Prandtl–Meyer expansion fan - final flow density

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 final static properties are a function of the final flow Mach number M2 and can be related to the initial flow conditions as shown for the density

Related formulas

Variables

ρ2initial flow density (kg/m3)
ρ1final flow density (kg/m3)
γratio of specific heat capacities (dimensionless)
M1initial flow Mach number (dimensionless)
M2final flow Mach number (dimensionless)