Rocket nozzles

Reading about rocket nozzle design you quickly find the declaration that the throat in the nozzle is required to achieve supersonic exhaust. But why is this the case? I tried many times to find this out, and didn't really have any luck. Then ChatGPT happened, and I decided to see how it would answer this question:

Can you explain why a rocket nozzle requires a throat to achieve supersonic flow?

A rocket nozzle requires a throat to achieve supersonic flow because of how gases behave when they expand and accelerate under pressure. This is best explained using the principles of compressible fluid dynamics and specifically the behavior of gases in a converging-diverging nozzle (often called a de Laval nozzle), which is the type of nozzle used in rockets.

Key Concepts:

  1. Subsonic and Supersonic Flow:

    • In subsonic flow (flow with velocity less than the speed of sound), if you decrease the cross-sectional area of the nozzle (like in a converging section), the velocity of the gas increases.
    • In supersonic flow (flow with velocity greater than the speed of sound), the opposite is true. To increase the velocity further, the cross-sectional area must increase, which is achieved in the diverging section of the nozzle.

  2. Choked Flow and the Throat:

    • The throat is the narrowest part of the nozzle. When gas flows through the nozzle, as the area decreases in the converging section, the gas velocity increases. If the gas is moving fast enough and the pressure drop is sufficient, the gas will accelerate to the speed of sound (Mach 1) at the throat. This condition is known as choked flow.
    • Once the gas reaches Mach 1 at the throat, further acceleration can only happen if the gas is allowed to expand. This is where the diverging section of the nozzle comes in.

  3. Converging-Diverging Nozzle:

    • In the converging section before the throat, the area decreases, causing the gas to accelerate (for subsonic flow). At the throat, the flow becomes sonic (Mach 1).
    • In the diverging section after the throat, the area increases, allowing the now-sonic flow to expand and accelerate further into supersonic speeds. Without the diverging section, the flow cannot become supersonic because the gas would not have enough room to expand and accelerate beyond Mach 1.

Why the Throat is Necessary:

  • Pressure-velocity relationship: At subsonic speeds, decreasing the nozzle's area increases the gas velocity, while increasing the pressure. As the area continues to decrease toward the throat, this effect continues until the gas reaches Mach 1.
  • Once the flow is choked at the throat, any further decrease in pressure can only cause the gas to expand and accelerate in a diverging section, which reduces pressure and increases velocity beyond the speed of sound.
  • The throat is the point where the transition occurs between the subsonic and supersonic flow. It allows the nozzle to control the gas velocity and ensure it can reach supersonic speeds after the throat.

Without the throat, the gas would not achieve the necessary Mach 1 speed at the transition point, and thus supersonic flow in the nozzle's diverging section would not occur. The combination of a converging section, throat, and diverging section allows for optimal acceleration of the gas, maximizing the rocket's exhaust velocity and thrust.

Why does choked flow result in a mach 1 limit?

Choked flow results in a Mach 1 limit because of the fundamental relationship between gas velocity, pressure, and the cross-sectional area of the nozzle in compressible fluid dynamics. Here's an explanation of why this happens:

1. Velocity, Pressure, and Area Relationship (Subsonic to Sonic Flow)

In subsonic (below the speed of sound) compressible flow, when gas flows through a converging nozzle (an area that narrows), the gas velocity increases as the area decreases. This happens because the pressure differential forces the gas to accelerate to maintain the conservation of mass (continuity equation) and momentum (Euler's equation).

At the same time, as the velocity increases, the pressure decreases, and the gas density also changes due to compressibility.

However, as the flow velocity approaches the speed of sound (Mach 1), something different happens: the compressibility of the gas becomes very significant. The flow cannot respond to further decreases in area in the same way as it does at subsonic speeds because the speed of sound represents a physical limit on how fast disturbances (pressure waves) can propagate in the gas.

2. Why the Limit Occurs at Mach 1:

  • Pressure Waves and Information Transfer: In subsonic flow, pressure disturbances (small changes in pressure that move through the gas and allow it to "know" about changes in the environment, like nozzle shape) travel at the speed of sound. As the gas approaches Mach 1, it becomes more and more difficult for these pressure disturbances to propagate upstream because the flow velocity is now comparable to the speed of sound.
  • At Mach 1, the gas is traveling at the same speed as any pressure disturbances. This means that the flow cannot "sense" further downstream changes (like additional narrowing of the nozzle), and therefore the flow cannot accelerate further in the converging section.

3. Critical Condition: Choked Flow

  • Choked Flow is a condition where the flow speed reaches Mach 1 at the narrowest point of the nozzle, the throat. At this point, the flow becomes "choked" because no additional increase in mass flow rate can occur, even if the pressure differential between the inlet and the outlet increases. The nozzle has effectively reached its maximum flow capacity.
  • Once Mach 1 is achieved at the throat, the gas cannot accelerate further unless it moves into a diverging section, where the nozzle expands. In the diverging section, the behavior of the gas is inverted: to accelerate to supersonic speeds (Mach > 1), the area must increase.

4. Mass Flow Rate and Conservation Laws

The mass flow rate ((\dot{m})) is constant through the nozzle and depends on the gas density, velocity, and area:

[ \dot{m} = \rho A v ]

  • As the flow area decreases, the velocity ((v)) must increase to maintain the same mass flow rate, given that the density ((\rho)) also changes because the gas is compressible.
  • When Mach 1 is reached, the velocity of the gas is at its maximum for a given area. Beyond this point, further area reduction will not increase the velocity because the flow is already "choked" at the sonic limit. Instead, reducing the area further increases pressure but not velocity.

5. How Supersonic Flow is Achieved (Beyond Mach 1)

To exceed Mach 1, the flow must move through a diverging section of the nozzle after reaching Mach 1 at the throat. In this diverging section, the flow can continue to accelerate, but only if it has already reached the sonic condition (Mach 1) at the throat. The diverging section allows the gas to expand and accelerate further, reaching supersonic velocities (Mach > 1).

In summary, choked flow occurs at Mach 1 because the flow's velocity matches the speed of sound, preventing further acceleration through a converging nozzle section alone. The flow becomes choked because pressure waves can no longer propagate upstream, meaning that the flow can't adjust to further changes in the nozzle's shape without expanding in a diverging section.

Which is hands-down the best explanation I have ever read. I can't say I fully understand it still, and I feel a lot of it is simply how numbers fall out of equations, and the limits of those equations. Like at some point you just have to trust the math, because intuition isn't useful anymore.