 So, flow through a convergent nozzle can be established in one of two ways, one by pulling the flow through the nozzle, the other one is by pushing the flow through the nozzle. What I mean by pulling the flow through the nozzle is that you know we for instance we have a convergent nozzle like this and we actually keep the stagnation conditions fixed the upstream stagnation conditions fixed and we vary the the downstream the ambient pressure downstream of the nozzle. So, that is what I mean by pulling the flow through the nozzle. Now, in the case of pushing the flow through the nozzle, we would keep the ambient pressure downstream ambient pressure fixed and vary the inlet stagnation condition ok. Let us now take a look at each one of this in turn starting with pulling the flow through the nozzle ok. So, we will label the inlet as station 1 and the exit as station 2 in this discussion ok. So, let us say that you know we have set the ambient pressure to be less than the inlet pressure. So, that flow is established in the in the nozzle. So, the inlet state is denoted 1 and as I said exit state is denoted as 2 and the flow through the nozzle is an isentropic process, but the pressure in this case is not low enough to accelerate the flow to the speed of sound ok. So, what we can do now is actually try to lower the pressure further. For example, we could lower the pressure ambient pressure to something like this in which case the flow would expand to this pressure and the exit state would be like this. And we could keep going like this until the pressure is low enough that we actually accelerate the fluid to the sonic speed at the exit of the nozzle. So, for example, if this were to be the ambient pressure then the exit state is the sonic state because the fluid accelerates to the speed of sound. So, that is how we establish the flow through the nozzle by pulling the flow by continuously or by continuously lowering the ambient pressure. And the inlet stagnation conditions notice that the inlet stagnation conditions have remained constant throughout this ok. So, the stagnation state, stagnation temperature pressure remain constant throughout ok. So, we keep lowering the ambient pressure and we establish a flow through the nozzle ok. So, here we have a situation where we finally reach the sonic state at the exit of the nozzle. So, the speed of sound here is equal to the speed of sound at that temperature ok. Now, what would happen if we were to lower the pressure ambient pressure furthermore ok. So, that is something that we will come back and discuss. Notice that at the exit now M is equal to 1 and the flow is subsonic throughout the convergent nozzle. Now, let us see how we establish a flow in the same convergent nozzle by pushing the flow through the nozzle. So, you may recall that here ambient condition, ambient pressure is fixed and inlet stagnation condition notably inlet stagnation pressure. So, let us say that to begin with we have kept the inlet stagnation pressure at a certain value and the ambient pressure is P equal to P ambient. So, that is represented by this isobar throughout ok. So, the fluid enters at a certain static pressure P1 and accelerates to the ambient pressure P2 in this case. Now, we raise the stagnation pressure again here the idea is continuously raise the stagnation pressure until the exit state becomes the sonic state or the speed of sound I am sorry the velocity at the exit is equal to the speed of sound ok. Now, remember isobars decrease in this direction. So, when we raise the stagnation pressure that means that we are moving on to an isobar that looks like this ok. So, stagnation pressure increases in this direction. So, this is the new isobar and state 1 the inlet static pressure is likely to be higher because the inlet stagnation pressure has become higher. What is that T0 is maintained constant in this particular in this particular case because we can change T0 also, but normally P0 is increased let us say this is static state 1 and now the fluid expands inside the nozzle it is still subsonic notice that M is equal to 1 is over here. So, the flow is still subsonic, but it gets I am sorry. So, it expands up to this pressure which is equal to the ambient pressure. So, now the acceleration inside the nozzle is more than what it was before. Remember this quantity V2 square over 2 this is V2 square over 2 Cp. So, you can see that V2 square over 2 Cp is higher in this case when compared to the previous case that means the flow is accelerating even more inside the nozzle. So, we keep going until we reach a value of stagnation pressure which is such that the fluid expands to the speed of sound at the exit and P2 is equal to P ambient in this case. Still the flow is subsonic in the nozzle reaching sonic state at the exit. So, state 0.0 since we are maintaining T0 constant state 0.0 keeps sliding along this line T0 equal to constant and because we are maintaining the ambient pressure fix state 0.2 which is the exit state point keeps sliding along this isobar corresponding to P equal to P ambient. So, that is what we have shown here. So, state 0.0 slides like this state 0.2 slides like this until we reach a value of P0 for which the exit state is the sonic state. So, once again here also the question of what happens when we increase the stagnation pressure in this case further what happens in this case and the previous case what we said was what happens if we lower the stagnation sorry the ambient pressure somehow. So, that would correspond to for example, the ambient pressure being something like this. So, this would be the new ambient pressure and in this case if we increase the stagnation pressure even more this would be the new stagnation pressure. So, in this case since the exit state 2 is already at the sonic state any further changes in downstream pressure cannot be communicated upstream and it is also not possible to increase the flow speed beyond the speed of sound in this case. So, what we would get is the same process line in the TS diagram there will be no change to this. So, the ambient pressure will be like this. So, what would happen in this case is that we would have the exit state continuing to be at the same sonic state, but now in this case the exit pressure would be greater than the ambient pressure. So, the flow will come out of the nozzle. So, this is the ambient which is at ambient pressure. So, the flow comes out of the nozzle at a pressure P2 which is greater than P ambient. So, the flow comes out at a higher pressure than the ambient pressure which means the flow has to expand some more and reduce its pressure in order to equilibrate with the ambient pressure. So, that is why when the exit pressure of the fluid is greater than the ambient pressure the flow is said to be under expanded meaning it needs to expand some more outside the nozzle in order to equilibrate in order to reduce its pressure and equilibrate with the ambient. Now in this case state 1 because we have changed the upstream condition remember we do not face the same difficulty as we faced in the previous case that the change in pressure cannot be communicated upstream because the flow is already moving with the speed of sound as the exit. In this case we are changing the stagnation pressure upstream of the nozzle. So, definitely the change there will be a change in the flow field inside the nozzle. So, in fact state 0.1 would be at a higher pressure compared to this remember. So, the new state point the static pressure corresponding to the new state point 1 will be higher than the previous one and the expansion inside the nozzle will again be up to the sonic state. So, this would be the process curve inside the nozzle. However, P2 in this case will be greater than P ambient like before. P1 will be greater than the earlier P1. However, exit state can still only be sonic and P2 will be greater than P ambient. Remember we wrote down the area Mach number relationship. So, dA over A was equal to m square minus 1 dV over V. So, in this case there is a discontinuity in the in the area nozzle area profile at the exit. So, it just ends abruptly. So, the in the in the case of the flow however in the case of the fluid it will reach accelerate to sonic speed at the exit always which is why we get the exit state to be sonic state always in this case because of the discontinuity in the in the profile and P2 in this case again will also be greater than the ambient pressure. But here the ambient pressure remains the same. The exit pressure has simply become higher than the ambient pressure. Exit pressure was equal to ambient pressure previously now because we have increased the stagnation pressure and this is being done upstream of the nozzle. The exit pressure is higher than the ambient pressure, but it is still sonic state. So, the process curve definitely changes in this case whereas the process curve remains the same in this case. So, those are the two differences. I am sorry that is the most important difference between the two cases, but P2 is greater than P ambient in this case also and the flow is under expanded in this case as well. The fluid has to expand further outside the nozzle. So, whenever the fluid is under expanded, further expansion takes place outside in the ambient. So, that the fluid can decrease its pressure and equilibrate with the ambient. So, typically what would happen is if I have let us say this is my nozzle. So, when the jet issues out of the nozzle, it has to undergo further expansion which means the jet swells. So, it becomes bigger in size. So, the jet swells in size. So, this is the jet boundary and what would happen is eventually the jet would sort of contract like this then again, eventually again swell and then contract and so on. So, it will take some distance to equilibrate with the ambient condition. It cannot be done instantaneously. So, the pressure inside the jet bounces up and down until after several bounces it equilibrates with the ambient pressure. If the exit pressure were exactly equal to the ambient pressure. So, for example, corresponding to this case let us say corresponding to this case the jet because its pressure is exactly equal to the ambient pressure. It will neither swell nor shrink, but it will come out at a constant diameter equal to the exit diameter when it is correctly expanded. Notice that state 2 would be termed correctly expanded in the exit pressure. The speed of when the exit speed is the speed of sound and the exit pressure is equal to the ambient pressure. Notice that even in the other cases the exit pressure is always equal to the ambient pressure, but we do not call that correctly expanded because the flow is subsonic. So, the speed the once the flow speed reaches the speed of sound then there is a possibility that its pressure could be different from the ambient pressure. Whenever the flow the exit state is subsonic state the exit pressure will always be equal to the ambient pressure. Now, the exit pressure can be different from ambient pressure only when the exit state is a sonic state. It can be different. For example, the exit pressure here is different from the ambient pressure. So, that is why in this case we particularly denote or say that the flow is correctly expanded because the speed is equal to the speed of sound and the pressure is equal to the ambient pressure. In this case the jet diameter is constant and equal to the exit nozzle exit diameter. The same is true in this case also. Notice that the flow would be correctly expanded just when it reaches the sonic speed and its pressure is equal to the ambient pressure. The exit pressure is always equal to the ambient pressure for the subsonic exit case and it becomes sonic and its pressure is equal to the ambient pressure it is called correctly expanded. After that if we decrease the pressure further then the flow is said to be over expanded. So, whenever the flow is under expanded the jet when it comes out it swells like this, it swells like this and then goes through several such bounces before it equilibrates with the ambient pressure. This can be actually seen very nicely in this picture. So, here is a picture of the SR-71 taking off from the Dryden Air Force Base. So, you can see these alternating structures here. So, these are called the shock diamonds. So, these are the, so this illustrates the bouncing of the pressure of the fluid as it equilibrates with the ambient pressure. So, depending upon how under expanded the flow it is, if the fluid is severely under expanded then the distance taken to equilibrate with the atmospheric pressure will be longer, you will see more shock diamonds. If it is slightly under expanded then you may see only one or two shock diamonds and the flow will equilibrate with the ambient pressure very, very quickly. But the important point is that it is possible to have the exit pressure of the fluid different from the ambient pressure in the case of the convergent nozzle when the exit speed is equal to the speed of salt. And the exit speed is less than the speed of salt, the exit pressure is always equal to the ambient pressure. Again, because information, any change in downstream pressure is communicated immediately upstream to the fluid. So, the fluid pressure is the same as the ambient pressure. Now, because part of the expansion takes place outside the nozzle as I have sketched here it is a sort of mandatory or essential in propulsion application to have the nozzle profile itself like this as a convergent-divergent nozzle profile. Because if the fluid expands in the atmosphere then the resulting force is exerted against the atmosphere and is not realized as thrust. Remember, we can realize the force due to expansion as a thrust only when the fluid expands against a nozzle or a metal surface. So, it has to be forced against a metal which is why in order not to lose too much of the thrust to expansion outside the nozzle, propulsion applications tend to have convergent-divergent nozzle if the loss outside is too much. Otherwise, it may just be sufficient to have a convergent nozzle itself because it is simpler in construction. The convergent-divergent nozzle is more difficult to operate it has other operational issues. So, a rocket nozzle for example would be convergent-divergent because there will be too much loss of thrust if you do not use the divergent part. Whereas, in an aircraft engine a convergent part itself is okay the flow may be under expanded but not severely under expanded. So, convergent nozzle is sufficient for an aircraft engine because after extracting the enthalpy or after converting the enthalpy of the fluid to turbine work the only the remaining part is converted to thrust in the case of an aircraft engine. So, the inlet stagnation pressure is not as high as it is in the case of a rocket engine because there is no turbine in the case of a rocket engine the entire enthalpy of the high temperature fluid has to be converted to kinetic energy or thrust okay. So, if you use only a convergent nozzle in the case of a rocket nozzle rocket engine then there will be tremendous loss of thrust.