Prandtl–Meyer expansion fan - turn angle

Description

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, is a centred expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point.

Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible for the flow to turn through a single “shock” wave because this would violate the second law of thermodynamics.

Since the flow turns in small angles and the changes across each expansion wave are small, the whole process is isentropic. This simplifies the calculations of the flow properties significantly. Since the flow is isentropic, the stagnation properties like stagnation pressure P0, stagnation temperature T0 and stagnation density rho0 remain constant.

The Mach number after the turn (M2) is related to the initial Mach number (M1) and the turn angle (theta) by the shown formula.

On the shown fomrula, the Prandtl–Meyer function for mach number can be calculated from the Prandtl–Meyer function.

See also

Prandtl–Meyer expansion fan – Prandtl–Meyer function

Related formulas

Variables

θturn angle (radian)
v(M1)Prandtl–Meyer function for the initial Mach number (radian)
v(M2)Prandtl–Meyer function for the final Mach number (radian)