Prandtl–Meyer Calculator
Expansion around a convex corner, solved from the turn or from either Mach number, at any γ.
Type into any white box. Upstream values set the incoming flow and hold the turn; downstream values set the outgoing flow and recompute the turn.
Results
| Quantity | Upstream before the corner | Downstream after the fan |
|---|
What this is computing
Turn a supersonic flow away from itself and it cannot do so across a single wave — it expands through a fan of Mach waves, each one infinitesimally weak. Nothing is discontinuous, so entropy is unchanged: an expansion is isentropic, which is exactly what a shock is not.
ν is the angle a flow must already have turned, starting from M = 1, to reach M. It is not a property you can measure at a point — only differences of it are physical.
A turn is therefore read off as a subtraction. The fan runs between Mach angles μ₁ and μ₂, and since M rises through it the fan opens rather than converges — which is why it never coalesces into a shock.
ν → M is the direction that matters
You are normally given the corner, not the answer: a wall that turns 15° sets θ, and M₂ has to come back out of ν. That inverse has no closed form and most calculators simply omit it, which is why this page solves it.
The turn runs out
ν rises towards a finite ceiling, so there is a largest angle any flow can turn through: from M₁ = 2 in air, 104.07° remains before the flow would need infinite Mach number. Turn a wall further than that and the flow cannot follow it — it separates, and a vacuum region opens between flow and wall.
γ moves the ceiling a long way
νmax is 130.45° for air, 159.20° for combustion products at γ = 1.3, and 89.97° for a helium tunnel at γ = 1.667. The gas decides how far a flow can be turned at all, not just by how much it accelerates.