Questions: Compressible Flow Basics

Question 1

An aircraft cruises at 250 m/s at high altitude where the air temperature is −57°C (216 K) and the speed of sound is approximately 295 m/s. What is the Mach number, and which flow equations are appropriate?

• A. Ma ≈ 0.85; compressible flow equations are required — density changes are significant above Ma 0.3
• B. Ma ≈ 0.73; incompressible equations are fine — the flow is subsonic
• C. Ma ≈ 0.85; incompressible Bernoulli applies — only supersonic flows require compressibility corrections
• D. Ma ≈ 0.73; compressible equations are always needed at high altitude regardless of Mach number

Answer: (A)

Ma = V/a = 250/295 ≈ 0.85. Since Ma > 0.3, density changes exceed 5% and the incompressible assumption introduces significant errors. Note that the speed of sound (295 m/s) is substantially lower than at sea level (≈340 m/s) because it depends on temperature: a = √(γRT). The same airspeed produces a higher Mach number at altitude than at sea level — which is why Mach number, not airspeed, is the correct parameter for compressibility.

Question 2

An engineer uses the incompressible Bernoulli equation (P + ½ρV² = const) to predict stagnation pressure in an airflow at Mach 0.9. What kind of error results?

• A. No error — Bernoulli's equation is exact for all adiabatic flows regardless of Mach number
• B. A negligible underestimate — density changes at Ma 0.9 are too small to matter in practice
• C. A significant underestimate of stagnation pressure — the incompressible form ignores the coupling between kinetic energy and internal energy that compressibility introduces
• D. A significant overestimate — the incompressible form predicts higher dynamic pressure than actually occurs

Answer: (C)

At Ma = 0.9, the incompressible Bernoulli equation underestimates stagnation pressure by roughly 6–7%. The incompressible form treats density as constant, so it captures only the mechanical pressure-velocity tradeoff. At high Mach numbers, compressing a gas also heats it (converting kinetic energy to internal energy), which is not captured by the simple ½ρV² term. The correct form requires integrating the compressible energy equation, yielding the isentropic stagnation relations.

Question 3

The local speed of sound decreases as air temperature decreases, so at high altitude an aircraft flying at the same airspeed as at sea level will have a higher Mach number.

• T. True
• F. False

Answer: True

a = √(γRT), so speed of sound is proportional to the square root of absolute temperature. At 12 km altitude, temperature drops to about 216 K compared to ~288 K at sea level, reducing the speed of sound from ~340 m/s to ~295 m/s. An aircraft at 250 m/s is at Ma ≈ 0.73 at sea level but Ma ≈ 0.85 at altitude — closer to sonic conditions and well into the regime where compressibility corrections are significant.

Question 4

A compressible flow should be turbulent, because the density fluctuations that define compressibility require chaotic velocity patterns.

• T. True
• F. False

Answer: False

Compressibility and turbulence are independent phenomena. Compressibility refers to significant density variation in response to pressure changes — it is a Mach number effect. Turbulence refers to chaotic, irregular velocity fluctuations — it is a Reynolds number effect. A laminar supersonic flow in a converging-diverging nozzle is compressible but not turbulent. A turbulent low-speed airflow over a car is turbulent but essentially incompressible. The two can occur together, but neither requires the other.

Question 5

Why does the incompressible form of Bernoulli's equation become invalid at high Mach numbers, and what physical process does it fail to account for?

Model answer: Incompressible Bernoulli assumes density is constant, so it only models the exchange between pressure and kinetic energy (½ρV²). At high Mach numbers, accelerating a gas to near-sonic speeds compresses it — density decreases and temperature drops — and decelerating it to stagnation conditions heats it. This coupling between kinetic energy and internal energy (thermodynamic energy) is not captured by the simple ½ρV² term. The correct compressible treatment couples the momentum, continuity, and energy equations and uses isentropic relations.

The practical threshold is Ma ≈ 0.3: below this, density changes are less than 5% and incompressible Bernoulli is accurate enough. Above Ma 0.3, the incompressible form systematically underestimates stagnation pressure — an error that matters for pitot tube calibration, nozzle design, and turbomachinery.