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.

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.

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.

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.

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.

Wave diffraction due to a step change in bottom topography is considered for the case of two superimposed fluids of different, but constant, densities. The interface lies below the upper surface of the step. Shallow water theory is shown to be applicable only if the ratio of a non-dimensional frequency parameter to the departure of the density ratio from unity is sufficiently small. An approximate solution of the full equations, obtained by a method applied by Miles (1967) to surface wave diffraction, yields results limited only by the condition that the frequency parameter be small.

Under the assumption that the horizontal scales of the flow of a stratified fluid are much greater than the vertical scale, it can be shown that the pressure distribution in the fluid is nearly hydrostatic and that the solution for steady flows can be reduced to the solution of a non-linear partial differential equation with only horizontal co-ordinates as the space variables. The theory built on the basic assumption is the shallow-water theory for stratified fluids. Transformations are explicitly given with which a class of solutions for steady three-dimensional flows of a fluid of arbitrary stratification, continuous or discontinuous, issuing from a large reservoir can be found from a corresponding solution for a homogeneous fluid, provided a free surface is present and the shallow-water theory is applicable. A few examples of exact solutions according to the shallow-water theory are given and the parallel flow in a horizontal canal issuing from a large reservoir with the same horizontal bottom, which has some bearing on previous works on stratified flows, is discussed. But it is emphasized that the class is a very special one and that there are other solutions not belonging to this class. The conditions under which a solution belonging to this class is valid are discussed.

The results of experiments designed to investigate the shock-induced boundary layer on a semi-infinite flat plate are described.

Those for the laminar boundary layer are shown to be in agreement with a theory due to Lam & Crocco (1958) which describes two distinct domains, one near the shock where the flow is quasi-steady in a shock-fixed co-ordinate system and an unsteady region in which the flow characteristics approach the familiar steady state asymptotically. Experimental results are also presented for the non-laminar boundary layer. In particular the transition to turbulence in this unsteady boundary layer is discussed in some detail.

‘Establishment times’ for steady boundary layers are given for both laminar and turbulent flows, and their relevance to the testing times available in shock tubes is discussed. The measured heat transfer rates are compared with existing theories.

The linearized problem of the instability of a layer of liquid flowing down an inclined plane was formulated by Yih (1954) and was solved by Benjamin (1957). It was found that the instability of such a film flow is initially due to long surface waves of infinitesimally small amplitudes. In the present study, a closed-form expression for the non-linear development of these long surface waves is obtained. It is shown that in the neighbourhood of the neutral curve an exponentially growing infinitesimal disturbance may develop into supercritically stable wave motion of small but finite amplitude if the surface tension of the liquid is sufficiently large. Theoretically obtained amplitudes of such waves are consistent with Kapitza's (1949) observation. The approach used in this analysis is a modification of the method used by Reynolds & Potter (1967), who extended the method of Stuart (1960) and Watson (1960) in their study of the non-linear instability of plane Poiseuille and Poiseuille-Couette flow.

The two-dimensional stratified flow over an obstacle placed in a channel of finite height is examined to determine the extent to which Long's model provides an adequate description of real flows. A simple numerical method of solving Long's model for obstacles of arbitrary shape is used to calculate predicted streamline patterns which are compared with experimental observations of the flow over two bluff obstacles. If only a few lee-wave modes are excited there is qualitative agreement between theory and experiment, but, if the flow is subcritical with respect to several lee-wave modes, the effects of turbulence become dominant and the inviscid model is no longer useful. The theory predicts that the drag on an obstacle can increase with decreasing speed owing to the momentum transfer to lee-wave motion. Direct measurement of drag indicates that there are conditions under which the drag does increase with decreasing speed, but under these conditions the wake is dominated by turbulence and no lee waves can be detected.

A new method for treating boundary-value problems in gas-kinetic theory has been developed. The new method has the advantage of reproducing the bulk or asymptotic flow properties accurately whilst giving a realistic description of the behaviour of the molecular distribution function in the neighbourhood of a wall. As an example, the Kramers, or slip-flow, problem is solved for a general specular-diffuse boundary condition and some new expressions for the slip coefficient, flow speed and molecular distribution function at the surface are derived.

A brief discussion of the eigenvalue spectrum of the associated Boltzmann equation is given and its physical significance pointed out.

Certain analogies between this problem and the Milne problem in neutron transport theory are demonstrated.

An experiment is described which indicates a flow reversal due to fluid elasticity in the case of a rotating-plate experiment. An attempt to predict this reversal analytically on the ‘infinite’ plate assumption fails and recourse is made to a numerical procedure to solve the ‘finite’ plate problem. It is shown that the reversal is due to edge effects.

A formal algebraic analogy is drawn between meteorological parameters, such as the Richardson number, and the parameters describing the effect of rotation or streamline curvature on a turbulent flow. The analogy between the phenomena is a good first approximation. Semi-quantitative use of the analogy to apply meteorological data to curved shear layers shows that the effects of curvature on the apparent mixing length are appreciable if the shear-layer thickness exceeds roughly 1/300 of the radius of curvature; larger effects may occur in compressible flow. Application of the Monin-Oboukhov formula considerably improves the agreement between prediction and experiment in boundary layers on curved surfaces.

In considering the onset of instability of an electrically conducting fluid between rotating permeable perfectly conducting cylinders in an applied axial magnetic field, it is found that oscillatory axially symmetric modes occur at large values of Hartmann number, in addition to the usual stationary modes. Results are presented showing the effect of the oscillatory modes on the criterion of onset of instability.

The asymptotic behaviour of the stability criterion is considered in the limit of very large radial Reynolds number, and also in the limit where both the radial Reynolds number and the Hartmann number are large.