Higher approximations in boundary-layer theory Part 1. General analysis

Prandtl's boundary-layer theory is embedded as the first step in a systematic scheme of successive approximations for finding an asymptotic solution for viscous flow at large Reynolds number. The technique of inner and outer expansions is used to treat this singular-perturbation problem. Only analytic semi-infinite bodies free of separation are considered. The second approximation is analysed in detail for steady laminar flow past plane or axisymmetric solid bodies. Attention is restricted to low speeds and small temperature changes, so that the velocity field is that for an incompressible fluid, the temperature field being calculated subsequently. The additive effects are distinguished of longitudinal curvature, transverse curvature, external vorticity, external stagnation enthalpy gradient, and displacement speed. The effect of changing co-ordinates is examined, and the behaviour of the boundary-layer solution far downstream discussed. Application to specific problems will be made in subsequent papers.

On the theory of hypersonic flow past plane and axially symmetric bluff bodies

The ‘Newtonian-plus-centrifugal’ approximate solution (Busemann (1933) and Ivey (1948)) for hypersonic flow past plane and axially symmetric bluff bodies in gases with the ratio of the specific heats λ constant and equal to unity is rederived using ‘boundary layer’ techniques together with the von Mises variables x and ψ. A method of successive approximations then gives a closer approximation to this solution for ε (λ − 1)/(λ + 1) small and the free-strea Mach number infinite. Formulae for the streamlines, shock shape and pressure distribution are determined to this approximation. These formulae are valid for any plane or axially symmetric shape, giving the ‘stand-off’ distance of the shock wave from the body as ½εlog(4|3ε) and ε times the nose radius of curvature for plane and axially-symmetric flows respectively. Particular results are computed for a number of special shapes. For certain shapes, the theory has a singular point where the first approximation to the pressure vanishes (θ = 60° for a sphere). Actually, the theory is not applicable where the pressure becomes too small. The corresponding theory for gases of general thermodynamic properties is deduced, the approximation being valid provided the total energy of the gas is large compared with the energy contained in the translational modes of the gas molecules.

The linearised theory of conical fields in supersonic flow, with applications to plane aerofoils

SummaryIn many important problems of supersonic flow, either for the whole field of flow or a part of it, the velocity components are constant on straight lines through a fixed point. Such velocity fields are called conical fields. In the usual theory of the linearised perturbations of a steady supersonic flow, the velocity is assumed to differ only slightly from a uniform undisturbed velocity, and in the defining equations and boundary conditions all non-linear terms in the components of the perturbation velocity (and their space derivatives) are neglected. In this paper the equations of linearised supersonic conical fields, and their general solution, are set out both for the region inside and for the region outside the Mach cone of the origin in the conical field. The results are applied to flow past plane triangular aerofoils with straight edges downstream of the vertex (of which there are six cases), to flow past those plane aerofoils of more extended shape and finite span for which the solution may be obtained by combining a finite number of conical fields, and to the problem of plane triangular vanes in semi-infinite free jets. In the applications, the velocity fields are determined in considerable detail, but a main purpose in setting them out was to exhibit the mathematical methods used and the physical considerations that enter in determining the mathematical solutions.