Prandtl–Meyer expansion fan - Maximum turn angle
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 Prandtl–Meyer function v(M) determines the angle through which a sonic flow (M = 1) must turn to reach a particular Mach number (M)
As Mach number varies from 1 to ∞ , v takes values from 0 to v_max, where v_max is given by the shown formula. Then the maximum turn angle can be calculated by the Prandtl–Meyer expansion fan – turn angle formula, with substituting v(M2) with the calculated v_max
Prandtl–Meyer expansion fan – turn angleRelated formulas
|νmax||Prandtl–Meyer function (radian)|
|γ||ratio of specific heat capacities (dimensionless)|