Page 133 - 48Fundamentals of Compressible Fluid Mechanics

5.4. SHOCK TUBE 95

zones. In zone 3 is original material that was in high pressure chamber but is in
the same pressure as zone 2. Zone 4 is where the gradual transition between
original high pressure to the low pressure occurs. The boundaries of the 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, 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 (5.10)). In zone 5
the velocity is zero and the pressure is in its original value.
The properties in the different zones have different relationship. The re-
lationship between zone 1 and 2 zones is of the moving shock in to still medium
(again this is the case of suddenly open chamber that was discussed in the previ-
ous section). The material in zone 2 and 3 is moving in the same velocity (speed)
but the temperature and the entropy are different, while the pressure in the two
zone is the same. The pressure and the temperature amount other properties in
zone 4 isn’t constant and are continuous between the conditions at zone 3 to con-
ditions at zone zone 5. The expansion front wave velocity is larger then the velocity
at the back front expansion wave velocity. The zone 4 is expanding during initial
stage (until the expansion reach to the wall).
and the typical velocity
thus the whole process take only
The shock tube has relatively small length
a few milliseconds or less. Thus, this kind of experiments require fast recoding

devises (relatively fast camera and fast data acquisitions devises.). A typical design
is in the range of speed of sound,
problem of shock tube is to find the pressure to a achieve the desired temperature

or Mach number. The relationship of the different properties were discussed earlier
and because it is a common problem and it provides the a review of the material
so far.
The following equations were developed earlier and are repeated here to
clarifies the derivations. The pressure ratio between the two sides of the shock is
(5.68)

Rearranging equation (5.68) becomes

(5.69)

Or expressing the velocity is

,
(5.70)

And the velocity ratio between the two sides of the shock is

,
(5.71)

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