Compressible Flow

Why it matters

Compressible flow is crucial in civil engineering for understanding the behavior of gases in high-speed applications, such as wind effects on structures and the design of ventilation systems. It is also essential for the analysis of gas pipelines and the performance of various hydraulic machines.

Key ideas

• Compressible Flow: This refers to fluid flow where the fluid density changes significantly. It is typically associated with gases rather than liquids.
• Mach Number (M): A dimensionless quantity representing the ratio of the speed of the flow to the speed of sound in the medium. It is a key parameter in compressible flow.
• Subsonic, Transonic, Supersonic, and Hypersonic Flows: These are classifications based on the Mach number. Subsonic (M < 1), Transonic (M ≈ 1), Supersonic (1 < M < 5), and Hypersonic (M > 5).
• Isentropic Flow: A flow that is both adiabatic and reversible, often used as an idealization in compressible flow analysis.
• Shock Waves: These occur in supersonic flows and are characterized by abrupt changes in pressure, temperature, and density.
• Choked Flow: A condition where the flow rate is limited by the sonic condition at the throat of a nozzle.

Formulas

• M = v / a
  • M: Mach number (dimensionless)
  • v: flow velocity (m/s)
  • a: speed of sound in the medium (m/s)
• For a calorically perfect ideal gas: 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)·M1^2)^(γ/(γ-1))
  • p0: isentropic stagnation (total) pressure (Pa)
  • p: local static pressure (Pa)
  • M1: local Mach number (dimensionless)
  • γ: specific heat ratio (dimensionless)

Worked example

Given:

• Local Mach number, M1 = 2.0
• Specific heat ratio, γ = 1.4

Find: Isentropic stagnation-to-static pressure ratio, p0/p

1. Use the isentropic flow relation for pressure: p0/p = (1 + ((γ - 1)/2)·M1^2)^(γ/(γ-1))
2. Substitute the given values: p0/p = (1 + ((1.4 - 1)/2)·2.0^2)^(1.4/(1.4-1))
3. Calculate: p0/p = (1 + 0.2·4)^(3.5) p0/p = (1 + 0.8)^(3.5) p0/p = 1.8^(3.5) p0/p ≈ 7.82445

Final Answer: 7.82445 (dimensionless). This is not an arbitrary downstream/upstream pressure ratio and does not apply across a shock.

Common mistakes

• Confusing the Mach number with velocity; remember that Mach number is a ratio.
• Incorrectly assuming incompressible flow conditions for gases at high speeds.
• Misapplying isentropic relations to flows with shock waves.

For GATE CE

• Questions often involve calculating Mach numbers, pressure ratios, and identifying flow regimes (subsonic, supersonic, etc.).
• Practice problems on isentropic flow relations and shock wave calculations.
• Understand the implications of choked flow in nozzles and ducts.

Quick check

1. What is the Mach number if the flow velocity is 340 m/s and the speed of sound is 340 m/s?
2. Define choked flow.
3. What happens to the pressure across a normal shock wave?

Answers: 1. 1 (dimensionless) 2. A condition where the flow rate is limited by the sonic condition at the throat of a nozzle. 3. The pressure increases across a normal shock wave.

Reference

NASA: Isentropic flow equations.