Prandtl-Meyer Expansion Calculator
Mach 2 air expanding round a 10° corner accelerates to Mach 2.38489, with ν rising from 26.38° to 36.38°.
Air 1.4; hot combustion gases about 1.3; helium 1.667.
2.38488715459307
Full answer, then cut to fewer decimals (not rounded):
- 2.38488715459307
- 2.3848871545930
- 2.384887154593
- 2.38488715459
- 2.3848871545
- 2.384887154
- 2.38488715
- 2.3848871
- 2.384887
- 2.38488
- 2.3848
- 2.384
- 2.38
- 2.3
- ν₁
- 26.37976081 °
- ν₂ = ν₁ + θ
- 36.37976081 °
- Pressure ratio p₂/p₁
- 0.5479687313
- Temperature ratio T₂/T₁
- 0.8420905495
Step-by-step solution
- ν₁ = ν(2) = 26.37976081°
- ν₂ = ν₁ + θ = 36.37976081° → M₂ = 2.384887155
- p₂/p₁ = (p₂/p₀)/(p₁/p₀) = 0.5479687313 (isentropic expansion)
Prandtl-Meyer Expansion Formula
ν(M) = √((γ+1)/(γ−1)) arctan √((γ−1)(M²−1)/(γ+1)) − arctan √(M² − 1)
expansion by θ: ν(M₂) = ν(M₁) + θ, isentropic
Worked Example
Mach 2 air expanding round a 10° corner:
- ν₁ = ν(2) = 26.37976081°
- ν₂ = ν₁ + θ = 36.37976081° → M₂ = 2.384887155
- p₂/p₁ = (p₂/p₀)/(p₁/p₀) = 0.5479687313 (isentropic expansion)
ν₂ = ν₁ + θ: 36.37976081 ° · Pressure ratio p₂/p₁: 0.5479687313 · Temperature ratio T₂/T₁: 0.8420905495
GATE tip
The same function gives compression turns only for very small angles; larger compressions form shocks. Under-expanded nozzle exits turn the jet outward through a Prandtl–Meyer fan.
More Compressible Flow Calculators
-
Isentropic Flow Calculator
T₀/T = 1 + (γ − 1)M²/2, p₀/p = (T₀/T)^(γ/(γ−1))
A/A* = (1/M)[(2/(γ+1))(1 + (γ−1)M²/2)]^((γ+1)/(2(γ−1)))
Example: Air (γ = 1.4) at Mach 2: Pressure ratio p/p₀ = 0.127805.
-
Normal Shock Calculator
M₂² = (1 + (γ−1)M₁²/2) / (γM₁² − (γ−1)/2)
p₂/p₁ = 1 + 2γ(M₁² − 1)/(γ + 1); ρ₂/ρ₁ = (γ+1)M₁²/((γ−1)M₁² + 2)
Example: A normal shock in air at M₁ = 2: Downstream Mach number M₂ = 0.57735.
-
Oblique Shock Calculator
tan θ = 2 cot β (M₁² sin²β − 1) / (M₁²(γ + cos 2β) + 2)
normal component M₁ sin β obeys the normal-shock relations; M₂ = Mₙ₂ / sin(β − θ)
Example: Mach 2 flow turned by a 10° wedge: Shock wave angle β (weak) = 39.3139 °.
-
Nozzle Flow Calculator
ṁ = A* p₀ √(γ/(RT₀)) (2/(γ+1))^((γ+1)/(2(γ−1)))
exit Mach from Aₑ/A* (supersonic branch); vₑ = Mₑ√(γRTₑ); F = ṁvₑ + (pₑ − pₐ)Aₑ
Example: A rocket nozzle with 7 MPa, 3000 K chamber, 100 cm² throat and expansion ratio 10, in vacuum: Mass flow rate ṁ = 44.3031 kg/s.
Frequently asked questions
Why is an expansion fan isentropic?
It is a continuous series of Mach waves, each infinitesimally weak, so no entropy is produced, unlike a shock.
What is the maximum turning angle?
ν_max = 90°(√((γ+1)/(γ−1)) − 1) = 130.45° for air, the turn that would expand the flow to vacuum.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Prandtl-Meyer Expansion Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/prandtl-meyer-calculator/
IEEE
Scientific Converters, "Prandtl-Meyer Expansion Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/prandtl-meyer-calculator/