Questions: Compressible Flow and Isentropic Flow Analysis

Question 1

An engineer needs to accelerate air from a subsonic inlet to a supersonic exit velocity. What nozzle geometry is required?

• A. A converging nozzle, because reducing the cross-sectional area always accelerates the flow
• B. A diverging nozzle, because supersonic flow requires expanding area to maintain momentum
• C. A converging-diverging nozzle: converge to reach M = 1 at the throat, then diverge to continue accelerating into the supersonic regime
• D. A constant-area duct, because once flow is initiated at the inlet velocity, it maintains that speed without geometric forcing

Answer: (C)

The area-velocity relationship in compressible flow reverses at M = 1. For subsonic flow (M < 1), a converging passage accelerates the flow — just as in incompressible flow. But for supersonic flow (M > 1), a converging passage decelerates it. To continuously accelerate from subsonic to supersonic, you need to pass through M = 1 at the throat (minimum area), then diverge to continue accelerating. This counterintuitive behavior arises because at supersonic speeds, density falls so rapidly with increasing velocity that a larger area is needed to pass the same mass flow rate. Option A is the classic error: assuming 'squeeze = accelerate' always holds.

Question 2

Downstream pressure is progressively reduced below the inlet pressure of a converging nozzle. Mass flow rate initially increases with each pressure reduction, but eventually further pressure reductions produce no additional flow. What causes this limit?

• A. The nozzle reaches its structural pressure limit, and the walls flex to reduce the effective flow area
• B. The flow reaches M = 1 at the throat (choked flow); once the throat is sonic, reducing downstream pressure cannot propagate upstream to increase mass flow
• C. Viscous losses at high velocities cause boundary layer separation, reducing the effective cross-section of the nozzle
• D. The fluid becomes incompressible at high flow rates, and incompressible flow cannot accelerate beyond a fixed maximum

Answer: (B)

Choked flow is the key. In subsonic flow, pressure information travels upstream at the speed of sound, allowing downstream conditions to influence the mass flow. When the throat reaches M = 1, the flow there equals the speed of sound — the local speed at which disturbances propagate. Any pressure change downstream cannot travel upstream past the sonic point. The throat is already at the condition that maximizes mass flow for the given upstream stagnation conditions; reducing downstream pressure further only changes the flow pattern downstream of the throat without altering what happens at or above it.

Question 3

A normal shock wave raises static pressure and static temperature across the shock but leaves stagnation temperature unchanged.

• T. True
• F. False

Answer: True

Across a normal shock, mass, momentum, and energy are conserved. Energy conservation means the total enthalpy (and thus stagnation temperature T₀) is unchanged — kinetic energy is converted to thermal energy but nothing is added or removed. Static temperature and pressure both jump sharply upward as the flow decelerates. However, the shock is irreversible (entropy increases), so stagnation pressure P₀ decreases across it, even though T₀ does not. This distinction — T₀ conserved, P₀ not — is important for engineering: lost stagnation pressure means lost work capacity, which is why inlets are designed to minimize shock strength.

Question 4

Reducing the exit pressure of a choked converging nozzle below the critical pressure ratio will increase the mass flow rate through the nozzle.

• T. True
• F. False

Answer: False

Once a converging nozzle is choked (M = 1 at the exit), the mass flow rate is fixed by the upstream stagnation conditions — it cannot be increased by further reducing downstream pressure. The sonic condition at the throat acts as a one-way valve: no downstream pressure signal can propagate upstream past the M = 1 point to increase velocity or mass flow there. Reducing downstream pressure beyond the choking condition only changes the flow structure downstream of the nozzle exit (e.g., forming expansion fans or oblique shocks), not the mass flow rate through it.

Question 5

Why does a converging passage accelerate subsonic flow but decelerate supersonic flow, and what does this imply about the nozzle geometry needed to achieve supersonic exit conditions?

Model answer: In subsonic flow, the continuity equation (ρAV = constant) behaves approximately as in incompressible flow: smaller area → higher velocity. In supersonic flow, density decreases so rapidly with increasing velocity that the density drop overwhelms the area decrease — to maintain constant mass flow, the area must actually increase as velocity increases. A converging passage therefore decelerates supersonic flow and accelerates subsonic flow. To go from subsonic to supersonic, the flow must pass through M = 1 at a minimum-area throat; beyond the throat, the passage must diverge to allow further supersonic acceleration. Hence a converging-diverging nozzle is required.

This is the central counterintuitive result of compressible flow. Students expecting 'squeeze = speed up' are correct for subsonic conditions but wrong for supersonic. The switchover at M = 1 is the critical point — not just mathematically but physically: it is the speed at which the way area changes translate to velocity changes reverses. Recognizing this reversal and its design implication (converging-diverging geometry) is the key to understanding compressible nozzle analysis and the design of rocket engines, supersonic wind tunnels, and jet inlets.