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Shock Tube
The shock tube is a study tool with very little practical purposes.
It is used in many cases to understand certain phenomena.
Other situations can be examined and extended from these phenomena.
A cylinder with two chambers connected by a diaphragm.
On one side the pressure is high, while the pressure on the other side
is low.
When the diaphragm is ruptured the gas from the high pressure
section flows into the low pressure section.
When the pressure is high enough, a shock is created that it travels
to the low pressure chamber.
This is the same case as in the suddenly opened valve case described
previously.
At the back of the shock, expansion waves occur with a reduction of
pressure.
The temperature is known to reach several thousands degrees in a very
brief period of time.
The high pressure chamber is referred to in the literature is the
driver section and the low section is referred to as
the expansion section.
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.
Figure 5.20:
The shock tube schematic with a pressure "diagram."

The properties in the different zones have different
relationships.
The relationship between zone 1 and zone 2 is that of a moving shock
into still medium (again, this is a case of sudden opened valve).
The material in zone 2 and 3 is moving at the same velocity (speed)
but the temperature and the entropy are different, while
the pressure in the two zones are the same.
The pressure, the temperature and their properties in zone 4 aren't
constant and continuous between the conditions in zone 3 to
the conditions in zone 5.
The expansion front wave velocity is larger than the
velocity at the back front expansion wave velocity.
Zone 4 is expanding during the initial stage (until the expansion
reaches the wall).
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 RankineHugoniot relationship and the perfect gas
model, the following is obtained:
By utilizing the isentropic relationship for zone 3 to 5 results in
Solution
Next: Shock with Real Gases
Up: Normal Shock
Previous: Workedout Examples for Shock
Index
Created by:Genick BarMeir, Ph.D.
On:
20071121