Understanding Area Change in Supersonic Nozzles

Nozzles are devices designed to increase the velocity of a fluid. The way the area of a nozzle changes along the direction of flow depends on whether the flow is subsonic (velocity less than the speed of sound) or supersonic (velocity greater than the speed of sound), and whether the flow is accelerating or decelerating.

The Area-Velocity Relation

For compressible flow, the relationship between the change in area ($\frac{dA}{A}$) and the change in velocity ($\frac{dV}{V}$) is given by the area-velocity relation:

\begin{equation} \frac{dA}{A} = \frac{dV}{V}(M^2 - 1) \end{equation}

Where:

• $dA$ is the change in area.
• $A$ is the area.
• $dV$ is the change in velocity.
• $V$ is the velocity.
• $M$ is the local Mach number.

Area Change in Accelerating Flow

We are interested in accelerating flow, which means the velocity is increasing along the flow direction. Therefore, $dV$ must be positive ($dV > 0$). Now let's consider the effect of the Mach number ($M$):

• Subsonic Flow ($M < 1$): In this case, $M^2 - 1$ is negative ($M^2 - 1 < 0$). Since $dV > 0$, for the equation to hold, $dA$ must be negative ($dA < 0$). This means the area must decrease to accelerate a subsonic flow. This is the principle of a convergent nozzle.
• Supersonic Flow ($M > 1$): In this case, $M^2 - 1$ is positive ($M^2 - 1 > 0$). Since $dV > 0$, for the equation to hold, $dA$ must be positive ($dA > 0$). This means the area must increase to accelerate a supersonic flow. This is the principle of a divergent nozzle.

A nozzle designed to accelerate flow from subsonic to supersonic speeds is called a convergent-divergent nozzle. The flow converges to accelerate up to Mach 1 at the throat (minimum area) and then diverges to accelerate into the supersonic regime.

The question specifically asks about the supersonic section of a nozzle that has accelerating flow. In the supersonic section, the Mach number is greater than 1 ($M > 1$). As explained by the area-velocity relation, to achieve acceleration ($dV > 0$) in the supersonic regime, the area must increase ($dA > 0$) along the flow direction.

Therefore, in the supersonic section of a nozzle with accelerating flow, the area along the flow direction increases.