Formula & Calculator
Critical Mach Number Relation
Freestream Mach number at which local flow first reaches sonic speed somewhere on the airfoil surface.
Interpretation
Critical Mach number relation: C_p,cr = (2/(γ·M_cr²))·(((1+((γ−1)/2)·M_cr²)/(1+(γ−1)/2))^(γ/(γ−1)) − 1). It gives the pressure coefficient at the critical Mach number where sonic flow first appears. Example: γ=1.4, M_cr≈0.72 → C_p,cr calculated from formula.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| C_p,cr | Critical pressure coefficient | |
| γ | Ratio of specific heats | |
| M_cr | Critical Mach number |
What it means
The critical Mach number is the free‑stream Mach number at which the local flow first reaches sonic speed (M=1) on the airfoil surface. The relation gives the corresponding pressure coefficient C_p,cr at that point. This is important for determining the drag divergence onset and for designing supercritical airfoils. The formula is derived from isentropic flow relations and the condition that the local Mach number equals 1. Understanding the critical Mach number helps engineers delay shock formation and reduce wave drag. The relation is used in airfoil design and in performance analysis of transonic aircraft.
Worked example
Critical Mach Number – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| M_cr | 0.6 |
| Parameter | Value |
|---|---|
| M_cr | 0.75 |
Common mistakes
- Critical Mach number relation: C_p,cr = (2/(γ·M_cr²)) · (((1+((γ−1)/2)·M_cr²)/(1+(γ−1)/2))^(γ/(γ−1)) − 1).
- Critical Mach number M_cr: When local flow reaches sonic (M=1) somewhere on the airfoil.
- C_p,cr: Pressure coefficient at that point.
- Determines onset of compressibility effects (drag rise).
- Solve iteratively – no closed‑form for M_cr.
Applications
The critical Mach number relation defines the Mach number at which sonic flow first appears on an airfoil. It is a key parameter for transonic aircraft design, as it marks the onset of shock waves and drag rise. Engineers use it to set cruise Mach numbers, to design supercritical airfoils that delay drag rise, and to predict performance penalties. The relation involves the critical pressure coefficient, which depends on the airfoil shape. By understanding critical Mach number, aerospace engineers can optimise wing design to achieve higher cruise speeds with acceptable drag penalties, improving fuel efficiency and range.
- Transonic airfoil design for commercial and military aircraft
- Determination of cruise Mach number for efficiency
- Drag divergence prediction and mitigation
- Design of swept wings to delay critical Mach effects
- Performance trade‑offs between speed and fuel consumption
Frequently Asked Questions
It gives the freestream Mach number at which the local flow first reaches sonic speed somewhere on the airfoil surface. This is important for determining when compressibility effects become significant.
Cp,cr = pressure coefficient at the point where M=1 locally
γ = specific heat ratio
Mcr = critical Mach number
It marks the onset of local sonic flow, which can lead to shock waves, wave drag, and buffet. Aircraft are often designed to have a high Mcr to delay these effects.
Given the incompressible pressure distribution (or Cp,cr), solve the relation iteratively. Alternatively, use the Prandtl‑Glauert correction to estimate.
- Confusing critical Mach number (onset of local sonic flow) with drag‑divergence Mach number (onset of large drag rise).
- Using the wrong value of γ.
- Assuming Mcr is independent of angle of attack.
For a given airfoil, the maximum suction peak has incompressible Cp,min = −0.5. Find Mcr using the Prandtl‑Glauert correction: Cp,cr = −0.5 / √(1−Mcr²). Also, the isentropic relation gives Cp,cr = (2/(γMcr²))·(((1+((γ−1)/2)Mcr²)/(1+(γ−1)/2))^(γ/(γ−1))−1). Solve iteratively; for γ=1.4, Mcr ≈ 0.7.
Thicker airfoils have higher suction peaks (more negative Cp), so Mcr decreases. Thin airfoils have higher Mcr.
Drag divergence occurs at a Mach number slightly higher than Mcr (typically MDD ≈ Mcr + 0.05).
They have flatter upper‑surface pressure distributions, reducing the suction peak and delaying the onset of sonic flow, thus increasing Mcr.