Prandtl-Meyer Expansion Calculator

Isentropic supersonic expansion around a convex corner. Computes M₂ via the Prandtl-Meyer function, isentropic property ratios, and Mach angles.

Compressible Flow · Expansion

Gas presets — click to fill γ

Inputs

Upstream Mach number M₁ (dimensionless)

Specific heat ratio γ (dimensionless)

c_p/c_v — use gas presets above

Flow deflection (turning) angle θ (degrees)

Theory

Prandtl-Meyer function ν(M):

ν(M) = √((γ+1)/(γ−1)) × arctan(√((γ−1)/(γ+1)×(M²−1))) − arctan(√(M²−1))

ν₂ = ν₁ + θ [expansion relation]

ν_max = 90°×(√((γ+1)/(γ−1)) − 1) [vacuum limit]

Isentropic property ratios (P₀, T₀ conserved):

T₂/T₁ = (1+(γ−1)/2·M₁²) / (1+(γ−1)/2·M₂²)

P₂/P₁ = (T₂/T₁)^γ/(γ−1)

ρ₂/ρ₁ = (T₂/T₁)^1/(γ−1)

Where:

ν(M) = Prandtl-Meyer function [degrees] — angle through which a flow at M = 1 must expand to reach M
θ = flow turning (deflection) angle at the convex corner [degrees]
μ = arcsin(1/M) — Mach angle [degrees]; the expansion fan spans from μ₁ to μ₂
ν_max = 90°×(√((γ+1)/(γ−1))−1) — for air ≈ 130.45°; represents expansion to vacuum (P → 0)

Prandtl-Meyer table — air (γ = 1.4):

M	ν [deg]	μ [deg]	P/P₀	T/T₀
1.0	0.000	90.00	0.5283	0.8333
1.5	11.91	41.81	0.2724	0.6897
2.0	26.38	30.00	0.1278	0.5556
2.5	39.12	23.58	0.0585	0.4444
3.0	49.76	19.47	0.0272	0.3571
4.0	65.78	14.48	0.0066	0.2381
5.0	76.92	11.54	0.0019	0.1667
∞	130.45	0.00	0	0

Prandtl-Meyer expansion fans occur at convex corners in supersonic flow where the flow turns away from itself. The process is isentropic — total pressure and total temperature are conserved, so no entropy is generated. This contrasts with oblique shocks (concave corners) which are non-isentropic. The expansion always increases both M and the Mach angle span of the fan.