Prandtl–Meyer Expansion Calculator (supersonic turn: ν, M₂, p₂/p₁)

Understanding the Prandtl–Meyer expansion calculator — concept and principle

This calculator finds what happens when a supersonic flow turns away from itself around a convex corner: the Prandtl–Meyer angles before and after, the new Mach number and the pressure, temperature and density ratios. It also gives the Mach angles and the largest further turn possible. It is used in supersonic aerodynamics and nozzle design courses.

Turning away from the flow opens an expansion fan: the gas speeds up while pressure and temperature fall.

How it is calculated

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

ν₂ = ν₁ + θ; M₂ solves ν(M₂) = ν₂ numerically

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

p₂/p₁ = (T₂/T₁)^(γ/(γ−1)); ρ₂/ρ₁ = (T₂/T₁)^(1/(γ−1))

Mach angle μ = arcsin(1/M)

The principle

When supersonic flow turns away from itself, it expands smoothly through a fan of weak Mach waves instead of a shock. Each wave turns the flow a tiny bit and speeds it up. The Prandtl–Meyer function is the total angle a sonic flow must turn to reach Mach M, so adding the wall's turning angle gives the new ν and hence the new Mach number. The process is isentropic, so pressure, temperature and density follow the standard isentropic relations. ν has a maximum, about 130.45° for γ = 1.4, corresponding to expansion to vacuum.

Worked example

Default inputs: M₁ = 2, θ = 10°, γ = 1.4.

ν₁ = ν(2) = 26.38°; ν₂ = 26.38 + 10 = 36.38°.
Solving ν(M₂) = 36.38° gives M₂ = 2.3849.
T₂/T₁ = 1.8 ÷ (1 + 0.2 × 2.3849²) = 0.8421.
p₂/p₁ = 0.548; ρ₂/ρ₁ = 0.6507.
Mach angles: μ₁ = 30°, μ₂ = 24.791°; largest further turn = 104.074°.

Each point is the calculator re-run with only “Turning angle of the wall θ (°)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Turning angle of the wall θ (°)	Downstream Mach number M₂
2.5	2.0919
5	2.1864
7.5	2.284
10	2.3849
12.5	2.4896
15	2.5984
20	2.8306

How to read the result

Expansion lowers pressure and temperature and raises Mach number, the opposite of a shock. Expansions are reversible, so unlike shocks they lose no total pressure. On a diamond airfoil at small angle, expansion fans on the rear faces and shocks on the front faces together produce lift and wave drag.

Common mistakes

Using the Prandtl–Meyer relation for compression turns, which form shocks.
Entering degrees where radians are needed in hand calculations.
Expecting the formulas to hold for subsonic flow.

Keep in mind: The relations assume a perfect gas and two-dimensional flow. Real nozzle and vehicle design needs full analysis by qualified engineers.

Frequently asked questions

Why does flow speed up around a convex corner?

The turning lets the flow expand, converting pressure into speed.

Is there a limit to the turn?

Yes. Beyond the maximum Prandtl–Meyer angle the flow would expand to vacuum.

Does an expansion fan lose total pressure?

No. It is isentropic.

Why are both Mach angles shown?

The fan is bounded by the Mach lines at the start and end of the turn.

Sources: NASA Glenn Research Center, Beginner's Guide to Aeronautics, "Prandtl-Meyer Angle" and "Expansion Fan"

Checked 9 Nov 2026. NASA Glenn Research Center, Beginner’s Guide to Aeronautics, “Prandtl-Meyer Angle” and “Expansion Fan” (ν function; ν₁ = ν₀ + turning angle; solve for the downstream Mach number). Isentropic ratios T₀/T = 1 + (γ−1)M²/2: standard formula.

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