What Is Prandtl-Meyer Expansion?
Prandtl-Meyer expansion is an isentropic (entropy-preserving) process in which supersonic flow turns around a convex corner, accelerating to a higher Mach number. Unlike oblique shock waves that form at concave corners and decelerate the flow, expansion fans spread out in a centered wave pattern and cause the flow to speed up while pressure and temperature decrease. The process was first described by Ludwig Prandtl and Theodor Meyer in 1908.
This phenomenon is fundamental to supersonic aerodynamics. It explains how supersonic flow accelerates around convex surfaces on aircraft wings, in rocket nozzle divergent sections, and around blunt body shoulders. The expansion fan consists of infinite infinitesimal Mach waves spreading from the corner, each turning the flow by an infinitesimal angle. The total turning angle determines the downstream Mach number through the Prandtl-Meyer function.
The Prandtl-Meyer Function
The Prandtl-Meyer function gives the angle (in radians or degrees) corresponding to a given Mach number. The downstream Mach number is found by adding the turning angle to the upstream PM angle, then inverting the function numerically.
Prandtl-Meyer Angle vs Mach Number (γ = 1.4)
| Mach | ν (degrees) | Mach Angle (degrees) |
|---|---|---|
| 1.0 | 0.00 | 90.00 |
| 1.5 | 11.91 | 41.81 |
| 2.0 | 26.38 | 30.00 |
| 3.0 | 49.76 | 19.47 |
| 5.0 | 76.92 | 11.54 |
| ∞ | 130.45 | 0.00 |
Frequently Asked Questions
What is the maximum turning angle?
For air (gamma = 1.4), the maximum Prandtl-Meyer angle is 130.45 degrees, corresponding to infinite Mach number. In practice, real gas effects, boundary layers, and flow separation limit the achievable turning angle to well below this theoretical maximum. Typical aerospace applications involve turning angles under 30 degrees.
Is the expansion process isentropic?
Yes, Prandtl-Meyer expansion is ideally isentropic (no entropy increase). Unlike shock waves where entropy increases and total pressure decreases, expansion fans preserve total pressure and total temperature. This makes the process reversible and thermodynamically efficient, which is why supersonic nozzles use expansion to accelerate flow efficiently.
How does this differ from an oblique shock?
Oblique shocks form at concave corners (flow turning into itself) and decelerate the flow with entropy increase. Expansion fans form at convex corners (flow turning away from itself) and accelerate the flow isentropically. Together, they form the complete toolkit for analyzing supersonic flow around bodies and through channels.