The necessary conditions of Clebsch and Weierstrass and of the multiplier rule in the calculus of variations, which arise from the study of the first variation of a function, are summarised. A further necessary condition associated with the second variation is stated. The latter condition is applied to two problems: (i) the determination of the thrust-time programme which maximises the altitude of a sounding rocket, (ii) the determination of the thrust direction programme for a rocket with a known propellant expenditure programme which yields a maximum range. In both problems it is found that the additional necessary condition is satisfied.

A review of existing work in the field of linearised characteristics suggests that the hypersonic approximation introduced by Chapkis should be adequate in many cases of practical interest; furthermore, of the methods available, this approximation requires least computation for the calculation of pressure distributions on a given shape at given speed. The application of this method to sharp-edged conical shapes shows good agreement with measured zero-incidence pressure distributions for a variety of shapes and a wide range of hypersonic parameter. The agreement is less good at incidence, possibly due to flow separations in the real flow.

The problem of predicting the turbulent supersonic base pressure in axisymmetric flow is treated by an extension of a method of solution to the two-dimensional problem given in Ref. 1. The solution consists principally in tracing the boundary-layer development from upstream of the base to downstream of the recompression region for a given base pressure. A unique solution is obtained by specifying the shape of the rehabilitated boundary-layer velocity profile.

A comparison with experiment in the case of the step-down cylinder problem (the sting-support problem) yields some very favourable results. It is pointed out that, while it has not been found possible to obtain a solution to the problem of a vanishingly small sting, the base pressure does not vary appreciably while the sting is decreased from about 0-3 of the base diameter down to zero. It would appear that the present analysis is capable of giving accurate results down to sting/diameter ratios of the order of 0·3.

There have been, in the past, many fatigue tests carried out on a variety of materials and components. These all indicate a wide scattering in the lives (measured by the number of stress cycles to failure) endured by nominally identical components subjected to nominally identical forces before failure occurs. To interpet this scattering several equations have been suggested as representing the statistical distribution functions that fit the lives obtained for individual types of component. Of these functions the log normal distribution function has been perhaps the most widely used. For the central regions of the probability distribution, i.e. about the mean, the log normal distribution and several others represent experimental results very closely indeed, but engineers and designers of all kinds dare not design on the mean fatigue life. They are concerned with specifications that either exclude the possibility of failure or admit only a very small probability of failure. It is thus with the accuracy with which the “lower tail” of the probability distribution curve fits the experimental results that they are concerned.

To assess the fit at this lower end for one type of component, a large number (about 1,000) of aluminium specimens have been tested and the corresponding lives plotted. The results are very interesting. They show clearly that the log normal distribution for this type of component and material is pessimistic for a probability of failure of less than 0·3. This result is felt by the authors to be of very great importance. It has further been shown that the use of the “one-sided censored distribution function”, used previously by one of the authors, gives a curve that will fit the lower results better than the complete log normal distribution would do.

It is with the testing procedure adopted, the specimens used, the distribution functions considered and the conclusions obtained therefrom that this paper is concerned.

The author has previously discussed the flow in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted. Yuan has given a solution for high injection in which certain derivatives of the axial velocity become infinite at the centre of the channel. Physically this means that there must be a viscous layer at the centre of the channel and that Yuan’s solution is ignoring the shear layer. In this paper the complete solution for large injection is obtained by the method of inner and outer expansions and therefore includes the viscous layer. The resulting series solutions are confirmed by numerical results.

The intake is conceived as a device that generates a shock system and a subsonic patch within a supersonic field. The object of the paper is to obtain a qualitative description of this flow structure. The analytical procedure is a simple but fairly powerful mapping technique. Several examples are discussed and these show how the flow on the intake alters under the influence of gradually changing boundary conditions.

This paper deals with an unsteady-state three-dimensional heat conduction problem in a prismatic body which has a complicated boundary shape. Numerical techniques such as a relaxation method must be used to solve a problem of this type; this method, however, requires unusually large amounts of labour. The method used in this paper consists of the application of conformal mapping and numerical solution of a partial differential equation and is applied first to a prismatic body of square cross section for which the exact solution is well known. As a second application, the approximate solution for a problem involving a star-shaped cross section is presented.

This paper is concerned with the effectiveness of reinforcing a hole in a plate on one surface only. The particular problem considered is that of a circular hole in a plate of infinite extent, subjected to axially symmetrical tension at infinity, when the reinforcement consists of an annular plate on one surface. Because of the interaction between bending and stretching, the problem is highly non-linear and it has only been possible to solve it for certain limiting cases. The stress concentration factor depends on a dimensionless parameter proportional to the tension at infinity. It is shown that the stress concentration factor is increased, because of the eccentricity of the reinforcement, if the parameter is small, but for values of the parameter greater than a certain minimum the stress concentration factor is decreased. The results also make it quite clear that there is no possibility of obtaining reasonably accurate results by linearisation of the equations.

This paper presents a theoretical investigation of the interactions which occur when a laminar boundary layer on a flat surface encounters an expansive corner followed by a shock or other compressive agency. The method consists basically of two equations; the first relates deflections in the external streamlines to pressure changes and the second relates pressure changes to thickening or thinning of the boundary layer. For simplicity it is assumed that the pressure changes and streamline deflections are small.

A comparison is made with the only available experimental results, under conditions which are far outside the range of applicability of the theory. In these circumstances the agreement between the predicted and experimental pressure distributions is reasonable.

An attempt has been made to determine the natural frequency of fundamental flexural mode of a cantilever beam with uniform taper by the Galerkin method. The method suggested considerably reduces the calculations as compared with the other methods available and the results are checked with the correction factors derived by Martin.

A method of interpolating low aspect ratio lifting surface mode deflections for use in aeroelastic calculations is described. It is based on the spline curve, the numerical analogue of the draughtsman’s spline which is a thin flexible beam used for drawing a smooth curve through a set of given coordinate points. The present problem is one of surface fitting and a matrix scheme using two-way spline curves is adopted. The method of treating the case in which the data points are randomly placed, whilst retaining an exact fit at the points, is discussed, as are the results of a practical example.

An analytical discussion on the effect of linear damping on flutter speed of systems having three or more degrees of freedom is described. The study is based on energy considerations introduced in a previous paper, where binary systems were discussed.

A general solution is obtained by using potential flow theory for the design of two-dimensional suction slots for parallel channels. These slots are formed by free streamlines and are so arranged that the channel walls are free from adverse pressure gradient. Where suction slots are intended for the control of boundary layers, the present solution can only be used as a basis for experiment. It is interesting to note that the results also suggest a new method for the design of diffusers by suction.

Certain three-dimensional surfaces studied by Townend have the property that at every streamwise section a supersonic air stream is compressed isentropically in the form of a reversed Prandtl-Meyer flow. If all the compressions are centred on points lying on one straight line the flow is conical. Such a surface may be of use as an engine intake. In this paper an approximate boundary-layer calculation for such a surface is made for laminar flow with zero heat transfer in the conical case. It is found that under certain circumstances a useful amount of compression can be obtained before separation occurs (a pressure ratio of 6, for instance, for an initial Mach number of 10). Limiting streamlines are drawn and displacement thickness calculated for a typical case, and the Mach number at separation is determined for a range of initial Mach numbers and angles of sweep. Instability due to sweep is considered.

Results obtained from previous studies of the three-shock confluence are applied to a simple wedge intake with the objective of obtaining a qualitative description of the flow for given initial conditions (M0, δ0). A simple mapping procedure is used, so that when M0, δ0 varies slowly and continuously the resulting effect on the shock geometry can be discussed in a systematic fashion. A number of special curves are computed to help classify various regions of initial conditions.

Experiments have shown that the drag of a body of revolution with a blunt base can be substantially reduced by mounting behind the body a disc of rather smaller diameter than the body. With a single disc the base drag was reduced by about 35 per cent and with two discs a reduction of 55 per cent was obtained. When the cavity between the body and the best single disc was filled in to form a solid boat-tailed afterbody there was an even greater reduction of drag.

A single disc mounted nearer to the body than the position giving low drag gave a large increase of drag with severely unsteady flow. The nature of the flow in both the low-drag and high-drag régimes is discussed.

A study of the effect of linear damping on flutter speed is described. The energy exchange at flutter is considered in each degree of freedom of a general n-degrees-of-freedom system and an energy compatibility equation is formed. An analytical discussion of binary systems, based on the energy compatibility equation, is then presented.

This paper is concerned with half-ring wing-body combinations with prescribed wing loadings at supersonic speeds. Configuration shapes, their pressure distributions and their associated flow fields are discussed.

At flight speeds above about M=2 the cross-sectional area of the fully-expanded jet nozzle tends to be larger than the intake streamtube. The principle of integrating this increase in area of the propulsion unit with the usable volume in order to reduce the after-body drag of the combination is discussed and calculations are made to establish the order of the effect. It is deduced that successful integration of this type can produce worthwhile increases in lift/drag ratio.

A method for the detection of the important modes participating in flutter is presented in this paper. The method is based on energy considerations and is illustrated by numerical examples.