Prandtl–Meyer Function Calculator

The Prandtl–Meyer function ν(M) for a calorically perfect gas. The first calculator replaces Anderson’s Appendix C for a single Mach number. The second turns a supersonic flow through θ isentropically, either as an expansion around a convex corner, where ν(M₂) = ν(M₁) + θ, or as a gradual isentropic compression on a concave wall, where ν(M₂) = ν(M₁) − θ, and gives the pressure, density and temperature ratios. The γ in the top bar applies to every value on the page.

Appendix C · Prandtl–Meyer function and Mach angle

ν and μ for one Mach number

Enter one known quantity

Section 9.6 · Prandtl–Meyer expansion and isentropic compression

Turning the flow through θ

Enter any two of the three inputs

Reference

Relations used

Prandtl–Meyer function

ν rises monotonically from 0 at M = 1 toward ν_max = (π/2)(√((γ+1)/(γ−1)) − 1) as M → ∞ (130.45° for γ = 1.4), so a turn larger than ν_max − ν(M₁) would expand the flow to zero pressure. The calculator inverts ν by bisection.

Across the turn

The turn is isentropic, so the total conditions are unchanged and the static ratios follow from the isentropic relations at M₁ and M₂. For a compression θ enters with the opposite sign: ν(M₂) = ν(M₁) − θ, so M₂ < M₁ and p₂ > p₁. Since ν(1) = 0, a compression can turn the flow through at most ν(M₁) before it reaches M = 1.

The compression is isentropic only if the wall turns gradually. Its Mach waves converge and coalesce into an oblique shock away from the wall; at a sharp concave corner the shock forms at the wall itself, and the Oblique Shock calculator applies.

Background