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Flow in the Laval Nozzle
Figure 2.5:
Left: Explanation of Laval nozzle. Right: The relation between the cross-section
and the flow velocity
.
![\begin{figure}\centering\leavevmode
\epsfxsize =.45\columnwidth \epsfbox{eps/S.ps}\hfil \epsfxsize =.45\columnwidth \epsfbox{eps/v-S.ps}\end{figure}](img474.png) |
Consider a tube whose cross-section,
, changes gradually, which is called Laval nozzle.
Assuming the flow is steady
and is essentially one-dimensional,
the continuity equation (2.1) is rewritten as
![\begin{displaymath}
\rho u S = {\rm constant},
\end{displaymath}](img476.png) |
(2.85) |
or
![\begin{displaymath}
\frac{1}{\rho}\frac{\partial \rho}{\partial x}+\frac{1}{u}\f...
...ial u}{\partial x}+\frac{1}{S}\frac{\partial S}{\partial x}=0.
\end{displaymath}](img477.png) |
(2.86) |
Equation of motion (2.2) becomes
![\begin{displaymath}
u \frac{\partial u}{\partial x} = - \frac{1}{\rho }\frac{\pa...
...x} = - {{c_s^2 } \over \rho }\frac{\partial \rho}{\partial x},
\end{displaymath}](img478.png) |
(2.87) |
where we used the relationship of
![\begin{displaymath}
{{\partial p} \over {\partial x}} = \left( {{{\partial p} \o...
...r {\partial x}} = c_s^2 {{\partial \rho } \over {\partial x}}.
\end{displaymath}](img479.png) |
(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
![\begin{displaymath}
\left( {{{u^2 } \over {c_s^2 }} - 1} \right){1 \over {u }}{...
...\partial x}}
= {1 \over S}{{\partial S} \over {\partial x}},
\end{displaymath}](img481.png) |
(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 cross-section increases (
).
On the other hand, in the subsonic regime, the velocity decreases (
)
while the cross-section increases (
).
See right panel of 2.5.
If
at the point of minimum cross-section (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 cross-section increases.
Next: Steady State Flow under
Up: Super- and Subsonic Flow
Previous: Super- and Subsonic Flow
  Contents
Kohji Tomisaka
2007-07-08