For a given Prandtl-Meyer expansion, the upstream Mach number is 3 and the pressure ratio across the wave is \(p_{2} / p_{1}=0.4\). Calculate the angles of the forward and rearward Mach lines of the expansion fan relative to the free-stream direction.

Prandtl-Meyer Expansion:

1. \(M_{2}>M_{1}\). An expansion corner is a means to increase the flow Mach number.
2. \(p_{2} / p_{1}

3. The expansion fan itself is a continuous expansion region, composed of an infinite number of Mach waves, bounded upstream by \(\mu_{1}\) and downstream by \(\mu_{2}\) (see Fig. 4.32), where \(\mu_{1}=\arcsin \left(1 / M_{1}\right)\) and \(\mu_{2}=\arcsin \left(1 / M_{2}\right)\).
4. Streamlines through an expansion wave are smooth curved lines.
5. Since the expansion takes place through a continuous succession of Mach waves, and \(d s=0\) for each Mach wave, the expansion is isentropic.

Figure 4.32:

Transcribed Image Text:

My Pi 1 Forvard Mach line Rearward Mach line P2 102 H T