Flow in the Laval Nozzle

Figure 2.5: Left: Explanation of Laval nozzle. Right: The relation between the cross-section $S(x)$ and the flow velocity $v_x$.

Consider a tube whose cross-section, $S(x)$, changes gradually, which is called Laval nozzle. Assuming the flow is steady $\partial /\partial t=0$ and is essentially one-dimensional, the continuity equation (2.1) is rewritten as

$$\rho u S = {\rm constant},$$ (2.85)

or

$$\frac{1}{\rho}\frac{\partial \rho}{\partial x}+\frac{1}{u}\f... ...ial u}{\partial x}+\frac{1}{S}\frac{\partial S}{\partial x}=0.$$ (2.86)

Equation of motion (2.2) becomes

$$u \frac{\partial u}{\partial x} = - \frac{1}{\rho }\frac{\pa... ...x} = - {{c_s^2 } \over \rho }\frac{\partial \rho}{\partial x},$$ (2.87)

where we used the relationship of

$${{\partial p} \over {\partial x}} = \left( {{{\partial p} \o... ...r {\partial x}} = c_s^2 {{\partial \rho } \over {\partial x}}.$$ (2.88)

When the flow is isothermal, use the isothermal sound speed $c_{is}^2$ instead of the adiabatic one. From equations (2.86) and (2.87), we obtain

$$\left( {{{u^2 } \over {c_s^2 }} - 1} \right){1 \over {u }}{... ...\partial x}} = {1 \over S}{{\partial S} \over {\partial x}},$$ (2.89)

where the factor ${\cal M}=u/c_s$ is called the Mach number. For supersonic flow ${\cal M}>1$, while ${\cal M}<1$ for subsonic flow.

In the supersonic regime ${\cal M}>1$, the factor in the parenthesis of the lhs of equation (2.88) is positive. This leads to the fact that the velocity increases ($d u/d x>0$) as long as the cross-section increases ($d S/d x>0$). On the other hand, in the subsonic regime, the velocity decreases ($d u/d x<0$) while the cross-section increases ($d S/d x>0$). See right panel of 2.5.

If ${\cal M}=1$ at the point of minimum cross-section (throat), two curves for ${\cal M}<1$ and ${\cal M}>1$ 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.