The stability of natural convection of a viscous fluid in a vertical slot having isothermal side walls of different temperatures is investigated analytically. Both the conduction and boundary-layer régimes are found to be unstable with respect to stationary disturbances in the form of multicellular secondary flows. Theoretical predictions of the critical Rayleigh number and of the form of the secondary flow are verified by experimental measurements.

Measurements are reported of the fluctuating lift acting on a sphere and the moment acting about the centre of a sphere at supercritical Reynolds numbers (R > 4 × 105). The lift and moment fluctuations are random functions of time which scale with the free-stream dynamic pressure and sphere dimensions. The power spectra of the lift and moment also scale with the above parameters and with the Strouhal number, nd/U. The spectra contain a maximum spectral density at very low frequencies (nd/U < 0·0003) and do not reveal appreciable effects of vortex shedding at discrete frequencies.

Hot wire anemometers were placed near the surface but outside the boundary layer along a great circle in the meridian plane in which the lift was measured. The fluctuating velocity component near the surface on the upstream hemisphere in this meridian plane is highly correlated with the fluctuating lift in the same meridian plane. The correlation betweeen the lift and tangential velocity near the surface suggests that the fluctuating lift is produced by the component of fluctuating bound vorticity about the sphere that is normal to the meridian plane in which the lift force is measured. The fluctuating moment measured about an axis passing through the centre of the sphere and perpendicular to the above meridian plane is almost perfectly correlated with the fluctuating lift (the measured correlation coefficients were 0·99 and 1·00). The fluctuating moment coefficient is very small ($\sqrt{C^2_m}\simeq 5\times 10^{-4}$) compared to the fluctuating lift coefficient ($\sqrt{C^2_L}\simeq 6\times 10^{-2}$). The exceptional correlation between the random lift and moment suggests that the unsteady moment about the sphere centre (which can be produced only by shear stress fluctuations) is caused by the fluctuations of bound vorticity (residing in the boundary layer and wake) that are responsible for the unsteady lift.

Numerical calculations show that the flow of viscous incompressible fluid in a circular pipe is stable to small axisymmetric disturbances at all Reynolds numbers. These calculations are linked with known asymptotic results.

The asymptotic behaviour for large Rayleigh number and Prandtl number of O(1) of two-dimensional convection cells in a fluid between horizontal plates heated from below has been discussed by Pillow (1952) and more recently by Robinson (1967). The flow models derived by Pillow and Robinson differ from each other. The purpose of the present paper is to point out the likelihood of an inconsistency in Robinson's flow model, and to discuss qualitatively a few hitherto unnoticed features of the flow.

The flow under discussion represents an idealization of the bath-tub vortex; distortions of the free surface, finite sink size, and all rigid boundaries have been eliminated from the problem in order to isolate the effect of the non-uniform stretching of vortex lines produced by the sink flow. A boundary-layer type of approximation is made about the axis, which requires that the meridional Reynolds number (N) be large, and since the problem is still intractable, an expansion is made in powers of K = R2/N (where R is the swirl Reynolds number), which measures the strength of the interaction between the swirl and meridional velocity fields. In the limit of zero K the flow is a modified Burgers vortex whose radius decreases to zero at the sink. For non-zero K, the interaction is not restricted to the vortex core, because the presence of the vortex modifies the outer irrotational flow, inducing a radial mass flux into the core, whose dependence on the axial co-ordinate is calculated to the first order in K. The structure of the core is obtained, again to the first order in K, from two co-ordinate expansions, one near the stagnation point on the axis, and the other near the sink, although only the first few terms of the latter can be determined explicitly. It is shown how the methods can be extended not only to higher orders in K, but also to any other narrow viscous vortex in which the vortex lines are stretched non-uniformly away from an internal stagnation point.

In gas-liquid flows a certain sudden change of the flow structure may occur, which can be described as a transition from ‘jet flow’ to ‘froth flow’ accompanied by energy dissipation and pressure build-up. Upstream of this phenomenon the gas is the continuous phase; downstream the liquid is the continuous phase. The phenomenon, which has been called ‘mixing shock’, shows some similarity and also some differences with the plane shock wave in gasdynamics. In the first part of this paper the mixing shock is treated as a one-dimensional macroscopic process. With the aid of the laws of conservation of mass, momentum and energy, expressions are obtained for the pressure and entropy change across the mixing process. In addition the stability of the mixing shock in a cylindrical flow channel is treated. Next, a theory that explains the gas entrainment mechanism in the mixing shock is proposed. As an experimental tool a water-air ejector with the water as a driving medium was used. The experiments confirm the macroscopic and the microscopic theory. In the last section of this paper theoretical and experimental evidence is combined to construct a model of the processes that play a role in the shock.

It is shown that Yih's exact solution for the non-diffusive flow of a non-homogeneous fluid into a sink in a confined porous medium is equivalent to a class of diffusive flows with isopycnic lateral boundaries.

Internal gravity waves in a stratified atmosphere of unbounded inviscid incompressible fluid are considered. A class of non-linear waves is found in the Boussinesq approximation, whereby the inertial variation of density is neglected but the buoyancy is not, by reduction of the equations of motion to ordinary differential equations. To find similar non-linear waves when the atmosphere is slightly stratified, i.e. when the inertial variation of density is small but not entirely negligible, the equations of motion are first expressed in Lagrangian variables and derived from a variational principle. The Lagrangian variables are transformed to systematize the Boussinesq approximation. Finally, the properties of the non-linear waves in a slightly stratified atmosphere are found by Whitham's method of averaging. In terms of the transformed Lagrangian variables, strong non-linearity affects the linearized solution only by adding to the pressure a term proportional to the square of the wave amplitude. It follows that the amplitude of the waves is inversely proportional to the density of the atmosphere, even where the amplitude is not small and the linear approximation becomes invalid.

An experimental study was made of the steady-state natural convection induced in enclosures by a small hot spot centrally located on the floor. Enclosures of rectangular and circular floor plan were employed, with height equal to one-half the major dimension of the floor plan. The movement of air within the chambers was made visible by adding metaldehyde dust particles and illuminating them with an intense light beam. The Grashof number (Gr) based on hot-spot temperature and enclosure height ranged from 8 × 105 to 1 × 1010. Laminar flows were observed for Gr [lsim ] 1.2 × 109. The experimental flows in the circular chamber are compared in a companion paper with theoretically calculated flows (Torrance & Rockett 1969). In the region of laminar flows the agreement was excellent. The present paper notes certain similarities in the flows in rectangular and circular geometries. The disturbing effect of a slight heating of one wall of the rectangular enclosure was also investigated. Measurements were made of heat transfer from the hot spot to the air in the chamber.

The paper describes an investigation of the internal waves that are produced in a stratified fluid having constant Brunt—Väisälä frequency by a cylinder which executes small vibrations at a lower frequency. Explicit solutions are found for slender cylinders having arbitrary cross-sections. When the cross-sectional area of the cylinder varies with time it is found necessary in calculating the surface pressures and power output to take account of terms in the governing equations that are significant only at distances from the cylinder comparable to or larger than the scale height of the density variations. For this case a simple expression for the power output is obtained in terms of the rate of change of the cross-sectional area of the cylinder.

When the vibrating cylinder is rigid its cross-sectional area is independent of time and then the expression for the power output is very similar to von Kármán's expression for the drag of a body of revolution in supersonic flow.

In both the above cases it is found that one quarter of the power is radiated in each of the four directions that are inclined at a particular angle to the horizontal.

An experimental investigation has been carried out on the nature of disturbance amplification in the laminar natural convection boundary layer formed on a vertical flat surface with uniform heat flux input. A pair of calibrated hot-wire anemometers were used to obtain amplitude and phase profiles of disturbance velocities and to measure the distance amplification rate as a function of disturbance frequency. An interferometer was used to measure the amplitude and phase profiles of disturbances in the temperature field. The relative amplification of a temperature disturbance was measured as it was convected downstream. For this latter measurement a technique employing interferometric moiré patterns was used. All of these measurements are in good agreement with the theory. The results of this investigation further strengthen the conclusions drawn in an earlier theoretical investigation, that the frequencies which amplify fastest and lead to transition and turbulence occupy a band much higher than the frequencies which begin to amplify first. These results also support stability theory in the very close agreement between calculated and measured distributions of disturbance quantities.

Numerical solutions to the Boussinesq equations containing a temperature-dependent viscosity are presented for the case of axisymmetric buoyancy-driven convective flow in a cylindrical cell. Two solutions, one with upflow and the other with downflow at the centre of the cell, were found for each set of boundary conditions that were considered. The existence of these two steady-state régimes was verified experimentally for the case of a cylindrical cell having rigid insulating lateral boundaries and isothermal top and bottom planes.

Using a perturbation expansion it is also shown that only one of these solutions remains stable in the subcritical régime. This, however, seems to be confined to a very narrow range of Rayleigh numbers, beyond which, according to all the evidence presently at hand, both solutions are equally stable for those values of the Rayleigh and Prandtl numbers for which axisymmetric motions occur.

Finally, certain fundamental differences between the problem considered here and that of thermal convection in a layer of infinite horizontal extent are briefly discussed.

Many of the known analytic solutions of the equation for neutral disturbances to a stably stratified, inviscid, parallel shear flow are shown to belong to a wider family of solutions when a transformation to the hypergeometric differential equation is possible. Two particular cases in which the transformation can be made are examined in some detail and the solutions are expressed in a simple analytical form. A number of novel solutions are presented as examples.

It is shown that the principal non-linear interaction of inviscid inertial modes does not produce a resonant response in the steady geostrophic circulation. The rectified geostrophic flow manifested in a closed rotating container by a periodic excitation seems to result from a combination of viscous and non-linear effects within the boundary layers.

An analytical study was made of the natural convection induced in an enclosure by a small hot spot centrally located on the floor. The enclosure was a circular cylinder, vertically oriented, with height equal to radius. A Prandtl number of 0.7 (air) was assumed; the Grashof number (Gr) was based on cylinder height and hot spot temperature. The equations of fluid flow in axisymmetric cylindrical co-ordinates were simplified with the Boussinesq approximation. The equations were solved numerically with a computationally stable, explicit method. The computation, starting from quiescent conditions, proceeded through the initial transient to the fully developed flow. Solutions were obtained for Gr from 4 × 104 to 4 × 1010. The theoretical flows are in excellent agreement with experimentally observed laminar flows (Gr [lsim ] 1.2 × 109) which are discussed in a companion paper, Torrance, Orloff & Rockett (1969). Turbulence was observed experimentally for Gr [gsim ] 1.2 × 109. When the theoretical calculations were extended to Gr = 4 × 1010, a periodic vortex shedding developed, suggestive of the onset of laminar instability. The theoretical results reveal a √Gr scaling for the initial flow transients and, at large Gr, the velocities and heat transfer rates.

A linearized stability analysis of the flow of water in a channel with a loose bed and straight banks is described. It is assumed that the wavelength of the perturbations, which develop into meanders or braids, is longer than the width of the channel. It is therefore long compared with the ripples or dunes which cover the bed of such a channel and whose wavelength is shorter than the width of the channel. The latter need be allowed for only as roughness elements creating resistance. The variation of resistance to flow and rate of transport of bed material with velocity are discussed briefly and taken into account. Instability is interpreted as leading to a meandering or braided channel and it is shown that all practicable channels are unstable. Wavelengths calculated for channels expecte to meander are compatible with those given by Inglis's empirical rule and wavelengths calculated for channels which become braided are approximately the same as those observed.

The stability of plane Couette flow is examined for liquids in which the viscosity varies with depth. Under suitable conditions, the flow may be stable or unstable at small Reynolds numbers for disturbances with wavelengths long compared with the liquid depth.

A mechanism which may be stabilizing or destabilizing is found to derive from the role of diffusion in the neighbourhood of a ‘critical layer’, where the liquid velocity equals the phase velocity of a wave-like disturbance. This mechanism, which requires a viscosity gradient, is quite distinct from that found by Yih (1967) for the case of a viscosity discontinuity. An example is considered, for which these two mechanisms may be of comparable importance.

The uniform motion of a closed, axisymmetric body along the axis of an unbounded, rotating, inviscid, incompressible fluid is considered on Long's hypotheses that: the flow is steady; the flow is uniform far upstream of the body; the inertial waves excited by the body cannot propagate upstream. The appropriate similarity parameters are k, an inverse Rossby number based on the body length, and δ, the slenderness ratio of the body. It is conjectured that an upper bound to the parametric régime in which the solution implied by Long's hypotheses remains valid, say kδ ≡ k < kc, is determined by the first occurrence, with increasing k, of a local reversal of the flow.

A general solution for the stream function is established in terms of an assumed distribution of dipoles along the axis of the body. The disturbance upstream of the body is found to be proportional to the product of k2 and the dipole moment (total dipole strength) and to fall off only as the inverse distance, as compared with the inverse cube of the distance for a potential flow. The corresponding wave drag is found to depend on the power spectrum of the dipole distribution in the axial wave-number interval (0, k) and to be a monotonically decreasing function of the axial velocity for fixed angular velocity. Asymptotic solutions for prescribed bodies are established in the following limits: (i) K → 0 with δ fixed; (ii) δ → 0 with k fixed; (iii) k → ∞ with kδ fixed. Both the upstream disturbance and the wave drag in the limit (i) depend essentially on the dipole moment of the body with respect to a uniform, potential flow. The limit (ii) is analogous to conventional slender-body theory and yields a dipole density that is proportional to the cross-sectional area of the body. The limit (iii) leads to a singular integral equation that is solved to determine kc and the dipole moment for a slender body.

The results are applied to a sphere and a slender ellipsoid. The upstream axial velocity and the drag coefficient based on Stewartson's results for a sphere are found to differ significantly from Maxworthy's (1969) measurements, presumably in consequence of viscous separation effects. Maxworthy's measured values of upstream axial velocity are found to agree with the theoretical values for an equivalent ellipsoid, based on the sphere plus its upstream wake, for k [lsim ] kc.

Experimental tests of an axisymmetric jet of air impinging on both water and wet cement were performed and analyzed. Since the cavities formed on the water were unsteady and irregular, cavities were formed on wet fast-setting cement and allowed to set with the jet impinging. In this way, detailed measurements of the solidified cavity shape were made and shown to agree well with theory. This correlation of the data with the theory indicates that little gas was entrained in the liquid and that the influence of liquid viscosity and surface tension was small for the experimental conditions tested. A simplified analysis is also presented for an incompressible axisymmetric gas jet impinging normally on a liquid surface. The analysis was effected by combining the following physical conditions and assumptions: (i) the stagnation pressure corresponding to the centreline conditions of the jet at the bottom of the cavity is equal to the hydrostatic pressure, wherein an empirical turbulent jet decay law is used to predict the variation of stagnation pressure with distance from the nozzle; (ii) the force on the liquid is equal to the total change in normal momentum, which is equal to the weight of the displaced liquid; (iii) the shape of the cavity is a paraboloid.

A conference on turbulent boundary-layer prediction was held at Stanford, California, from 18 to 23 August 1968. The meeting was sponsored by the Mechanics Division of the U.S. Air Force Office of Scientific Research and by the industrial sponsors of the Internal Flow Program of the Mechanical Engineering Department at Stanford University. Attendance was by invitation, and there were 75 participants. The following is a brief account of the organization and content of the meeting; the full proceedings are available in two volumes.These have been reproduced by direct photo offset process and may be obtained from the Mechanical Engineering Department, Stanford University, Stanford, California 94305, for $6.00 each, or $11.00 for the set of two, plus postage (U.S. 50c, Europe $1.00); hard covers $1.00 extra per volume.