Prandtl–Meyer expansion fan

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 ${\displaystyle {\mu }_1=\arcsin \left(\frac{1}{M_1}\right)}$ with respect to the flow direction, and the last Mach line is at an angle ${\displaystyle {\mu }_2=\arcsin \left(\frac{1}{M_2}\right)}$ 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 (${\displaystyle p_0}$), stagnation temperature (${\displaystyle T_0}$) and stagnation density (${\displaystyle {\rho }_0}$) remain constant. The final static properties are a function of the final flow Mach number (${\displaystyle M_2}$) and can be related to the initial flow conditions as follows, where ${\displaystyle \gamma }$ is the heat capacity ratio of the gas (1.4 for air):

${\displaystyle \begin{array}{cc}\frac{T_2}{T_1}& =\left(\frac{1+\frac{\gamma -1}{2}{M}_1^2}{1+\frac{\gamma -1}{2}{M}_2^2}\right)\\ \frac{p_2}{p_1}& ={\left(\frac{1+\frac{\gamma -1}{2}{M}_1^2}{1+\frac{\gamma -1}{2}{M}_2^2}\right)}^{\frac{\gamma }{\gamma -1}}\\ \frac{{\rho }_2}{{\rho }_1}& ={\left(\frac{1+\frac{\gamma -1}{2}{M}_1^2}{1+\frac{\gamma -1}{2}{M}_2^2}\right)}^{\frac{1}{\gamma -1}}.\end{array}}$

The Mach number after the turn (${\displaystyle M_2}$) is related to the initial Mach number (${\displaystyle M_1}$) and the turn angle (${\displaystyle \theta }$) by,

${\displaystyle \theta =\nu (M_2)-\nu (M_1)}$

where, ${\displaystyle \nu (M)}$ is the Prandtl–Meyer function. This function determines the angle through which a sonic flow (M = 1) must turn to reach a particular Mach number (M). Mathematically,

${\displaystyle \begin{array}{cc}\nu (M)& =\int \frac{\sqrt{M^2-1}}{1+\frac{\gamma -1}{2}{M}^2}\frac{dM}{M}\\ & =\sqrt{\frac{\gamma +1}{\gamma -1}}\arctan \sqrt{\frac{\gamma -1}{\gamma +1}\left(M^2-1\right)}-\arctan \sqrt{M^2-1}.\end{array}}$

By convention, ${\displaystyle \nu (1)=0.}$

Thus, given the initial Mach number (${\displaystyle M_1}$), one can calculate ${\displaystyle \nu (M_1)}$ and using the turn angle find ${\displaystyle \nu (M_2)}$. From the value of ${\displaystyle \nu (M_2)}$ one can obtain the final Mach number (${\displaystyle M_2}$) and the other flow properties. The velocity field in the expansion fan, expressed in polar coordinates ${\displaystyle (r,\varphi )}$ are given by

${\displaystyle v_r=\sqrt{2(h_0-h)-c^2},v_{\varphi }=c,\text{where}\varphi =-\int \frac{d(\rho c)}{\rho \sqrt{2(h_0-h)-c^2}},}$

${\displaystyle h}$ is the specific enthalpy and ${\displaystyle h_0}$ is the stagnation specific enthalpy.