A Prandtl-Meyer expansion fan is a two-dimensional simple wave that occurs when a supersonic flow turns around a sharp convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. Prandtl-Meyer expansions are a subset of centered waves, where all the Mach waves can be extended to meet at a point, regardless of whether that point is along the wall. 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. Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure, temperature and density decrease. Since the process is isentropic, the stagnation properties (e.g. the total pressure and total temperature) remain constant across the fan. The theory was described by Theodor Meyer on his thesis dissertation in 1908, along with his advisor Ludwig Prandtl, who had already discussed the problem a year before.
Flow properties The expansion fan consists of an infinite number of expansion waves or Mach lines. The first Mach line is at an angle μ 1 = arcsin ( 1 M 1 ) {\displaystyle \mu _{1}=\arcsin \left({\frac {1}{M_{1}}}\right)} with respect to the flow direction, and the last Mach line is at an angle μ 2 = arcsin ( 1 M 2 ) {\displaystyle \mu _{2}=\arcsin \left({\frac {1}{M_{2}}}\right)} with respect to final flow direction. 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 ( p 0 {\displaystyle p_{0}} ), stagnation temperature ( T 0 {\displaystyle T_{0}} ) and stagnation density ( ρ 0 {\displaystyle \rho _{0}} ) remain constant. The final static properties are a function of the final flow Mach number ( M 2 {\displaystyle M_{2}} ) and can be related to the initial flow conditions as follows, where γ {\displaystyle \gamma } is the heat capacity ratio of the gas (1.4 for air):
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