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Flow in the Laval Nozzle
Figure 2.5:
Left: Explanation of Laval nozzle. Right: The relation between the crosssection and the flow velocity .

Consider a tube whose crosssection, , changes gradually, which is called Laval nozzle.
Assuming the flow is steady
and is essentially onedimensional,
the continuity equation (2.1) is rewritten as

(2.85) 
or

(2.86) 
Equation of motion (2.2) becomes

(2.87) 
where we used the relationship of

(2.88) 
When the flow is isothermal, use the isothermal sound speed instead of the adiabatic one.
From equations (2.86) and (2.87),
we obtain

(2.89) 
where the factor
is called the Mach number.
For supersonic flow , while for subsonic flow.
In the supersonic regime ,
the factor in the parenthesis of the lhs of equation (2.88) is positive.
This leads to the fact that the velocity increases () as long as the crosssection increases ().
On the other hand, in the subsonic regime, the velocity decreases ()
while the crosssection increases ().
See right panel of 2.5.
If at the point of minimum crosssection (throat), two curves for and have an
intersection.
In this case, gas can be accelerated through the Laval nozzle.
First, a subsonic flow is accelerated to the sonic speed at the throat of the nozzle.
After passing the throat, the gas follows the path of a supersonic flow, where the velocity is accelerated as long as
the crosssection increases.
Next: Steady State Flow under
Up: Super and Subsonic Flow
Previous: Super and Subsonic Flow
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Kohji Tomisaka
20070708