PrandtlMeyer

Prandtl-Meyer expansion describes the isentropic expansion of a supersonic flow as it turns around a convex corner. It contrasts with oblique shocks and is used to predict flow properties in ducts, nozzles, and inlets where the flow deflects without losing total pressure.

For a perfect gas with constant γ, the Prandtl-Meyer function is defined for Mach numbers M > 1 by: ν(M) = sqrt((γ+1)/(γ-1)) * arctan( sqrt((γ-1)/(γ+1) * (M^2 - 1)) ) − arctan( sqrt(M^2 - 1) ).

ν(M) is measured in radians (or degrees if converted). It is zero at M = 1 and increases monotonically with M. The maximum value as M → ∞ is ν_max = (π/2)[ sqrt((γ+1)/(γ-1)) − 1 ]. For air, γ ≈ 1.4, ν_max ≈ 2.28 rad (about 130 degrees). The inverse function M(ν) is obtained numerically or from lookup charts.

In ductwork or nozzle flows, the deflection angle Δθ produced by an isentropic expansion from Mach M1 to M2 satisfies Δθ = ν(M2) − ν(M1). The expansion occurs at convex corners and along walls in supersonic flow and is isentropic. It is a fundamental tool in the design of converging-diverging nozzles, diffusers, and hypersonic inlets.

The concept is named for Ludwig Prandtl and the engineer Meyer, who contributed to the development of the expansion fan theory underlying this relation.