Prandtl Meyer Expansion Calculator

What Is a Prandtl-Meyer Expansion?

When supersonic flow passes over a surface that turns outward, the flow expands through a fan of waves called an expansion wave. Ludwig Prandtl and Theodor Meyer developed the theory that links upstream Mach number, deflection angle, and downstream Mach number for this isentropic process.

Prandtl-Meyer Function

$$\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}} \arctan\sqrt{\frac{(\gamma-1)(M^2-1)}{\gamma+1}} - \arctan\sqrt{M^2-1}$$

For an expansion wave, the downstream Prandtl-Meyer function equals the deflection angle plus the upstream value: $\nu(M_2) = \theta + \nu(M_1)$. The Mach angle is $\mu = \arcsin(1/M)$.

Example

For $M_1 = 1.5$, $\theta = 15°$, and air ($\gamma = 1.4$), the downstream Mach number is about 2.0. Expansion waves are isentropic, so total pressure stays constant through the fan.

Related tools: Mach Number Calculator.

Frequently Asked Questions

What is the Mach angle?

The Mach angle is arcsin(1/M). For M = 1.5 it is about 41.8 degrees. It defines the boundary of the Mach cone for a supersonic source.

Why must M be greater than 1?

Prandtl-Meyer expansion applies only to supersonic flow. Subsonic flow cannot sustain the expansion fan described by this theory.

What value of gamma should I use?

Use 1.4 for air at standard conditions. Hot exhaust gases or other fluids may need a different specific heat ratio.

Is an expansion wave isentropic?

Yes. Unlike shock waves, expansion fans are continuous and isentropic. Total pressure and entropy stay constant through the wave.

Where are expansion waves seen?

They appear at rocket nozzle exits, over expanded nozzles, and anywhere a supersonic stream turns away from a solid surface.

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What Is a Prandtl-Meyer Expansion?

Prandtl-Meyer Function

Example

Frequently Asked Questions

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