The partial differential equations which describe steady flow of fluid in saturated homogeneous permeable solid material under non-isothermal conditions are stated. From these are derived the equations for flow of liquid (in particular, water) using suitable approximations and making use of empirical laws when necessary.

It is then postulated that the only ‘ponderomotive’ (i.e. massmoving) forces present are those due to thermal expansion effects. Free convection results. An approximate solution of the equations is attempted for plane flow by means of classical perturbation methods, the temperature and stream-function variables being represented by power series in a convection parameter proportional to the Rayleigh number.

A numberical example of the method, with boundary conditions based on a geothermal area at Wairakei, New Zealand, is given. The results show features which are in fair agreement with temperature measurements made in the area, and it appears that the convection parameter η is of the order of 10.

In the first part of the paper the dimensional laws governing the processes of heat and momentum transport from an infinite rough plane are assembled and their consequences set out. In the second part, the detailed equations for the turbulent energy, the mean square temperature fluctuation and the covariance of temperature and vertical velocity are used, together with some speculative assumptions concerning the dissipative action of the turbulence, to derive a series of relations between the turbulent intensities and the Austausch coefficients. One of these relations indicates that the flux form of the Richardson number cannot exceed a critical value which is about 0·15. It follows that in highly stable conditions the buoyancy forces have little direct effect on the turbulent energy balance, their action being primarily to cause a reduction in the scale of the motion and some change in its structure.

The supply of water to the outside of a long vertical tube was adjusted until the water film was seen to be only just disturbed by a train of travelling waves. Under these conditions the Reynolds number, defined as the discharge per unit width divided by the kinematic viscosity, was 4·4. The wave train was slightly irregular, and average values of the length and velocity of the waves were about 0·45 in. and 5½ in./sec, the temperature being 19° C.

In this paper, two-dimensional problems of the diffraction and stability of shock waves are investigated using an approximate theory in which disturbances to the flow are treated as a wave propagation on the shocks. These waves carry changes in the slope and the Mach number of the shock. The equations governing the wave propagation are analogous in every way to the non-linear equations for plane waves in gas dynamics, and their solutions can be deduced by the same mathematical techniques. Since the propagation speed of the waves is found to be an increasing function of Mach number, waves carrying an increase in Mach number will eventually break and form what we may call a ‘shock’, corresponding to the breaking of a compression wave into a shock in the ordinary plane wave case. Such a ‘shock’ moving on the shock is called a shock-shock. The shock-shock is a discontinuity in Mach number and shock slope, and it must be fitted in to satisfy the appropriate relations between these are interpreted as the trace of cylindrical sound waves in the flow behind the shock. In particular a shock-shock is the trace of a genuine shock in the flow behind, and thus corresponds to Mach reflection.

The general theory of the wave propagation is set out in § 2. The subsequent sections contain applications of the theory to specific problems, including the motion of a shock along a curved wall, diffraction by a wedge, stability of plane shocks and the instability of a converging cylindrical shock.

Consider a fluid of constant viscosity μ. If the fluid is incompressible, we suppose it to move in a conservative field in which the force per unit mass is - ∇Ω. If the fluid is compressible, we suppose the field of force to be absent. Both cases are included if we write V = ερΩ, where ε = 1 for an incompressible fluid, and ε = 0 for a compressible fluid. The equation of continuity

This paper deals with an experimental investigation of a turbulent low-speed jet of air spreading out radially over a flat smooth plate: a flow which has been discussed by Glauert (1956) in his theory of the wall jet.

The aim of the experiments has been to determine the mean velocity distribution and rate of growth of the jet. It is found, within the experimental range and accuracy, that the velocity profiles are similar and that the rate of change of velocity and width of the jet can be expressed by simple power-laws.

The damping of surface waves of small amplitude in liquid contained in cylinders has been calculated. Viscous dissipation in an assumed laminar boundary layer was taken to be the primary cause of damping. Experimental results were obtained for the logarithmic decrement as a function of the ratio of liquid height to cylinder radius for several water-filled cylinders. Theory and experiment were found to be in good agreement.

The decay of small perturbations on a plane shock wave propagating along a two-dimensional channel into a fluid at rest is investigated mathematically. The perturbations arise from small departures of the walls from uniform parallel shape or, physically, by placing small obstacles on the otherwise plane parallel walls. An expression for the pressure on a shock wave entering a uniformly, but slowly, diverging channel already exists (given by Chester 1953) as a deduction from the Lighthill (1949) linearized small disturbance theory of flow behind nearly plane shock waves. Using this result, an expression for the pressure distribution produced by the obstacles upon the shock wave is built up as an integral of Fourier type. From this, the shock shape, ξ, is deduced and the decay of the perturbations obtained from an expansion (valid after the disturbances have been reflected many times between the walls) for ξ in descending power of the distance, ζ, travelled by the shock wave. It is shown that the stability properties of the shock wave are qualitatively similar to those discussed in a previous paper (Freeman 1955); the perturbations dying out in an oscillatory manner like ζ−3/2. As before, a Mach number of maximum stability (1·15) exists, the disturbances to the shock wave decaying most rapidly at this Mach number. A modified, but more complicated, expansion for the perturbations, for use when the shock wave Mach number is large, is given in §4.

In particular, the results are derived for the case of symmetrical ‘roof top’ obstacles. These predictions are compared with data obtained from experiments with similar obstacles on the walls of a shock tube.

This paper deals theoretically with a problem of hydrodynamic stability characterized by small values of the Reynolds number R. The primary flow whose stability is examined consists of a uniform laminar stream of viscous liquid running down an inclined plane under the action of gravity, being bounded on one side by a free surface influenced by surface tension. The problem thus has a direct bearing on the properties of thin liquid films such as have important uses in chemical engineering.

Numerous experiments in the past have shown that in flow down a wall the stream is noticeably agitated by waves except when R is quite small; on a vertical water film, for instance, waves may be observed until R is reduced to some value rather less than 10. The present treatment is accordingly based on methods of approximation suited to fairly low values of R, and thereby avoids the severe mathematical difficulties usual in stability problems at high R. The formulation of the problem resembles that given by Yih (1954); but the method of solution differs from his, and the respective results are in conflict. In particular, there is dis-agreement over the matter of the stability of a strictly vertical stream at very small R. In contrast with the previous conclusions, it is shown here that the flow is always unstable: that is, a class of undamped waves exists for all finite values of R. However, the rates of amplification of unstable waves are shown to become very small when R is made fairly small, and their wavelengths to become very large; this provides a satisfactory explanation for the apparent absence of waves in some experimental observations, and also for the wide scatter among existing estimates of the ‘quasi-critical’ value of R below which waves are undetectable. In view of the controversial nature of these results, emphasis is given to various points of agreement between the present work and the established theory of roll waves; the latter theory gives a clear picture of the physical mechanism of wave formation on gravitational flows, and in its light the results obtained here appear entirely reasonable.

The conditions governing neutral stability are worked out to the third order in a parameter which is shown to be small; but a less accurate approximation is then justified as an adequate basis for an easily workable theory providing a ready check with experiment, This theory is used to predict the value of R at which observable waves should first develop on a vertical water film, and also the length and velocity of the waves. These three predictions are compared with the experimental results found by Binnie (1957), and are substantially confirmed.

A first-order relationship between changes in area and shock strength is derived for the case of a shock moving through a small area change in a channel. By integration of this relationship the area of the channel is obtained as a function of the shock strength in closed form. This result is interpreted as giving the average strength of a shock at a given time as it moves along a channel of arbitrary shape.

By suitable choices of the shape of the channel, descriptions of converging cylindrical and spherical shocks are obtained. These descriptions are found to be in close agreement with the similarity solutions valid near the points of collapse of the shocks. The reason for such good agreement is examined.

If, in a given solution of the boundary layer equations, the position of the wall is varied, then additional solutions of the boundary layer equations may be deduced. The theorem considers the nature of such solution, for the general case of time-dependent three-dimensional compressible flow.

Applications of the theorem arise in several different fields, and it is shown that useful quantitative results can often be obtained with the minimum of calculation. In this paper, chief attention is focused on the case of a rotating circular cylinder, and explicit formulae are developed for the skin friction, valid for sufficiently low rotational speeds. The important results which the theorem gives for slip flow have been noted by previous extenions to these previous treatments are made. Other applications of the theorem are briefly mentioned.

The finite difference method due to Lax (1954) is used to solve the equations of motion for a cylindrically symmetric flow of a compressible fluid. In particular, a converging cylindrical shock is found to increase in strength in agreement with the formula of Chisnell (1957). The artificial diffusion introduced by the method causes the pressure to remain finite at the axis, but a reflected diverging shock is obtained.

A proof is given that, in spite of certain singularities in the oblique area distributions, the supersonic area rule can be used to determine correctly the wave drag, according to linearized theory, of thin symmetrical rectangular wings at zero incidence. The proof could be extended to cover the case of swept wings with supersonic edges, for which similar singularities also exist.

The results of measurements of the vertical distribution of airborne particles, released usually at a height of 500 ft., and sampled for periods of about 30 minutes at downwind distances of 100, 300 and 500 metres, are presented and discussed.

At all distances the frequency distributions of the particle elevation with respect to the site of release are closely similar in shape and size to the frequency distribution of wind inclination at the site of release. This is interpreted as showing that, despite the uncorrelated effects of small eddies, high correlation was maintained in the motion of a particle, for periods of 1 minute or more, by the dominant action of persistent eddies which contributed heavily to the turbulent energy. In contrast to this result, the wind observations showed that the Eulerian auto-correlation coefficient (Rr) fell to about 0.2 in 10 seconds.

Detailed measurements have been made of the temperature field in natural convection above a heated horizontal surface in air, with and without a cold upper boudary. The object was to provide experimental material for the testing of existing theories of convection and the development of new ones. Using a variety of experimental techniques, it was possible to measure the heat transfer, the mean temperature profiles, mean squares of the temperature fluctuations and the autocorrelation functions of the temperature fluctuations. These results are considered in relation with the ‘similarity’ theory of Priestley and the ‘neutral stability’ theory of Malkus. Both these theories lead to the conclusion that, for convection between parallel planes at high Rayleigh numbers, nearly the whole of the mean temperature variation occurs in comparatively thin surface layers whose mutual interaction is small. The present experiments confirm this in some detail, but the exact form of the variation of mean temperature cannot be completely reconciled with either of the theories.

Isolated masses of buoyant fluid were released in a water tank. Their width, 2r, and the distance travelled, z, were measured as functions of time and were found to follow roughly the laws $r = nz,\;\;\;\;\; w = C(g \overline {B}r)^ \frac{1}{2},$ where w is the vertical velocity, $\overline {B}$ the mean buoyancy, and n and C are constants. These equations are predicted by dimensional analysis, assuming viscosity to be negligible, and the constants appear to be independent of the Reynolds number. It is found that C [eDot ] 1·2 and n is in the neighbourhood of 4.

Since the Froude number relating the buoyancy and inertia forces is the same as for isolated masses of buoyant air in the atmosphere, it is concluded that the constants will have the same value in this latter case. This is confirmed roughly by observation of cumulus cloud towers.

Some of the characteristics of the motion observed in the experiments are described and comparison is made with vortex rings.

In an earlier paper the uniform shear flow past a sphere was studied, by investigating how vortex lines are deformed by the ‘primary flow’ (flow in the absence of shear), and deducing the ‘secondary’ vorticity field (first approximation for small shear). In another paper the image system associated with each element of secondary vorticity was found, whence the Biot-Savart law can be used to determine the secondary flow field by integration. The integration is here carried out for the ‘downwash’ (secondary flow component perpendicular to the undisturbed flow, down the velocity gradient) on the dividing streamline. Difficulties due to the infinite domain of integration and singularities of the integrand are overcome by selecting variables of integration carefully and using known analytical properties of the secondary vorticity. From the computation of downwash is inferred the first approximation (for small shear A) to the ‘displacement’ δ (displacement of the dividing streamline, up the velocity gradient, far upstream of the sphere). If U is the upstream flow velocity and a the radius of the sphere, the computed value of lim (Uδ/Aa2) is 0·9. $A \rightarrow 0$ Details of the calculation show that the secondary trailing vorticity is not an important contributor to the displacement, The downwash is due almost entirely to vorticity upstream of the sphere (Hall's earlier simplified theory gave good results, e.g., 1.24 instead of 0.9, because it concentrated on the effect of local vorticity in producing downwash); further, this produces displacement principally through its image vorticity.

The relation between theories for a sphere and experimental results on Pitot tubes (beginning with Young & Maas 1936) is discussed. Theoretical evidence on tertiary- and quartary-flow effects is used here in the light of recent work which renders the successive-approximation sequence uniformly valid at infinity. The conclusion is that the theories, taken together, are not inconsistent with the experimental evidence that (i) at values of the ‘shear parameter’ Aa/U at which the displacement is measurable the ratio δ/a seems to have asymptoted to an approximately constant value, and (ii) displacement is greatly reduced in supersonic flow (Johannesen & Mair 1952) or when ‘sharp-lipped’ tubes are used (Livesey 1956).