 So, let us now look at flow through a convergent-divergent nozzle. So, here also it is possible to establish the flow either by pulling the flow or pushing the flow. So, we are going to look at pulling the flow through a convergent-divergent nozzle. So, once again our notation would be like this. So, inlet state is denoted 1 and the exit state is denoted 2 and this is the ambient and this is of course, the throat. So, here P0, T0 fixed and P ambient varies. So, let us say that we initially establish a flow in the nozzle by reducing the ambient pressure to let us say a value just below the inlet stagnation pressure let us say. So, we have flow through the convergent-divergent nozzle. So, the flow accelerates from the inlet up to the throat as you can see here and then because it is subsonic and this is a diverging passage the flow decelerates in the divergent portion and leaves at the ambient pressure. So, P2 equal to P ambient the flow is entirely subsonic because the ambient pressure is not sufficiently low to accelerate the flow to sonic speed in the throat. So, we have established a sort of a weak flow in the nozzle. Let us say that we reduce the ambient pressure some more to a level where the fluid accelerates to the speed of sound at the throat that is illustrated here. So, P star equal to P throat, but the fluid because the ambient pressure is sufficiently high the fluid is still decelerated in the divergent part of the nozzle. So, when the flow accelerates to the sonic state at the throat however it encounters high enough back pressure or ambient pressure that the flow decelerates in the divergent portion of the nozzle as you can see here. So, the flow is subsonic everywhere except at the throat where it is sonic. Now, let us say that we lower the ambient pressure some more. Now, as I mentioned earlier if you lower the ambient pressure some more the so as you can see here let us just catch a nozzle. So, here we have already reached the sonic state. So, if I lower the ambient pressure some more. So, the information can propagate upstream because the flow is subsonic here still the information can propagate upstream up to the throat only because the fluid is already moving at the speed of sound at the throat or it has attained the sonic state at the throat any information that tries to propagate upstream from here will be carried away by the fluid will be prevented from propagating upstream by the fluid which is already moving at the speed of sound. So, the information is trying to propagate upstream with the local speed of sound, but the fluid itself is moving at the local speed of sound. So, the information cannot proceed beyond the sonic state. You may recall that this is one of the points that we that I mentioned when we discussed reference state or sonic state as a reference state that sonic state divides regions of fluid which we can control and regions which we cannot control. So, that is very clearly visible here as well as in the case of the convergent nozzle. So, what this means is that the flow in the convergent portion of the nozzle will remain the same as before will remain the same as this once the fluid has actually accelerated reached sonic speed at the throat and the flow only in the divergent part of the nozzle can be changed. Now, because I have lowered the ambient pressure some more, the fluid actually has a tendency to accelerate beyond the speed of sound in the throat. It accelerates to the speed of sound in the throat and it accelerates beyond the speed of sound in the just at the beginning of the convergent portion because the exit pressure is somewhat lower as you can see here. So, the flow accelerates to supersonic Mach numbers beyond the throat because the exit pressure has been lowered somewhat, but then the exit pressure is still not low enough. So, it accelerates to supersonic Mach numbers, but still because the exit pressure is still not low enough a normal shock is triggered. So, XY represents a normal shock and once a normal shock is triggered remember MY is less than 1. This is MX is greater than 1 subsonic Mach number across a normal shock as you know is I am sorry downstream Mach number across a normal shock is subsonic. So, the fluid becomes subsonic and then it decelerates in the divergent portion because it is a divergent portion it decelerates in the divergent portion of the nozzle and then reaches the ambient pressure. Remember it has the flow is subsonic in the divergent portion which means that the pressure P2 is equal to the ambient pressure. Now, if I lower this ambient pressure somewhat then the flow will accelerate for a longer portion in the divergent nozzle, but will still encounter a normal shock. So, basically this state would be pushed down. So, if I lower the ambient pressure somewhat. So, if I lower the ambient pressure somewhat then the flow will reach a higher value of supersonic Mach number here, but then there will be a normal shock stronger normal shock and it will become subsonic then reach the ambient pressure. Remember the higher the upstream Mach number in the case of a normal shock wave and the stronger the shock will be in the lower the exit the lower the downstream Mach number. So, now we have lowered the ambient pressure somewhat and the flow accelerates to a higher supersonic Mach number, but then the irreversibility as you can see from here is also more because the shock is stronger. Notice that the irreversibility here is not that much. So, this is delta S not very high this is delta S loss of stagnation pressure across the shock wave which is quite high. So, the loss of stagnation pressure will be quite high in this case and since the flow is subsonic in the divergent portion it comes out with the pressure P2 equal to P ambient. So, basically what has happened is so here we have a normal shock at this location and in this case let me just complete this picture notice that here. So, this is m equal to 1 this is m less than 1, m greater than 1, m less than 1 and in the previous case I have already written that. So, in this case again this is m equal to 1 and the shock has moved further downstream. So, m less than 1, m equal to 1, m greater than 1 and that m less than 1. So, as we can see as we keep lowering the ambient pressure the shock moves further and further downstream in the nozzle and we next come to a limiting case when the shock stands just at the exit of the nozzle that is this state here. So, the loss of stagnation pressure is the maximum in this case and the normal shock stands at the exit because it is accelerating to the highest possible Mach number. The flow is accelerating fully in the nozzle reaching the highest possible Mach number and then a normal shock is triggered. So, that means that the loss of stagnation pressure is the highest in this case and the Mach number after the shock is still subsonic. So, p2 is still equal to p ambient. So, this may be sketched like this. So, normal shock stands just at the exit and in this case m is equal to 1, m is less than 1, m is greater than 1 here and it is less than 1 just past the normal shock. Now, if I lower the ambient pressure some more beyond this case then the normal shock will leave the nozzle and will typically stand outside the nozzle. We will discuss that a little bit later, but if I lower the pressure even a little bit then the flow in the entire nozzle will be isentropic and completely shock free which is illustrated here. Notice that the ambient pressure would be like this, the ambient pressure would be greater than the exit pressure which is also called the design pressure. So, p2 is greater than the ambient pressure which means the flow in this case is going to be under expanded I am sorry no this whole thing is wrong. So, p2 in this case is less than p ambient and in contrast to what we saw in the case of conversion nozzle here p2 is less than p ambient. So, the flow is said to be over expanded. So, the flow after it comes out of the nozzle will now have to undergo a compression process to equilibrate with the ambient pressure. So, this is something that will not be seen in a conversion nozzle because the Mach number at the exit of the conversion nozzle can only be 1 it cannot become supersonic. So, you never see or never encounter over expanded flow in the case of a conversion nozzle whereas in the case of a conversion divergent nozzle you encounter over expanded flow when p2 is less than p ambient. Notice that the flow is subsonic in the conversion portion sonic at the throat supersonic at the in the divergent portion completely shock free and hence isentropic. So, any change in the ambient pressure from now onwards will not change the flow in the inside the nozzle, but it will definitely change the flow outside the nozzle. So, let us put all these things together. Now, before we do that well we can do it in a slightly different way. So, here what we are plotting in this plot? So, what we have done so far is a TS diagram describing the process inside the nozzle. What we will do next is sketch the variation of p over p0 static pressure at any location divided by the inlet stagnation pressure. This is p0 inlet. So, we establish a weak flow first that is corresponding to A diagram A. So, each one of the variation that we are going to show now is linked to this each one of the case here. So, this corresponds to case A that was shown before. Now, we lower the exit pressure a little bit more so that the flow accelerates to sonic speed in the throat, but in this case remember the flow field in the entire nozzle is changed as we can see here. So, this corresponds to case B that we saw before. Now, the Mach number at the throat is equal to 1 because p over p0 is equal to 0.53. And notice that the flow in the entire nozzle is different from what it was before. Now, if you lower the ambient pressure some more the flow accelerates. So, the flow in this portion the convergent portion of the nozzle remains the same. Now, the flow accelerates to supersonic Mach number. So, this is m less than 1 this is m equal to 1 flow accelerates to supersonic Mach number in the divergent portion for a short distance and then encounters a normal shock and becomes subsonic and then equilibrates with the ambient pressure. If I lower the ambient pressure some more then I get this the shock moves further downstream and the flow accelerates even more in the divergent portion of the nozzle. So, the Mach number becomes even higher ahead of the shock wave. So, the shock wave becomes stronger pressure rise across the shock wave as you can see increases and loss of stagnation pressure across the shock wave also increases. So, limiting case the normal shock stands at the exit. So, this corresponds to case E that I illustrated before. So, this is the limiting case the Mach number before the shock wave is the highest and the loss of stagnation pressure is also the highest. Now, if the pressure is lowered let us say any further then the flow becomes completely shock free. So, it is subsonic here sonic at the throat and supersonic and shock free. So, in this case the exit pressure P exit is usually known as the design pressure and the supersonic exit Mach number is usually known as the design Mach number. So, the nozzle is designed to produce supersonic flow at this Mach number that is why it is called the design Mach number. The purpose of a convergent divergent nozzle is to accelerate the flow to supersonic speed. So, whatever expansion was taking place outside the nozzle in the case of a convergent nozzle now takes place inside the divergent portion of the convergent divergent nozzle that is why this supersonic Mach number is always called the design Mach number. Let us put all these things together like this what is that the exit Mach number is less than 1 for all these cases and so the P exit is equal to P ambient. Once I lower the pressure below the 1 corresponding to E for some up to some pressure ambient pressure the flow is over expanded. So, when I drop the pressure slightly from E like this the flow inside the nozzle now becomes the same as what we saw for Kcf. So, the exit pressure is less than the ambient pressure but I can continue to reduce the ambient pressure like this and the flow inside the nozzle will remain the same it will not change because this fluid is already moving at supersonic speed. So, no change can be communicated upstream. So, as I keep lowering this lowering the ambient pressure the flow inside the nozzle remains the same but the level of over expansion keeps decreasing because the fluid the exit pressure of the fluid or the ambient pressure of the fluid keeps approaching the exit pressure of the fluid. I am sorry the ambient pressure keeps approaching the exit pressure of the fluid. So, the level of over expansion keeps reducing. So, the level of over expansion is a maximum when the exit pressure is just less than this value and when we reach this value when the ambient pressure becomes equal to the design pressure the flow is correctly expanded. So, correctly expanded and then if I lower the pressure ambient pressure if I continue to lower the ambient pressure then the flow comes out of the pressure which is greater than the ambient pressure. So, the flow becomes under expanded just like what we saw in the case of the convergent nozzle. So, the flow goes from a state where p exit is equal to p ambient then it becomes over expanded when p exit is less than p ambient then correctly expanded p exit equal to p ambient at the design condition then p exit greater than p ambient under expanded. So, nozzle convergent divergent nozzle exhibits rich variety of exit conditions and remember for all the conditions below this value of exit pressure the flow is supersonic at the exit. Now, we have already seen what the jet what an under expanded jet looks like when it comes out of the nozzle because it initially because it is under expanded it has to undergo further expansion. So, it swells and then it shrinks and then it sort of bounces around like that whereas in the case of the over expanded flow the situation is slightly different let us see what that looks like. So, if you you may remember that the normal shock is just pulled out of the nozzle in this case. So, that we drop the pressure a little bit so that the normal shock is just pulled outside the nozzle. So, what that happens is the normal shock actually becomes something called an oblique shock. Remember in this case the fluid is coming out at a pressure which is less than the ambient pressure which means the fluid has to be compressed to reach the ambient pressure. So, the jet instead of swelling the jet actually shrinks like this this is the jet boundary and the compression of the fluid takes place through through a pair of oblique shock. So, the normal shock which was present like this is now pulled out and becomes something like this let us choose a slightly different color. So, the normal shock is pulled out and it becomes a pair of oblique shocks like this. So, this results in a compression process inside the jet and the entire jet shrinks like this but it does not remain like this like before the equilibration cannot take place instantaneously. So, the jet then after it swells it then expands then it again contracts and expands then contracts like that for a few diameters until it eventually equilibrates with the ambient. The processes inside the jet here can actually be calculated to a reasonable extent the expanding and swelling of the I am sorry the swelling and contraction of the jet can still be sort of tracked reasonably accurately using one-dimensional gas dynamics or rather gas dynamics itself without going to complicated CFD. And this is usually done in texon propulsion, aircraft propulsion. You can consult my book on aircraft propulsion where I have worked out a complete example going for about 4, 3, 4 shock diamonds or so. So, you can see that the flow through a convergent divergent nozzle is much more complicated in compared to the flow through a convergent nozzle. Establishing the flow through a convergent divergent nozzle is much more complicated. Typically, there is a lot of loss of stagnation pressure whenever you have a normal shock which is why I said before you make a decision to go to a convergent divergent nozzle you must ensure that it is worthwhile. When a considerable amount of expansion takes place outside the nozzle then you resort to a convergent divergent nozzle because this will have a lot of starting difficulties which we have not really discussed in this course. The starting difficulties associated with the nozzle or due to this normal shock presence of the normal shocks and such details are usually discussed in greater detail in propulsion courses. You can actually look at the lectures from my propulsion course or also read the theory given in my propulsion book. I have video lectures on aircraft propulsion also. You can actually look at those video lectures which are available in NPTEL. You can look at those also. So, what we will do next is to work out an example which illustrates these concepts and then move on to flow of steam through nozzle. So, far we have looked at flow of calorically perfect through nozzles and we are able to do a lot of things but when we have steam as I said you know we would not be looking at normal shock in steam and so on. So, we will basically be looking at isentropic flow of steam through convergent nozzle or convergent divergent nozzle.