Questions: Thermodynamic Relations in Compressible Flow

Question 1

A gas accelerates through an adiabatic converging nozzle, increasing from M = 0.3 to M = 0.7. What happens to the static temperature of the gas?

• A. It increases, because faster-moving gas carries more thermal energy
• B. It stays the same, because temperature is a state property independent of velocity
• C. It decreases, because kinetic energy increases at the expense of internal thermal energy
• D. It equals the stagnation temperature throughout the nozzle

Answer: (C)

The key is the conservation of stagnation enthalpy h₀ = h + V²/2 = cₚT + V²/2. Because the flow is adiabatic, h₀ (and therefore T₀) is constant. As the gas accelerates, V²/2 increases, so cₚT must decrease — static temperature falls. This is counterintuitive to students who associate 'more energy' with 'higher temperature,' but kinetic energy and thermal energy are both forms of energy that trade off within a fixed total budget. Option A reverses the trade-off; option D confuses static and stagnation temperature.

Question 2

Why can pressure disturbances not travel upstream in supersonic flow?

• A. Supersonic flow compresses the gas so much that pressure waves cannot form
• B. Viscosity increases at high Mach numbers, damping out all disturbances
• C. The flow velocity exceeds the local speed of sound, so pressure waves propagating upstream are swept downstream faster than they can advance
• D. Shocks at the inlet absorb all pressure information before it can propagate

Answer: (C)

Pressure disturbances propagate as sound waves at speed a relative to the fluid. Upstream propagation requires the wave to move against the flow. In subsonic flow (V < a), a wave traveling at speed a in the upstream direction still makes net upstream progress, so information reaches all parts of the flow. In supersonic flow (V > a), the flow sweeps any wave downstream faster than the wave can travel upstream — net upstream speed is negative. The flow is literally outrunning its own pressure signals. This is why supersonic flow cannot 'feel' downstream conditions and behaves fundamentally differently, leading to phenomena like shock waves.

Question 3

Stagnation temperature remains constant along a streamline in an adiabatic nozzle, even as the local flow velocity changes dramatically.

• T. True
• F. False

Answer: True

This is the direct consequence of energy conservation for adiabatic, no-work flow. The steady-flow energy equation gives h₀ = h + V²/2 = const along a streamline when there is no heat transfer and no shaft work. Because h = cₚT for a calorically perfect gas, T₀ = T + V²/(2cₚ) is also constant. The stagnation temperature represents the total thermal equivalent of the flow's energy — it can be partitioned between internal energy (static temperature) and kinetic energy differently at different locations, but the total is conserved.

Question 4

At M = 1 (sonic flow), the static temperature equals the stagnation temperature because most kinetic energy has been converted into thermal energy at the throat.

• T. True
• F. False

Answer: False

This is exactly backwards. At M = 1, the flow is moving at the speed of sound — it has significant kinetic energy. The isentropic relation gives T/T₀ = 2/(γ+1), which for air (γ = 1.4) yields T/T₀ = 0.833. Static temperature is about 17% lower than stagnation temperature at the sonic point. Static temperature equals stagnation temperature only at M = 0 (a fluid at rest), where all energy is internal. As M increases, an increasing fraction of the total energy is kinetic, so static temperature progressively drops below stagnation temperature.

Question 5

Why is Mach number — rather than flow speed alone — the key parameter governing compressible flow behavior?

Model answer: Mach number M = V/a compares flow speed to the local speed of sound, which is the speed at which pressure information propagates through the fluid. What matters physically is not how fast the gas moves in an absolute sense, but whether it moves faster or slower than its own pressure signals. Below M = 1, pressure disturbances can travel in all directions, the flow 'knows' about downstream conditions, and gradual adjustments occur. Above M = 1, no pressure information travels upstream, the flow is blind to what lies ahead, and fundamentally different phenomena (shock waves, expansion fans) emerge. Two flows at the same speed but different temperatures have different sound speeds and thus different Mach numbers — and completely different physics.

A concrete example: air at 300 m/s and 300 K has a ≈ 347 m/s, so M ≈ 0.86 (subsonic). The same 300 m/s air at 200 K has a ≈ 283 m/s, so M ≈ 1.06 (supersonic) — with shocks and discontinuous pressure changes. The speed is identical, but the physics is completely different because the Mach number crosses 1. This is why all the governing equations of compressible flow are written in terms of M, not V.