Problem 3.11

Air at a temperature of ^oC flows through a supersonic wind tunnel. A Schlieren photograph of the flow reveals weak waves originating at imperfections on the walls. These weak waves are at an angle of ^o to the flow. Find the air velocity in the wind tunnel.

Given:
\( T= \) -10^oC = 263 K,
Mach angle \( = \mu = \) 35^o.

To calculate: \( V=? \)

The problem description is schematically shown in Fig. 1.

Fig. 1: Schematic of the problem description

The Mach angle is given by the equation,

\[ \sin\left(\mu\right)=\frac{1}{M} \]

\[ M=\frac{1}{\sin\left(\mu\right)}=\frac{1}{\sin\left(35\right)} \]

\[ \boxed{M=1.74}\ . \]

The speed of sound ahead of the wedge can be calculated as,

\[ a=\sqrt{\gamma\,R\,T}=\sqrt{1.4\times287\times318} \]

The velocity can be calculated as,

\[ V=M\,a=1.74\times\sqrt{1.4\times287\times318} \]

\[ \boxed{V = 566.8\,\text{m/s}}\ . \]

Problem 3.11:

Solution: