RD-V: 0520 Double Oblique Shock

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

図 2. Problem Description for Double Oblique Shock

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 $\frac{g}{m{m}^3}$

図 3. Boundary Condition for Double Oblique Shock

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 take a look at pressure element velocity and density once steady-state is reached.

図 4. Contour for Elemental Velocity

図 5. Contour for Elemental Pressure

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 5^th power and the legend changed to greyscale with black at the minimum and white for the maximum value.

図 6. Schlieren Output to the Power of (5/density)

The shock front shown in 図 7 divides the flow into four states as each shock front slows down and recompresses.

図 7. Shock Fronts and Density Map Showing the different States

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.

図 8. Deflection of Supersonic Flow through an Oblique Shock

First, determine the angle of the shock $\beta$, from the flow Mach number $M_1$ and the wedge angle $\theta$, by solving:(7) $\frac{\text{1}}{\tan \left(\theta \right)}=\left(\frac{\gamma +1}{2}\frac{M_1{}^2}{M_1{}^2\sin {\left(\beta \right)}^2-1}-1\right)\tan \left(\beta \right)$

Rankine-Hugoniot Jump formulas for the oblique shock are then used to obtain the values in post-shock state:(8) $\frac{P_2}{P_1}=\frac{\text{2}\gamma M_1{}^2\sin {\left(\beta \right)}^2-\gamma +1}{\gamma +1}$ (9) $\frac{{\rho }_2}{{\rho }_1}=\frac{\left(\gamma +1\right)M_1{}^2\sin {\left(\beta \right)}^2}{\text{2}+\left(\gamma -1\right)M_1{}^2\sin {\left(\beta \right)}^2}$

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 $M_{\text{4}}$.

$\frac{P_{\text{3}}}{P_{\text{0}}}=\left[\frac{\text{2}\gamma }{\gamma \text{+1}}{M}_{\text{0}}{}^2\sin {\left({\beta }_3\right)}^2-\frac{\gamma -1}{\gamma +1}\right]$

Oblique shock of angle ${\beta }_{\text{3}}$ between State 0 and State 3.

図 9. Mach Value Contour . w/o possible node paths going through every state of the flow

$\frac{P_{\text{4}}}{P_{\text{2}}}={\left(\frac{\text{1+}\frac{\gamma \text{-1}}{\text{2}}{M}_{\text{2}}{}^2}{\text{1+}\frac{\gamma \text{-1}}{\text{2}}{M}_{\text{4}}{}^2}\right)}^{\frac{\gamma }{\gamma -1}}$

Isentropic verification process from State 2 to State 4.

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

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.

図 10. Pressure Profile in Steady-State. with the value of the relative error to the analytical value

図 11. Mass Density Spatial Profile in Steady-State. with the value of the relative error to the analytical value

Analytical and numerical values for the flow Mach number in each state of the flow are compared.