# RD-V: 0510 Prandtl-Meyer Expansion Fan

An expansion wave appears in a supersonic flow as it turns around a convex corner.

The Prandtl-Meyer expansion fan is a classic example of incompressible fluid mechanics. It involves a supersonic flow propagating along with a convex corner. As the flow turns around the corner, it gradually loses pressure and accelerates. This process is considered isentropic and happens through an infinity of small steps forming a visible fan. The numerical results obtained for the thermodynamics variables in each state can be compared to analytical values obtained using compressible fluid mechanics theory.

## Model Description

A supersonic wind tunnel (80 mm length, 20 mm height) is filled with a supersonic flow (M=3) of air.

The lower wall includes a convex corner with a downward angle of 8 degrees.

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.

Units: mm, ms, g, N, MPa

### Model Method

The fluid domain is meshed with quads elements with an average mesh size of 0.12*0.12 mm. This accounts for around 130 000 elements.

The material law used for air is a hydrodynamic fluid (/MAT/LAW6) in combination with an ideal gas equation of state (/EOS/IDEAL-GAS).

The hydrodynamic fluid (/MAT/LAW6) is used within a multifluid law (/MAT/MULTILFUID) which uses a Finite Volume solver. The fluid is considered non-viscous.

Air is defined with the following characteristics:
/MAT/LAW6
Initial density
$\text{1}\text{.225}×{\text{10}}^{-6}\frac{g}{m{m}^{3}}$
/EOS/IDEAL-GAS
Heat capacity ratio
1.4
Initial pressure
0.1 MPa
Initial density
$\text{1}\text{.225}×{\text{10}}^{-6}\frac{g}{m{m}^{3}}$

Since there is only one fluid (air), the multi-material law only integrates the air defined using LAW6, with a volume fraction of 1.

To indicate that it is an EULERIAN material, /EULER/MAT should be defined for the /MAT/MULTIFLUID material.

The /ALE/MUSCL option activates a full second-order integration scheme in time and space. It should be added for more precision.

### Boundary Conditions

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) $a=338\frac{m}{s}$

When using /MAT/MULTIFLUID, the default boundary conditions are sliding walls.

A non-reflecting boundary is defined using /EBCS/INLET/VP. Input density and pressure are defined for the air in the multi-material law.

A non-reflecting boundary is defined with /EBCS/FLUXOUT.

### Engine Control

As of the Radioss 2019.0 release, the critical time step scale factor for all ALE and EULER elements default to 0.5.

It can be modified using the keyword /DT/ALE:
/DT/ALE
0.5                  0.000000

## Results

It is interesting to look at the pressure once a steady-state condition is reached. The pressure decreases as the air expands.
The mass density and temperature see a similar decrease through the fan, while the velocity and the flow Mach number increase (図 5).

Values of all thermodynamics quantities (pressure, mass density, etc.) in the final state after the expansion fan can be computed, and the numerical results compared to analytical values.

### Analytical Analysis

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) $\phi \text{=}{\mathrm{sin}}^{-1}\left(\frac{1}{M}\right)$
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) $v\left(M\right)=\sqrt{\frac{\gamma +1}{\gamma -1}}\mathrm{arctan}\sqrt{\frac{\gamma +1}{\gamma -1}\left({M}^{2}-1\right)}-\mathrm{arctan}\sqrt{{M}^{2}-1}$
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) $\frac{{\rho }_{2}}{{\rho }_{1}}={\left(\frac{1+\frac{\gamma -1}{2}{M}_{1}{}^{2}}{1+\frac{\gamma -1}{2}{M}_{2}{}^{2}}\right)}^{\frac{\text{1}}{\gamma -1}}$

Initial State Final State
Mach number 3 3.4519
Pressure [MPa] 0.180 0.0516
Mass density $\left[\frac{g}{m{m}^{3}}\right]$ 1.225E-06 7.6337E-07

### Results Comparison with Analytical Values

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

To draw node paths, the elemental results visible on the animation have to be averaged at the nodes, using the averaging method simple in HyperView.

The spatial profile for pressure and mass density on this node path are visible in 図 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) $\frac{{P}_{2}}{{P}_{1}}=\frac{\text{2}\gamma {M}_{1}^{2}\mathrm{sin}{\left(\beta \right)}^{2}-\gamma +1}{\gamma +1}$ (13) $\frac{{\rho }_{2}}{{\rho }_{1}}=\frac{\left(\gamma +1\right){M}_{1}^{2}\mathrm{sin}{\left(\beta \right)}^{2}}{\text{2+}\left(\gamma -\text{1}\right){M}_{1}^{2}\mathrm{sin}{\left(\beta \right)}^{2}}$
The relative error between numerical and analytical values in the final state for pressure, mass density, and Mach number is less than 1%.

Values Analytical Simulation Percent Error
Mach number 3.4519 3.4518 0.004
Pressure [MPa] 0.05157 0.05158 0.004
Mass density $\left[\frac{g}{m{m}^{3}}\right]$ 7.6337E-07 7.6334E-07 0.003

### Conclusion

Supersonic flow is modeled using a /MAT/LAW6 hydrodynamic fluid law with an ideal gas equation of state (/EOS/IDEAL-GAS). This material is then referenced by the multi-fluid material /MAT/LAW151 (MULTIFLUID) which uses the finite volume solver. An expansion fan forms as the flow goes around a convex corner.

The analytical approach shows a good correlation between theoretical and numerical results.

The expansion process is correctly simulated. The use of the multi-fluid material allows for very accurate modeling of the dynamics of supersonic flows.

1 R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, 1998