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Next: Shock Tube Up: The Moving Shocks Previous: Partially Closed Valve   Index

## Worked-out Examples for Shock Dynamics

Solution

It can be noticed that the gas behind the shock is moving while the gas ahead the shock is still. Thus it is the case of shock moving into still medium (suddenly open valve case). First, the Mach velocity ahead the shock has to calculated.

Utilizing POTTO-GDC or that Table (5.4) one can obtain the following table

Shock Dynamics Input: My′ k = 1.3
Open valve
Mx Mx′ My My′ Ty/Tx Py/Px P0y/P0x
2.41794 0 0.501926 1.296 1.80862 6.47856 0.496948

Using the above table, the temperature behind the shock is

In same manner it can be done for the pressure ratio as following

The velocity behind the shock wave is obtained (for confirmation)

Solution

The first thing which is needed to be done is to find the prime Mach number . Then, the prime properties can be found. At this stage the reflecting shock velocity is unknown.

Simply using the Potto-GDC provides for the temperature and velocity the following table:

Shock Dynamics Input: Mx′ k = 1.4
Close valve
Mx Mx′ My My′ Ty/Tx Py/Px P0y/P0x
2.04445 1.2961 0.569957 0 1.72395 4.70974 0.700101

Or if you insist on doing the steps yourself find the upstream prime Mach, to be 1.2961. Then using the Table (5.2) you can find the proper . If this detail is not sufficient enough then simply utilize the iteration procedure described earlier and obtain

Shock Dynamics Input: Mx′ k = 1.4
Close valve
i Mx My TyTx PyPx Myp
0 2.2961 0.534878 1.94323 5.98409 0
1 2.04172 0.5704 1.72169 4.69671 0
2 2.04454 0.569942 1.72402 4.71016 0
3 2.04445 0.569957 1.72395 4.70972 0
4 2.04445 0.569957 1.72395 4.70974 0

The table was obtained by utilizing Potto-GDC with the iteration request.

Solution

The ratio can be obtained from Table (5.3). It can also be obtained from the stationary normal shock wave table. Potto-GDC provides for this temperature ratio the following table

 2.3574 0.52778 2.0000 3.1583 6.3166 0.55832
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This means that the required and using this number in the moving shock table provides

 2.3574 0.52778 0.78928 0.0 2.000 6.317 0.55830
<>

Solution

Refer to the section (5.3.5) for the calculation procedure. Potto-GDC provide the solution of the above data

 1.1220 0.89509 0.40000 0.20000 1.0789 1.3020 0.99813
<>

If the information about the iterations are needed see following table.

 0 1.4000 0.73971 1.2547 2.1200 0.20000 1 1.0045 0.99548 1.0030 1.0106 0.20000 2 1.1967 0.84424 1.1259 1.5041 0.20000 3 1.0836 0.92479 1.0545 1.2032 0.20000 4 1.1443 0.87903 1.0930 1.3609 0.20000 5 1.1099 0.90416 1.0712 1.2705 0.20000 6 1.1288 0.89009 1.0832 1.3199 0.20000 7 1.1182 0.89789 1.0765 1.2922 0.20000 8 1.1241 0.89354 1.0802 1.3075 0.20000 9 1.1208 0.89595 1.0782 1.2989 0.20000 10 1.1226 0.89461 1.0793 1.3037 0.20000 11 1.1216 0.89536 1.0787 1.3011 0.20000 12 1.1222 0.89494 1.0790 1.3025 0.20000 13 1.1219 0.89517 1.0788 1.3017 0.20000 14 1.1221 0.89504 1.0789 1.3022 0.20000 15 1.1220 0.89512 1.0789 1.3019 0.20000 16 1.1220 0.89508 1.0789 1.3020 0.20000 17 1.1220 0.89510 1.0789 1.3020 0.20000 18 1.1220 0.89509 1.0789 1.3020 0.20000 19 1.1220 0.89509 1.0789 1.3020 0.20000 20 1.1220 0.89509 1.0789 1.3020 0.20000 21 1.1220 0.89509 1.0789 1.3020 0.20000 22 1.1220 0.89509 1.0789 1.3020 0.20000
<>

Solution

The procedure described in the section. The solution is

 1.2380 0.81942 0.50000 0.80000 1.1519 1.6215 0.98860
<>

The complete iteration is provided below

 0 1.5000 0.70109 1.3202 2.4583 1 1.2248 0.82716 1.1435 1.5834 2 1.2400 0.81829 1.1531 1.6273 3 1.2378 0.81958 1.1517 1.6207 4 1.2381 0.81940 1.1519 1.6217 5 1.2380 0.81943 1.1519 1.6215 6 1.2380 0.81942 1.1519 1.6216
<>

The time it takes the shock to reach the end of the cylinder is

Solution

The stationary difference the two sides of the shock are:

Solution

This situation is open valve case where the prime information is given. The solution is given by equation (5.66) and it is the explicit analytical solution. For this case the following table easily be obtained from Potto-GDC for the left piston

 1.0715 0.93471 0.0 0.95890 1.047 1.173 0.99959 40.0 347.
<>

While the velocity of the right piston is

 1.1283 0.89048 0.0 0.93451 1.083 1.318 0.99785 70.0 347.
<>

The time for the shocks to collide is

Next: Shock Tube Up: The Moving Shocks Previous: Partially Closed Valve   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21