Prandtl-Meyer Expansion Calculator — ν(M) Function

🌊 Expansion Fan Schematic

📝 Configuration

🌬️ Gas Selection

Gas Preset:

🌊 Expansion Conditions

Upstream Mach Number ($M_1$):

Turning Angle ($\delta$) [deg]:

Specific Heat Ratio ($\gamma$):

📐 Upstream State

Upstream Pressure ($p_1$) [Pa]:

Upstream Temperature ($T_1$) [K]:

Gas Constant ($R$) [J/kg·K]:

Key Equations:

Prandtl-Meyer function:
$\nu(M) = \sqrt{\dfrac{\gamma+1}{\gamma-1}}\;\arctan\!\sqrt{\dfrac{\gamma-1}{\gamma+1}(M^2-1)}\;-\;\arctan\!\sqrt{M^2-1}$

Expansion fan:
$\nu_2 = \nu_1 + \delta$

Maximum turning:
$\nu_{\max} = \dfrac{\pi}{2}\left(\sqrt{\dfrac{\gamma+1}{\gamma-1}} - 1\right)$

Isentropic ratios:
$\dfrac{p_2}{p_1} = \left[\dfrac{1 + \frac{\gamma-1}{2}M_1^2}{1 + \frac{\gamma-1}{2}M_2^2}\right]^{\!\gamma/(\gamma-1)}$

📊 Results & Visualization

Configure inputs and click Calculate to view results.

ℹ️ About Prandtl-Meyer Expansion

When a supersonic flow encounters a convex corner (expansion corner), it accelerates through a continuous, isentropic expansion fan — a centered wave of infinite Mach lines diverging from the corner.

Key properties:
- The process is isentropic: total pressure and temperature are conserved.
- The downstream Mach number is always higher than upstream.
- Static pressure, temperature, and density all decrease.
- The Prandtl-Meyer function ν(M) is a monotonically increasing function used to relate upstream and downstream Mach numbers.
- A maximum turning angle ν_max exists (≈ 130.45° for γ = 1.4) corresponding to M → ∞.

📘 Calculation Methodology

Mathematical Model & Theory

The Prandtl-Meyer function describes the relationship between Mach number and the turning angle through an isentropic expansion fan. It is derived from the characteristic equations of supersonic flow.

$$\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\arctan\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \arctan\sqrt{M^2-1}$$ $$\nu_2 = \nu_1 + \delta \quad \Rightarrow \quad M_2 = \nu^{-1}(\nu_2)$$ $$\nu_{\max} = \frac{\pi}{2}\left(\sqrt{\frac{\gamma+1}{\gamma-1}} - 1\right)$$ $$\frac{d\nu}{dM} = \frac{\sqrt{M^2-1}}{M\left(1 + \frac{\gamma-1}{2}M^2\right)}$$

Inversion of ν(M₂) = ν₂ is performed via Newton-Raphson iteration using the analytical derivative above (typically < 10 iterations to machine precision).

Worked Engineering Example

Problem:
Air (γ = 1.4) at M₁ = 2.0 turns through a 10° expansion corner. Find M₂ and the isentropic pressure ratio.

Step-by-step:
1. Compute ν₁ = ν(2.0):
$\nu_1 = \sqrt{6}\;\arctan\!\sqrt{\tfrac{3}{6}} - \arctan\!\sqrt{3} = 26.3798°$

2. ν₂ = 26.3798° + 10° = 36.3798°

3. Invert ν(M₂) = 36.3798° via Newton-Raphson:
$M_2 = 2.3854$

4. Isentropic pressure ratio:
$\frac{p_2}{p_1} = \left[\frac{1 + 0.2(4)}{1 + 0.2(5.690)}\right]^{3.5} = 0.5685$

Result: M₂ = 2.385, p₂/p₁ = 0.569 (pressure drops, flow accelerates).

Assumptions & References

Assumptions: Steady, two-dimensional flow. Calorically perfect gas (γ = const). Isentropic process (no shocks within the fan). Sharp convex corner (centered expansion). Uniform upstream conditions.

References:

Anderson, J. D. Modern Compressible Flow, McGraw-Hill, 4th ed. — Ch. 4, §4.14 (Prandtl-Meyer Expansion Waves).
Shapiro, A. H. The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1, Wiley, 1953.
Gupta, S. C. Applied Computational Fluid Dynamics — §7.11, Eq. 7.175.
NACA Report 1135 — Equations, Tables, and Charts for Compressible Flow, 1953.

ThermoFluidCalc Report

🌊 Prandtl-Meyer Expansion Calculator — ν(M) Function