Prandtl-Meyer Expansion Calculator

About the Prandtl-Meyer Expansion Calculator

A Prandtl-Meyer expansion fan occurs when supersonic flow turns around a convex corner. Unlike shocks, which compress and generate entropy, expansion fans are isentropic and smoothly accelerate the flow while lowering pressure, temperature, and density. That makes the Prandtl-Meyer relation the standard tool for turning-angle calculations in supersonic flow. It is the standard way to turn a wedge angle into a Mach-number change and estimate the new flow state. That is the practical basis for nozzle turning calculations.

The Prandtl-Meyer function ν(M) relates turning angle to Mach number. For an expansion through angle θ, the downstream state satisfies ν₂ = ν₁ + θ, and the new Mach number is found by inverting the function numerically.

This calculator handles that inversion, then reports downstream Mach number, Mach angle, and isentropic property ratios. It is useful for supersonic nozzle contours, expansion around corners and trailing edges, rocket plumes, and shock-expansion approximations in compressible-flow analysis.

When This Page Helps

Use this calculator when you need to move from a supersonic turning angle to the downstream Mach state without manually inverting the Prandtl-Meyer function.

It is useful for nozzle design, supersonic aerodynamics, and classroom compressible-flow problems where the main difficulty is the function inversion, not the interpretation of the result.

How to Use the Inputs

1. Select expansion (convex turn) or isentropic compression mode.
2. Enter the upstream Mach number (> 1).
3. Enter the specific heat ratio γ (1.4 for air).
4. Enter the wall deflection angle in degrees.
5. Read the downstream Mach number and isentropic property ratios.
6. Use the reference table for standard ν(M) values.

Example Calculation

Tips & Best Practices

• The Prandtl-Meyer function is monotonically increasing — higher M always gives higher ν.
• For small deflections (θ < 5°), linear theory gives good approximations without the PM function.
• An centered expansion fan spreads over an angular range equal to the change in Mach angle: Δµ = µ₁ − µ₂.
• In real flows, boundary layers and viscosity smear out expansion fans, but the inviscid prediction is excellent.
• The PM function starts at ν = 0 for M = 1 — all supersonic flow comes from a sonic condition.

Practical Guidance

Prandtl-Meyer analysis is most useful for inviscid supersonic flow where the turning is smooth and the process stays isentropic. It gives a compact way to estimate how much the Mach number rises and how far the thermodynamic state drops as the flow expands.

Common Pitfalls

The most common mistake is applying expansion-fan logic to subsonic flow or to turns that actually create shocks. Another is forgetting that the isentropic property changes follow from the new Mach number after inversion, not just from the angle alone. If viscosity, boundary-layer growth, or strong geometric complexity matter, this result is only the first approximation.

Frequently Asked Questions

• Why are expansion fans isentropic but shocks are not?

  Expansion fans are continuous, gradual processes with no entropy generation. Shock waves are discontinuous — they have finite thickness where viscous dissipation generates entropy irreversibly.

• What is the maximum PM angle?

  For γ=1.4: ν_max = 130.45° at M→∞. This means supersonic flow can turn at most 130.45° through an expansion fan. In practice, the limit is the vacuum condition (zero pressure).

• Can expansion fans occur in subsonic flow?

  No. Expansion fans only occur in supersonic flow. In subsonic flow, the pressure field communicates upstream, allowing smooth gradual acceleration without discrete wave structures.

• What is shock-expansion theory?

  It combines oblique shocks (for compression surfaces) with PM expansions (for expansion surfaces) to analyze 2D supersonic flows around simple shapes like diamonds and wedges. It gives exact results for inviscid flow.

• How does γ affect PM expansion?

  Lower γ (more molecular degrees of freedom) increases ν_max. For γ=1.3 (CO₂ or high-temperature air), ν_max = 152.8°. For γ=1.667 (monatomic), ν_max = 103.1°.

• Where do expansion fans occur on an airfoil?

  On a supersonic airfoil, expansion fans form where the surface turns the flow away from itself, such as around trailing edges or convex sections after a compression region. They are the opposite of compression shocks and usually follow places where the surface bends outward.