The wave drag at zero lift of slender delta wings and similar configurations

Abstract

Ward's slender-body theory of supersonic flow is applied to bodies terminating in either (i) a single trailing edge at right angles to the oncoming supersonic stream, or (ii) two trailing edges at right angles to one another as well as to the oncoming stream, or (iii) a cylindrical section with two or four identical fins equally spaced round it. The wave drag at zero lift, D, is given by the expression $\frac {D}{\frac {1}{2}\rho U^2} &=& \frac {1}{2\pi}\int^l_0 \int^l_0 log\frac{1}{|s-z|}S^{\prime \prime}(s)S^{\prime \prime}(z)dsdz - \\ &-& \frac{S^\prime (l)}{\pi}\int^l_0 log \frac {l}{l-z}S^{\prime \prime}(z)dz + \frac{S^{\prime 2}(l)}{2\pi} \{ log \frac{l}{(M^2-1)^{1|2}b}+k \} $ where l is the length of the body, b the semi-span of the trailing edge (or length of trailing edge of a single fin), and S(z) is the cross-sectional area of the body at a distance z behind the apex. The constant k depends on the distribution of trailing-edge angle along the span for each trailing-edge configuration. In case (i) it is 1·5 for a uniform distribution of trailing-edge angle and 1·64 for an elliptic distribution. In case (ii) it is 1·28 for a uniform distribution and 1·44 for an elliptic distribution. Study of case (iii) indicates that interference effects due to the presence of the body reduce the drag of the fins. For example, with a uniform distribution of trailing-edge angle, k for two fins falls from 1·5 in the absence of a body to 1·06 when the body radius equals the trailing-edge semi-span, while k for four fins falls from 1·28 to 0·45 under the same conditions.

Where ordinary finite-wing theory is applicable, the present method must agree with it for small $(M^2-1)^{1|2}b|l$, and this is confirmed by two examples (§3), but within the limit imposed by slenderness the present method is of course more widely applicable, as well as simpler, than finite-wing theory.

It is not known experimentally whether slender-body theory gives accurate predictions of drag at zero lift, for the shapes here discussed, under the conditions for which on theoretical grounds it might be expected to do so. It should be noted that, although tests have not yet been made on ideally suitable bodies, no clear the drag is therefore twice that of a wing made up of two of them. The final stages of the process cannot be represented by slender-body theory, but the initial trend may well be indicated fairly accurately.