RD-V: 0510 Prandtl-Meyer Expansion Fan

The flow turns around the corner and expands. Due to the nature of supersonic flows, its pressure, mass density, and temperature decrease, while its velocity and Mach number increase.

Figure 2. Problem Description for the Prandtl-Meyer Expansion Fan

Air is defined with the following characteristics:

/MAT/LAW6

Initial density

$\text{1}\text{.225}\times {\text{10}}^{-6}\frac{g}{m{m}^3}$

/EOS/IDEAL-GAS

Heat capacity ratio

1.4

Initial pressure

0.1 MPa

Initial density

$\text{1}\text{.225}\times {\text{10}}^{-6}\frac{g}{m{m}^3}$

Figure 3. Boundary Conditions for Prandtl-Meyer Expansion Fan

The fluid is given an initial velocity along the z-axis, defined with /INIVEL/FVM as:(1) $V_x=0\frac{m}{s}$ (2) $V_y=0\frac{m}{s}$ (3) $V_z=\text{1014}\text{.18510}\frac{m}{s}$

The value of velocity $V_z$ corresponds to a speed of Mach 3, with the Mach number $M$ defined as:(4) $M=\frac{V_z}{a}$

Where, $a$ is the sound velocity, which is computed as:(5) $a=\sqrt{\frac{\gamma P}{\rho }}$ (6) $a=338\frac{m}{s}$

It can be modified using the keyword /DT/ALE:

/DT/ALE
0.5                  0.000000

It is interesting to look at the pressure once a steady-state condition is reached. The pressure decreases as the air expands.

Figure 4. Pressure Contour of Fluid Domain

The mass density and temperature see a similar decrease through the fan, while the velocity and the flow Mach number increase (Figure 5).

Figure 5. Mach Number Contour of Fluid Domain

Figure 6. Density Contour of Fluid Domain

As the flow turns around the corner, it goes through an infinite number of expansion waves, which together form the expansion fan. Each expansion wave causes a decrease in pressure, as well as an acceleration of the flow. The first Mach line is at an angle ${\phi }_{\text{1}}$ with respect to the flow direction, and the last Mach line is at an angle ${\phi }_{\text{2}}$ inferior to ${\phi }_{\text{1}}$. Since the flow turns in small angles and the changes across each expansion wave are small, the whole process is considered isentropic. $\phi$ is called the Mach angle, and defined as:(7) $\phi \text{=}{\sin }^{-1}\left(\frac{1}{M}\right)$

Figure 7. Deflection of a Supersonic Flow through an Expansion Fan

The Mach number $M^2$ of the flow in the final state is obtained from the Prandtl-Meyer angle $v\left(M\right)$ of the incident flow. The Prandtl-Meyer angle of a flow is a function of its Mach number, defined as:(8) $v\left(M\right)=\sqrt{\frac{\gamma +1}{\gamma -1}}\arctan \sqrt{\frac{\gamma +1}{\gamma -1}\left(M^2-1\right)}-\arctan \sqrt{M^2-1}$

The Prandtl-Meyer angle of the incident flows and the angle of the corner $\theta$ (in deg) are bound by this relation:(9) $v\left(M_{\text{2}}\right)=v\left(M_{\text{1}}\right)\text{+}\theta$

Once the value of the Prandtl angle in the final state $v\left(M_{\text{2}}\right)$ is determined, the equation for the Prandtl-Meyer angle is reversed to determine the Mach number $\left(M_{\text{2}}\right)$. Pressure and mass density in the final state are determined using the isentropic flow relations:(10) $\frac{P_2}{P_1}={\left(\frac{1+\frac{\gamma -1}{2}{M}_1{}^2}{1+\frac{\gamma -1}{2}{M}_2{}^2}\right)}^{\frac{\gamma }{\gamma -1}}$ (11) $\frac{{\rho }_2}{{\rho }_1}={\left(\frac{1+\frac{\gamma -1}{2}{M}_1{}^2}{1+\frac{\gamma -1}{2}{M}_2{}^2}\right)}^{\frac{\text{1}}{\gamma -1}}$

Numerical results for the thermodynamic variables can be read by drawing a node path along a current line going through the fan, as:

Figure 8. Two Possible Node Paths Going Through Every State of the Flow

The spatial profile for pressure and mass density on this node path are visible in Figure 9. The two flats on the curves correspond to the initial and final states. The value of the thermodynamic quantities from the model are compared to those determined analytically.(12) $\frac{P_2}{P_1}=\frac{\text{2}\gamma M_1^2\sin {\left(\beta \right)}^2-\gamma +1}{\gamma +1}$ (13) $\frac{{\rho }_2}{{\rho }_1}=\frac{\left(\gamma +1\right)M_1^2\sin {\left(\beta \right)}^2}{\text{2+}\left(\gamma -\text{1}\right)M_1^2\sin {\left(\beta \right)}^2}$

Figure 9. Pressure and Mass Density Spatial Profile in Steady State. with the value of the relative error to the analytical value

The relative error between numerical and analytical values in the final state for pressure, mass density, and Mach number is less than 1%.