Experiments have been done on the effects of heat transfer on wall-pressure distributions through separated regions with both laminar and turbulent boundary layers at a free-stream Mach number of about 3. The temperature of the flat plate on which the boundary layer was formed could be varied from about − 35° C to + 75° C. According to theory, this variation should have produced appreciable alterations at a laminar separation point in either the pressure or the pressure gradient, but no sign of this appeared in the overall pressure distributions, which, for laminar layers, remained unaffected by wall temperature. A possible explanation is given for this apparent discrepancy between theory and experiment. With turbulent layers, the variations in wall temperature did produce small changes in the pressure distributions. However, for most practical purposes such changes could be ignored. Hence the convenient conclusion is suggested that in supersonic separating flow with either a laminar or a turbulent boundary layer the pressure distributions are not significantly affected by moderate variations in wall temperature.

A theory is initiated for the generation of waves upon a water surface, originally at rest, by a random distribution of normal pressure associated with the onset of a turbulent wind. Corrlations between air and water motions are neglected and the water is assumed to be inviscid, so that the motion of the water, starting from rest, is irrotational. It is found that waves develop most rapidly by means of a resonance mechanism which occurs when a component of the surface pressure distribution moves at the same speed as the free surface wave with the same wave-number.

The development of the waves is conveniently considered in two stages, in which the time elapsed is respectively less or greater than the time of development of the pressure fluctuations. An expression is given for the wave spectrum in the initial stage of development (§ 3.2), and it is shown that the most prominent waves are ripples of wavelength λcr = 1·7 cm, corresponding to the minimum phase velocity c = (4gT/ρ)1/4 and moving in directions cos-1(c/Uc) to that of the mean wind, where Uc is the ‘convection velocity’ of the surface pressure fluctuations of length scale λcr or approximately the mean wind speed at a height λcr above the surface. Observations by Roll (1951) have shown the existence under appropriate conditions, of waves qualitatively similar to those predicted by the theory.

Most of the growth of gravity waves occurs in the second, or principal stage of development, which continues until the waves grow so high that non-linear effects become important. An expression for the wave spectrum is derived (§ 4.1), from which follows the result $\overline {\xi ^2} \sim \frac {\overline{p^2}t} {2 \surd 2\rho ^2 U_c g},$ where $\overline {\xi ^2}$ is the mean square surface displacement, $\overline {p ^2}$ the mean square turbulent pressure on the water surface, t the elapsed time, Uc the convection speed of the surface pressure fluctuations, and ρ the water density. This prediction is consistent with published oceanographic measurements (§ 4.3).

It is suggested that this resonance mechanism is more effective than those suggested by Jeffreys (1924, 1925) and Eckart (1953), and may provide the principal means whereby energy is transferred from the wind to the waves.

An analytical expression for the fringe shift in the N-wave is derived from the improved linearized theory for a slender supersonic projectile. From this expression an approximate mapping function is found which gives a simple test for N-wave flow. The validity of the fringe-shift expression is quantitatively confirmed by measuring an interferogram of the flow around a sphere. N-wave flow is shown to exist around a small sphere for r greater than about 70 diameters. Measurements of the shock waves from a sphere and a cone-cylinder show that the shocks assume their asymptotic positions for r > 14 diameters for the sphere, and r > 7 diameters for the cone-cylinder.

This paper describes the results of an experimental investigation of the turbulent boundary layer on a flat plate with zero pressure gradient, carried out at the Laboratoire de Mécanique de l'Atmosphère (I. M. F. Marseille). Transition to turbulent flow was obtained either by increasing the preturbulence upstream of the plate by means of a grid, or by a series of emery paper roughness elements beginning at the leading edge. The measurements were of the space-time double correlations, i.e. double velocity correlations with both spatial separation and time delay, from which isocorrelation lines for optimum delay can be drawn, and of the spectrum function, all for the longitudinal component of the velocity fluctuation.

Atmospheric dissociation will be appreciable in the neighbourhood of projectiles travelling at speeds greater than 2 km/sec. This introductory paper on possible effects of dissociation on the airflow, and hence on the drag, stability and aerodynamic heating of such projectiles, is intended mainly as a source of ideas for later, more comprehensive investigations.

A nearly uniform shear flow, was obtained in the working section of a wind tunnel by inserting a grid of parallel rods with varying spacing.

The function of such a grid is to impose a resistance to the flow, so graded across the working section as to produce a linear variation in the total pressure at large distances downstream without introducing an appreciable gradient in static pressure near the grid. A method of calculating a suitable arrangement of the rods is described. Although this method is strictly applicable only to weakly sheared flows, an experiment made with a grid designed for a shear parameter as large as 0·45 gave results in close agreement with the theory. There was no evidence from the experiment of any large-scale secondary flow accompanying the shear–a danger inherent in an empirical attempt to grade the resistance of the grid–nor was any tendency observed for the shear to decay with increasing distance from the grid.

In the Munk-Jones cross-flow theory for slender bodies of revolution (Munk 1924; Jones 1946), the cross force on an inclined body is obtained by replacing the three-dimensional flow by a non-steady two-dimensional flow, and by equating the cross-force to the rate of change of cross-flow momentum on a transverse lamina moving past the body with the free stream velocity U0. The result obtained for the lift force L on an element of the body is, for small angles of attack α, $dL|dx = \frac {1}{2} \rho U^2_0(dA|dx)2 \alpha$ where A is the cross-sectional area of the body, and, by integration $L = \frac {1}{2} \rho U^2_0 A_B 2 \alpha$ where AB is the base area of the body.

This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.

The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.

The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

The purpose of this note is to comment on the calculation of equilibrium gas dynamic parameters behind strong shock waves. The writer has been aroused by the appearance of a paper by Guman (1956) presenting a generalized computing scheme for ionizing shock waves in monatomic gases. In that paper the reader is not cautioned about including excited states in the Saha equation for the computation of the degree of ionization behind shock fronts at appropriate temperatures and densities. The same paper treats the adiabatic exponent γ = cp/cv as constant across strong shocks when at the same time it is implied that the computing scheme is of general validity. Hence, the unwary reader might attempt to apply the scheme in a regime where γ is not only no longer constant but is no longer a useful quantity for characterizing the shock conditions. Other authors (see, for example, Glass, Martin & Patterson (1953)) characterize flows in which a shock has excited internal degrees of freedom in terms of a variable specific heat ratio when in fact one cannot use this quantity in calculating shock front conditions.

This paper is a discussion of recent experiments in shock-wave refraction which have clarified a special type of shock outflow process appearing to have relevance to other shock interactions, and notably to shock reflection from an oblique wall. For certain incident shock strengths and angles of incidence α, the air/methane refraction problem simulates closely the situation in the trouble-some range of the reflection problem, in which α lies between the value αe at which the theoretical solutions terminate and the value α0 that marks the onset of Mach reflection, and in which the flow deflections cannot be reconciled with theoretically permissible reflected shock strengths. In the analogous refraction cases, the reflected shock is observed to increase in strength along its length to a maximum value at the intersection point, and to be followed by a subsonic rarefaction zone which also increases in severity near the intersection. In fact, this zone appers to coalesce into a subsonic discontinuity, just at the intersection point—a feature which would contradict one of the basic assumptions of the regular reflection and refraction theories. Other refraction experiments suggest that a similar process is relevant to the Mach reflection configuration, and may account for the discrepancies in the three-shock theory for weak incident shocks.

An exact solution is found in a fairly simple form for two-dimensional progressive waves of arbitrary amplitude on a fluid of unlimited depth, when only surface tension and not gravity is taken into account as the restoring force. The calculated wave forms are exhibited graphically for various amplitudes, and the relation between wave velocity and amplitude is plotted. The wave of greatest height occurs when the vertical distance between trough and crest is 0·730 wavelengths (compared with 0·142 for gravity waves). Higher waves are prevented from appearing by the enclosing of air bubbles in the troughs.

A new problem in hydrodynamic stability is investigated. Given two contiguous viscous incompressible fluids, the fluid on one side of the plane interface being bounded by a solid wall and that on the other side being unbounded, the problem is to determine the hydrodynamic stability when the fluids are in steady unidirectional motion, parallel to the interface, with uniform rate of shear in each fluid. The mathematical analysis, based on small disturbance theory, leads to a characteristic value problem in a system of two linear ordinary differential equations. The essential dimensionless parameters that appear in the present problem are the viscosity ratio m, the density ratio r, the Froude number F, and the Weber number W, as well as the parameters α, R (which is proportional here to the flow rate of the inner fluid) and c, that occur in the study of hydrodynamic stability of a single fluid. The results obtained are presented graphically for most fluid combinations of possible interest. The neutral stability curve in the (α, R)-plane is single-looped, as in the boundary layer case. The calculated critical Reynolds numbers are higher than the values observed in liquid film cooling experiments. (In these experiments, the outer fluid is usually a turbulent gas, in which the thickness of the laminar sublayer is of the same order of magnitude as the liquid film thickness.) General agreement between the theoretical and experimental values exists for all critical quantities except the Reynolds number. Gravity and surface tension are found here to have a destabilizing effect on the flow, in agreement with experimental evidence. Semi-infinite plane Couette flow is a special case of the present problem and the known stability of this flow is recovered. The linear velocity profile of two adjacent fluids with the same viscosity, but different densities, is shown to be unstable for high enough Reynolds numbers. The Reynolds stress distribution for a neutral oscillation in the general case is discussed qualitatively.

A circular cylinder is rotating in a uniform stream of viscous liquid, the circumferential velocity of the cylinder being much greater than that of the uniform stream. It is shown that the effect of the uniform stream can be regarded as a small perturbation of the rotary flow due to the cylinder, and that a uniformly valid first approximation to the flow field can be obtained in this way. This approximate solution has the interesting property that the circulation about any circular contour concentric with the cylinder is the same. The lift on the cylinder is shown to be that given by the classical formula, and the drag force is zero to the order of the approximation considered.

For flows of jet type, the assumption of a coefficient of eddy kinematic viscosity in turbulent flow leads to the possibility of combining in one the equations for laminar and turbulent motion. An approximation to the solution of these equations is found for the flow of compressible fluid issuing from a narrow slit, far from the slit. The stream function is expanded in a power series in squares of the Mach number. Bickley's solution (1937) for the corresponding problem in incompressible flow is used to start the iterative process by which successive terms of the power series are obtained. In order to find an analytical form for the second term of the series, it has been assumed that the Prandtl number is unity, that the viscosity varies as the nth power of the absolute temperature, and that the stagnation temperature of the jet is the same as that of the surrounding gas. The solution found differs only slightly from that of Howarth (1948) and Illingworth (1949) when laminar flow is considered; only the ‘change of scale’ effect (arising from a distortion of the coordinates in Bickley's solution) is of importance. In turbulent flow the effect of the second term of the series is as important as the ‘change of scale’ effect. The effect of compressibility on the width of the mixing region is discussed for both laminar and turbulent jet flow far from the orifice.

The intensity of turbulent motion in the wake of square mesh grids of bars has been examined experimentally. The measurements show that there are considerable departures from homogeneity over surfaces normal to the mean stream. Large variations of intensity have been observed at a distance of 80 meshlengths from the grid, where they are diminishing slowly, if at all. This inhomogeneity is believed to be the cause of the discrepancies between various sets of measurements of energy decay in the ‘initial period’. There is also a possibility of error in other measurements made in grid turbulence and analysed on the assumption of homogeneity. Observations with a grid of 2 in. mesh show that v2 is significantly less than u2, which is consistent with results from other laboratories.

The problem is to determine the downwind concentration of non-buoyant smoke emitted from a continuous elevated pointsource in a turbulent airstream. The velocity and eddy diffusivity coefficients are represented by related powers of the height above ground, and are independent of the position of the source. Exact solutions are obtained for the zero and second moments of the concentration distribution along lines lying in the cross-wind direction at ground level. In special cases, these moments may also be determined along lines at general height. In one such case the concentration is determined exactly (rather than just the two moments) and it is found that the cross-wind distribution always has a Gaussian form. If it is assumed that in all cases the cross-wind profile is Gaussian, a formulation for the concentration can be given purely in terms of the known zero and second moments. When the source is moving with constant velocity across the wind, the first moment as well as the zero and second moments is exactly determined, and under a similar assumption a formula for the concentration is found.

This paper concerns a narrow basin of uniform cross-section open to the sea at one end and closed at the other. An incident long wave of prescribed general form is supposed to enter from the sea and to represent the combination of tide and surge as generated in the sea. The solution of the linear terms of the equations of continuity and motion gives the reflection of this wave at the head of the estuary. This paper gives the next approximation when the non-linear terms are retained, the second-order motion being made determinate by the condition that, at the mouth, it reduces to a progression towards the sea.

The chief results relate to the surface elevation at the head of the estuary. When the first order elevation there increases steadily to a maximum, the effect of the ‘shallow water terms’ is to make high water higher and earlier, while the effect of the ‘frictional term’ is to make high water lower and later. For a short estuary, the interaction of the tide on a surge, due to a given sequence of meteorological conditions over the sea, is to make it higher when its maximum occurs at the time of tidal high water than when its maximum occurs at the time of tidal low water. This is directly opposite to the corresponding result when the estuary is of infinite length.

This paper describes a simple model which can be used to investigate the transport of water vapour by thermal plumes in the atmosphere. For an approximate treatment of these plumes, it is assumed (as in a previous paper) that the vertical velocity, temperature and specific humidity are constant across the ascending column, and that the inflow velocity due to mixing at the edge of the plume is proportional to the vertical velocity within the plume. The behaviour of the rising air is then investigated by means of equations representing the conservation of mass, momentum, heat and water vapour, and numerical solutions are obtained for representative cases.

It is shown that in a stably stratified atmosphere the plume air will become saturated if the source is sufficiently strong, and that the height of this saturation level can be determined in terms of dimensionless parameters representative of the source and the environment. Above the saturation level the analysis is continued by taking into account both vapour and liquid phases of water, and an approximate treatment is given for the behaviour of small plume clouds. It is found that there is a critical size for these; smaller clouds remain characteristic of the plume, but larger ones extend upwards to heights which do not depend on the previous parameters.

Certain two-dimensional, laminar bounday layers are considered whose streamlines are closed. The speed at the solid boundary is supposed uniform, the bounday outline being stationary, and the speed in the boundary layer is supposed to differ only slightly from that of the boundary. A formal solution is then obtained for the motion in the boundary layer. The analysis confirms that a closed boundary layer may exist and yields a condition needed to determine the inviscid motion. The condition is extracted in a simplified but approximate form and two examples of its use are given. A further class of closed boundary layers, namely those for which the pressure is uniform, is also considered. For this class the condition needed to determine the inviscid motion may be derived in a form both simple and exact.

An analogue method for solving certain quasi-linear hyperbolic partial differential equations is presented. The characteristic lines are formed by scanning electronically the screen of a cathoderay tube. The boundary conditions are introduced in the form of an opaque mask. The solution appears as a picture on the screen of a second cathode-ray tube. The experiments demonstrate the feasibility of the approach, but the development of the machine has not been carried beyond this stage.