Measurements of the velocity, pressure, and turbulence behind a series of normal plates in the uniform test section of an air tunnel are described, the oscillation of the wake being prevented in all but one test through use of symmetrically located tail plates. By a combination of experimental and computational techniques, details of the pattern of flow over a wall on a plane boundary in an infinite fluid are closely approximated. A significant difference is indicated between the characteristics of such a flow and those of the flow past an isolated plate with oscillating wake.

The recent work on equilibrium (i.e. self-preserving) turbulent boundary layers in adverse pressure gradients is compared with theoretical predictions based on current generalizations about the turbulent shear flow. Using only assumptions of similarity, it is possible to show that an equilibrium layer can exist only if the free stream velocity varies as a power of distance downstream with an exponent greater than −$-\frac{1}{3}$ and if the velocity defect from the free stream is small. Assuming further that the effective eddy viscosity is independent of distance from the wall over the outer part of the layer, most of the properties of equilibrium layers may be computed from the known behaviour of layers in zero pressure gradient. The predicted values of skin friction and the predicted shape and magnitude of the mean velocity distribution are in fair agreement with the observations of Clauser. Finally, the modifications that are necessary if the velocity defect is not small are discussed briefly.

A programme of shock tube experiments has been conducted to study the refraction of plane shock waves at interfaces between two gases. Shocks of strength ζ = 0·85 (weak) and ζ = 0·30 (fairly strong) were allowed to impinge, at various angles of incidence, on interfaces between air/CO2 and air/CH4, and the resulting configurations were photographed through a Mach-Zehnder interferometer. From the interferograms, measurements were made of the strengths of the reflected waves, and of the angles of refraction, and the values were compared with the theoretical calculations of Polachek & Seeger (1951). Within the range of parameters for which the refraction model assumed by the theory is applicable–the so-called ‘regular refraction’ region–the observations were in excellent agreement with the theoretical predictions.

When the study was extended to ranges of the parameters for which the theory is clearly inadequate, a succession of rather complex ‘irregular refraction’ patterns was observed. Although these configurations were highly interesting qualitatively, each of them involved curved shocks, non-uniform regions of flow, and other less simple processes which discouraged any formal theoretical analysis. On a less rigorous basis, however, it could be shown that these patterns were internally consistent, and that each represented a distortion of a regular refraction process which was reasonable under the prevailing aerodynamic conditions.

Certain observations in these refraction experiments appear to be of some significance outside the specific problem. (i) The sensitivity of strong shock refractions to the values of the specific heat ratio λ for the two gases suggests a possible technique for the measurement of λ and its temperature dependence. (ii) Two of the irregular refraction patterns display a transition process which would be equally appropriate to the onset of the Mach configuration in the shock reflection problem. (iii) Some irregular refractions can be considered as special cases in the problem of the interaction of a shock and a boundary layer.

The flow at high Mach number past a body with a rounded nose is considered. Viscosity and heat conduction are neglected, and the body is assumed to be two-dimensional and symmetrical about an axis parallel to the incident stream.

An exact solution is first obtained in the case λ → 1, M → ∞, where λ is the adiabatic index and M is the Mach number. This solution is then used as the basis of a double expansion in δ = (λ - 1)/(λ + 1) and M−2, after the exact solution has been modified to make the series expansion converge near the body. The expansion is carried out as far as the terms of order (δ + M−2)2.

The results are displayed for various values of δ and M−2; typical results are as follows. With M−2 = 0, δ = 1/6 (λ = 1.4), and for a parabolic body having unit radius of curvature at the nose, the shock is approximately a parabola with radius of curvature 1.822 at the nose. The distance between the body and the shock along the axis of symmetry is 0.323, and the height of the sonic point from this axis is 0.744 both on the shock and on the body. The actual pressure distribution on the body is shown in figure 4, and agrees well with experiment. The pressure falls to zero at a distance 0.86 downstream from the nose of the body, measured along the axis of symmetry. On the assumption that the pressure remains negligible beyond this point, the total drag is 1.39 ρV2, where ρ0 is the density andV is the velocity of the incident stream.

The second-order slender-body solution is derived for an unyawed elliptic cone in supersonic flow. The result is used as the basis for a critique of various approximations in compressible flow theory: slender-body, linearized, first- and second-order thin-wing theories; edge corrections; and the method of linearized characteristics.

This paper is concerned with the motion in water of air bubbles whose equivalent spherical radii are in the range 0.5-4.0 mm. These bubbles are not spherical but are, approximately, oblate spheroids; and they may rise steadily in a vertical straight line, or along a zig-zag path, or in a uniform spiral. The rectilinear motion occurs when the radius is less than about 0.7 mm, and the other motions occur for larger bubbles. There is disagreement in the literature as to whether it is the zig-zag or the spiral motion that occurs. It was found experimentally that, when the bubbles are produced in the manner described in this paper, only the zig-zag motion occurs when the radius of the bubble is less than about 1 mm, but bubbles of larger radius either zig-zag or spiral depending upon various factors.

The spiralling bubble is treated theoretically by assuming that the flow near the front of the bubble is inviscid (the Reynolds number of the motion is several hundred) and considering the distribution of pressure over the front surface. Equations are obtained relating the geometrical parameters of the spiral, the shape of the bubble and the velocity of rise. The analysis is simplified by assuming that the pitch of the spiral is large compared with its radius, and the velocity of rise and shape of the bubble are determined as functions of the radius. The experimental and theoretical values are compared, and fair agreement found. Reasons to account for the disagreement are proposed.

A modification of the theory is proposed to take account of the presence of impurities or surface-active substances in the water, and the velocities of rise thus predicted are in agreement with the experimental observations.

The zig-zag motion is treated in a similar way, and the analysis leads to an equation which determines the stability of the rectilinear motion. The value of the Weber number at which the rectilinear motion. The value of the Weber number at which the rectilinear motion becomes unstable is deduced, and is found to be in fair agreement with experiment. The experimental evidence on the wake behind solid bodies is described briefly, and reasons are given for suggesting that the zig-zag motion is due to an interaction between the instability of the rectilinear motion and a periodic oscillation of the wake.

This paper considers the flow at high Mach number behind the curved shock formed when a supersonic stream impinges on an axisymmetrical body with a rounded nose.

The solution is obtained as a double expansion in $\delta = (\gamma -1)|(\gamma +1),$ where γ is the adiabatic index, and M−2; the expansion is developed to within terms of order (δ+M−2)3.

Expressions are obtained for the distance between the body and the shock, the radius of curvature of the shock compared with that of the body, and the pressure distribution on the body.

Results of an experimental investigation of the heat loss of fine heated wires immersed in a supersonic stream at right angles to the flow direction are presented. The measurements show that the heat loss of the wire is independent of the free-stream Mach number for values of the latter between 1.3 and 4.5. Since the wire is always in the wake of a detached shock wave, the streamlines in the neighbourhood of the wire pass through a normal shock wave. The Reynolds number Re2 based on conditions behind a normal shock, thus becomes the characteristic parameter for the heat transfer. The measurements covering a Reynolds number range of 3 to 220 show the existence of two flow regimes. For Re2 > 20 the Nusselt number is a linear function of the square root of the Reynolds number, and the equilibrium temperature is nearly independent of Re2 < 20 the Nusselt number decreases more slowly, and the equilibrium temperature rises sharply with decreasing Reynolds number.

This paper contains a theoretical investigation of the displacement effect of a pitot tube in a shear flow. Viscosity is neglected throughout so that the vorticity field alone is considered.

It is first shown that a two-dimensional approach does not produce a large enough displacement effect because it does not include the stretching of vortex tubes that takes place around a three-dimensional pitot tube. Then the three-dimensional problem is considered. A solution is obtained in the plane of symmetry for a sphere in a shear flow. This solution is found by making an assumption about the rate of stetching of vortex tubes perpendicular to the plane of symmetry and then considering the shear flow as a small perturbation of a uniform flow. A solution in the plane of symmetry is sufficient to obtain the displacement effect, which is found to be of the same order as the experimental result obtained by Young & Maas (1936) for a conventional pitot tube. The sphere may be considered to represent a conventional pitot tube (of slightly smaller diameter), so it is concluded that a large part of the displacement effect of a pitot tube may be accounted for without the inclusion of viscosity, i.e. by consideration of the vorticity field alone.

To a first approximation, the vorticity in the plane of symmetry is found to depend only on the distance from the centre of the sphere.

An outline of shear flows past some two-dimensional bodies is given in an appendix. The bodies considered are a circular cylinder and a two-dimensional ‘pitot-tube’ consisting of two parallel semi-infinite plates.

An analytical study is made of the resurgent wave motion induced by pressure distributions moving parallel to a straight coast line. The resurgence is shown to consist of an infinite number of edge wave modes; expressions for these modes are given and the wavelengths, frequencies and amplitudes are shown to be in agreement with experimental results. The effects of a Gaussian pressure distribution are analysed. For large-scale disturbances off the east coast of the United States only the fundamental mode is excited. Conditions, such as storm size, speed and distance from shore, for maximum induced wave amplitudes are derived.

The ‘Newtonian-plus-centrifugal’ approximate solution (Busemann (1933) and Ivey (1948)) for hypersonic flow past plane and axially symmetric bluff bodies in gases with the ratio of the specific heats λ constant and equal to unity is rederived using ‘boundary layer’ techniques together with the von Mises variables x and ψ. A method of successive approximations then gives a closer approximation to this solution for ε (λ − 1)/(λ + 1) small and the free-strea Mach number infinite. Formulae for the streamlines, shock shape and pressure distribution are determined to this approximation. These formulae are valid for any plane or axially symmetric shape, giving the ‘stand-off’ distance of the shock wave from the body as ½εlog(4|3ε) and ε times the nose radius of curvature for plane and axially-symmetric flows respectively. Particular results are computed for a number of special shapes. For certain shapes, the theory has a singular point where the first approximation to the pressure vanishes (θ = 60° for a sphere). Actually, the theory is not applicable where the pressure becomes too small. The corresponding theory for gases of general thermodynamic properties is deduced, the approximation being valid provided the total energy of the gas is large compared with the energy contained in the translational modes of the gas molecules.

This paper proposes a theory of collisions between small drops in a turbulent fluid which takes into account collisions between equal drops. The drops considered are much smaller than the small eddies of the turbulence and so the collision rates depend only on the dimensions of the drops, the rate of energy dissipation ε and the kinematic viscosity v. Reasons are given for believing that the collision rate due to the spatial variations of turbulent velocity is shown to be $N = 1\cdot 30(r_1+r_2)^2n_1 n_2(\epsilon|v)^{\frac {1}{2}}$, valid for $r_1|r_2$ between one and two. A numerical integration has been performed using this expression to show how an initially uniform distribution will change because of collisions. An approximate calculation is then made to take account also of collisions which occur between drops of different inertia because of the action of gravity and the turbulent accelerations.

The results are applied to the case of small drops in atmospheric clouds to test the importance of turbulence in initiating rainfall. Estimates of ε are made for typical conditions and these are used to calculate the initial rates of collision, the change in mean properties and the rate of production of large drops. It is concluded that the effects of turbulence in clouds of the layer type should be small, but that moderate amounts of turbulence in cumulus clouds could be effective in broadening the drop size distribution in nearly uniform clouds where only the spatial variations of velocity are important. In heterogeneous clouds the collision rates are increased, and the effects due to the inertia of the drop soon become predominant. The effect of turbulence in causing collisions between unequal drops becomes comparable with that of gravity when ε is about 2000 cm2 sec−3.

A method is presented for treating problems of the propagation and ultimate decay of the shocks produced by explosions and by bodies in supersonic flight. The theory is restricted to weak shocks, but is of quite general application within that limitation. In the author's earlier work on this subject (Whitham 1952), only problems having directional symmetry were considered; thus, steady supersonic flow past an axisymmetrical body was a typical example. The present paper extends the method to problems lacking such symmetry. The main step required in the extension is described in the introduction and the general theory is completed in §2; the remainder of the paper is devoted to applications of the theory in specific cases.

First, in §3, the problem of the outward propagation of spherical shocks is reconsidered since it provides the simplest illustration of the ideas developed in §2. Then, in §4, the theory is applied to a model of an unsymmetrical explosion. In §5, a brief outline is given of the theory developed by Rao (1956) for the application to a supersonic projectile moving with varying speed and direction. Examples of steady supersonic flow past unsymmetrical bodies are discussed in §6 and 7. The first is the flow past a flat plate delta wing at small incidence to the stream, with leading edges swept inside the Mach cone; the results agree with those previously found by Lighthill (1949) in his work on shocks in cone field problems, and this provides a valuable check on the theory. The second application in steady supersonic flow is to the problem of a thin wing having a finite curved leading edge. It is found that in any given direction the shock from the leading edge ultimately decays exactly as for the bow shock on a body of revolution; the equivalent body of revolution for any direction is determined in terms of the thickness distribution of the wing and varies with the direction chosen. Finally in §8, the wave drag on the wing is calculated from the rate of dissipation of energy by the shocks. The drag is found to be the mean of the drags on the equivalent bodies of revolution for the different directions.

This note advocates a model of the steady flow about a bluff body at large Reynolds number which is different from the classical free-streamline model of Helmholtz and Kirchhoff. It is suggested that, although the free-streamline model may be a proper solution of the Navier-Stokes equation with μ = 0, it is unlikely to be the limit, as μ → 0, of the solution describing the steady flow due to the presence of a bluff body in an otherwise uniform stream. The limit solution proposed here is one which gives a closed wake.

A closed wake contains a standing eddy, or eddies, whose general features can be inferred from the results of an earlier investigation of steady flow in a closed region at large Reynolds number. In all cases, the drag (coefficient) on the body tends to zero as the Reynolds number tends to infinity. The proccedure for finding the details of the closed wake behind two-dimensional and axisymmetrical bodies is described, although no particular case has yet been worked out.

This paper considers the changes that occur in the character of short-crested sea waves when they are refracted by a shallowing depth of water. Besides a change in mean wave-length and direction there is also a change (usually an increase) in the mean length of the crests. If the waves approach obliquely they become skew, that is, the crests become staggered one behind another.

When a short-crested sea is superposed on a long-creasted swell, refraction tends to amplify the longer waves more than the shorter ones. This also produces an increase in the mean length of the crests.

Numerical examples are given.

The irrotational motions induced by, but outside of, the selfpreserving turbulent wake and jet are examined. It is found that there is a mean flow towards the centre of the jet, although there is no such flow in the case of the wake. Phillips's (1955) results on the nature of fluctuating irrotational flows are found to be largely unaffected by the introduction of simple inhomogeneities into the boundary conditions, but another mode of fluctuation with movement along the lines of mean flow is also shown to be likely. It is pointed out that, contrary to a statement of Corrsin & Kistler (1954), it is possible for the vorticity-free fluid between bulges of turbulent fluid to partake of the mean velocity of the turbulent fluid.

An inequality is demonstrated involving the rate of production S of mean-square vorticity in isotropic turbulence and a factor γ which may be said to allow for intermittency or for the non-vanishing of fourth-order cumulants. An extreme state, corresponding to equality of this relationship, occurs if S = 0·64 and γ = 2·14. The experimental values are S = 0·4 and γ = 4. Another kinematical relation shows that the mechanism of vorticity production resembles collision between fluid particles rather than the swirling of contracting jets.

Sir Charles Darwin has advocated a study of the ‘drift’ of material surfaces in the classically investigated irrotational flows past bodies. This suggestion is followed up and given further support in the present paper. In particular, it is shown how secondary flows can be evaluated by use of the ‘drift function’ t for the primary flow. This is a function (§1) such that material surfaces initially at right angles to the stream drift into shapes expressible by equations t=constant.

The analysis leads to a simple expression (§4) for the secondary velocity field in the flow past an infinite cylinder of any cross-section, with the upstream velocity normal to its axis and increasing linearly with distance along the axis-a problem in which only the secondary vorticity field was previously known. The drift past a sphere is computed and illustrated (§6), and the secondary vorticity field in shear flow past a sphere is tabulated (§7). There is also a detailed study (§3) of the asymptotic form of the secondary velocity field in flow past any body, based on a result of Darwin concerning ‘hydrodynamic mass’.

The dynamical theory of the motion of a body through an inviscid and incompressible fluid has yielded three relations: a first, due to Kirchhoff, which expresses the force and moment acting on the body in terms of added masses; a second, initiated by Taylor, which expresses added masses in terms of singularities within the bòdy; and a third, initiated by Lagally, which expresses the forces and moments in terms of these singularities. The present investigation is concerned with generalizations of the Taylor and Lagally theorems to include unsteady flow and arbitrary translational and rotational motion of the body, to present new and simple derivations of these theorems, and to compare the Kirchhoff and Lagally methods for obtaining forces and moments. In contrast with previous generalizations, the Taylor theorem is derived when other boundaries are present; for the added-mass coefficients due to rotation alone, for which no relations were known, it is shown that these relations do not exist in general, although approximate ones are found for elongated bodies. The derivation of the Lagally theorem leads to new terms, compact expressions for the force and moment, and the complete expressions of the forces and moments in terms of singularities for elongated bodies.

The compression of a perfect gas between a uniformaly moving piston and a rigid wall is discussed in the one-dimensional case. If the piston moves with a finite speed, it will initiate a shock in the gas which will reflect successively from rigid wall and piston and cause the compression process to deviate from a reversible adiabatic process. Expressions are derived for the relative changes in pressure and density at each shock reflection. Then values of density and pressure after any number of shock reflections are computed relative to their initial values, and, in terms of these, the corresponding values of temperature and entropy, as well as shock speeds, are determined. The limiting value of the entropy change, as the number of reflections goes to infinity, is obtained as a function of the ratio of specific heats of the gas and the strength of the initial shock. Hence it is possible to estimate an upper limit to the deviation of the shock compression process from a reversible adiabatic process. Some illustrative numerical examples are given.