Values of the pressure coefficient Cp between +1 and -1 can be measured directly in incompressible flow by comparison with the known pressures on a circular cylinder placed in the tunnel working section. The development of the method and the apparatus required are described. The method gives more consistent and accurate readings than can easily be obtained with normal manometer measurements, particularly when the pressure differences are small and the tunnel speed not very steady. Since the coefficients depend on Reynolds number only, less precise measurement and control of tunnel speed are required, unless, the coefficients to be measured are subject to critical scale effects. Possible ways of extending the range of coefficients which can be measured are suggested.

The solution to the general problem of transferring a rocket between two terminals in space with minimum fuel expenditure is explained and the results obtained when application is made to a number of particular problems of space navigation are described. The mathematical techniques which may usefully be employed in the calculation of optimum rocket trajectories are exemplified by a method of solving the problem of obtaining maximum range from a rocket missile over the Earth's surface.

By a slight modification to den Hartog’s classical theory of harmonically forced vibration off resonance it is shown that, with such distorted forcing wave forms as might be encountered in the stalling or choking flutter of an aerofoil in cascade, the influence of damping on the amplitude of motion can still be assessed with considerable accuracy by the assumption of an equivalent harmonic excitation. The result is valid for any amount of Coulomb damping at which motion is significant in the presence of any “ velocity ” damping coefficient up to that defined by a logarithmic decrement of about 1-3, the formal conclusion of the argument being stated in Section 4.

Pressure measurements were made at one section of an aerofoil of constant chord spanning a wind tunnel. A flat plate was mounted in the tunnel upstream of the aerofoil, so that the wake from the plate produced a spanwise velocity gradient at the aerofoil. By traversing the plate across the tunnel the position of the wake could be altered relative to the pressure holes in the aerofoil, so that the complete pressure distribution over the aerofoil in the neighbourhood of the wake could be found. The results showed that, because of the secondary flow produced by the interaction between the aerofoil and the wake, the variation of pressure across the wake for a given chordwise position was very small, except close to the leading edge. Thus the lift coefficient, based on the velocity outside the wake, was nearly constant across the wake. The form drag coefficient, based on the same velocity, was considerably reduced in the wake. Even when the form drag coefficient was based on the local velocity of the stream, a considerable reduction was found in the wake.

Details of the design, construction and use of an electrical resistance network for solving certain of the problems encountered in the flexure and extension of flat plates are given. Some typical experiments have been carried out with the aid of this network and the method of operation of the analogy is described. The results of these experiments are presented and they are compared with the corresponding theoretical values. The comparison is as good as could be expected in view of the limitations imposed by the very coarse mesh which was used; essentially the same results would have been obtained by the exercise of a relaxation computation, using the same size of net. No practical difficulties would be encountered in building a large network having a very fine mesh.

A transient sound wave is generated by discharging a spark in an air stream, the velocity and temperature of which are required. After a time interval of about 25 microseconds a shadowgraph photograph of the sound wave is taken; a second photograph is obtained after a further known time interval of the same order. The two exposures, both on the same negative, show the propagation of the sound wave in the air stream and from this the Mach number, the true velocity, the local velocity of sound, and hence the temperature, can be calculated.

Using this method, measurements of Mach numbers of the order of 0·5 gave values between 98 per cent, and 100 per cent, of those calculated from pressure measurements. Typical examples of the sound wave photographs are shown. With further development along the lines indicated in the paper greater accuracy should be possible. The local speed of sound was measured to an estimated accuracy of ±1·5 per cent.

Since only a very short time interval is needed to obtain the photographs, the method appears promising for investigating explosions of brief duration, or dealing with flows of pulsating character.

The theory developed in Part I is applied to a number of problems of aeronautical interest, the most important of which is to the setting of “ streamline “ walls about a symmetrical aerofoil placed in the centre of a channel. It is shown how the position of the streamline wall can be deduced from the (experimentally determined) position of a constant pressure wall. This theory is applicable to symmetrical aerofoils of any given shape, and makes allowance for the presence of the aerofoil's wake. To illustrate the theory, and to test it by an extreme example, the flow is calculated about a circular cylinder, with a diameter about half the tunnel height, for both straight and constant pressure walls. The solid blockage is calculated in each case and compared with the standard first order theory. For this extreme example the standard theory fails badly for straight walls, but is reasonably accurate for constant pressure walls.

For many conventional aircraft, the thickness of the wing and fuselage can be represented approximately by source distributions in the plane of the wing and on the axis of the fuselage. Such aircraft may be considered as having essentially planar thickness distributions. However, missiles with cruciform wings, biplane arrangements, “ ring ” wings and so on, require non-planar source distributions to represent the wing and fuselage thickness. For this reason, the term “ spatial thickness distribution ” is used.

To simplify the investigation of spatial thickness distributions, a singularity representing an element of volume is introduced. It is shown that the optimum distribution of such elements in a prescribed space gives rise to a minimum wave drag value consistent with that obtained for a Sears- Haack optimum body of revolution.

A method of estimating the fatigue properties of notched test pieces when cyclic loads are applied in the presence of mean stresses is described. The method takes into account the local yielding which occurs near the base of the notch when the nominal stresses are sufficiently high. The general form of the predicted R/M diagrams closely resembles those obtained experimentally in previously published work.

The equations of thin plate theory are expressed in polar co-ordinates and transformed using the Mellin transform. Problems involving discontinuous and isolated normal loadings may then be solved in the case of the built-in or freely supported wedge-shaped boundary.

Slender-body theory is used to derive the ogive of curvature approximation for very slender, pointed, convex head shapes at supersonic speeds. Results of application of this approximation, together with the λ-method for circular arc ogives, to a variety of non-slender head shapes show very good agreement with the method of characteristics, van Dyke's second-order theory and experiment. Good agreement with the method of characteristics and with experiment is obtained even in cases when the stagnation pressure losses across the nose shock wave are not negligible.

Ward’s slender-body theory is extended to derive first approximations to the external forces on slender bodies of general cross section with discontinuous profile slope. Two classes of body are considered: bodies whose profile (typified by the local radius) is continuous between the nose and base, and certain bodies whose profile is discontinuous, such as bodies with annular or side air intakes and wing-bodies on which the wing has an unswept leading edge. (Where air intakes are concerned, it is assumed that they are sharp-edged and that there is no “ spillage ” of the internal flow).

The following conclusions apply to the former class of bodies. The variation of drag with Mach number is found to depend only on the discontinuities in the longitudinal rate of change of the cross-sectional area, and is thus independent of cross-sectional shape. The drag itself is unchanged if the direction of the flow is reversed. The expressions for lift and moment assume the same forms as for smooth pointed bodies, the lift depending only on conditions at the base of the body.

The general theory is applied to winged bodies of revolution with an unswept wing leading edge: the results bear a marked resemblance to those obtained by Ward. The results for wings alone are seen to be applicable, with one modification, to subsonic as well as to supersonic speeds.

Equations for the calculation of the subsonic flow of an inviscid fluid through given two-dimensional channels (the “ direct” problem), and for the design (the “ indirect” problem) of such channels are derived. The method is based on conformal mapping, and in the special case of channels with walls made from a number of straight sections, or with wall pressure prescribed as step-functions, yields the same results as the well-known Schwarz-Christoffel mapping theorem technique. However, it is more general than this latter method, since it is capable of dealing with curved walls or continuously varying wall pressures. The compressibility of the fluid is allowed for only approximately, the ideal gas being replaced by a Kàrmàn-Tsien tangent gas.

In Part II the theory is applied to various problems of aeronautical interest, perhaps the most important of which is to the setting of “ streamlined ” walls about a symmetrical aerofoil placed in the centre of the channel.

Special forms of the receptance functions of mechanical systems are shown to indicate the existence of nodes. these are associated with horizontal straight lines in receptance diagrams. the effects of this nodal condition are explained in terms of simple torsional and translational systems for the sake of simplicity; but the theory may be extended to any type of system and any set of generalised co-ordinates. a particular instance arises if principal co-ordinates are used. the paper is a sequel to an earlier paper.

A formula is derived for the buckling in compression of an orthotropic sandwich panel, with all edges simply-supported. Two special types of symmetrical sandwiches, that of a sandwich having zero core flexural stiffnesses and of a corrugated core sandwich, are considered, and curves giving the critical buckling end load per inch are presented. An approximate theory is also given for the bending of sandwich panels under the action of an axial compressive load and a uniform lateral pressure. Curves are presented giving the maximum bending moments in a corrugated core sandwich panel.

The possible use of jet thrust as a source of lift, either by deflection of the force from the normal propulsion axis, or by some other means, raises a new stability problem under conditions of hovering flight and flight at speeds for which the aerodynamic damping becomes insignificant. At the same time, despite the small dynamic pressure, the direct air loads cannot be neglected, for, due to the very large positive or negative incidences peculiar to the motion, the associated coefficients are of a compensatingly high order. Consequently, the problem retains an aerodynamic element, but is principally one of automatic stabilisation, amenable to treatment by ordinary mechanical or electrical circuit theory. The investigation has been restricted to the longitudinal-symmetric motion with virtually the only workable form of automatic control, namely, one responsive to the angular displacement and velocity in pitch, but these conditions undoubtedly embrace all the essentials of any practical application, e.g. in take-off and landing by the new technique. The solution to the stability in a gust, or in a succession of disturbances, may then be easily obtained in closed analytical form, but the more general case of unsteady motion, when the displacements are no longer purely oscillatory, necessitates, in part, the use of step-by-step integration. This arises from the added complexity of the aerodynamic terms, due to the appearance of significant changes of incidence, and to the very large and variable values of the resistance derivatives at the angles involved, so that, in the present state of knowledge, the forces determining the motion cannot be expressed functionally. The labour which such calculations entail may be considerably shortened, however, by the application of a procedure based on mean deviations, but it is also found that the conclusions to be drawn from the hovering case remain valid in a number of important respects for the general problem. This feature is demonstrated in the discussion at the end of the paper on a number of examples designed to illustrate the characteristics of the automatic control and the conditions necessary for stability in a variety of circumstances, both when the aircraft is hovering and when its displacement increases continuously with time.

Experiments have been made to find the effect of the ratio of sting to base diameter on the base pressure of an axially symmetric body at zero incidence in a supersonic stream. The Mach number of the flow was 1·994 and the model boundary layer was turbulent. The model used was a one inch diameter circular cylinder without boat-tailing. It passed through and was supported upstream of the nozzle throat. This method of support allowed measurements to be made in the important (and hitherto unexplored) case of zero sting diameter.

As the sting to base diameter ratio was increased from 0 to 0·85, the base pressure decreased. The minimum value reached was approximately 0·8 of the value it would have at the base of a two-dimensional body with a similar ratio of boundary layer thickness to base height. The base pressure coefficient rose rapidly to zero as the ratio was further increased to unity.

Under the conditions of the experiments, with a sting to base diameter ratio of 0·4 the base pressure coefficient differed from that without a sting by approximately ten per cent. With the more modest ratio of 0·2, the difference was approximately three per cent.

Ward's slender-body theory is extended to examine the unsteady motion of a slender body through a fluid at rest. The cross-sectional area and shape of the body, its forward velocity and its lateral motion are all arbitrary functions of time, but are subject to the restrictions of small disturbances. The length of the body is fixed. An approximate velocity potential is obtained by the joint use of two integral transforms, and the accuracy of this potential is discussed in some detail. General expressions for the aerodynamic forces acting on the body are derived in terms of a co-ordinate system which moves forward with the body in the mean direction of motion, but is fixed laterally. These expressions are then transformed, for the particular case of an oscillating rigid body moving forward with variable velocity, to expressions in terms of co-ordinates referred to the body axes. (The former co-ordinate system makes possible a fairly compact general treatment, whereas the latter is more convenient if the solution of a particular rigid-body problem is required.) Finally, arguments are advanced to justify the application of the slender-body theory to wings of slender plan form whose trailing edges are perpendicular to the direction of motion, and results are given for an oscillating wing moving forward with variable velocity.

A generalised conical flow theory is used to deduce an integral equation relating the velocity potential on a delta wing (with subsonic leading edges) to the given downwash distribution over the wing. The complete solution of this integral equation is derived. This complete solution is composed of two parts, one being symmetric and the other anti-symmetric with respect to the span wise co-ordinate; each part represents a velocity potential. For example, if y is the spanwise co-ordinate and x is measured in the free stream direction, then a downwash of the form w= - α11 Ux|y| is symmetric and will give rise to a symmetric potential, whereas w= - α11 Ux|y| sgn y is anti-symmetric and gives rise to an anti-symmetric potential. The velocity potentials of such flows are given in the form of Tables for all downwashes up to and including homogenous cubics in the spanwise and streamwise co-ordinates. Table III gives similar formulae in the limiting case when the leading edges become transonic; these are compared with results given elsewhere and serve as a check on the results of Tables I and II.