Questions: Isentropic Flow with Area Change and Nozzles

Question 1

A supersonic flow (M = 2.0) enters a diverging duct section. What happens to the flow velocity?

• A. Velocity decreases — the diverging geometry slows the flow, as in an incompressible diffuser
• B. Velocity increases — at supersonic speeds, density falls so rapidly that mass continuity requires higher velocity in a larger area
• C. Velocity stays constant — the Mach number is already above 1, so the area change has no effect
• D. Velocity increases only if the flow is choked at a throat immediately upstream

Answer: (B)

At supersonic speeds (M > 1), the factor (M² − 1) in the area-velocity relation dA/A = (M² − 1)dV/V is positive, so area and velocity change in the same direction. A diverging duct increases area, which increases velocity. The physical reason: at supersonic speeds, density drops so steeply with velocity that even though each fluid parcel is moving faster, the lower density means more volume is needed to carry the same mass flux — hence the flow spreads into a larger cross-section. This directly contradicts low-speed intuition where narrowing accelerates flow.

Question 2

In a converging-diverging (de Laval) nozzle, where can the Mach number equal exactly 1 (sonic conditions)?

• A. At the inlet, where the flow velocity is lowest relative to the downstream section
• B. At any point in the converging section, depending on the inlet pressure ratio
• C. Only at the throat — the minimum-area cross-section — where dA = 0
• D. At the exit plane, where static pressure matches ambient and the flow is fully expanded

Answer: (C)

The area-velocity relation requires dA = 0 when M = 1 (since the factor becomes zero). This means sonic conditions can only occur at a local area extremum — in practical nozzles, this is the throat (minimum area). Flow at M < 1 throughout the converging section cannot reach M = 1 until it reaches the throat, and only does so if the pressure ratio is sufficient to choke the flow. The exit plane operates at some supersonic Mach number greater than 1 in a properly operating nozzle.

Question 3

A converging duct usually accelerates compressible flow, regardless of whether the incoming flow is subsonic or supersonic.

• T. True
• F. False

Answer: False

A converging duct accelerates subsonic flow (M < 1) because the area-velocity factor (M² − 1) is negative, making area decrease correspond to velocity increase. But for supersonic flow (M > 1), the factor is positive, so a converging duct decelerates the flow — the opposite effect. This counterintuitive result is one of the central insights of compressible flow theory and is why rocket nozzles use a converging-diverging geometry rather than a simple converging one to reach supersonic speeds.

Question 4

For a given area ratio A/A* in isentropic flow, there are exactly two possible Mach number solutions — one subsonic and one supersonic.

• T. True
• F. False

Answer: True

The isentropic area-Mach relation A/A* = f(M) is not monotonic — it decreases from infinity at M = 0, reaches a minimum of 1 at M = 1 (the throat), then increases again toward infinity as M → ∞. So any area ratio greater than 1 corresponds to two Mach numbers: one subsonic (on the decreasing branch) and one supersonic (on the increasing branch). Which solution applies in a physical nozzle depends on the downstream pressure conditions, not just the geometry. This duality is a critical design consideration.

Question 5

Why does supersonic flow accelerate in a diverging duct, contrary to everyday experience with water in a funnel or air in a subsonic diffuser?

Model answer: At subsonic speeds, density changes are negligible, so continuity (mass flux = density × velocity × area = constant) requires velocity to increase when area decreases (and vice versa). At supersonic speeds, the density drops steeply as velocity increases — in fact, the density effect dominates over the velocity effect. To maintain constant mass flux when velocity rises, the density falls so much that the flow must spread into a larger cross-sectional area to carry the same mass per second. Consequently, area and velocity change in the same direction at supersonic speeds.

Mathematically, this is captured by the sign of (M² − 1) in the area-velocity relation. Below M = 1 it is negative (area and velocity oppose each other); above M = 1 it is positive (they reinforce each other). The sonic condition M = 1 is a singular point where small area changes cause finite velocity changes — which is why the throat is the only location where M = 1 can be achieved.