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Next: Shock with Real Gases Up: Normal Shock Previous: Worked-out Examples for Shock Index
Initially, the gas from the driver section is coalescing from small shock waves into a large shock wave. In this analysis, it is assumed that this time is essentially zero. Zone 1 is an undisturbed gas and zone 2 is an area where the shock already passed. The assumption is that the shock is very sharp with zero width. On the other side, the expansion waves are moving into the high pressure chamber i.e. the driver section. The shock is moving at a supersonic speed (it depends on the definition, i.e., what reference temperature is being used) and the medium behind the shock is also moving but at a velocity, , which can be supersonic or subsonic in stationary coordinates. The velocities in the expansion chamber vary between three zones. In zone 3 is the original material that was in the high pressure chamber but is now the same pressure as zone 2. Zone 4 is where the gradual transition occurs between original high pressure to low pressure. The boundaries of zone 4 are defined by initial conditions. The expansion front is moving at the local speed of sound in the high pressure section. The expansion back front is moving at the local speed of sound velocity but the actual gas is moving in the opposite direction in . In fact, material in the expansion chamber and the front are moving to the left while the actual flow of the gas is moving to the right (refer to Figure (5.20)). In zone 5, the velocity is zero and the pressure is in its original value.
The shock tube is a relatively small length and the typical velocity is in the range of the speed of sound, thus the whole process takes only a few milliseconds or less. Thus, these kinds of experiments require fast recording devices (a relatively fast camera and fast data acquisition devices.). A typical design problem of a shock tube is finding the pressure to achieve the desired temperature or Mach number. The relationship between the different properties was discussed earlier and because it is a common problem, a review of the material is provided thus far.
The following equations were developed earlier and are repeated here for clarification. The pressure ratio between the two sides of the shock is
Rearranging equation (5.82) becomes
Or expressing the velocity as
And the velocity ratio between the two sides of the shock is
The fluid velocity in zone 2 is the same
From the mass conservation, it follows that
After rearranging equation (5.88) the result is
On the isentropic side, in zone 4, taking the derivative of the continuity equation, , and dividing by the continuity equation the following is obtained:
Since the process in zone 4 is isentropic, applying the isentropic relationship ( ) yields
From equation (5.90) it follows that
Equation (5.92) can be integrated as follows:
The results of the integration are
Or in terms of the pressure ratio as
As it was mentioned earlier the velocity at points and 3 are identical, hence equation (5.95) and equation (5.89) can be combined to yield
After some rearrangement, equation (5.96) is transformed into
Or in terms of the Mach number,
Using the Rankine-Hugoniot relationship and the perfect gas model, the following is obtained:
By utilizing the isentropic relationship for zone 3 to 5 results in
Next: Shock with Real Gases Up: Normal Shock Previous: Worked-out Examples for Shock Index Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21 include("aboutPottoProject.php"); ?>