# Prandtl–Meyer Expansion Calculator

URL: https://speedycalculator.com/calculators/physics/prandtl-meyer-expansion/
Last updated: 2026-10-30

> The Prandtl–Meyer expansion calculator finds the downstream Mach number, Prandtl–Meyer angles, Mach angles and pressure, temperature and density ratios when supersonic flow turns away from itself through an expansion fan.

## Inputs

- Upstream Mach number M₁
- Turning angle θ (°, rad, arcmin, arcsec, mrad)
- Heat capacity ratio γ

## Results

- Prandtl–Meyer angle ν₁
- Prandtl–Meyer angle ν₂
- Downstream Mach number M₂
- Upstream Mach angle μ₁
- Downstream Mach angle μ₂
- Temperature ratio T₂/T₁
- Pressure ratio p₂/p₁
- Density ratio ρ₂/ρ₁
- Maximum turning angle

## Formula

```
ν(M) = √((γ+1)/(γ−1))·atan(√((γ−1)/(γ+1)·(M² − 1))) − atan(√(M² − 1)),  ν₂ = ν₁ + θ
p₂/p₁ = [(1 + (γ−1)/2·M₁²)/(1 + (γ−1)/2·M₂²)]^{γ/(γ−1)}
```

## Explanation

When supersonic flow turns around a convex corner it expands smoothly through a fan of Mach waves, speeding up and losing pressure – the opposite of a shock. The Prandtl–Meyer function relates the turning angle to the Mach number.

### How to use the Prandtl–Meyer Expansion Calculator

- **Upstream Mach number M₁**
- **Turning angle θ**
- **Heat capacity ratio γ**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the upstream Mach number, the turning angle and γ. M₂ is found by solving the Prandtl–Meyer function numerically.

### Inputs explained

FieldWhat to enterUpstream Mach number M₁A numberTurning angle θUnits: °, rad, arcmin, arcsec, mradHeat capacity ratio γA number

### What each result means

- **Prandtl–Meyer angle ν₂** = [Prandtl–Meyer angle ν₁] + [Turning angle θ]
- **Downstream Mach number M₂** = [m12]
- **Upstream Mach angle μ₁** = deg(asin(1÷[Upstream Mach number M₁]))
- **Downstream Mach angle μ₂** = deg(asin(1÷[Downstream Mach number M₂]))
- **Temperature ratio T₂/T₁** = (1 + ([Heat capacity ratio γ] - 1)÷2×[Upstream Mach number M₁]^2)÷(1 + ([Heat capacity ratio γ] - 1)÷2×[Downstream Mach number M₂]^2)
- **Pressure ratio p₂/p₁** = [Temperature ratio T₂÷T₁]^([Heat capacity ratio γ]÷([Heat capacity ratio γ] - 1))
- **Density ratio ρ₂/ρ₁** = [Temperature ratio T₂÷T₁]^(1÷([Heat capacity ratio γ] - 1))
- **Maximum turning angle** = deg(pi÷2×(sqrt(([Heat capacity ratio γ] + 1)÷([Heat capacity ratio γ] - 1)) - 1)) - [Prandtl–Meyer angle ν₁]

### Example values

How the main result changes when **Upstream Mach number M₁** changes and everything else stays at the example values above:

Upstream Mach number M₁Downstream Mach number M₂1.51.84122.38492.52.967433.578344.881

## Worked example

Inputs — Upstream Mach number M₁: 2; Turning angle θ: 10 °; Heat capacity ratio γ: 1.4.

Results — Prandtl–Meyer angle ν₁ = 26.37976 °; Prandtl–Meyer angle ν₂ = 36.37976 °; **Downstream Mach number M₂ = 2.38489**; Upstream Mach angle μ₁ = 30 °; Downstream Mach angle μ₂ = 24.7908 °; Temperature ratio T₂/T₁ = 0.84209; Pressure ratio p₂/p₁ = 0.54797; Density ratio ρ₂/ρ₁ = 0.65072; Maximum turning angle = 104.0743 °.

## Key terms

- **Expansion fan**: A continuous series of Mach waves turning supersonic flow.

## FAQ

**Is an expansion fan isentropic?**
Yes – unlike shocks, it involves no loss of total pressure.

**What is the maximum turning angle?**
About 130.5° from Mach 1 for γ = 1.4 (expansion to vacuum).

**What is a Mach angle?**
μ = arcsin(1/M), the angle of Mach waves.

**Where do expansion fans occur?**
Around the rear of supersonic wings and nozzle exits.


## Sources

- [NASA Glenn – Beginner's Guide to Aeronautics](https://www.grc.nasa.gov/www/k-12/airplane/)
- [NACA Report 1135 – Equations, tables and charts for compressible flow](https://ntrs.nasa.gov/citations/19930091059)