THE DERIVATIVES Cw Cmy, Chey

The derivations for these three derivatives are exactly the same as for CDy above, and the results are exactly the same as (7.8,7) except that CD is replaced by the appropriate coefficient.

The Mach number effect on these three derivatives can he calculated from aerodynamic theory for both subsonic and supersonic flow. It is quite sensitive to the shape of the wing, high-aspect ratio straight wings being most affected by M, and highly-swept and delta wings being least affected. An upper limit is obtained by considering two-dimensional flow. For subsonic edges, the Prandtl-Glauert theory! and simple sweep theory combine to give for an infinite wing of sweepback angle Л

a, a

(1 – M2 cos2 A) a where a( is the lift-curve slope in incompressible flow. Whence

In level flight, with L = W, M2GL is a constant, so that M dCLjdM varies as 1/(1 — M2 cos2 Л). The theory of course breaks down at M ~ sec Л where an infinite value would be predicted, but nevertheless large values of M dGj-Jd M may be expected near that Mach number. At supersonic speeds, two-dimensional theory for swept wings gives the result

After differentiating with respect to M, the result obtained is again (7.8,8), which therefore applies for infinite yawed wings at both subsonic and supersonic speeds. The results given above derive from a linear theory that predicts proportional changes in the pressure distribution when M is changed—i. e. the pressure distributions remain unaltered in form, but changed in magnitude. Hence the results for Cm and would be of the same form, i. e.

M9Cm M2 cos2 A

ЭМ 1-М2 cos2 A m‘

M3Cft M2 cos2 A

3M 1-М2 cos2 Л ы> f A. M. Kuethe and J. D. Schetzer, Foundations of Aerodynamics, Secs. 11.6, 11.14.

The vanishing of dCrJd M will hold only for truly subsonic and truly super­sonic flows. In the transition region between them there is a very important redistribution of pressure, such that the center of pressure on two-dimensional wings moves from,25c in subsonic flow to,50c in supersonic flow. This would lead to a negative дСт/дМ, possibly of large magnitude, in the transonic range. The vagaries of transonic flow are such that test results are the only way to get reasonably reliable results in this speed range.

No general rules can be given for the derivatives with respect to pA or GT. Aeroelastic analysis or wind-tunnel testing must be used to find these. By way of example, we can calculate the contribution to dCmldpa associated with the fuselage bending treated in Sec. 7.4. We found there that the lift coefficient of the tail is given by

The pitching moment contributed by the tail is (6.3,8)

cmt = – vHcLt

Hence (^t = —V„ (7.8,12)

Wtaii H dp„

When (7.8,11) is differentiated with respect to pd and simplified, and the resulting expression is substituted into (7.8,12), we obtain the result

The corresponding contribution to Cm is

All the factors in this expression are positive, except for Cm, which may be of either sign. The contribution of the tail to Cmy may therefore be either positive or negative. The tail pitching moment is usually positive at high speeds and negative at low speeds. Therefore its contribution to 0OTfr is usually negative at high speeds and positive at low speeds. Since the dynamic pressure occurs as a multiplying factor in (7.8,14), then the aerolastic effect on Gmy goes up with speed and down with altitude.

Figure 7.6 shows the large effects of thrust coefficient on CL, cD, cm and values of the associated derivatives dCLldCT etc. can be found from data in this form.