Prandtl-Meyer Expansion Fan Calculator

📝 Configuration

Solve For Unknown:

Upstream Mach number ($M_1$):

Deflection Angle ($\theta$) [degrees]:

Downstream Mach number ($M_2$):

Specific Heat Ratio ($\gamma$):

Upstream Temp ($T_1$) [K] — Optional:

Upstream Pressure ($P_1$) [kPa] — Optional:

Theoretical Limits:
• Max turning angle ($\nu_{max}$): 130.45°
• Supersonic flow required ($M > 1.0$)

Quick Presets:

📊 Results & Diagrams

Configure inputs and click Calculate Expansion Fan to view results.

📘 Calculation Methodology

Mathematical Model & Theory

Prandtl-Meyer expansion fans occur when supersonic flow turns around a convex corner. The flow expands isentropically through a continuous fan of Mach lines. The Prandtl-Meyer function $\nu(M)$ represents the angle through which a sonic flow ($M=1$) must expand to reach Mach $M$:

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

The downstream state is determined by the relation:

$$\nu(M_2) = \nu(M_1) + \theta$$

Since the relation is implicit, a **Newton-Raphson iteration** is used to find $M_2$ from $\nu(M_2)$ using the derivative:

$$\frac{d\nu}{dM} = \frac{\sqrt{M^2-1}}{M \left(1 + \frac{\gamma-1}{2} M^2\right)}$$

Academic References

Anderson, John D.: Modern Compressible Flow: With Historical Perspective, McGraw-Hill.
Zucrow, M. J., Hoffman, J. D.: Gas Dynamics, Wiley.

Worked Engineering Example

Problem Statement:
Supersonic air flow ($\gamma = 1.4$) at $M_1 = 2.0$ expands over a corner deflection of $\theta = 15^\circ$. Compute the downstream Mach number $M_2$, static pressure ratio $P_2/P_1$, and static temperature ratio $T_2/T_1$.

Step-by-step Solution:
1. Calculate the upstream Prandtl-Meyer angle $\nu(M_1)$:
$$\nu(2.0) = \sqrt{6} \arctan\left(\frac{\sqrt{3}}{\sqrt{6}}\right) - \arctan(\sqrt{3}) = 26.38^\circ$$ 2. Calculate the target downstream Prandtl-Meyer angle $\nu(M_2)$:
$$\nu(M_2) = \nu(M_1) + \theta = 26.38^\circ + 15^\circ = 41.38^\circ$$ 3. Invert the Prandtl-Meyer function numerically for $\nu(M_2) = 41.38^\circ$ (converted to radians) using Newton-Raphson:
$$M_2 = 2.60$$ 4. Apply isentropic relation ratios:
$$\frac{T_2}{T_1} = \frac{1 + 0.2 \times (2.0)^2}{1 + 0.2 \times (2.60)^2} = 0.7658$$ $$\frac{P_2}{P_1} = (0.7658)^{3.5} = 0.3931$$ Final Results:
• Downstream Mach: 2.60
• Temperature Ratio: 0.7658
• Pressure Ratio: 0.3931