The Streamline Analogy (Version III): Gothert’s Rule

Gothert’s rule states [3] that the slope of a profile in a compressible flow pattern is larger by the factor 1 / 1 — M2 than the slope of the corresponding profile in the related incompressible flow pattern. But

if the slope of the profile at each point is greater by the factor 1 yj 1 — M2, it is also true that the camber (f) ratio, angle of attack (a) ratio, the thickness (t) ratio, must all be greater for the compressible aerofoil by the factor 1 / 1 — M2.

Thus, by Gothert’s rule we have:

ainc finc tinc

a f t

Compute the aerodynamic coefficients for this transformed body for incompressible flow. The aerody­namic coefficients of the given body at the given compressible flow Mach number are given by:

C^=C^=C^= 1

CPlnc Cl1iic Cmiuc 1 — M2 •

The application of Gothert’s rule is much more complicated than the application of version I of the P-G rule. This is because, for finding the behavior of a body with respect to M2, we have to calculate for each M2 at a time, whereas by the P-G rule (version I) the complete variation is obtained at a time. However, only the Gothert rule is exact with the framework of linearized theory and the P-G rule is only approximate because of the contradicting assumptions involved.

Now, we can see some aspects about the practical significance of these results. A fairly good amount of theoretical and experimental information on the properties of classes of affinely related profiles in incompressible flow, with variations in camber, thickness ratio, and angle of attack is available. If it is necessary to find the CL of one of these profiles at a finite Mach number M2, either theoretically or experimentally, we first find the lift coefficient in incompressible flow of an affinely related profile. The camber, thickness and angle of attack are smaller than the corresponding values for the original profile

by the factor "Мд. Then, by multiplying this CL for incompressible flow profile by 1/(1 — M^),

we find the desired lift coefficient for the compressible flow.

This method of collecting data for incompressible flow is cumbersome, since the data is required for a large number of thickness ratios. It would be more convenient in many respects to know how Mach number affects the performance of a profile of fixed shape. The direct problem, discussed in Subsection 9.13.2, yields information of this type.