When supersonic flow turns around a convex corner it expands smoothly through a fan of Mach waves, speeding up and losing pressure – the opposite of a shock. The Prandtl–Meyer function relates the turning angle to the Mach number.
How to use the Prandtl–Meyer Expansion Calculator
- Upstream Mach number M₁
- Turning angle θ
- Heat capacity ratio γ
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the upstream Mach number, the turning angle and γ. M₂ is found by solving the Prandtl–Meyer function numerically.
Inputs explained
| Field | What to enter |
|---|---|
| Upstream Mach number M₁ | A number |
| Turning angle θ | Units: °, rad, arcmin, arcsec, mrad |
| Heat capacity ratio γ | A number |
What each result means
- Prandtl–Meyer angle ν₂ = [Prandtl–Meyer angle ν₁] + [Turning angle θ]
- Downstream Mach number M₂ = [m12]
- Upstream Mach angle μ₁ = deg(asin(1÷[Upstream Mach number M₁]))
- Downstream Mach angle μ₂ = deg(asin(1÷[Downstream Mach number M₂]))
- Temperature ratio T₂/T₁ = (1 + ([Heat capacity ratio γ] – 1)÷2×[Upstream Mach number M₁]^2)÷(1 + ([Heat capacity ratio γ] – 1)÷2×[Downstream Mach number M₂]^2)
- Pressure ratio p₂/p₁ = [Temperature ratio T₂÷T₁]^([Heat capacity ratio γ]÷([Heat capacity ratio γ] – 1))
- Density ratio ρ₂/ρ₁ = [Temperature ratio T₂÷T₁]^(1÷([Heat capacity ratio γ] – 1))
- Maximum turning angle = deg(pi÷2×(sqrt(([Heat capacity ratio γ] + 1)÷([Heat capacity ratio γ] – 1)) – 1)) – [Prandtl–Meyer angle ν₁]
Example values
How the main result changes when Upstream Mach number M₁ changes and everything else stays at the example values above:
| Upstream Mach number M₁ | Downstream Mach number M₂ |
|---|---|
| 1.5 | 1.841 |
| 2 | 2.3849 |
| 2.5 | 2.9674 |
| 3 | 3.5783 |
| 4 | 4.881 |
Formula
ν(M) = √((γ+1)/(γ−1))·atan(√((γ−1)/(γ+1)·(M² − 1))) − atan(√(M² − 1)), ν₂ = ν₁ + θ
p₂/p₁ = [(1 + (γ−1)/2·M₁²)/(1 + (γ−1)/2·M₂²)]^{γ/(γ−1)}
Worked example
Inputs — Upstream Mach number M₁: 2; Turning angle θ: 10 °; Heat capacity ratio γ: 1.4.
Results — Prandtl–Meyer angle ν₁ = 26.37976 °; Prandtl–Meyer angle ν₂ = 36.37976 °; Downstream Mach number M₂ = 2.38489; Upstream Mach angle μ₁ = 30 °; Downstream Mach angle μ₂ = 24.7908 °; Temperature ratio T₂/T₁ = 0.84209; Pressure ratio p₂/p₁ = 0.54797; Density ratio ρ₂/ρ₁ = 0.65072; Maximum turning angle = 104.0743 °.
Key terms
- Expansion fan
- A continuous series of Mach waves turning supersonic flow.
Frequently asked questions
Is an expansion fan isentropic?
Yes – unlike shocks, it involves no loss of total pressure.
What is the maximum turning angle?
About 130.5° from Mach 1 for γ = 1.4 (expansion to vacuum).
What is a Mach angle?
μ = arcsin(1/M), the angle of Mach waves.
Where do expansion fans occur?
Around the rear of supersonic wings and nozzle exits.