 Good morning. So, since till the last class, we have been discussing the shaped nozzle. The last class, what we have done is, we have talked about how do we design a shaped nozzle using method of characteristics. I have said that, first of all the initial condition we choose slightly ahead of the throat. Then initial curvature is fixed rather we choose that initial curvature, so that we get a required Mach number. After that, then we make a computational grid consisting of multiple points and then similarly, we have the Mach lines emitting from the valve. So, these are all Mach lines emitting from different locations. Then first we solve in this domain till we get to the point I, where the Mach number is our design exit Mach number. Once we get to design exit Mach number, we drop a Mach cone from here based on this Mach number and we get the length of the nozzle now P. Then we get we have to solve for this domain. So, essentially this is the flow across when the two Mach lines cross. First of all initially, we can get this domain by looking at this Mach number and how much turning it is given. Then we have crossing of Mach numbers Mach lines based on that we can get the conditions here and how much turning is required to get parallel uniform flow. That is how we solve for this entire contour. So, by doing this we can solve for a two dimensional flow field of course, for the entire nozzle. Now, the solution gives us theta d theta d m d p d rho d t d u etcetera variation in all the properties. First of all since d theta is a coming from the solution, the wall which is one of the stream lines is can be obtained from here. So, the wall contour can obtained from here. Then the variation in all the properties can be obtained. So, this is how we get the solution by making the grid. However, one of the and another point is that the accuracy of this prediction depends on the fineness of our grid. If you go to a very fine grid, the accuracy will be better. If you go to a coarse grid, the accuracy will be compromised. But this is a very good initial guess of our shape. Why I call it initial guess? Because of the fact that along this stream, this is a stream line. So, we have a velocity. It is not the wall. It is a stream line. So, we have a velocity. But we know that in reality this all the fluids are viscous. Little bit of viscosity is there. So, at the wall they have to come to no slip boundary condition. We cannot have slip boundary condition in this case. So, therefore, there should be no slip boundary condition at the wall, which essentially means that if I look at the wall, let me take any wall not like this. If the flow is going like this, at the wall, the velocity should be 0. And then we have a boundary layer. Across this boundary layer, there is a gradual increase in the flow velocity till it reaches the free stream velocity here. So, if I draw the velocity profile, it should look like this. So, at the edge of the boundary layer only, it reaches the free stream velocity. However, for this method, we have considered everything as a free stream flow. We have not considered this boundary layer. And that can be to certain extent bring in lot of errors. Essentially, by not considering the boundary layer, we have completely eliminated the skin friction losses. But I have said in the previous lectures that the losses are because of skin friction as well as because of the 3D effect. So, this method will somehow because when we went from one dimensional flow to the multidimensional flow, there were some losses that will be eliminated by using this method. But it does not even consider the skin friction losses. So, therefore, skin friction something that needs to be addressed. So, let us now first start from addressing the skin friction. Let us look at the effect of friction. First of all, I would like to point out here that from this point to this point, the flow is accelerating. So, as the flow is accelerating, the velocity is increasing, then the pressure is decreasing. So, this is an accelerated flow. The pressure is decreasing according to Euler's equation which we have derived before. As the pressure is decreasing, del p between the inlet and the exit is greater than 0. We have higher pressure at the inlet, lower pressure at the exit. So, del p which I am defining as d i minus p e is greater than 0. So, what we have is a favorable pressure gradient. This is called a favorable pressure gradient. Because the pressure gradient is favorable, the chances of flow separation are less. First of all, secondly, because of the presence of favorable pressure gradient and the velocities are relatively high, not relatively very high at the exit, we can have Mach number 6, 7, etcetera, pretty high velocities. So, if the velocities are high and we have favorable pressure gradient, the boundary layer is thin. Boundary layer is quite thin. What do I mean by thin boundary layer? That the extent of this boundary layer is much less compared to the extent of free stream. So, the extent of the boundary layer where this change is occurring when the flow is coming to 0 velocity, that is much less compared to the extent of the free stream. That is why we called it as a thin boundary layer. Now, because of this thinness of the boundary layer, for all practical purposes, we can assume that the boundary layer is non-existent. So, then essentially instead of going like this, what we can do is we start from this. Now, at the edge of the boundary layer, I have just said that the velocity is equal to the free stream velocity and that is what we are assuming here. So, essentially what we are saying is that this wall is our edge of the boundary layer, which is quite thin compared to the free stream. Therefore, the thickness of the boundary layer can be neglected and I can work with this. So, therefore, on other words what we are saying is that the screen fission losses are quite small, which is a assumption. But in reality, is this assumption going to give us good results? We are assuming boundary layer is thin, but even this thin boundary layer, how does it affect? So, let us look at the effect of boundary layer. First of all, what does this boundary layer do? We are saying that across this boundary layer, as you can see that the velocity is changing. Beyond this we have a uniform flow, but across the boundary layer the velocity is changing. So, now this boundary layer effectively what it does is that it reduces effective flow area. That is the area in which the flow is uniform is now reduced. So, let me take an example of a pipe flow. Let us say the pipe is quite large. So, now I have a boundary layer here, a boundary layer here. It is not a fully developed flow. So, the boundary layers do not merge. Now if I look at the flow here, say at one location, the flow is uniform here. It will be uniform here also, but the question is are we going to have the same velocity here and here? Let us say I call it u 1, I call it u 2. Are we going to have the same velocity? The answer is no, because of the fact actually this u 1 is here at the entry. Because of the fact that the here the velocity has gone down. If I draw the velocity profile here close to the wall the velocity has gone down in between it is uniform. Now our mass flow rate m dot is integral rho u dot n d a over the surface of s a. So, this is the area in which over this area. So, mass flow rate is given like this. If you consider the density is constant, in that case the integral essentially mass flow rate is proportional to integral u dot n d a. Now let us look at this and this. At this point the velocity is uniform. It is covering the entire area. So, the effective flow area is more whereas, at this point only through this you have uniform flow here the velocity has reduced. Now if from the continuity equation there is no accumulation of mass here. So, the mass flow rate should remain constant. So, the mass flow rate here and here should be same. In order to achieve that since there is a slowing of velocity here this has to move faster. Therefore, our u 2 is greater than u 1. So, the boundary layer kind of squeezes the flow and accelerates it. So, same thing is going to happen here. Since it is going to reduce the effective flow area the flow will be accelerated. So, this is called displacement thickness. So, it is seems like effectively the wall is displaced. So, this is called the effective displacement thickness where the actual flow area or effective flow area is reduced. So, the consequence is that velocity is going to increase. Either we increase the velocity or mass flow rate, but mass flow rate is coming from some source which will not be affected. So therefore, velocity is going to increase whereas, no because the mass flow rate is essentially dictated by your throat condition. So, we have the throat before it. The throat condition and the combustion chamber conditions completely fixes the mass flow rate that will not change. So, if the mass flow rate is not change the only thing is that your velocity is going to change velocity is going to increase. As the velocity increases your Mach number increases then every property is going to change because we have defined those things as a function of the Mach number, but now your wall is fixed. Now your wall is not changing you have already designed it. So, now it is no longer isentropic the assumption that you have made because that was based on a isentropic assumption there was a fixed for a fixed contour there was a fixed flow property. Now the flow velocity. So, you have designed for a given Mach number or given velocity which has now increased because of the presence of boundary layer. So, it is no longer going to exactly follow your calculations. Therefore, your theta requirement is different, but theta you cannot change. So, the net effect will be all the flow properties are going to change. So, this is something that one of the effect of boundary layer. Secondly, if I look at again this one if I look at this two conditions the flow is slowing down here right because of that there is a net deficit of the momentum. Momentum goes down right and we have seen that the rate of change of momentum is the parameter that will dictate how much thrust we will produce. So, now the momentum is decreasing because of the slowing down of the flow here. So, because of that there is a net deficit of momentum which lead to skin friction losses. Skin friction losses essentially is net deficit of momentum which essentially means that thrust reduces. The thrust is going to decrease. Third one of the most difficult conditions are the complex conditions in rockets particularly rocket nozzles is now let us understand what is happening. We have a nozzle designed like this. We have a boundary layer coming here let us say because of this boundary layer the flow is no longer isentropic. Let us assume that we had designed an ideal nozzle right, but now it is no longer ideal or conditions have changed. So, it may so happen that we get a over expanded nozzle. If we get a over expanded nozzle what will happen there will be a shock wave. If the extent of over expansion is such that there is a shock wave that sits as the exit in the previous classes we have got those conditions. So, let us say now we have a shock wave sitting at the exit or shock wave can go in right. So, there is a shock wave that goes in as this shock wave goes in this shock wave interacts with this boundary layer. So, we have a shock boundary layer interaction. This is one scenario let us look at the other scenario once again let us look at a flat plate. We have a boundary layer the velocity is going like this right it comes to 0 velocity here this flow is supersonic this flow is subsonic. How will a supersonic flow turn to a subsonic flow? There is no change in area here right there is no work done nothing. So, the only possibility is going through a shock wave. So, there is going to be a shock wave here right only then the continuity of the flow will be maintained. Otherwise how rather shock wave is a discontinuous process but in the molecular level it is continuous. So, there is going to be a shock wave here close to the wall right. So, even if there is no shock wave entering from the outside or because of the condition of the over expansion there will be a shock wave close to the boundary layer because the flow has to become subsonic before it reaches the this condition and you do not have a throat right. So, there is going to be a shock wave here and now this shock wave is going to interact with the boundary layer. The shock boundary layer interaction is something that is a very complex subject because shock waves inherently are unstable and they are source of lot of energy. So, essentially it is like local vaporization it will just push the boundary layer out it is possible that the shock wave will just push the boundary layer out and we get to that condition where this boundary layer separates because of the interaction with the shock wave. So, there are two things first of all certain ambient conditions there can be a shock wave entering that is a different phenomena where it is going to interact with the boundary layer and create some nasty problems. In other cases also because of this flow has to become subsonic in this zone there can be a shock wave and because of that the shock boundary layer interaction can occur which may lead to separation. Now, what is another point that is important to notice here is that when we are in the boundary layer the flow is subsonic right in the boundary layer the flow is subsonic and we have a diverging passage. In the subsonic flow diverging passage what happens pressure will increase. So, now we have another problem that we do not have same pressure in the boundary layer and in the free stream we have a pressure mismatch and then we have a adverse pressure gradient in the boundary layer which may lead to flow separation. So, all this can lead to flow separation. So, even though we have a favorable pressure gradient in the free stream in the boundary layer we may not have favorable pressure gradient that may lead to flow separation. The shock waves can interact with the boundary layer which will enhance this flow separation because shock wave will make the boundary layer further subsonic right. So, all this lead to a possibility of flow separation. So, even though the free stream is accelerating is a very nice clean flow boundary layer can bringing lot of problems. So, these are something that needs to be addressed when we are designing a rocket nozzle. So, therefore, what we have to do is once we have this initial contour. Now, we can do a full navier stroke solution and see how the flow behaves with this initial contour and then we can refine our design to take care of this effects. So, that another numerical iteration will be required to take care of all these effects which is an essential part because otherwise you can get into huge troubles. Now, this is the effect of friction let me look at another point now. If I look at the exit of the nozzle I have ambient pressure here P a this is my back pressure. Now, I have designed a rocket nozzle for a given condition, but we are not going to operate it at that condition always. Then how does it affect the flow field? How does it going to affect our performance that we want to discuss little bit. So, next topic we are going to discuss is the effect of back pressure. Next is effect of back pressure. First of all like to point out here that this back pressure is not going to affect our flow rate because our nozzle is choked. It is a converging diverging nozzle nozzle is choked. So, flow rate is not going to be affected, but it is going to affect the flow inside the diverging portion. The rockets on the ground at sea level experiences a different back pressure and as it goes up the pressure drops as height increases the pressure drops and there is a huge variation in pressure. So, if it goes to say about 1415 kilometer the pressure will drop to say few 100 from 1 bar it will drop to say 5000 Pascal or 10000 Pascal or so I do not know exact numbers, but there is a huge variation in pressure. When we go to 100 kilometer altitude typically a lower thawed bit or altitude then the pressure is even much lower and as we go up it almost reaches vacuum outer space is almost vacuum. So, there is a massive variation in pressure. Now, we cannot design for this variation performance of a rocket nozzle is the rocket nozzle is designed for a particular back pressure. Now, the thing is that if you have designed for a particular back pressure P b and you have to operate at other back pressure how the performance is going to be affected it has a significance effect. For example, if our exit condition this is fixed by our design if this is greater than P b right this is an under expanded nozzle that we have discussed under expanded. Now, for if we have under expanded nozzle what happens let us say this is our nozzle exit we have an under expanded nozzle this is P e this is P b and P e is greater than P b. So, now, we have to have some further expansion outside so that finally, the mechanical equilibrium is satisfied. So, let us say that the jet plume let me draw it here this is my jet plume this is how it is coming out. Now, we have to further expand for that first thing that will happen is that there is going to be some expansion fan at the edge. Now, this expansion fan will propagate like this because of this expansion fan there is going to be a drop in pressure right. So, there is going to the pressure is here is going to drop less than P e, but this expansion fan will cross each other and then they reflect back as a compression wave there is going to be a compression wave. This compression waves then again reflect back from the edge as an expansion fan again a compression wave like this it continues as long as there is a pressure difference when this two pressures equal to the P b P equal to P b then this continuation stops. So, essentially what we see is that when we have an under expanded nozzle first we get an expansion fan followed by a compression wave then expansion fan like that there will be series of expansion fans and compression waves that is how it will finally, come to the balance condition for. So, therefore, this is one thing that happens for under expanded nozzle on the other hand if P e is less than P b this is called an over expanded nozzle. So, once again this is my nozzle exit this is P b P e this is P b P e this is P b and P e is less than P b. So, it is expanded more than that is required, but finally, it has to match this pressure. So, what will happen now the pressure has to increase pressure will increase first through a shock wave. So, again this is my boundary first there will be a oblique shock wave a meeting from here when they cross they will go into yeah first like this is that a lambda shock actually here sorry there is a mistake it is a continues as an expansion fan two expansion fans crossing each other from here it reflects as a shock wave from the way age it reflects as a shock wave. So, it is a shock wave then again expansion fan like that these are called lambda waves shape of the waves are like lambda. So, here now we have shock wave this shock wave now reflects as an expansion fan then expansion fan reflects as a shock wave like that it continues once again this pressure is going to gradually rise till we get a matching that the pressure is equal to P b. Now, if you look at the plume of rockets operating particular under expansion these are shock waves the flow is slowing down you get the plume looks like this bright regions light regions bright regions like this. There are some bright regions light regions well after the shock wave because the flow is velocity is low you can see lot of light there then light like that. So, it will be like rings the plume is light rings this is because of this formation till it finally, and it continues for a while it finally, reaches the back pressure. So, the bottom line is that that they exit we do not have the back pressure it will not be experiencing the back pressure some other pressure. So, therefore, and this angle of the shock wave that is formed here is dictated by the exit velocity m e and also the pressure required pressure ratio for significantly higher value of P a by P c when P when this value is very very high then the angle may be as high as 90 degree that is a normal shock wave. So, when the exit pressure is very very low compared to the ambient pressure. So, exit pressure is low compared to the ambient pressure we can have a normal shock wave sitting here and that condition we had already derived what will be the condition for that where we can have a normal shock wave sitting there and another point is that across the normal shock across the shock waves there is going to be lot of shock induced losses. So, when we have the shock waves the losses increase and we can see the shock wave is present here also right here also we have shock waves here also we have shock waves. So, finally, the thrust is going to decrease we have discussed that in detail that is why when we have ideal expansion then the thrust is maximum on both sides there is going to be losses. Now, as we have said that if this pressure is very very low we can have a shock wave sitting here if the pressure is even lower than this then it will go in right or in other words if the ambient pressure is higher shock will go in if the ambient pressure is lower we will have an expansion fan here right. So, now once the shock goes in we will have shock boundary layer interaction that we have discussed then there is going to be a performance drop because of the losses reduced by the shock wave all those things can greatly affect the performance of the nozzle. Now, let us come to the practical approach we have to design the rocket for a particular mission it has to run say for 20 minute 20 second 30 second whatever should we design it for this point or this point or this point where should we put our ideal because we designed for the ideal case. If I do it ideal here at the sea level as it goes up the pressure exit pressure remains like this which is our sea level pressure now the ambient pressure is decreasing. So, we have an under expanded nozzle continuously under expanded. So, the pressure is dropping we have continuously under expanded nozzle. So, it will never reach our optimum condition if we design for this where exit pressure is equal to the pressure here then from here to here we have a over expanded nozzle we get into this problems. So, once again that is something we do not want again we never get optimum and the flow we can have separations and lot of bad things happening. So, if we design it for somewhere here then what happens initially. So, our design pressure is P e. So, this is our P e initially this pressure is higher. So, we have a over over expanded nozzle initially, but as we go up this pressure is dropping P b is dropping. So, we are initially over expanded let us over expanded as we go up the thrust is increasing because P b is dropping we are coming close to ideal. At one point we need ideal beyond this now the pressure is further decreasing and we have this condition. So, we have like this. So, we have an increase and then decrease in thrust. Now, depending on where do you want to depending on the mission requirement we design it like this. So, this is the best possible scenario for everything else we have a continuously dropping thrust as we go up. Therefore, for all practical applications the nozzles are not designed for either sea level or at the maximum height. It is designed for somewhere in between to have the ideal condition. So, that it grows across the optimum condition while operating that is one way is a passive in the design state itself we can make sure that we get an optimum condition, but are there any other ways. Yes, what we can do is we can have a variable area nozzle. So, we can change the exit area and we can play with the variation in pressure. So, area keeps on changing we can maintain the pressure. So, or another thing is that we design in such a way that we get advantage of not having to deal with all these things shock waves and all. So, that is the other designs which are preferred particularly for higher stages of rockers which are plug nozzles or expansion deflection nozzles. So, let me discuss little bit not much above those nozzles. So, first is plug nozzles. What did we say then our design philosophy is that we design the rocket at a higher altitude. So, during the launch then it is over expanded. So, we have initial over expansion and we know that this over expansion is going to bring in some losses. We want to minimize this loss for that we can design it for the we can make some changes to offset thrust loss associated with over expansion. What we do is instead of having a confined jet we actually play with the jet boundary. This is a plug the flow comes here goes uniformly outside from here and this is called a plug and this is called a plug. At the exit of the nozzle if I look from the front we have a body like this. This is called a plug of course, this body is contoured and this contour can also be obtained using method of characteristic. So, the flow is coming here it is a uniform flow coming here. Now, what we have is a jet boundary going like this. So, we have jet boundary on both the sides like this. So, what is happening is that the flow coming from the combustor or in the nozzle is going through an annular passage now. It is no longer the straight open passage it is going through an annular passage and what we have is we have two corners here here and here and these two corners are not at the same location. Now, when I drew this there was a jet boundary here at the edge it is equal to the atmospheric pressure it was equal to p a. So, at the corner it will take the atmospheric pressure. So, here also then actually like this it takes the atmospheric pressure because of the presence of this plug and this plug can be designed in such a way that there is a gradual expansion. So, therefore, this shock formation and that are the yeah initial shock formation all can be eliminated. This is typically what is used for supersonic intakes for aircrafts. If you look at a fighter aircraft or as a supersonic missile they have a nose coming out it is like a plug that plug essentially serves two purpose. First of all whatever shock wave is there it is generated from here and that shock wave now can be controlled by proper designing. So, you can have a weaker shock that will reduce the losses. Secondly, the distance here will determine how much will be the reflected shock wave and that can also be then controlled with the design. Thirdly this plug can be moved up and down also. So, we can change the area we can change the shock location depending on our requirement. So, therefore, the losses induced by the shock wave can be reduced to a certain extent by providing this plug and that offsets the thrust loss because of the over expansion during the initial stage of the rocket operation. So, that is one of the ways. Another possibility is instead of putting a contour plug we can have something like a bluff body. So, that is called expansion deflection nozzle. This is also essentially in the plug your plug was protruding outside the nozzle. In expansion deflection it does not protrude outside it is all inside. So, you have inside body like this. So, flow once again comes annually to this and then it flows like this and we have the edge here inside. So, here we have atmospheric condition, here we have the jet. So, now depending on this location we create the shock wave or whatever it is required and this is the ambient. So, by that time it comes to the end here all the bad phenomena occurs here and finally, when it comes to the exit it has almost matched the ambient pressure. So, all the shock formation and all the expansion waves everything will be occurring here because this is my jet boundary and then therefore, all the effects are confined inside, but not directly affecting the flow field. This is safe what is happening is in this passage. So, this is something that expansion this is called expansion deflection this also that. So, the flow here also expands along a corner and because of this expansion it can expand to the actual atmospheric pressure particularly this is the atmospheric pressure. So, it expands to the actual atmospheric pressure therefore, the over expansion is eliminated. So, if you eliminate the over expansion the shock induced losses once again retro reiterate will be reduced and because of that the performance will be better. So, for the over expanded case we were having a continuous drop like this now that can be eliminated we can have a uniform or an increase thrust like this. So, therefore, this is a better way of putting a nozzle or rather designing a nozzle rather than having a straight contour because a straight contour essentially you have to be very careful. And as we have said particularly this over expansion case is more dangerous because of the fact that you can have shock wave going in and that will have huge losses because there will be shock induced losses there will be shock boundary layer interaction leading to flow separation. So, there can be very I will say devastating phenomena occurring in the flow which is something we do not want to have that is why all this both this plug nozzle as well as the expansion deflection nozzles are primarily for the over expansion case under expansion is still must safer. So, this is what we wanted to discuss about the nozzle flows with this we come to an end on our discussion on nozzle flows. So, if I can just conclude what we have said is that we started with defining the performance properties of a rocket we defined the thrust coefficient we defined the characteristic velocity we have seen that the thrust coefficient essentially is the function of nozzle design. So, we discuss the nozzle in detail starting from the area variation area rule then how does the performance effect then over expansion ideal expansion under expansion then we go into the shape nozzle we discussed conical nozzle we discussed shape nozzle we discussed how do we design a shape nozzle using method of characteristics then little bit of we discussed what is a plug nozzle what is an expansion deflection nozzle we have also discussed how does the friction or boundary layer effect the performance we discussed how the back pressure effects the performance. So, essentially all this part was focused on the thrust coefficient maximization. So, the entire nozzle description or this discussion was focused on that another parameter which we had discussed during the performance of chemical rocket is characteristic velocity. And we have shown that the I S P which is the final important parameter for us is the product of this two or thrust is the product of this two thrust coefficient and characteristic velocity. So, now we I think we should have a fairly good handle on the thrust coefficient. So, next what we are going to talk about is the characteristic velocity we have shown that the characteristic velocity depends on chamber pressure and chamber temperature. So, now next thing what we are going to talk about chamber pressure also is something as I have said it comes from the pump and all we want to maximize it, but there is a limit we do not want to go beyond that. So, next thing what we talk about is now the temperature another parameter that appeared again and again in all our derivations where gamma coefficient rather ratio of specific heats that depends on the composition. So, the temperature as well as the value of gamma and r both of them essentially is the output from the chemical reaction or combustion that is occurring inside our combustor. We have talked about designing of combustors also by the way, but at the time we have assumed that the value of gamma is known T C naught is known. Next thing what we are going to do now is to see how we get those values how do we get the composition how do we get the final temperature. So, the next topic we are going to start from the next class essentially is the combustion part. We will not go into combustion dynamics or kinetics you start with combustion chemical equilibrium. So, we will see that if you take two type of fuel and oxidizer when they react what is going to be the final product how we get that final product because we consider equilibrium chemistry. Secondly, once we have the final product how do we get the temperature these are the two things that we are going to focus on. Once we have done with that then we talk little bit about the propellants, but again in the propellants I am not going to much of detail. So, after the propellants are done then that completes our discussion on chemical rockets. I would like to tell you that there is a very good book by professor Ram Utti on rocket propulsion that describes the particularly the chemical rockets solid propellants liquid propellants etcetera in a very nice way you should read that. At the same time you can also follow his lectures in NPTEL on rocket propulsion where he even talks about various components of rockets how they are designed how they are chosen. So, in this course we are primarily focusing on the theoretical aspect of rockets not going into the component details. So, we will focus only on the nozzle and the basic combustion characteristics. So, I will stop here now in the next class I will start discussing combustion.