The generation of Aeolian tones is interpreted in terms of the theroy of aerodynamic sound. To do this, the intensity of the radiated sound is expressed in terms of the fluctuations in force upon a moving rigid body using the approximation that, at low mach numbers, these forces can be calculated assuming incompressibility of the flow. The fluctuations in lift and drag upon a circular cylinder at Reynolds numbers between 40 and 160 are calculated by integrating the fluctuations in momentum in the eddying wake, using the experimental data of Kovasznay (1949). It is found that the fluctuating lift per unit length is approximately $f_l = 0\cdot 38\rho U^2dcos2\pi nt$ where ρ is the density of the fluid and t the time, and that the magnitude of the fluctuations in drag is about 10% of this.

The axial length scale of these force fluctuations was found by an auxiliary experiment to be very large for Reynolds numbers below 100. When 100 < R < 160, the lenth scale is approximately 17d, the transition apparently occurring as a result of the instability to three-dimensional disturbances of the laminar eddying wake. Using this datum, the mean square acoustic pressure generated by the motion about the cylinder at these Reynolds numbers is found to be $\overline{P^2}(r) \sim 0\cdot 27 cos^2 \theta \frac {\rho ^2 ldU^6 S^2}{a^2r^2}$ where θ is the angle between the direction of observation and the incident stream, l the length of the cylinder and a the velocity of sound in the medium. Experiments undertaken to test this result directly, using sound intensity measurements from a whirling arm apparatus in an accoustically quiet room, gave very good agreement.

At higher Reynolds numbers, when the cylindg wake is turbulent, the theory leads to a, similar expression for $\overline{P^2}$ but with a smaller numerical factor. Analysis of previous experimental data on this basis gives good agreement, with a value of about 0·037 for the numerical constant. In view of this, it is likely that some of the earlier data which had appeared to give a Mach number dependence of M4, not M6, have been misinterpreted. Finally, mention is made of the conditions when the frequency of vortex generation is the same as the natural frequency of the wire, and the greatly increased intensity of the sound is seen to be the result of the greatly increased axial length scale of the force fluctuations.

The relative simplicity of this phenomenon makes possible perhaps the least ambiguous confirmation yet found of some of the essential ideas in the theory of aerodynamic sound.

Consider a wing-jet combination placed in a wind tunnel; the measured thrust on the wing due to a high speed jet emerging from it will require to be corrected to give the infinite stream value. This paper provides a theory of these wind tunnel corrections, and incidently establishes that the ideal thrust is almost independent of the jet exit angle.

Frictionless flows with finite voticity are usually made determinate by the imposition of boundary conditions specifying the distribution of vorticity ‘at infinity’. No such boundary conditions are available in the case of flows with closed streamlines, and the velocity distributions in regions where viscous forces are small (the Reynolds number of the flow being assumed large) cannot be made determinate by considerations of the fluid as inviscid. It is shown that if the motion is to be exactly steady there is an integral condition, arising from the existence of viscous forces, which must be satisfied by the vorticity distribution no matter how small the viscosity may be. This condition states that the contribution from viscous forces to the rate of change of circulation round any streamline must be identically zero. (In cases in which the vortex lines are also closed, there is a similar condition concerning the circulation round vortex lines.)

The inviscid flow equations are then combined with this integral condition in cases for which typical streamlines lie entirely in the region of small viscous forces. In two-dimensional closed flows, the vorticity is found to be uniform in a connected region of small viscous forces, with a value which remains to be determined—as is done explicitly in one simple case—by the condition that the viscous boundary layer surrounding this region must also be in steady motion. Analogous results are obtained for rotationally symmetric flows without azimuthal swirl, and for a certain class of flows with swirl having no interior boundary to the streamlines in an axial plane, the latter case requiring use of the fact that the vortex lines are also closed. In all these cases, the results are such that the Bernoulli constant, or ‘total head’, varies linearly with the appropriate stream function, and the effect of viscosity on the rate of change of vorticity at any point vanishes identically.

Two concentric spheres are supposed to rotate about the same axis with almost the same angular velocity, so that the viscous stresses over the surfaces of the spheres induce a flow which may be represented by a small perturbation superimposed upon a rigid body rotation of the fluid as a whole. The governing equations are therefore linearized in the magnitude of the perturbation, and it appears that the validity of this linearization is independent of the Reynolds number of the primary rotation. Attention is then restricted to the case in which the Reynolds number is large, the principal object of the note being to exemplify some of the properties of rotating systems at large Reynolds numbers in terms of a particularly simple mathematical model.

It is found that the cylindrical surface that touches the inner sphere (the axis being the axis of rotation) is a singular surface in which velocity gradients are very large. Everywhere outside this cylinder, the fluid rotates as a rigid body with the same angular velocity as the outer sphere. Inside the cylinder, the velocity distribution in the central (inviscid) core of the motion is shown to be determined by the velocity distribution in the boundary layers over the spheres, and explicit solutions are obtained for all these velocity distributions. The mechanics of the cylindrical shear layer itself is also discussed, though no explicit solution is obtained in this case.

The speed of free floating bodies on the surface of a water stream in a sloping channel has been found to be sensibly the same as the mean speed of the layer of water in which the bodies are floating, contrary to some recorded opinions.

After an extensive survey of mean-velocity profile measurements in various two-dimensional incompressible turbulent boundary-layer flows, it is proposed to represent the profile by a linear combination of two universal functions. One is the well-known law of the wall. The other, called the law of the wake, is characterized by the profile at a point of separation or reattachment. These functions are considered to be established empirically, by a study of the mean-velocity profile, without reference to any hypothetical mechanism of turbulence. Using the resulting complete analytic representation for the mean-velocity field, the shearing-stress field for several flows is computed from the boundary-layer equations and compared with experimental data.

The development of a turbulent boundary layer is ultimately interpreted in terms of an equivalent wake profile, which supposedly represents the large-eddy structure and is a consequence of the constraint provided by inertia. This equivalent wake profile is modified by the presence of a wall, at which a further constraint is provided by viscosity. The wall constraint, although it penetrates the entire boundary layer, is manifested chiefly in the sublayer flow and in the logarithmic profile near the wall.

Finally, it is suggested that yawed or three-dimensional flows may be usefully represented by the same two universal functions, considered as vector rather than scalar quantities. If the wall component is defined to be in the direction of the surface shearing stress, then the wake component, at least in the few cases studied, is found to be very nearly parallel to the gradient of the pressure.

The method of characteristics for steady supersonic flow problems in three dimensions, due to Coburn & Dolph (1949), is extended so that flow with shocks and entropy changes may be treated. Equations of motion based on Coburn & Dolph's characteristic coordinate system are derived and a scheme is described for solving these by finite differences.

A linearized method of characteristics is developed for calculating perturbations of a given three-dimensional field of flow. This is a generalization of the method evolved by Ferri (1952) for perturbations of plane flow and conical flow.

This paper considers the flow due to a jet spreading out over a plane surface, either radially or in two dimensions. Solutions of the boundary layer equations are sought, according to which the form of the velocity distribution across the jet does not vary along its length. For laminar flow, such a similarity solution is obtained explicitly. For turbulent flow, an eddy viscosity is introduced, and it is eventually seen that complete similarity is not attainable, but that confident predictions can nevertheless be made about the nature of the velocity distribution and the rate of growth of the wall jet.

Ward's slender-body theory of supersonic flow is applied to bodies terminating in either (i) a single trailing edge at right angles to the oncoming supersonic stream, or (ii) two trailing edges at right angles to one another as well as to the oncoming stream, or (iii) a cylindrical section with two or four identical fins equally spaced round it. The wave drag at zero lift, D, is given by the expression $\frac {D}{\frac {1}{2}\rho U^2} &=& \frac {1}{2\pi}\int^l_0 \int^l_0 log\frac{1}{|s-z|}S^{\prime \prime}(s)S^{\prime \prime}(z)dsdz - \\ &-& \frac{S^\prime (l)}{\pi}\int^l_0 log \frac {l}{l-z}S^{\prime \prime}(z)dz + \frac{S^{\prime 2}(l)}{2\pi} \{ log \frac{l}{(M^2-1)^{1|2}b}+k \} $ where l is the length of the body, b the semi-span of the trailing edge (or length of trailing edge of a single fin), and S(z) is the cross-sectional area of the body at a distance z behind the apex. The constant k depends on the distribution of trailing-edge angle along the span for each trailing-edge configuration. In case (i) it is 1·5 for a uniform distribution of trailing-edge angle and 1·64 for an elliptic distribution. In case (ii) it is 1·28 for a uniform distribution and 1·44 for an elliptic distribution. Study of case (iii) indicates that interference effects due to the presence of the body reduce the drag of the fins. For example, with a uniform distribution of trailing-edge angle, k for two fins falls from 1·5 in the absence of a body to 1·06 when the body radius equals the trailing-edge semi-span, while k for four fins falls from 1·28 to 0·45 under the same conditions.

Where ordinary finite-wing theory is applicable, the present method must agree with it for small $(M^2-1)^{1|2}b|l$, and this is confirmed by two examples (§3), but within the limit imposed by slenderness the present method is of course more widely applicable, as well as simpler, than finite-wing theory.

It is not known experimentally whether slender-body theory gives accurate predictions of drag at zero lift, for the shapes here discussed, under the conditions for which on theoretical grounds it might be expected to do so. It should be noted that, although tests have not yet been made on ideally suitable bodies, no clear the drag is therefore twice that of a wing made up of two of them. The final stages of the process cannot be represented by slender-body theory, but the initial trend may well be indicated fairly accurately.

A converging channel of area ratio 34 : 1 has been developed to produce strong shock waves in a tube. Shock waves of speeds M1=3−7·5 in oxygen and M1 = 5−10 in nitrogen have now been studied with an interferometer, and values of the relaxation time τ for the approach to vibrational equilibrium behind the shocks have been measured. The value of τ (atmospheric) varies from 54μsec at 800° K to 1·3 μsec at 3000° K for oxygen and from 19 μsec at 3500° K to 5 μsec at 5500° K for nitrogen. For oxygen, the graph of log τ against T−1/3 is not quite the straight line predicted by the Landau-Teller theory. The density ratios across the shocks were measured and compared with values calculated by the Bethe-Teller method for variable specific heats. Agreement between the measured and calculated values is satisfactory. Experiments were also performed on oxygen-nitrogen mixtures to determine the effect of nitrogen on the approach to equilibrium of the oxygen. It was found that O2 and N2 collisions at approximately 2000° K are 40% as effective in transferring energy to the oxygen as O2 and O2 collisions. A device that detects a shock with a time lag of less than 1μsec, consisting of an evaporated gold film which changes its resistance when heated by the shock, was also developed.

A previous paper drew attention to the collective importance of three physical quantities Q, R, S associated with ideal fluid flow in a horizontal channel. Invariability of these quantities at different cross-sections of the flow implies respectively conservation of flow rate, energy and momentum; and their values determine a wave-train uniquely. The properties of Q, R, S are recalled in the present paper to account for the various effects of lowering a rigid obstacle into a stream. The conditions giving rise to dissimilar types of flow are examined; in particular, the circumstances causing stationary waves on the downstream side are clearly distinguished from those under which the receding stream assumes a uniform ‘supercritical’ state. A well-known result in the theory of the solitary wave is shown to apply to the receding stream even when the extreme conditions for the wave are exceeded; although it fails to account for a region close to the obstacle where the curvature of the streamlines becomes large. In passing, a feature of the theory is shown to bear on the practical problem of producing a uniform stream. Precise calculations are made for the flow under a vertical sluice-gate and under an inclined plane. To account for the region near the bottom edge of the sluice-gate, a method based on conformal transformation is used whereby an unknown curve in the hodograph plane is approximated by an arc of an ellipse. The accuracy of the results is more than sufficient for practical purposes, and they compare favourably with solutions previously obtained by relaxation methods. A number of experiments with water streams are described.

In this paper the spatial variations and spectral structure of steady-state turbulent shear flow in channels are investigated without the introduction of empirical parameters. This is made possible by the assumption that the non-linear momentum transport has only stabilizing effects on the mean field of flow. Two constraints on the possible momentum transport are drawn from this assumption: first, that the mean flow will be statistically stable if an Orr-Sommerfeld type equation is satisfied by fluctuations of the mean; second, that the smallest scale of motion that can be present in the spectrum of the momentum transport is the scale of the marginally stable fluctuations of the mean. Within these two constraints, and for a given mass transport, an upper limit is sought for the rate of dissipation of potential energy into heat. Solutions of the stability equation depend upon the shape of the mean velocity profile. In turn, the mean velocity profile depends upon the spatial spectrum of the momentum transport. A variational technique is used to determine that momentum transport spectrum which is both marginally stable and produces a maximum dissipation rate. The resulting spectrum determines the velocity profile and its dependence on the boundary conditions. Past experimental work has disclosed laminar, ‘transitional’, logarithmic and parabolic regions of the velocity profile. Several experimental laws and their accompanying constants relate the extent of these regions to the boundary conditions. The theoretical profile contains each feature and law that is observed. First approximations to the constants are found, and give, in particular, a value for the logarithmic slope (von Kármán's constant) which is within the experimental error. However, the theoretical boundary constant is smaller than the observed value. Turbulent channel flow seems to achieve the extreme state found here, but a more decisive quantitative comparison of theory and experiment requires improvement in the solutions of the classical laminar stability problem.

This paper presents an approximate solution of the problem of the onset of convection between plane-parallel plates heated from below when the fluid between them absorbs and emits thermal radiation. A complete solution to this problem would be extremely difficult, and the equation of radiative transfer is therefore developed in two approximate forms, one appropriate to an opaque medium, the other to a transparent medium. This equation is then combined with the dynamical equations of the problem.

The initial static state is investigated by use of the Milne-Eddington approximation, and it is shown that there can be very large variations of temperature near to the boundaries.

The conditions for marginal stability are investigated both for motions which are restricted to the temperature boundary layer, and for motions which take place in the body of the fluid. In the former case it is found that a complete solution is provided by the approximate form for a transparent medium, and in the latter case a reasonable interpolation has to be made between results for the two approximate forms in order to complete the solution.

The effect of radiative transfer both on the initial static state and on the dynamical equations is such that the fluid is stabilized. This stabilization could probably be detected in the laboratory under certain conditions. In the earth's atmosphere the critical Rayleigh number for large scale motions may be increased by a factor 105, while at the surface of the solar photosphere the factor may be as large as 1012.

In this paper the problem is considered of long gravity waves approaching a semi-infinite barrier which extends parallel to the wave crests, the whole system being in rotation. It is well known that, when the rotation is zero, there is a ‘shadow region’ behind the barrier in which the disturbance diminishes rapidly with distance from the edge. However, it is shown that the rotation gives rise to an additional wave in the shadow region. The crests of this wave are at right-angles to the incident wave, and it travels along the barrier without attenuation in that direction. The amplitude falls off exponentially with distance from the barrier, as in a Kelvin wave. The amplitude at the barrier may exceed that of the incident waves.

The problem arises in connexion with the propagation of tides and storm surges in the ocean.

The flow rate of liquid metals is commonly measured by electromagnetic flowmeters. In these the fluid moves through a region of transverse magnetic field, inducing a potential difference between two electrodes on the walls of the pipe. The ratio of signal to flow rate is dependent on the velocity profile, and this is affected by electromagnetic forces.

In this paper the ultimate steady velocity profile and its associated pressure gradient and induced potential are calculated for the case of laminar flow in a circular pipe whose walls are conducting but without contact resistance. Laminar flow is encouraged by a transverse field. When the fluid conductivity and field strength are sufficiently high, boundary layers occur with a thickness inversely proportional to normal field intensity. The induced potential difference is then 0·926 of the value corresponding to the case of uniform velocity if the walls are non-conducting.

The distance the fluid must travel after entering the transverse field before the steady state is reached is next estimated by a Rayleigh approximation. The inlet velocity is taken to be uniform and effects which occur at the edge of the field are neglected. The process falls into two stages, first a boundary-layer growth and then an adjustment of the velocity away from the walls, occupying a much greater length of pipe. The entry length is shorter than it is in the case of flow in a rectangular pipe, but is still too long for appreciable distortion of the velocity profile to occur within practical flowmeters except at low flow rates. The pressure drop associated with the adjustment of the velocity profile is found to be independent of field strength, if this is high, and about one-eighth of the drop which occurs in the non-conducting case.

Experiments are described in which steady-state pressure gradients and induced potential differences were measured in mercury flowing along Perspex pipes of 0·5 and 0·25 in. bore in transverse fields up to 14500 gauss. The results confirmed the steady-state theory within the limitations of experimental accuracy and the assumption in the theory of high conductivity and an intense field. The experiments also covered the entry region in many cases, and showed that the theoretical entry lengths were correct in order of magnitude but over-estimated. However, the exact entry condition was uncertain, and steady readings were difficult to obtain in the entry region.

This paper considers the two-dimensional laminar boundary layer on an infinite flat plate normal to an oncoming stream, the plate making transverse oscillations in its own plane. The exact solution is shown to depend on a single ordinary differential equation, containing the frequency of oscillation as a parameter. Series methods are employed to evaluate the solution for small and large values of the frequency, and enough terms are calculated to give the solution with satisfactory accuracy over the whole frequency range. It is observed that the solution satisfies the full Navier-Stokes equations.

For a cylinder of arbitrary section making transverse or rotational oscillations, it is shown how the results for a flat plate can be used to describe the boundary layer in the neighbourhood of the front stagnation point. Difficulties which arise in extending the solution to cover the remainder of the cylinder are discussed, and an estimate is made of the fluctuating torque on a circular cylinder making transverse oscillations.

Propagation of spherical shock waves through self-gravitating polytropic gas spheres such as stars, caused by an instantaneous central explosion of finite energy E, is discussed theoretically. The problem is characterized by two lengths R0, L, where $R_0 = \left(\frac {E}{4\pi p_0}\right)^{1|3},\;\;\;\;\; L = \left(\frac {3C^2_0}{2\pi \rho_0G} \right)^{1|2}$p0 and C0 are the values of pressure, density and velocity of sound at the centre of the equilibrium pre-explosion state, and G is the constant of gravitation. R0 and L are scales connected with the power of the explosion and the dimensions of the star respectively, and their ratio A = R0/L has a fundamental significance. A solution especially suitable in the case of A = O(1) is developed in the form of power series in R/R0 (R is the distance between the shock front and the centre) by a method similar to that used in previous papers by the present author (1953, 1954). An approximation to this solution is carried out up to the term in R3. In particular, the velocity of the shock wave U is found to be $\frac {U}{C_0} = 1\cdot30 \left(\frac{R}{R_0}\right)^{-3|2} \{1 +0\cdot41A^2\left(\frac{R}{R_0}\right)^2 + 0\cdot 57\left(\frac{R}{R_0}\right)^3 +\ldot\}$ for the case of λ = 1.4, where λ is the ratio of specific heats.

Small spheroidal particles suspended in a sheared viscous liquid are sometimes observed to take up slowly preferred orientations, relative to the motion of the undisturbed liquid, which are independent of the initial conditions of release. These obsevations cannot be accounted for by the solution, obtained by Jeffery (1922), of the linearized Navier-Stokes equations. It is shown in this paper that the effect of the inertia of the liquid is to alter slowly the orbit of the particle in accordance with Jeffery's hypothesis that the particle ultimately moves in such a way that the dissipation of energy is a minimum, but that this effect is orders of magnitude too small to account for any of the experimental observations.

It is suggested that non-Newtonian properties of the liquid account for the observations. It is shown that the rate of orientation of a particle would then be independent of its size, and this prediction is verified experimentally. Other experimental evidence in support of this suggestion is also described.

Some remarks are also made about the possible effect of collisions between the particles when more than one particle is present.