# RD-V: 0520 Double Oblique Shock

A supersonic flow encounters two successive wedges and oblique shock are formed.

The double oblique shock is a classic example in compressible fluid mechanics. It involves a supersonic flow propagating along a wall with two unequal wedges. Each wedge slows down and compresses the flow with the formation of a shock wave. These two shocks then merge into a stronger, single one away from the wall, and a contact wave is formed inside the flow between the shocked states. The numerical results obtained for the thermodynamics quantities in each state of the flow can be compared to analytical values obtained with compressible fluid mechanics theory.

## Model Description

A supersonic wind tunnel (100mm length, 50 mm height) is filled with a supersonic flow Machine number (M=3) of air.

The lower wall includes two wedges of 8 degrees and 28 degrees.

The two wedges slow the flow down with the formation of shock waves.

Units: mm, ms, g, N, MPa

### Model Method

The fluid domain is meshed with 2D quads elements with an average mesh size of 0.15*0.15 mm. This accounts for around 160,000 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
1.225E-6 $\frac{g}{m{m}^{3}}$
/EOS/IDEAL-GAS
Heat capacity ratio
1.4
Initial pressure
0.1 MPa
Initial density
1.225E-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 volumer 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 inlet 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 defaults to 0.5.

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

## Results

It is interesting to take a look at pressure element velocity and density once steady-state is reached.
The Schlieren contour highlights the shock fronts. To improve the view of the shock waves from the Schlieren output, advanced math was used in HyperView. The /H3D/ELEM/SCHLIEREN output was raised to the 5th power and the legend changed to greyscale with black at the minimum and white for the maximum value.
The shock front shown in 図 7 divides the flow into four states as each shock front slows down and recompresses.

Fluid particles close to the wall go through two shock fronts, while those away from the wall see only one. However, it is stronger, so flow quantities in State 3 differ from State 2. State 2 and State 4 are separated by a verification wave, with allows for a pressure equilibrium between States 3 and 4. All other quantities are otherwise discontinuous between State 3 and State 4, as the two are separated by a contact wave.

Results obtained from the simulation can be compared to the values obtained analytically.

### Analytical Analysis

The shocks are created by the two wedges. These two obstacles slow down and recompress the flow as it changes direction and becomes parallel to the wedge. With supersonic flows, this process happens through an oblique shock wave, a stationary discontinuity which separates a pre-shock state (State 1), where the fluid is in this problem moving along the z-axis, from a post-shock state (State 2), with increased pressure and mass density, but a lower velocity, as well as a new direction, parallel to the wedge.

The theoretical values of thermodynamic quantities in every state can be determined analytically.

First, determine the angle of the shock $\beta$, from the flow Mach number ${M}_{1}$ and the wedge angle $\theta$, by solving:(7) $\frac{\text{1}}{\mathrm{tan}\left(\theta \right)}=\left(\frac{\gamma +1}{2}\frac{{M}_{1}{}^{2}}{{M}_{1}{}^{2}\mathrm{sin}{\left(\beta \right)}^{2}-1}-1\right)\mathrm{tan}\left(\beta \right)$
Rankine-Hugoniot Jump formulas for the oblique shock are then used to obtain the values in post-shock state:(8) $\frac{{P}_{2}}{{P}_{1}}=\frac{\text{2}\gamma {M}_{1}{}^{2}\mathrm{sin}{\left(\beta \right)}^{2}-\gamma +1}{\gamma +1}$ (9) $\frac{{\rho }_{2}}{{\rho }_{1}}=\frac{\left(\gamma +1\right){M}_{1}{}^{2}\mathrm{sin}{\left(\beta \right)}^{2}}{\text{2}+\left(\gamma -1\right){M}_{1}{}^{2}\mathrm{sin}{\left(\beta \right)}^{2}}$

All the physical properties in State 1 and State 2 can thus be determined.

The determination of State 3 and State 4 is more complex. Stage 3 is separated from State 0 by a single oblique shock wave. However, this shock wave is not generated by a single wedge of known angle, but rather by the combination from the two other shocks.

The pressure in State 2 and 3 is not the same because the flow went through two successive oblique shock waves, instead of through one stronger shock front. A verification process will happen after State 2 until the pressure has decreased to its value in the adjacent State 3.

State 3 and 4 needs to be determined together. State 3 is obtained from a flow initially in state 0 that encounters an oblique shock (which angle ${\beta }_{\text{3}}$ in unknown) and shares the same pressure that State 4. The State 4 pressure is obtained after an isentropic verification from State 2, in which the Mach number of the flow changes from ${M}_{\text{2}}$ to ${M}_{\text{4}}$.

The value of pressure in these two states is thus obtained by finding solutions of the following system (with ${P}_{\text{4}}$ = ${P}_{\text{3}}$), and the unknowns are ${\beta }_{\text{3}}$ and ${M}_{\text{4}}$.
$\frac{{P}_{\text{3}}}{{P}_{\text{0}}}=\left[\frac{\text{2}\gamma }{\gamma \text{+1}}{M}_{\text{0}}{}^{2}\mathrm{sin}{\left({\beta }_{3}\right)}^{2}-\frac{\gamma -1}{\gamma +1}\right]$
Oblique shock of angle ${\beta }_{\text{3}}$ between State 0 and State 3.
$\frac{{P}_{\text{4}}}{{P}_{\text{2}}}={\left(\frac{\text{1+}\frac{\gamma \text{-1}}{\text{2}}{M}_{\text{2}}{}^{2}}{\text{1+}\frac{\gamma \text{-1}}{\text{2}}{M}_{\text{4}}{}^{2}}\right)}^{\frac{\gamma }{\gamma -1}}$
Isentropic verification process from State 2 to State 4.

With ${\beta }_{\text{3}}$ and ${M}_{\text{4}}$ determined, the other thermodynamics variables in State 4 are determined considering an isentropic rarefaction process from State 2 to State 4, and for State 3, by considering an oblique shock with an angle.

The solution of the double wedge problem using the shock polar. 1

State 1 State 2 State 3 State 4
Shock angle [deg] 25.611 41.582 48.586 no shock
Mach number 2.603 1.722 1.537 1.746
Pressure [MPa] 0.180 0.595 0.574 0.574
Mass density $\left[\frac{g}{m{m}^{3}}\right]$ 1.85E-06 4.15E-06 3.70E-06 4.04E-06

### Numerical Results with Analytical Values Comparison

Numerical results for the thermodynamic variables can be read by drawing two node paths along two current lines, one close to the wall and going through State 1, 2 and 4, and the other further away from the wall, going from State 0 to State 3 (図 9).

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

The spatial profile for pressure and mass density on these two nodes paths are visible on 図 10 and 図 11. Each flat on the curves corresponds to one the flow states described above. The value of the thermodynamic quantities obtained from the model can then be compared to those determined analytically.
Analytical and numerical values for the flow Mach number in each state of the flow are compared.

Analytical Numerical Relative Error (%)
State 1 2.603 2.607 0.14
State 2 1.722 1.717 0.32
State 3 1.537 1.529 0.56
State 4 1.746 1.746 0.05

The relative error between numerical and analytical values for pressure, mass density and Mach number are for every state inferior to 1%.

### Conclusion

Supersonic flow is modeled using a /MAT/LAW6 hydrodynamic fluid law, associated with an ideal gas equation of state (/EOS/IDEAL-GAS), within a multi-fluid material /MAT/LAW151 to use the finite volume solver.

As the flows encounters two wedges, three stationary shock waves and a contact wave are formed within the flow. The value of the flow physical parameters in each state can be determined analytically.

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

The shocks and contact wave are rendered well on the simulation. The use of /MAT/MULTIFLUID allows for an accurate modeling of the dynamics of supersonic flows.

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