The paper considers a body, or space vehicle, of arbitrary shape, with conductivity K and specific heat c. It is first assumed that K and c are invariant with time and the body, with a given initial temperature, is assumed to be heated by a distribution of heat sources that varies with space co-ordinates and time. It is shown that a rigorous analytical solution to the problem is possible, provided that the distribution function satisfies prescribed regularity conditions.

The same problem is then considered, but with a heat-source distribution which is also a function of temperature. In this case the problem can be reduced to a Volterra-type integral equation amenable to step-by-step solution. Convective heating and radiation are specific examples of this second case.

The third case considered is that of a body whose geometry and thermal coefficients K and c are all time-dependent, and this problem is reduced to the fixed-geometry case by the addition of a hypothetical heat-source distribution which is a function of the velocity of ablation, of temperature and of its time derivative; an integral equation of Volterra type again results.

Finally, an example is given of the application of the analysis to the ablation of a conductive spherical shell.

A three-dimensional laminar-boundary-layer calculation is carried out over the area concerned. The external flow is simplified, being calculated by slender-body theory assuming conical flow, with two point vortices above the wing, their positions and strength being determined by experiment. Attempts are made to draw transition fronts both for two-dimensional and sweep instability from this calculation. The combination of these gives fronts similar to those observed in some experiments. Because there is little or no pressure gradient over the area in question it is suggested that it is a region where distributed suction might usefully be applied in order to maintain laminar flow and reduce drag.

The paper presents a solution to the buckling under shear stress of infinitely long plates which are reinforced by both transverse stiffeners and longitudinal stiffeners. Each family of stiffeners is assumed to consist of equally spaced stiffeners. Both simply-supported and clamped edge conditions are examined. Numerical results are obtained for the case of a plate with transverse stiffeners and a central longitudinal stiffener and relationships between the buckling stress and the flexural rigidity parameters of the stiffeners are provided for three different spacings of the transverse stiffeners.

Heat addition in the external flow around a supersonic aircraft may produce a field of increased pressure and thereby a propulsive and a lifting force. Less specifically, such effects could be produced by alternative processes involving the gradual expenditure of energy stored in an aircraft, and, according to the process, could be available at hyper-, super-, or subsonic speeds. A generalised study is made of associated effects on cruising range. Results are presented as performance “frameworks,” within which future experimental or analytic data should define regions of feasibility for particular systems.

The benefit to be obtained by using an interference fit between the pin and plate in a pin-jointed connection has already been established. An examination of the published results shows that some non-linearity occurs in the mechanism of load transference from the pin to the plate since, except at very high initial interference, doubling the load on the joint more than doubles the maximum shear stress in the plate. An examination of the stress-load relationship shows a distinct discontinuity, the load at which this discontinuity occurs being dependent upon both the initial interference and the coefficient of friction between the pin and the plate. It is shown that the results hitherto published correspond to a coefficient of friction between the pin and the plate of 0.3 and results for lower and higher coefficients are given.

The paper presents a solution to the buckling of infinitely long plates when they are reinforced by transverse stiffeners possessing both torsional and flexural rigidity. The cases of both edges being clamped and simply-supported are dealt with. Numerical results are presented for the ratio of torsional rigidity to flexural rigidity as obtained with a thin-walled circular tube. When the stiffeners are completely rigid, in which case the individual panels are clamped along the transverse edges, the results obtained are in agreement with existing solutions for isolated rectangular plates.

Previous papers have examined the implications of an assumed polytropic expansion law on the performance of nozzles. In this paper a peculiar anomaly is examined when the convergent nozzle is exhausting into a vacuum. It is shown that, on the strict assumptions of a polytropic expansion law, and the assumption that the flow will maximise for a given throat area, the thrust per pound of air increases progressively with falling polytropic efficiency, becoming infinite when the polytropic efficiency is zero. The author suggests that this anomaly is either concerned with the assumption that the flow will maximise or means that polytropic expansion efficiencies below a certain limit are theoretically impossible, but he believes that there may be some other explanation.

A theory is presented describing the flow close to the surface of a circular cone at incidence to a supersonic stream, which reconciles the first order theory of Stone (on which the widely used M.I.T. Cone Tables are based) with the physically necessary condition that the entropy on the cone surface be constant.

However, the results of this “vortical layer” solution appear to invalidate the method used in Ref. 1 to apply a boundary condition at the cone surface, and so to cast doubt on the results obtained by this theory.

The problem considered is the buckling of a rectangular plate, tapered in thickness in a direction parallel to two sides, and uniformly compressed in that direction. Curves are presented showing the variation of buckling stress coefficient with the side ratio and the amount of taper, for each of two types of thickness variation, namely exponential and linear, and for each of two sets of boundary conditions, namely all edges simply-supported, and ends clamped and sides simply-supported. It is shown that, unlike the case of a uniform plate, there are no discontinuous changes of buckling mode.

A linearised, unsteady aerofoil theory is constructed which refers to an axis system which remains in the near vicinity of an aerofoil moving along an arbitrary flight path through a fluid at rest. The potential equation, the pressure equation and the boundary conditions at the aerofoil and in the wake are shown to be formally identical to those usually derived on the basis of an axis system translating uniformly through the fluid.

The description of a photoelastic investigation, initiated by the Structures Committee of the Royal Aeronautical Society, into the elastic stress concentration factors associated with shouldered shafts subjected to various simple forms of loading is completed in this part of the paper. A general description of the manufacture of the models, the photoelastic techniques employed to evaluate the s.c.f. and the results of the tests on shouldered shafts subjected to torsion and to bending have been given in the previous two parts.

The present paper includes the results of the tests on shafts subjected to axial load, together with details of a supplementary investigation into the limiting value of the s.c.f., for a particular shoulder radius, as the depth of the shoulder is increased to infinity. A summary of the general conclusions for all forms of loading is given at the end of the paper.

The stress concentration factors for a family of square holes, with varying degrees of curvature at the corners, and with varying amounts of compact uniform reinforcement on the boundary, have been computed for each of four loading cases. Poisson's ratio was assumed to be 1/3.

Results of a study of the accuracy attainable from various approaches to low aspect ratio wing deformational analysis are described. Seven model multi-web wings, representing a restricted range of sweep angles, aspect ratios and other geometric factors, were tested for deflection influence coefficients and various analytical approaches were applied in the prediction of these results. Part I gave the experimental data; Part II dealt with analyses based on elementary and plate bending theories. This part describes and applies certain discrete element idealisations common to matrix structural analysis. The merits and shortcomings of the theories studied herein are reviewed and other promising approaches are discussed.

Newtonian impact theory has been used to estimate the external pressure forces on ducted bodies of rectangular, circular and semi-circular cross section. With an allowance for skin friction (or other incidence-independent drag) it is shown that maximum lift/drag ratios of about three to four are possible with little effect of body geometry provided that short and very divergent, or tall and narrow rectangular ducts are avoided.

Large changes occur in the separate lift and drag at maximum lift/drag ratio and these have to be made compatible with the weight and thrust capacity of the configurations. The maximum lift/drag ratio of a duct can be improved by the addition of wings, especially for a duct with poor lift/drag ratio, but the best overall performance is with a duct of good lift/drag ratio and low drag.

It is anticipated that impact theory underestimates the pressure forces, and hence the lift/drag ratios, which would obtain at finite Mach numbers but the trends with geometrical changes should be reasonably reliable.

A method is described for the determination of the oscillatory aerodynamic coefficients for a helicopter rotor in forward flight. The method treats the blades and the recirculating wakes on a three-dimensional basis, but certain approximations are made so that the problem is tractable. In particular it is assumed that the downwash effect of a wake sheet beneath the rotor can be represented by concentrating the doublet distribution in the sheet in a single line of doublets.

The paper presents a solution to the buckling of infinitely long plates clamped along the edges, together with an extension of the solution obtained by Stein and Fralich for the case when the edges are simply-supported. It is shown that, as a consequence of increasing the edge support from that of a simple support to one in which both deflection and rotation are prevented, the rigidity which an intermediate transverse stiffener must possess in order to support the plate effectively is much reduced. Agreement between the theoretical relationships and existing experimental data is good.

A method of calculation is developed to compute the overall performance of a multi-stage axial compressor, from a knowledge of the individual stage characteristics, by a “stacking” technique. Compressor models are designed and their overall performance calculated. These results are compared to show, qualitatively, the effect of alterations in design and stage performance on overall performance and to find how compressors should be designed for optimum performance.

A simple method is presented for calculating the heat transfer rate through a compressible laminar boundary layer. The temperature of the wall is assumed to be uniform, the viscosity being proportional to the absolute temperature with a ratio which may vary in an appropriate manner with position along the wall; the Prandtl number is arbitrary but greater than about 0·5.

The method assumes, following Lighthill, that heat transfer rates are determined mainly by the form of the velocity field in regions close to the wall. After applying the Ulingworth-Stewartson transformation the velocity is expanded in powers of the distance Y normal to the wall, three terms being retained. The first term, proportional to the skin friction, is zero near to a position of boundary layer separation. The second term is proportional to the pressure gradient and also to the wall temperature. Accordingly it can become close to zero when the wall is sufficiently cooled. The third term becomes important only when the first two are simultaneously close to zero and is proportional to the heat transfer rate. Since the three terms cannot, in fact, ever be all zero at the same position they form a uniformly valid non-trivial approximation to the velocity close to the wall.

With this velocity profile, and following similarity arguments first given by Liepmann, it is possible to reduce the integrated thermal energy equation to a first-order ordinary differential equation for the heat transfer rate, which is easily solved.

A comparison with an accurate solution of Poots, for a particular pressure gradient and wall temperature, yields agreement to within 1 per cent at one third of the distance from leading edge to separation and 3 per cent at twice this distance, with the error rising to 16 per cent at separation where the heat transfer rate is, of course, very low. The modesty of this error is very encouraging.

An experimental investigation of the viscous decay of a steady, three-dimensional, helical vortex is presented. The vortex was generated by a rectangular, symmetrical half aerofoil (N.A.C.A. 0009) of 10·5 in. span, 3 in. chord, cantilevered from the wall of the circular test section of a low speed wind tunnel. Only the local total head and flow directions were measured with a five-hole pressure sensitive probe at one to eight chord lengths behind the trailing edge of the aerofoil, and the radial, tangential and axial velocity distributions in the trailing vortex were derived. The static pressure variation in the vortex flow and the difficulties in its accurate measurement are discussed. All tests were performed at a Reynolds number based on the free-stream velocity and chord length of the aerofoil of 10,000.

The various experimental velocity distributions in the vortex were in good agreement with a linearised, axi-symmetric, laminar, incompressible, viscous vortex due to Newman. The decay of velocity and geometry parameters, however, were found to be approximately eight to ten times faster than predicted by the analysis based on a laminar kinematic viscosity. It was also observed that the amount of circulation which actually rolled up in the trailing vortex was only about 58 per cent of that expected around the (N.A.C.A. 0009) aerofoil used for this investigation.

Results of a study of the accuracy attainable from various approaches to low aspect ratio wing deformational analyses are described. Seven model multi-web wings, representing a restricted range of sweep angles, aspect ratios and other geometric factors, were tested for deflection influence coefficients; Part I gave experimental data. This paper describes, applies, and compares certain elementary and plate bending theories. A subsequent paper will deal with discrete element idealisations commonly employed in matrix structural analysis.