Formula and Method for Prandtl-Meyer Expansion
When a supersonic flow turns around a convex corner, it accelerates and expands through a continuous fan of infinitesimally weak Mach waves — a process first analyzed by Ludwig Prandtl and his student Theodor Meyer in 1908. Because each wave in the fan is isentropic, the whole expansion is reversible, and the flow's turning angle is tied to its Mach number by a single function: the Prandtl-Meyer function ν(M). This calculator uses that exact function to find the downstream Mach number, angle, and pressure/temperature ratios after a given expansion turn.
The Prandtl-Meyer function
ν(M) = √[(γ+1)/(γ-1)] · atan(√[(γ-1)/(γ+1) · (M² − 1)]) − atan(√(M² − 1)), where γ is the ratio of specific heats (γ = 1.4 for air) and M is the local Mach number. ν(1) = 0 by definition — a sonic flow has "turned" through zero degrees — and ν increases monotonically with M, approaching a finite maximum ν_max as M → ∞. A flow that starts at M₁ and turns through an additional angle θ around a convex corner ends up at the Mach number M₂ that satisfies ν(M₂) = ν(M₁) + θ. That single relationship is enough to solve the entire expansion-fan problem.
Solving for the downstream Mach number
ν(M) cannot be inverted algebraically to solve for M given ν, so this calculator first computes ν(M₁) from your inputs, adds the turn angle θ to get the target ν(M₂), then finds M₂ numerically using bisection — repeatedly narrowing a bracket on M until ν(M) matches the target to within machine precision. The same inputs also give the Mach angles μ = arcsin(1/M) at each station, and, through the isentropic relations p/p₀ = (1 + (γ−1)/2 · M²)^(−γ/(γ−1)) and T/T₀ = (1 + (γ−1)/2 · M²)^(−1), the static pressure ratio p₂/p₁ and temperature ratio T₂/T₁ across the fan.
Domain of validity and common mistakes
- M₁ must be ≥ 1: the Prandtl-Meyer function is only defined for supersonic flow; it has no meaning below Mach 1.
- θ cannot exceed ν_max − ν(M₁): beyond that limit the flow would need to reach M₂ = ∞ — expansion into a vacuum — which is a physical limit, not a calculator error.
- This models expansion, not shocks: a concave corner instead produces an oblique shock, with entropy and total-pressure loss, which needs shock relations rather than this isentropic fan-expansion model.
- Degrees vs. radians: ν and θ are entered and reported in degrees here, but the formula itself is naturally in radians — mixing the two mid-calculation is the most common hand-calculation error.
Real-world applications
- Designing supersonic nozzle contours (minimum-length nozzles) that expand flow smoothly through a Prandtl-Meyer fan instead of a compression shock.
- Predicting local pressure and Mach number on the leeward (expansion) side of a supersonic airfoil or wedge.
- Modeling under- or over-expanded rocket-exhaust plumes as they meet ambient pressure at altitude.
- Building and checking the M–ν–μ compressible-flow tables used throughout aerospace and gas-dynamics coursework.