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Next: Piston Velocity Up: Moving Shock into Stationary Previous: Moving Shock into Stationary   Index

### General Velocities Issues

When a valve or membrane is suddenly opened, a shock is created and propagates downstream. With the exception of close proximity to the valve, the shock moves in a constant velocity (5.12(a)). Using a coordinates system which moves with the shock results in a stationary shock and the flow is moving to the left see Figure (5.12(b)). The upstream'' will be on the right (see Figure (5.12(b))).

 stationary coordinates moving coordinates
Similar definitions of the right side and the left side of the shock Mach numbers can be utilized. It has to be noted that the upstream'' and downstream'' are the reverse from the previous case. The upstream'' Mach number is

The downstream'' Mach number is

Note that in this case the stagnation temperature in stationary coordinates changes (as in the previous case) whereas the thermal energy (due to pressure difference) is converted into velocity. The stagnation temperature (of moving coordinates) is

A similar rearrangement to the previous case results in

The same question that was prominent in the previous case appears now, what will be the shock velocity for a given upstream Mach number? Again, the relationship between the two sides is

Since Msx can be represented by Msy theoretically equation (5.63) can be solved. It is common practice to solve this equation by numerical methods. One such methods is successive substitutions.'' This method is applied by the following algorithm:
1. Assume that Mx = 1.0.
2. \label{shock:item:openValve} Calculate the Mach number My by utilizing the tables or Potto--GDC.
3. Utilizing
Mx = sqrt(Ty / Tx) ( My + My′ )
calculate the new improved'' Mx.
4. Check the new and improved Mx against the old one. If it is satisfactory, stop or return to stage \eqref{shock:item:openValve}.
To illustrate the convergence of the procedure, consider the case of My′ =0.3 and My′ =1.3. The results show that the convergence occurs very rapidly (see Figure (5.13)). The larger the value of My, the larger number of the iterations required to achieve the same accuracy. Yet, for most practical purposes, sufficient results can be achieved after 3-4 iterations.

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Next: Piston Velocity Up: Moving Shock into Stationary Previous: Moving Shock into Stationary   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21