In addition to the enormous aerodynamical literature concerned with the disturbances to a uniform stream produced by the presence of obstacles, there is a very extensive litrature on disturbances to irrotational non-uniform streams. The smaller body of work on disturbances to rotational streams is concerned principally with two-dimensional disturbances to a parallel shear flow; these are fairly easy to treat because the vortex lines are straight; they remain parallel and do not get stretched at all during the motion.

Perturbations of a given flow are considered, and the equations which govern the one-dimensional, non-steady flow of an inviscid, ideal compressible gas are linearized in the neighbourhood of this known solution, assumed isentropic, by a formal perturbation expansion. The perturbed flow is not assumed isentropic. Explicit solutions are obtained for a basic flow which is uniform or a centred simple wave and for an arbitrary simple wave if the perturbed flow is isentropic.

The perturbation of a uniform shock and perturbations in a shock tube lead to functional equations of a particular type, and a discussion of their solution is given.

A similar analysis is used to discuss the flow in a tube of slowly varying cross-section.

The flow past a flat plate at Reynolds numbers in the range 0·1 to 10·0 is investigated by an analogue method. The solution gives the stream function and the vorticity in the flow field surrounding the plate. From these are obtained the local coefficient of friction, the pressure distribution along the plate, and the total drag coefficient. The drag coefficient approaches the analytical values of Haaser (1950) and of Tomotika & Aoi (1953) as the Reynolds number decreases toward 0·1. The drag coefficient approaches the Blasius solution as the Reynolds number increases. At Reynolds number 10·0 the drag coefficient is still above the Blasius value, but is below the value obtained experimentally by Janour (1951). The difference from the experimental result is attributed for the most part to truncation error.

Measurements of mean velocity, turbulent stress and static pressure were made in the mixing region of a jet of air issuing from a slot nozzle into still air. The velocity was low and the two-dimensional flow was effectively incompressible. The results are examined in terms of the unsimplified equations of fluid motion, and comparisons are drawn with the common assumptions and simplifications of free jet theory. Appreciable deviations from isobaric conditions exist and the deviations are closely related to the local turbulent stresses. Negative static pressures were encountered everywhere in the mixing field except in the potential wedge region immediately adjacent to the nozzle. Lateral profiles of mean longitudinal velocity conformed closely to an error curve at all stations further than 7 slot widths from the nozzle mouth. An asymptotic approach to complete self-preservation of the flow was observed.

This paper presents an experimental investigation of the displacement of the total-head profile in a shear flow from its true position when it is measured with a flat-ended round Pitot tube. The experiments showed that at least two forms of displacement may exist depending on the flow conditions.

In a wake the displacement of the effective centre of pressure towards the higher velocity was found to obey the law deduced theoretically by Hall (1956), provided that changes in flow direction were not large. In the presence of large eddies the behaviour of the Pitot is further modified by its characteristics in yaw. This introuduces a total-head error reducing the observed total head.

In a turbulent boundary layer the experiments showed that the displacements were an order of magnitude smaller than in a wake and could be regarded as negligible for a round Pitot, even when it was touching the wall.

An empirical method for correcting the observations made with a round Pitot which is of the same width as the wake is given in an appendix.

This paper is concerned with the rates at which atoms and molecules react in the air that flows over a body flying through the atmosphere at hypersonic speeds. Using air as a working fluid, a series of shock tube experiments were carried out to provide information about these rates. Mach angle measurements were made to determine the state of the gas in three situations of interest.

1. Flow over flat plates was used to determine the state of the gas behind the incident normal shock; temperatures in the gas that passed through the shock varied between 2000 and 6000°:K and densities between standard and 1/80 of standard density.

2. Flow over wedges was employed to decelerate the flow behind the incident shock to a small supersonic Mach number; here temperatures downstream of the oblique shock increased, at most, 2000°:K above the free stream value.

3. A Prandtl-Meyer expansion was used to cool rapidly the dissociated gas, so that the recombination process could be investigated; temperatures dropped at most 2500°:K and the densities varied between standard and 1/200 of the standard value. In some cases, the initial degree of dissociation of air was over 45%.

The results (figure 11) indicate that the dissociation and recombination relaxation times of the chemical species found in air are very fast, when compared to the time it takes a particle of gas to flow either around a blunt body in hypersonic flight or past smtill models in shock tubes. Thus the shock tube is shown to be an instrument capable of supplying air at high temperatures in thermodynamic equilibrium (figure 5).

In the case of a non-melting blunt body of about 1 ft. diameter flying through the atmosphere at hypersonic speeds, the present results imply that, when the gas behind the detached shock is in thermodynamic equilibrium, the flow will also be in equilibrium as it expands around the body, provided its speed is greater than 10 000 ft./sec at altitudes below 180 000 ft. (figure 12).

A small-scale experimental model has been constructed for the study of steady-state slow free convection of water in saturated sand. The convection field is confined between coaxial cylinders maintained at constant temperature difference, and the upper and lower boundaries consist of planes with thermal and fluid insulation. Measurements of the temperature distribution within the convection space have been obtained for average boundary temperature differences (T1 − T0) of (I) 18·40° A, (II) 32·70° A, and (III) 46·68° A. Theoretical temperature values predicted from perturbation theory have been fitted by least squares. The first-order estimates of η/(T1 − T0), where η is the convection parameter or modified Rayleigh number, do not differ significantly from a constant value (± 3% S. D.) for the three given values of T1 − T0, indicating good agreement between theoretical and experimental results. First-order estimates are made also of the temperature coefficient of thermal conductivity b of the sand-water mixture, and of the coefficient of radiation loss c at the upper insulated boundary, but these estimates are less reliable. Separate determinations of η/(T1 − T0), b, c by direct physical measurement are in good agreement with the least-squares estimates.

This paper describes theoretical and experimental studies of the effects on shock tube flows of a monotonic convergence at the diaphragm section. Systematic flow equations are developed for tubes of uniform bore and tubes having either a monotonic convergence or a convergence-divergence in the diaphragm section. Except across the shock front itself, isentropic processes and ideal-gas behaviour have been assumed. Simplified procedures are presented for predicting the ideal-flow parameters over a wide range of operating conditions, as well as for comparing straight and convergent tubes. Such comparisons made by other investigators are found to be incomplete or in error. The experiments described utilize a very simple device for altering the diaphragm section convergence and a multi-station measurement of shock velocity. The expected effect of convergence is verified over a wide range of Mach numbers. Even at Mach numbers where the processes of shock formation can no longer be ignored, it is found that the relative performance between a uniform and convergent tube is preserved.

In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r. At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O(v1/3), where v is the kinematic viscosity, and in it the induced angular velocity is O(v1/6) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O(v1/4). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O(v1/3).

The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.

An investigation is made of the effect of a small disturbance on the flow in a complete rarefaction wave, for example, the flow produced by the rupture of a membrane originally separating a compressible gas from a vacuum. The perturbation arises from a rigid boundary slightly inclined to the direction of flow. The growth of the perturbed region is studied and the pressure field is calculated for diatomic gases.

The nature of the expanding boundary of the perturbed region is investigated. Arguments are put forward which suggest that this boundary can be a weak shock in certain circumstances. A second shock may also appear in some cases, following the first and of greater strength.

In an appendix the solutions are extended to monatomic gases and to fluids with an adiabatic index of 2. The latter results are suitable for a comparison with hydraulic experiments.

This paper describes the results of further experimental investigation of the turbulent boundary layer with zero pressure gradient. Measurements of autocorrelation and of space-time double correlation have been made respectively with single hot-wires and with two hot-wires with the separation vector in any direction. Space-time correlations reach a maximum for some optimum delay. In the case of two points set on a line orthogonal to the plate, the optimum delay Ti is not zero. In the general case it is equal to the corresponding delay Ti, increased by compensating delay for translation with the mean flow. Taylor's hypothesis may be applied to the boundary layer at distances from the wall greater than 3% of the layer thickness. Space-time isocorrelation surfaces obtained with optimum delay have a large aspect ratio in the mean flow direction, even if they are relative to a point close to the wall (0·03δ); the correlations along the mean flow then retain high values on account of the large scale of the turbulence.

A flat strip, of infinitesimal thickness and infinite lenght, is located in a viscous incompressible fluid, and both the strip and fluid are at rest initially. The strip is abruptly set into steady motion parallel to its length. An unsteady uni-directional flow of the fluid results, and there is a variable skin friction on the strip which must be overcome to maintain its velocity. In the early stages of motion, the skin friction is large, with a local behaviour which resembles that for a flat strip of infinite width. The skin friction near each edge of the strip can be more accurately represented by referring to the semi-infinite configuration that is realized on displacement of the other edge to infinity. At this stage, the results depend on separate consideration of the two limiting configurations, and the way to further improvements is not clearly delineated. The object of this paper is to provide a formulation which contains the preceding information and allows a systematic evaluation of all additional refinements. Thus, it is shown that the total skin friction on a strip of width 2a, moving with velocity V in a fluid whose coefficients of viscosity and kinematic viscosity are μ, ν, takes the form $D = 2\pi V \left[\frac {2a}{\surd (\pi vt)} + 1 - \frac {2}{\pi}\int ^\infty _1 \frac{1}{u^2} \surd \left(\frac {u-1}{u+1}\right) \rm {erfc} \left(\frac {au}{\surd (vt)} \right)du \right].$ Here the first two terms stem from the infinite and semi-infinite strip distributions, and the integral, containing a complementary error function, furnishes all corrections of the lowest exponential order, in the initial stages of motion, when $a| \surd (vt) \gg 1$ 1. The calculation is based on a new integral equation which makes explicit the effects of an isolated edge, and by iteration provides the interaction effects between edges at a finite separation.

The instability of small gas bubbles moving uniformly in various liquids is investigated experimentally and theoretically.

The experiments consist of the measurement of the size and terminal velocity of bubbles at the threshold of instability in various liquids, together with the physical properties of the liquids. The results of the experiments indicate the existence of a universal stability curve. The nature of this curve strongly suggests that there are two separate criteria for predicting the onset of instability, namely, a critical Reynolds number (202) and a critical Weber number (1.26). The former criterion appears to be valid for bubbles moving uniformly in liquids containing impurities and in the somewhat more viscous liquids, whereas the latter criterion is for bubbles moving in pure, relatively inviscid liquids.

The theoretical analysis is directed towards an investigation of the possibility of the interaction of surface tension and hydrodynamic pressure leading to unstable motions of the bubble, i.e. the existence of a critical Weber number. Accordingly, the theoretical model assumes the form of a general perturbation in the shape of a deformable sphere moving with uniform velocity in an inviscid, incompressible fluid medium of infinite extent. The calculations lead to divergent solutions above a certain Weber number, indicating, at least qualitatively, that the interaction of surface tension and hydrodynamic pressure can result in instabilities of the bubble motion.

A subsequent investigation of the time-independent equations, however, shows the presence of large deformations in shape of the bubble prior to the onset of unstable motion, which is not compatible with the approximation of perturbing an essentially spherical bubble. This deformation and its possible effects on the stability calculation are therefore determined by approximate methods. From this it is concluded that the deformation of the bubble serves to introduce quantitative, but not qualitative, changes in the stability calculation.

The boundary layer on a hot flat plate which is fixed at zero incidence in a slow stream carrying a progressive sound wave is investigated. Formulae are obtained for the skin friction and the heat transfer in the extreme cases when the frequency is very low and very high. In addition, different methods of simplifying the boundary layer equations in unsteady compressible flow are briefly compared.

After a Modification, the interpolation formula of Mott-Smith (1951) for the shock wave problem is found to be a solution of the Boltzmann equation at large Mach number in a finite region of molecular velocity space. This modification gives a unique determination of the shock wave thickness, removing the ambiguity for this in Mott-Smith's formula.

In the following it will be shown that a simple argument based on the use of the energy integral equation of the laminar boundary layer permits the derivation of a heat transfer formula valid for non-uniform temperature distribution and non-zero pressure gradients. The formula is then shown to be identical in structure with Lighthill's (1950) well-known results. Lighthill obtained his formula by solving the boundary layer equations in the von Mises form using operational methods. An elegant way to obtain the same results using exact similarity consideration was given by Lagerstrom (not yet published). The derivation given here is probably the most simple-minded one and the method may be useful for other applications as well. Furthermore, it is shown that the approach can be slightly modified to permit application of the formula to flow near separation. The latter result is applied to the Falkner-Skan solution for just separating flows and is found to be in excellent agreement with the exact solutions.

The motion of a viscous fluid contained between two rotating, circular cylinders whose axes are set slightly apart is considered. The equations of viscous motion are linearized by expanding the stream function in the form $\sum\limits^\infty_{n=0}\psi _n \gamma^n$, where γ is a parameter which depends on the distance between the cylinder axes. The ensuing analysis appears to hold for all values of the fluid viscosity v, and in particular for small values of v.

The asymptotic behaviour of the solutions for small v is examined, attention being mainly confined to the first order stream function ψ1 and the corresponding component of vorticity ζ1. Outside the boundary layers, where, for small v, ψ1 may be expanded asymptotically as $\sum \limits ^\infty _{n=0} \psi^{(n)}_1 v^{n|2$, the terms ζ(n)1 of the corresponding expansions for the vorticity are shown to be uniform throughout the fluid. It is noted that the asymptotic expansions of ψ1 for the region of the boundary layers and for the region outside the boundary layers may be combined in a single expansion which holds in both regions. The leading terms of this expansion are calculated by boundary layer methods.

An estimate is given of the distribution of skin frictional force per unit length, and of displacement area, on the outside of a semi-infinite cylinder, of arbitrary cross-section, moving steadily in a direction parallel to its generators. A Pohlhausen method is employed with a velocity distribution chosen to yield zero viscous retarding force on the boundary layer approximations. (The smallness of the fluid acceleration far from the leading edge has been pointed out by Batchelor (1954).) Like the Rayleigh method, this method is expected to yield reasonable results at large distances from the leading edge. However, for a large class of cross-sections, which includes all convex cross-sections and locally concave cross-sections with re-entrant angles greater than ½π, the method yields the expected square root growth of the boundary layer at the leading-edge, with a fairly close approximation to the coefficient, and it is supposed that the skin-frictional force and displacement area are given with reasonable accuracy along the whole length of the cylinder.

Results for the elliptic cylinder and the finite flat plate are given in closed form, valid for the whole length of the cylinder, and are expected to be in error by at most 20%. In addition, some estimate is given of the effect of corners on skin frictional force and displacement area.

A solution is given for the sound waves generated when an infinite wedge passes at supersonic velocity through the plane interface between different media. The results are then applied to the particular case when the two media consist of the same perfect gas at different temperatures. It is observed that the flow can be deduced for any symmetrical aerofoil of infinite span by superposition, and the case of a double wedge is considered in detail.

When plane pressure waves in a duct reach an open end, they establish a complicated three-dimensional wave pattern in the vicinity of the exit which tends to readjust the exit pressure to its steady-flow level. This adjustment process is continually modified by further incident waves, so that the effective instantaneous boundary conditions which determine the reflected wave depend on the flow history. In the analysis of a nonsteady-flow problem by means of a wave diagram, it has been customary to assume that the steady-flow boundary conditions are instantaneously established. While this simplifying assumption appears reasonable, the resulting errors have been undetermined. It is the purpose of the present investigation to obtain improved boundary conditions. The results of a previous study of the reflection of shock waves from an open end have now been extended to other waves of finite amplitude. The reflected waves computed by means of the new procedure are in good agreement with experimental data observed in a shock tube for a variety of flow conditions. The pressure variations in a reflected wave lag behind those derived in the conventional manner by the time in which a sound wave travels about one or two duct diameters. Such lags are small, but may occasionally become significant. As a consequence of the lag, certain discontinuities of the incident wave do not reappear in the reflected wave. This improved understanding of the reflection process has made it possible to clarify some previously unexplained experimental observations.