Prandtl-Glauert Rule for Supersonic Flow: Versions I and II

In Section 9.13, we have seen the similarity rules for subsonic flows. Now let us examine the similarity rules for supersonic flows. We can visualize from our previous discussions on similarity rule for subsonic compressible flows that the factor K1 in the transformation Equation (9.79) should have the following relations depending on the flow regime:

K1 = J1 — М2 for subsonic flow

K1 = JМ2, — 1 for supersonic flow.

Therefore, in general, we can write:

K = y/11 — Ml. (9.92)

However, there is one important difference between the treatment of supersonic flow and subsonic flow, that is, we cannot find any incompressible flow in the supersonic flow regime.

9.14.1 Subsonic Flow

We know that for subsonic flow the transformation relations are given by Equation (9.79) as:

xlnc x, zlnc K1z, ф К2фІЖ.

The transformed equation is:

K2[(1 — ML) (Фхх)іпє + = 0.

and the condition to be satisfied by this equation in order to be identical to Equation (9.78) is:

K1 = л/Г—Ml.

For this case the above transformed equation becomes Laplace equation.

9.14.2 Supersonic Flow

The transformation relations for supersonic flow are:

x’ = x, z’ = K1z, Ф = K2 ф,

where the variables with “prime” are the transformed variables. The aim in writing these transformations is to make the Mach number Mx in the governing equation (9.77) to vanish.

With the above transformation relations, the governing equation becomes:

K2[(1 – МІ)ф’хх + K2ф[z] = 0.

For supersonic flow, MOT > 1, therefore the above equation becomes:

K2M – 1Wxx – K2ФУ = 0.

By inspection of this equation, we can see that the Mach number MOT can be eliminated from the above equation with:

K1 = V7 ML – 1.

The equation becomes:

4>’xx – <P’zz = 0. (9’93a)

Now we must find out as to which supersonic Mach number this flow belongs.

The original form of the governing differential equation for this kind of flow, given by Equation (9.77), is:

(m2 – 1^xx – Фzz = 0. (9.93b)

For Equations (9.93a) and (9.93b) to be identical, it is necessary that:

M^ = fil.

By following the arguments of P-G rule for subsonic compressible flow, we can show the following results for versions I and II of the Prandtl-Glauert rule for supersonic flow.