Compressible Flow

Why it matters

Compressible flow is essential in understanding the behavior of gases at high velocities, such as in jet engines, rockets, and supersonic aircraft. It helps engineers design efficient propulsion systems and predict aerodynamic performance.

Key ideas

• Compressibility: Unlike incompressible flow, compressible flow considers changes in fluid density. This is significant at high speeds (typically above Mach 0.3).
• Mach Number (M): A dimensionless quantity representing the ratio of the speed of the flow to the speed of sound in the medium. It categorizes flow regimes: subsonic (M < 1), transonic (M ≈ 1), supersonic (M > 1), and hypersonic (M > 5).
• Isentropic Flow: An idealized flow where entropy remains constant. It simplifies analysis by assuming no heat transfer and reversible processes.
• Shock Waves: Discontinuities in pressure, temperature, and density occurring in supersonic flows. They are crucial in understanding drag and heat transfer in high-speed vehicles.
• Choked Flow: A condition where the flow rate is limited by the sonic condition at the throat of a nozzle, such that further lowering downstream pressure does not increase mass flow when upstream stagnation conditions, throat area and flow model remain fixed.

Formulas

• M = v / a
  • M: Mach number (dimensionless)
  • v: flow velocity (m/s)
  • a: speed of sound (m/s)
• a = sqrt(γ·R·T)
  • a: speed of sound (m/s)
  • γ: specific heat ratio (dimensionless)
  • R: specific gas constant (J/(kg·K))
  • T: temperature (K)
• p0/p = (1 + ((γ-1)/2)·M^2)^(γ/(γ-1))
  • p0/p: local isentropic stagnation-to-static pressure ratio for a calorically perfect gas (dimensionless)
  • γ: specific heat ratio (dimensionless)
  • M: local Mach number (dimensionless)

For a normal shock with upstream Mach number M₁ > 1 in a calorically perfect gas, the static-pressure ratio is p₂/p₁ = 1 + 2γ/(γ+1). It is different from the isentropic stagnation ratio. Across the shock entropy rises and stagnation pressure falls; stagnation temperature remains constant in the adiabatic no-work model. See NASA normal-shock equations and isentropic relations.

Worked example

Given: A jet flying at 340 m/s at an altitude where the temperature is 250 K. Calculate the Mach number.

1. Calculate the speed of sound (a):

   • Formula: a = sqrt(γ·R·T)
   • Values: γ = 1.4 (for air), R = 287 J/(kg·K), T = 250 K
   • Calculation: a = sqrt(1.4·287·250) = sqrt(100450) ≈ 317 m/s

2. Calculate the Mach number (M):

   • Formula: M = v / a
   • Values: v = 340 m/s, a = 317 m/s
   • Calculation: M = 340 / 317 ≈ 1.07

Final Answer: 1.07 (dimensionless)

Common mistakes

• Ignoring changes in density at high speeds, leading to incorrect assumptions about flow behavior.
• Miscalculating the speed of sound by using incorrect values for γ or R.
• Confusing subsonic and supersonic flow characteristics, especially in mixed flow regimes.

For GATE ME

Questions often involve calculating Mach numbers, analyzing shock waves, and understanding isentropic flow relations. Practice problems on choked flow and nozzle design are also common.

Quick check

1. What is the Mach number if a flow velocity is equal to the speed of sound?
2. Define choked flow in terms of Mach number.
3. What happens to pressure across a normal shock wave?

Answers: 1. 1 (Mach 1), 2. Mach number equals 1 at the throat, 3. Pressure increases across a normal shock wave.