Expansion Waves Questions and Answers

Answer: A

The turning is spread over an infinite number of vanishingly weak Mach waves radiating from the corner, which together form a centred Prandtl-Meyer fan. It cannot be concentrated into a single front, because an expansion discontinuity would require the entropy of the gas to fall.

Answer: B

Each Mach wave in the fan is infinitesimally weak, and the entropy rise across a weak wave goes as the cube of its strength, so the total rise through the fan is negligible. The expansion is therefore isentropic to an excellent approximation, and the ordinary isentropic relations may be used straight through it.

Answer: C

An expansion accelerates the flow, converting thermal energy into kinetic energy, so the static pressure, temperature and density all drop while the Mach number climbs. It is the exact opposite of a shock in every respect except one: the stagnation pressure survives, because no entropy is generated.

Answer: D

This is the essential contrast with a shock. Because the expansion generates no entropy, both stagnation pressure and stagnation temperature come through unaltered. A supersonic flow can therefore be expanded and re-compressed with loss occurring only in the compression.

Answer: A

The function is defined so that it is zero at Mach one and increases with Mach number. Its value at any Mach number is the total turning needed to expand a sonic stream up to that speed, which is what makes turning angles simply additive in it.

Answer: B

Turning angles add directly in the Prandtl-Meyer function, which is precisely why the function is tabulated. Look up the function at the upstream Mach number, add the turn, and read the downstream Mach number back out of the same table. No iteration is needed.

Answer: C

The function is defined to vanish at Mach one, which sets it up as a measure of turning accumulated from sonic conditions. It rises steeply at first and then flattens, so most of the available turning is used up over the first few Mach numbers.

Answer: D

The first characteristic of the fan is a Mach wave in the undisturbed upstream flow, so it lies at the upstream Mach angle. The trailing characteristic lies at the downstream Mach angle to the downstream flow direction, and since the flow has accelerated, that angle is smaller.

Answer: A

The characteristics fan out because each successive Mach wave sits in faster flow and so lies at a smaller angle. This divergence is the geometric expression of the expansion being spread out, and it contrasts with a compression corner, where converging characteristics coalesce into a shock.

Answer: B

Each successive compression wave sits in slower flow and so lies at a larger Mach angle, so the characteristics converge. Where they cross, they merge into a shock. This is the mirror image of the expansion case and explains why compressions are concentrated while expansions are spread.