Three-Dimensional Flow: The Gothert Rule

9.17.1 The General Similarity Rule

The Prandtl-Glauert rule is approximate because it satisfies the boundary conditions only on the axis and not on the contour. But Gothert rule is exact and valid for both two-dimensional and three-dimensional bodies. The potential equation is (for < lor > 1):

For Mx < 1, the equation is elliptic in nature and for > 1, it is hyperbolic. Here also, we make transformation by which the transformed equation does not contain Mx explicitly any more. Let:

X = x, У = К1У, z’= K1Z, Ф = К2Ф.

With the above new variables, Equation (9.106) transforms into:

(1 – MXMv+ к2(фу, у,+ ф’„) = 0.

Mx vanishes from the above equation for:

K1 = j|1 – Ml. (9.107)

With Equation (9.107), the resulting potential flow equation for subsonic flow is:

ф’х’ X + ф’уу + фф’ = 0

and for supersonic flow:

ф’х’х’ – фУУ – фР’У = °.

Again, for subsonic flow, the equation is exactly the same as the Laplace equation. For supersonic flow, the equation is identical with the compressible flow equation [Equation (9.106)] with Mx = V2.

Now:

, дф’ к дф K

дх’ дх

, _ дф’ _ K2 дф _ к2
v = ду = К дУ = К v

‘ дф’ К2 дф К2

w = ді = К1 д, = кw

с _ p – Рх _ _2 _____ 2_ дф

Р = 1 pVX = Vx= Vx дх

ср = -2 —

p V"

with the assumption that Vx = Vy. This assumption really does not impose any restriction on the rule, because in supersonic flow, the velocity itself is not important (that is, V/a is more relevant than V). Introduction of Equation (9.108a) into Equation (9.110) results in:

Cp = —2K2—-

p 2 Vx

that is: