How this instrument works
The Prandtl-Meyer angle ν measures how far a supersonic stream has turned through smooth, isentropic expansion since it first crossed Mach 1. It is not the geometric angle of a wall or a corner — it is a property of the flow itself, produced by integrating the relation between Mach number and flow-turning along a Mach wave. Because that integral has a closed form only for a calorically perfect gas, the result comes out as the difference of two arctangent terms: one scaled by √((γ+1)/(γ−1)), the other the plain atan(√(M²−1)).
The second term is not arbitrary: atan(√(M²−1)) equals arccos(1/M), which is exactly 90° minus the Mach angle μ = asin(1/M), the angle a pressure disturbance's Mach cone makes with the flow. At M = 2, for instance, μ = 30° and that term works out to 60°. The first term grows faster, with γ built into its scaling factor, and the gap between the two is the net turn ν(M). Because the expression needs M²−1 to be non-negative, the formula — and supersonic flow itself — starts exactly at M = 1.
ν(1) is exactly zero: a sonic flow has not turned at all yet, which is the reference state the whole function is built from. At the other extreme, as M grows without bound the angle does not run away — it saturates. For air, with γ = 1.4, ν approaches 130.45° as M → ∞, the theoretical ceiling on how much any single expansion can turn a supersonic stream before it would have to expand into a vacuum. Real nozzle and corner designs stay well inside that limit, but it bounds every one of them.
- Enter the upstream Mach number in the Mach number field; it must exceed 1, since Prandtl-Meyer expansion is a purely supersonic phenomenon.
- Read the Prandtl-Meyer angle field in degrees — it is the cumulative turn the flow has made since Mach 1, not a wall or corner's physical angle.
- To size an expansion corner or nozzle contour, compute the Prandtl-Meyer angle at both the upstream and downstream Mach numbers and subtract them.
- That difference is the flow-deflection angle the wall geometry must supply to reach the target downstream Mach number.
Worked example — Mach 2.0 expanding around a corner
Enter M = 2.0, a common design Mach number for supersonic wind-tunnel test sections and inlet ramps, into the Mach number field. The instrument evaluates √6 · atan(√0.5) − atan(√3) = 1.507611 − 1.047198 = 0.460414 rad, reported in the Prandtl-Meyer angle field as 26.379761°, or about 26.4°.
That figure is what an aerospace engineer reaches for when sizing an expansion around a convex corner: turning flow from Mach 1.5 up to Mach 2.0 needs a wall deflection of ν(2.0) − ν(1.5) = 26.3798° − 11.9052° = 14.4746°, found by reading this same instrument twice and subtracting the two angles.
Questions
Why must the Mach number be greater than 1?
Because Prandtl-Meyer expansion only exists in supersonic flow. The formula needs √(M²−1), which is imaginary below Mach 1, and physically a subsonic stream has no Mach waves to define a turning angle against — it adjusts pressure through the whole field at once rather than along characteristic lines. The instrument enforces M > 1 and flags anything lower as invalid.
Is the Prandtl-Meyer angle the same as a corner's turning angle?
Only when the corner deflects the flow through exactly that much. ν(M) is a property of a single Mach number — the total turn a flow accumulates from Mach 1 up to M. To size an actual corner or nozzle section between two known Mach numbers, subtract: the deflection angle equals ν(M downstream) minus ν(M upstream), as in the Mach 1.5-to-2.0 example above.
Why does the formula use two arctangent terms instead of one?
Because the closed-form integral of the Prandtl-Meyer relation splits that way for a calorically perfect gas. The second term, atan(√(M²−1)), equals 90° minus the Mach angle asin(1/M); the first term, scaled by √((γ+1)/(γ−1)), carries the gas's specific-heat ratio. Their difference is the net angle — a consequence of how the integration resolves, not two separate physical effects.
What is the maximum possible Prandtl-Meyer angle?
For air, with γ = 1.4, ν approaches 130.45° as Mach number goes to infinity — the theoretical limit on how far a supersonic stream can turn before it would expand into a vacuum. No real nozzle or corner reaches that limit; it is a mathematical ceiling set by the formula's asymptote, useful mainly as a sanity check on extreme designs.
Does the ratio of specific heats change the result?
Yes. This instrument fixes γ at 1.4, the standard value for air and other diatomic gases near room temperature. Combustion products, monatomic gases, or air with vibrational modes excited at very high temperature carry a different γ, which shifts both the shape of ν(M) and its saturation angle — gases with a lower γ saturate at a smaller maximum turn than cold air does.
Who actually uses the Prandtl-Meyer function?
Aerospace engineers designing supersonic nozzles and expansion-fan flows use it routinely — the method of characteristics for a minimum-length nozzle contour is built on ν(M) values read at each point along the wall. It also explains why supersonic flow over a convex corner speeds up and cools smoothly rather than separating, the opposite of what a subsonic corner does.