Calculate Prandtl-Meyer expansion fan properties: downstream Mach number, isentropic pressure/temperature/density ratios, and PM function values.
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.
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.
ν(M) = √((γ+1)/(γ−1)) · arctan(√((M²−1)(γ−1)/(γ+1))) − arctan(√(M²−1)). Expansion: ν₂ = ν₁ + θ. Compression: ν₂ = ν₁ − θ. Isentropic: T₂/T₁ = (1 + (γ−1)/2·M₁²) / (1 + (γ−1)/2·M₂²). p₂/p₁ = (T₂/T₁)^(γ/(γ−1)). ρ₂/ρ₁ = (T₂/T₁)^(1/(γ−1)).
Result: M₂ = 2.385, p₂/p₁ = 0.574, T₂/T₁ = 0.837
ν₁(M=2) = 26.38°. After 10° expansion: ν₂ = 36.38°. Inverting: M₂ = 2.385. Isentropic ratios give p₂/p₁ = 0.574 (44% pressure drop).
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.
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.
Expansion fans are continuous, gradual processes with no entropy generation. Shock waves are discontinuous — they have finite thickness where viscous dissipation generates entropy irreversibly.
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).
No. Expansion fans only occur in supersonic flow. In subsonic flow, the pressure field communicates upstream, allowing smooth gradual acceleration without discrete wave structures.
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.
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°.
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.