Aviation Calcs

Prandtl–Meyer expansion

Prandtl–Meyer Expansion Calculator

Enter an upstream Mach number and a convex turning angle to get the downstream Mach after a Prandtl–Meyer expansion, plus the isentropic static pressure and temperature ratios. You can also evaluate the Prandtl–Meyer function ν(M) by itself. Total pressure and total temperature are constant across the fan.

I want to
degrees, convex / expansion

ν(M2) = ν(M1) + θ. The maximum turning from Mach 1 to M → ∞ is νmax = 130.45° for γ = 1.4. A compression (concave corner) is an oblique shock, not an expansion fan.

Downstream Mach

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The Prandtl–Meyer function

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

ν(1) = 0. An isentropic expansion through a turning angle θ raises Mach so that ν(M2) = ν(M1) + θ. For γ = 1.4, ν(2) is 26.380°, and a further 10° of expansion takes the flow to about M2 = 2.385. The maximum turning from Mach 1 all the way to M → ∞ is

νmax = 90° (√((γ+1)/(γ−1)) − 1) = 130.45° for γ = 1.4

Static ratios follow from the isentropic identities on this site’s isentropic flow calculator, with P0 and T0 constant across the fan, so P2/P1 = (P0/P1) / (P0/P2).

Sources

NACA Report 1135 (Prandtl–Meyer function); NASA Glenn; Anderson, Modern Compressible Flow. For a concave (compressive) turn, use the oblique shock calculator.

Frequently asked questions

What is ν(2) for air (γ = 1.4)?

26.380 degrees, the standard NACA 1135 / textbook value. This page’s function matches that figure, and inverting it recovers M = 2.

Can I use this for a compression turn?

No. A concave corner in supersonic flow produces an oblique shock, which is not isentropic. Use the oblique-shock calculator for that, and this page for a convex expansion.

What if I ask for more turn than ν_max allows?

The flow cannot expand past νmax (130.45° from Mach 1, or less if you start at a higher Mach). The page flags that case rather than inventing a vacuum solution.