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How converging diverging rocket nozzle works

For a rocket to create the thrust for lift-off, two conditions must be met. First, the rocket must expel a maximum amount of burned gas in a given time interval and the second, is that the speed of the gas being expelled should be as high as possible. If this is the case, then why do rockets have converging-diverging nozzles to create thrust? Shouldn't rockets only have converging nozzles? We know from the venture effect that when the gas is in the converging section the pressure drops and velocity increases, as seen in this animation. The flow reaches the maximum speed at the throat. In the diverging section, the velocity gradually drops and the pressure increases accordingly. Since the velocity is dropping in the diverging section, we should not use a converging-diverging nozzle; instead only use the converging nozzle for the rocket engines. However, the gas flow in a converging-diverging nozzle is not as simple as we explained in the Venturi effect.

To explain the complexity of the problem, we will assume that the pressure, p zero, is constant in the burning chamber. We will gradually reduce the exit pressure from a value close, but less than p zero toward to zero pressure. During this pressure reduction process, we will explain how the gas flow in the nozzle responds to this drop in the pressure without any mention to the math and physics behind it. This is because it is rocket science and many of us, including myself, are not rocket scientists.

Considering that the Mach number 1 is representing the speed of gas is equal to the speed of sound, the different supersonic or subsonic gas flow speeds can be expressed by a Mach number. For gas flow where maximum flow speed is less than a Mach number of 0.3, the flow can be considered as incompressible. This is the case for the animation shown here and in this case the Venturi effect is valid. As you can see, the velocity of the gas particles increase toward the throat of the nozzle. As soon as it passed through the throat, the speed of the particles gradually begins to decrease.

Let's begin to drop the exit pressure gradually.

For case A, shown here, the exit pressure is reduced such that the flow speed reaches 0.7 Mach at the throat of the nozzle. As you can see, the flow behaves almost identical to the initial case we just discussed. However, if we use the Bernoulli and conservation of mass equations for the flow speed calculations, the solution would not be accurate. This is because with Mach 0.7 the flow speed is no longer incompressible.

For case B, we will reduce the exit pressure further so that the flow speed reaches to Mach 1 at the throat of the nozzle. This is a crucial point of converging-diverging flow behavior and things begin to change from this point. When the exit pressure is reached to this condition we refer to the nozzle flow as choked. This is because it doesn't matter how much further we reduce the exit pressure; the mass flow passing through the throat of the nozzle will not change. Other than the flow being choked, the flow still behaves similarly to the Venturi tube. Toward the throat, the flow velocity will increase, it will reach Mach one and the flow speed will be decreased or will be subsonic after the throat. Therefore, if we design a nozzle with this exit pressure the nozzle will not be efficient, since exit flow speed is not fast enough.

For case C, we will further reduce the exit pressure to see how the nozzle will respond. Since the exit pressure is below the choked flow exit pressure, the converging section of the nozzle does not know anything about the diverging section of the nozzle. This means that things are not changing in the converging region. Also, the mass flow passing through the nozzle is exactly equal to the case B mass flow where the flow was choked. However, the flow velocity will show a drastically new behavior in the diverging section of the nozzle. Instead of the velocity decreasing after the throat, it will actually begin to increase and become supersonic. This is the first time we have obtained a speed above the speed of the sound, but the exit pressure will not be low enough to sustain this speed increase. Somewhere in the diverging nozzle, a normal shock wave will be created and the flow speed behind this shock wave will be subsonic. So again, for this exit pressure, the nozzle will be inefficient and we cannot use it. However, we have now learned something. Reducing the exit pressure below the point of the choked flow exit pressure creates supersonic flow inside the diverging section of the nozzle. So, let's see what happens when we reduce the exit pressure even further.

If we reduce the exit pressure further, we will observe that the supersonic region starting from the throat will begin to widen toward the exit. Also, the formation of the shock wave will move toward the exit nozzle. So for the case D we will adjust the exit pressure such that a normal shock wave will be generated at the exit of the nozzle. One may start to think that we have obtained the perfect nozzle, where inside the diverging section we have supersonic flow, but we know from experience that the flow after a normal shock wave is subsonic. This means that the exit flow from the nozzle will be subsonic and that we will have to reduce the pressure further to obtain a better working condition for the given nozzle.

As seen in case E, reducing the exit pressure further results in a strange behavior in the gas flow. During this operation condition, the exit pressure of the nozzle is below the ambient pressure. This makes exit flow converge and causes a strange shock wave formation after exiting the nozzle. These shock waves are not normal in this case and are more complex than any that we have encountered so far. However, the flow beyond the exit nozzle becomes a mixture of supersonic and subsonic flow, called overextended flow. During this operation, the nozzle is inefficient, but usually the rocket engines are designed in this mode. This interesting flow pattern can be seen in the following figure.

First figure show this pattern extending backward.
The second figure shows a test case of this flow pattern in the sea elevation, meaning during liftoff. Why we use inefficient nozzle design during the lift-off. The answer to this question is simple. The answer will be clear to you when we study the next case.

For case F, the exit pressure is reduced even further and the rocket nozzle works in its optimum efficiency. There is no shock wave formation in the flow and the gas flow gets out from the nozzle almost at the same diameter as the diameter of the exit nozzle. The reason we used an inefficient flow for case E rather than case F, which is optimal, becomes clearer now. Since the rocket is ascending after the lift-off, the outside pressure begins to fall from case E pressure towards case F pressure. This takes the rocket nozzle towards its optimum design condition, as shown here.

For case G, what happens if we further reduce the exit pressure of the nozzle? Since the gas coming from the exit nozzle is at a higher pressure than the surrounding gas, the gas coming from the nozzle will expand outward as soon as it leaves the nozzle. This flow behavior is also inefficient, since some thrust force goes outward from the rocket instead of going in the opposite direction of the rocket flight. This situation happens when the rocket is working in its optimum efficiency keeps ascending to the thinner part of the atmosphere. This flow pattern is called under expanded flow and can be seen in the Apollo Six test flights, in this image.

I hope this explanation helps you to better understand the inner workings of the converging diverging rocket nozzle.

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