Analogy Version I

For this case of invariant profile in supersonic flow:

1

л/МЇ-ї.

Compute the flow around the given body at MOT = v/2. For any other supersonic Mach number, the aerodynamics coefficients are given by:

1

уМЇ-ї ’

where Cp, Cl and Cm are at MOT = fl and Cp, Cl and Cm are at any other supersonic Mach number.

9.14.2.1 Analogy Version II

Here the requirement is to find a transformation for the profile, by which we can obtain a body, for which the governing equation is Equation (9.93a) with exactly the same pressure distribution as the actual body for which the governing equation is Equation (9.93b). For this:

K2 = 1.

The derivation of the above two results are left to the reader as an exercise. From the above results, we see that in supersonic flow Mx = fl plays the same role as Mx = 0 in subsonic flow.

For version II, we can write:

9.14.2.2 Analogy Version III: Gothert Rule

For any given body, at given Mach number MOT, find the transformed shape by using the rule:

7 = J = 7 = ^M^-T’ (995)

where a is the angle of attack, f and t are the camber and thickness of the given body, respectively. The primed quantities are for the transformed body and unprimed ones are for the actual body.

Compute the aerodynamic coefficients of the transformed body for MOT = V2. The aerodynamic coefficients of the given body at the given Mach number MOT follow from:

We can state the Gothert rule for subsonic and supersonic flows by using a modulus: 11 — M^ |.

From the discussion on similarity rules for compressible subsonic and supersonic flows, it is clear that, in subsonic flow, there is a ready made linearized solution for MOT = 0. Hence, for such cases we can use the Prandtl-Glauert rule. But for supersonic flow the linear theory equations are very simple and, therefore, we can conveniently use the Gothert rule.

Example 9.1

A given profile has, at MOT = 0.29, the following lift coefficients:

CL = 0.2 at a = 3°

Cl = — 0.1 at a = —2°,

where a is the angle of attack. Plot the relation showing dCL/da vs. MOT for the profile for values of MOT up to 1.0.

Solution At Mx = 0.29:

0.2 + 0.1 3 + 2 3.438/rad

dCL da

0.06/degree

1.094w/rad.

By the Prandtl-Glauert rule:

Therefore:

1.047w/radian.

For any other subsonic Mach number, by the Prandtl-Glauert rule:

dCL у da ) inc 1.047л

~da ~ уД-Mg ~ yr-ML.

Therefore, we	have the following variation:
M	0.1 0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
dCL da	1.05л 1.07л	1.10л	1.14л	1.21л	1.31л	1.46 л	1.74л	2.40л