, it would appear that two outer regions must be considered. Results for progressive waves are discussed only briefly, as they are qualitatively similar to the surface wave case.

A linearized analysis of the two-dimensional double vortex sheet model of a jet shows that inviscid jet instabilities occur over a wide range of frequencies at all jet Mach numbers. No particular frequency for maximum growth rate exists unless finite shear layer thickness effects are considered. It is suggested that the model describes the essential characteristics of a real jet disturbed by long wavelength perturbations. The idea is advanced that the jet flow constitutes a broad band amplifier of high gain. Disturbances can grow rapidly to a size when nonlinear effects bring about significant interaction with the mean flow. By seeding the jet with disturbances of a type that are highly amplified it is argued that gross features of the flow may be affected and that the jet may be rendered less noisy at high Mach number. It is argued that some of these ideas are supported by the observation that a supersonic jet diffuses at an unusually rapid rate when subject to the oscillatory condition known as ‘screech’.

Observations of temperature drop as a function of heat flow in Rayleigh–Bénard convection with curved density profiles show: (1) reversal of slope in the heating curve, (2) oscillations with time, (3) history dependence, and (4) an increase in critical Rayleigh number as the curvature of the density profile is increased. Some of the results are quite similar to the predictions of Busse.

A new method is presented for solution of the Boltzmann equation governing the dynamic behaviour of gases. The essence of the method is idealization of the problem domain into subdomains called finite elements. Then, the Galerkin assumed-mode technique is employed as the basis for discretization of the individual finite elements and also for the assembly of the resulting algebraic models for these finite elements to form an algebraic model for the complete problem. The procedure is cast in a systematic matrix notation that makes evident the broad application potential of the analysis method. An illustrative application is presented for the problem of one-dimensional, linearized Couette flow. Numerical predictions of macroscopic flow velocity and viscous shear stress based upon the subject finite element method are compared with alternative analytical and numerical results. Special attributes of the finite element method are discussed in the context of this example problem. Applications to practical problems governed by generalized forms of the Boltzmann equation are projected on the basis of concepts established herein.

The phenomenon of second harmonic resonance for capillary-gravity waves is reconsidered here by the asymptotic method of multiple time and space scales. The periodic finite amplitude waves of permanent form found by Wilton in 1915 which correspond to this configuration are shown to be no more than a special case of the more general resonant interaction theory, and owe their existence to a critical choice of initial conditions. It is further suggested that the influence of viscous dissipation will render this solution virtually undetectable in a real liquid.

The axisymmetric motion of an inviscid, rotating liquid over a prescribed stream surface, say S, is constructed from assumed values of the velocity and azimuthal vorticity on S. The hypothesis of unseparated flow, which implies continuity of the vorticity on S, is shown to imply that: (a) the azimuthal vorticity and azimuthal circulation (relative to the basic flow) must be simply proportional to the perturbation stream function in the exterior of S; (b) the exterior field exhibits a dipole behaviour far upstream of the body, thereby satisfying Long's hypothesis of no upstream disturbance.

The disturbance induced in the uniform flow of a viscous, rotating liquid by an axial point force −D is studied under the restrictions that the Ekman number, E = 2Ων/U2, be small and that D = O(1/logE) as E → 0. The method of matched asymptotic expansions is invoked to obtain inner and outer (with reference to the dimensionless axial co-ordinate x refered to the length U/(2Ω)) approximations to the solution of the Oseen equations as E → 0. The outer approximation, E → 0 with Ex fixed, is also an outer approximation to the solution of the Navier–Stokes equations. The mass flow across any transverse plane, which is equal to D/U for an oseenlet in a non-rotating flow, vanishes in this approximation. The corresponding inner limit yields a non-uniform, cylindrical flow far upstream of the force in the inviscid limit, E → 0, if and only if D ∝ 1/(log E + const.). This cylindrical flow is a one-term, inner approximation to the solution of the Navier–Stokes equations and suffices to show that separation implies the failure of Long's hypothesis of no upstream influence for inviscid, rotating flow past a finite body. A two-term inner representation of the solution is related to Stewartson's solution of the Oseen equations for a moving source in an inviscid, rotating fluid.

The two studies by Professor Miles (1970a, b) on the motion of a rotating fluid past a body raise the important question of the determinancy of such flows, by theoretical arguments, which it seems worth while making more precise. Suppose we have a fluid which when undisturbed has a uniform velocity U in the direction Ox and a uniform angular velocity Ω about Ox. It is slightly disturbed, the resulting motion having velocity components (u + U, v, Ωr + w) relative to cylindrical polar axes (x, r, θ), centre O and in which r measures distance from Ox, while θ is the azimuthal angle. Assuming that u, v, w are sufficiently small for their squares and products to be neglected, and are independent of θ, the equations governing their behaviour reduce to