How to Use Prandtl–Meyer Expansion Calculator
The Prandtl–Meyer Expansion Calculator handles isentropic supersonic expansion fans around convex corners, solving in both directions: from a given turn angle θ to find downstream Mach M₂, and from a target M₂ back to the required turn angle.
- Enter upstream Mach number M₁ — must be supersonic (M₁ > 1); the expansion always accelerates the flow, increasing M.
- Enter specific-heat ratio γ — 1.4 for air, 1.67 for monatomic gases (He, Ar), 1.3 for CO₂.
- Select solve mode — “given θ, find M₂” or “given M₂, find θ”.
- Enter θ or M₂ as required; θ is the angle the flow turns through (positive for a convex expansion corner).
- Read ν(M₁), ν(M₂) and the resolved M₂ (or θ), plus the maximum possible expansion angle ν_max − ν(M₁) before the flow reaches vacuum.
- Check isentropic ratios — the panel shows p₂/p₁, T₂/T₁ and ρ₂/ρ₁ using isentropic flow relations at M₂.
Formula & Theory — Prandtl–Meyer Expansion Calculator
The Prandtl–Meyer Expansion Calculator evaluates the Prandtl–Meyer function ν(M) using the closed-form expression from gas dynamics:
ν(M) = √((γ+1)/(γ−1)) · arctan(√((γ−1)/(γ+1)·(M²−1))) − arctan(√(M²−1))
ν(M₂) = ν(M₁) + θ
ν_max = (π/2) · (√((γ+1)/(γ−1)) − 1) (vacuum limit, M → ∞)| Symbol | Meaning | SI Unit |
|---|---|---|
| ν(M) | Prandtl–Meyer function | ° or rad |
| M₁, M₂ | Upstream / downstream Mach numbers | — |
| θ | Flow turn angle (expansion) | ° |
| γ | Specific-heat ratio | — |
| ν_max | Maximum turning angle at M → ∞ | ° |
For air (γ = 1.4): ν_max ≈ 130.45°. Representative values: M = 2, ν ≈ 26.4°; M = 3, ν ≈ 49.8°; M = 5, ν ≈ 95.6°. The expansion is isentropic because each Mach wave in the fan is an infinitesimal turn; total temperature and total pressure are preserved across the entire fan.
Use Cases for Prandtl–Meyer Expansion Calculator
- Supersonic nozzle exit design — calculate the required wall-deflection angle to produce a target exit Mach number for wind-tunnel nozzle or rocket nozzle design.
- Diamond-foil and double-wedge airfoils — apply the shock–expansion method (oblique shock on the forward face, PM fan on the rear) to estimate wave drag and pressure distribution.
- Shock-expansion aerofoil theory — combine oblique shock results with Prandtl–Meyer fans to trace surface pressure coefficients on supersonic aerofoils analytically.
- Multi-ramp inlet analysis — evaluate the expansion fans between successive oblique shocks on multi-ramp supersonic inlets.
- Gasdynamic laser design — compute the rapid isentropic expansion required to achieve population inversion in a nitrogen or CO₂ gasdynamic laser cavity.
- Compressible-flow coursework — solve expansion-fan problems numerically and verify against graphical Mach-angle charts in standard gas dynamics textbooks.