Application to Wings of Finite Span

Consider a wing planform transformation described here. Planform

Taper ratio: к’ = к
Aspect ratio: A’ = A 11 — Ml |

Sweep angle: cot ф = cot 11 — Ml
A’ tan ф’ = A tan ф.

For subsonic flow, the transformation decreases A and for supersonic flow, the transformation increases A. Note that ф is sweep angle here.

Profile

The profile is given by the relations:

а=f=t=i i1 – M-1- (9118)

Thus, for wings (three-dimensional bodies), the Gothert rule is still more complicated; we have to trans­form not only the profile but also the planform, for each MOT. But this is the only reasonable method for wing analysis. In subsonic flow, these similarity rules are of great importance; but in supersonic flow, they are not that much important because even in two-dimensional subsonic flow, the elliptical equation is very difficult to solve, but in supersonic flow, the hyperbolic equation can be easily solved.

After making the transformations with Equations (9.116) and (9.118), find CL, CM, etc. for the incom­pressible case and then the corresponding coefficients for compressible case will be determined by the relations [Equation (9.115)]:

ca = cl = Cm= 1

Cp CL CM |1 — Ml

But it is tedious to find the variation of Cp, CL, CM with Mx because for each Mx we have to make the above transformations.