In this paper the shock waves of a slender aircraft are considered at a great distance from the aircraft so that all the shocks have coalesced into either the bow or rear shocks, but not at such a great distance (asymptotic) that the two shocks are of equal strength. Lower bounds are determined for the pressure jump across the bow shock subject to various aircraft constraints, and the pressure jumps across both the bow and rear shocks are determined at off-design conditions. The work described is considered to be a preliminary investigation and the results are such that further work seems to be warranted.

In recent papers Squire has presented results for the shock shape and pressure distribution on the lower surface of unyawed, lifing, conical bodies with sharp leading edges. The work, a development of Messiter’s first-order correction to Newtonian theory, was successfully applied to wings of diamond and caret section. This paper shows how the method may be used to include the effects of yaw. Results are presented here for the flat wing and some biconvex sections. Comparisons are made with experiment for both the flat and biconvex wings and agreement is shown to be good.

This paper determines the sheet thickness variation near a circular hole to yield a constant hoop stress in a sheet subjected to uniform radial stress at infinity. A similar analysis is presented for a variable thickness sheet plus an annular reinforcement. The stresses due to the remote application of shear and uniaxial tension are also determined.

The mean flow of an incompressible fluid in a straight suction duct with porous walls is considered and relations between the inflow and the pressure difference required to produce it are established on the basis of an elementary momentum analysis. It is assumed that bom the velocity and the pressure across planes normal to the direction of the mean flow inside the duct are distributed uniformly everywhere and that the effect of obstructions, which do not alter significantly the area of the duct cross section at any position, can be described in terms of losses of total pressure which are proportional to the square of the local speed within the duct. A variety of cases is discussed and it is shown that calculation of the internal pressure is straightforward and in agreement with experiment, at least for the case of a pipe having uniform porosity.

A boundary-layer development was measured on the near of a wing swept at 61°. The measurements approximately followed an external streamline from the minimum pressure to the neighbourhood of the separation line. Unfortunately the flow was found to be surprisingly sensitive to traverse gear interference. Moreover, the constraint imposed by the wind tunnel walls was sufficient to throw grave doubts on the use of the assumption of constant spanwise velocity to compute the external flow behaviour from the measured pressure distribution.

Comparison of the measurements with calculations using the method proposed by Cumpsty and Head showed the growth of streamwise momentum thickness, form parameter and crossflow to be seriously underestimated. However, only a small adjustment to the spanwise velocity outside the boundary layer over the rear of the wing was sufficient to bring the results into tolerable agreement. The necessity for such an adjustment to the spanwise velocity may be plausibly explained by the effect of tunnel constraints.

The results of extensive measurements which are presented here show the presence of secondary flow in the laminar corner boundary layer which is opposite in rotation sense to that found in the turbulent corner flow. This secondary flow is associated with progressive changes of the form of the velocity profiles. These changes become marked prior to transition and may be linked with the transition process. The laminar corner flow separates readily under the influence of an adverse streamwise pressure gradient.

An approximate theory, which yields the minimum induced drag of a planar wing with end-plates in ground effect, is presented. As a check on the accuracy of the method, it is compared with the exact linearised theory in the case of end-plates of vanishing depth (planar configuration). This shows that, within the limitations of the linearised lifting-surface theory, the present method is accurate in the range of wing heights of interest. In Section 5 the theory is compared with experiment. In this comparison two interpretations of the theory are employed. The first assumes that the vortex trace is the projection of the trailing edge of the configuration onto a plane normal to the direction of motion when the wing incidence is zero. The second is different from this only in that no constraint is placed on the wing incidence. Of the two interpretations the first is found to be in better agreement with experiment than the second for the planar configuration. However, the opposite is found to be true when end-plates of sufficient depth are fitted. A reason for this is proposed in Section 5.

The nature of flapping torsion flutter of a helicopter blade in forward flight is discussed. The essential complication in the analysis is the presence of periodic coefficients in the equations of motion; approximate solutions are obtained by use of a perturbation procedure. An unusual behaviour of the flutter equations which occurs when the fundamental frequency of flutter is a half-integer multiple of rotational speed is studied. Two different instability mechanisms can be distinguished and are related to the two energy sources in the system, namely the rotation of the rotor and the forward speed of the helicopter. It is found that forward flight can have a significant stabilising influence on flutter and that, as the tip speed ratio increases, flutter occurs predominantly at half-integer frequencies. The results are confirmed by the use of a numerical method.

In this paper a generalised presentation for symmetrically stiffened orthotropic cylinders under compression is developed, based on a linear orthotropic stability theory of cylinders. Similarities and differences in the minimum weight behaviour of stiffened cylindrical shells and flat transversely-stiffened wide columns are investigated in some detail to provide a satisfactory physical picture. The concluding results provide a comparative evaluation of various forms of stiffening systems for cylindrical shells under compression.

The historical development and current status of compressive orthotropic stability is reviewed and a concise development of the linear theory for the compressive case is presented. Available test data on orthotropically stiffened cylinders are synthesised and correlated in terms of this theory. The correlation has been used to establish regions in which linear orthotropic theory appears to be valid in terms of currently available test data.

Relationships for the sectional drag of wings are derived by using the linearised wing theory. It is suggested that some of these results may prove useful for checking the accuracy of numerical lifting-surface theories and for formulating approximate theories. Two main problems are considered, namely the lifting (no thickness) case and the thickness (zero lift) problem. In both cases, general planform shapes are examined and it is shown that particularly simple results are achieved for planar wings with a spanwise axis of symmetry.

When an aircraft flying at a constant airspeed performs circular orbits relative to the ground, its path relative to the air has interesting features if a wind of uniform velocity is blowing. The locus, which is here called an aeroid, bears some resemblance to a prolate trochoid; its equation is obtained in parametric and intrinsic forms, and its curvature and other fundamental properties are derived.

In the final section of the paper some practical implications are considered, including the effect of wind on the fuel consumption per orbit of a circling aircraft and the influence of winds of different headings on the path for a landing approach.

The linearised theory of supersonic flow is applied to a plane wing with a pointed apex whose leading edge is subsonic near the apex but supersonic further outboard. The change in character of the leading edge may occur either where the edge is kinked or at a point where its slope is continuous. In each case, the theoretical pressure field due to incidence is found to be singular along the downstream, outward-going Mach line through the point at which the edge changes character. The form of the singularity is different for the two cases, and somewhat unexpected. Measurements of the pressure on a plane wing with a curved, mixed leading edge showed no abrupt variation near the Mach line on which the singular behaviour was predicted.

A detailed discussion on the difference between an initial value line and a line characterised by a boundary condition has been presented. Two types of boundaries are described and illustrated. To examine each boundary, several different calculations have been performed for a straight pipe. The results of the numerical calculations are compared with an analytical solution. It is shown that known pressure and velocity at the pipe ends give the most accurate results. Comparisons are also made between several practical types of calculations which give similar findings. The use of time-dependent boundaries can lead to errors as large as 40 per cent in derived results. It is shown that good accuracy can be restored by converting the boundaries into initial value lines. It is concluded that in general no more than one time-dependent boundary should be used in any calculation. Finally it is demonstrated that errors are not revealed by means of pressure diagrams alone.

The decay of a trailing vortex containing a turbulent core is examined according to the argument that the turbulence is partly sustained by, and interacts with, the irrotational flow through viscous diffusion of eddies across the core boundary. The region separating the core from the outer irrotational flow, in which the interaction between the turbulence and the mean flow occurs, expands with time like (time)1/2, characteristic of viscous diffusion.

On the basis of a simple representation of the radial distribution of mean circumferential velocity, the decay of the vortex is calculated and is found to behave in a way similar to what Squire predicted, on the assumption of a constant eddy viscosity, in 1954. The essential difference from Squire’s solution found here is that the apparent viscosity is not simply proportional to the initial circulation about the vortex as he proposed, but depends on the Reynolds number defined with respect to the circulation.

Wind tunnel and flight observations of vortex decay can be more closely reconciled than Squire’s theory allowed; those differences that exist are attributed to the vastly different ages of the vortices in the laboratory and flight experiments. The principal conclusion of practical interest is that the maximum circumferential velocity in a trailing vortex generated by an aeroplane in flight is

where Γ is the circulation about large circuits enclosing the vortex, v the kinematic viscosity of the air and t the (sufficiently great) age of the vortex.

Bradshaw’s method of calculating the development of two-dimensional turbulent boundary layers involves the simultaneous solution of partial differential equations of mean motion and turbulent kinetic energy. The present approach avoids the computational complexities of this procedure.

The use of Thompson’s two-parameter family of velocity profiles and associated skin-friction law enables the momentum integral equation to be satisfied, along with Bradshaw’s version of the turbulent kinetic-energy equation at a specified fraction of the boundary layer thickness. This fraction (y/δ = 0·5) is chosen as representing the position in the boundary layer where Bradshaw’s equation, which contains several empirical functions, is shown by comparisons with experiment to hold with greatest accuracy. Thus the present simplified approach leads not only to a reduction in computing time but also to an appreciable increase in the general accuracy of prediction.

It is shown that a pair of supersonic inviscid conical flow fields can exist on either side of a conical deflagration wave. The configuration is relevant to the base flow question and indicates how base drag may be alleviated by burning. Results of exact computations are presented as well as those derived from a slender-body theory.

Cylinders of non-circular cross-section which can be constructed by circular arcs and straight line segments are analysed. Differential equations and boundary conditions for the buckling of eccentrically stiffened, isotropic or orthotropic, circular panels of zero Gaussian curvature are derived through variation of the total potential. Eight equations of continuity along the generators of adjacent arc-segments are enforced simultaneously through the use of Lagrangian multipliers which, with the potential minimised with respect to the admissible displacement functions, constitute an approximate solution of the cylinder problem. Numerical examples include stiffened cylinders with oval and egg-shaped sections. The buckling strengths of these non-circular cylindrical shells under bending are compared with their circular counterparts on the basis of equal circumferential length.

The Dynamic Relaxation method is used to analyse the post-buckling of flat rectangular plates. Extensive results are obtained for two types of problem in which the transverse edges are unloaded; in one case the transverse edges remain straight, in the other they are free to wave. Comparisons are made with alternative experimental and theoretical results. The potentiality of this approach to post-buckling problems is demonstrated by considering a variable thickness plate with mixed boundary conditions.

An energy approach has been used to obtain approximations to the natural frequencies of uniform beams when the effects of shear flexibility and rotatory inertia are included. The characteristic functions of simple beams are used to describe the bending deflection and the necessity for an assumed shear deflection shape has been eliminated by the use of simple relationships which exist between the effects of shear flexibility and rotatory inertia. A comparison of the approximate frequencies with the exact frequencies shows good agreement even at the higher frequencies. The method has been applied to hinged-hinged, fixed-free, free-free, fixed-fixed, fixed-hinged and free-hinged beams.