 Good morning. So, in the last class, we discussed the flow through a conical nozzle. We have shown that what are the losses, how they come in because of the 3-D effect. And then we said that if we kind of reduce the half cone angle, we can reduce those losses. The ultimate reduction is if you can get the flow to be parallel to the axis. And then we discussed that why a shaped nozzle whose goal is to get a parallel flow at the exit will be more advantageous over a conical nozzle. So, today now we are going to discuss about design of the shaped nozzles. There are two ways of doing it. One is, so we are talking about shaped nozzle. First is, if is given a specified nozzle geometry, this actually is a roundabout way. First we choose a nozzle geometry and then we can set up a numerical solution of equations of motion for the whole nozzle. We can use a finite difference scheme, a finite volume scheme, a finite element scheme whatever. So, this the primary thing is that the shape must be specified. With the specified shape, then we solve the full Navier stokes. We can give proper boundary conditions to the wall. Even thermal boundary conditions, we can have a fluid structure interaction code also. So, we can solve for at present, we can solve for the full flow field. But the problem is that this is very time consuming and expensive. It will give us the full velocity field, full pressure field etcetera. First of all the solution itself is going to be very time consuming and expensive, because the Reynolds number is quite high. The flow is going to be turbulent and it is quite high temperature. It has to be compressible. So, it is a compressible turbulent flow modeling with the wall conditions, which are neither isothermal nor adiabatic. There is heat transfer through the wall. So, all this needs to be modeled. So, therefore, it is quite time consuming and expensive. And it will give only one solution for one geometry. Our goal is to design the shape. So, then what we have to do is, we have to try different shapes and then optimize. So, therefore, then try different shapes and optimize. So, essentially the entire program in coming up with the design of a curve nozzle or a shape nozzle, which will give us the parallel flow at the exit and required pressure, required velocity is going to be quite expensive, if you do it this way. The easier method then is something that gives us the shape as the solution. We specify the exit conditions or the required conditions. Something that gives the shape as the solution will be an easier method, much more cheaper. And that is where method of characteristics comes in. The point here is the method of characteristics is applicable only to supersonic flows. It is not applicable to subsonic flows. So, therefore, the method of characteristics of course, for supersonic flows is widely used, because first of all it is simple and secondly supersonic portion is of importance. We have discussed that in the previous class, that the subsonic portion, the converging portion or subsonic portion is not that important, because since we have favorable pressure gradient, we can take any simple shape and that will give us a proper flow. Supersonic portion is the important part, where we need to have the design. So, method of characteristics then will gives us the initial shape. At present the practice is the initial shape we get from method of characteristics. Then we go to this, do the full analysis and check the validity of the initial design. That is a much cheaper approach and gives us much better solution. So, now we will focus on this method of characteristics. Method of characteristics depend on the fact that in supersonic flow, the influence of a small pressure difference is limited in a specific region. So, if you have a small pressure perturbation, it does not influence the entire flow field. It is limited to a particular region and the method of characteristics is build upon that. So, let us understand what I mean. Let us consider that I have a small source of pressure disturbance sitting here and we have a flow coming. This is a supersonic flow at a velocity u. This is source of pressure disturbance. It can be a loudspeaker, it can be a small pulse generator or something. Now, the flow is coming at supersonic speed. What happens is that we know that the disturbances propagate sound, particularly the compression waves or particularly the pressure waves, propagates through the air or the medium through sound waves. So, initially if the flow is subsonic, then let me put it this way. First let us look at a subsonic. First let us look at there is no flow. We have a source here. When this source propagates, it will propagate uniformly all around the disturbance. So, we get a spherical disturbance going all around. So, it is like a spherical disturbance going all around. This is unbiased stationary flow, the stationary condition. If we give a flow here, that is a biased flow, biased condition. This will change this pattern. So, now what happens is that more will be propagating on this side, less will be propagating on this side. So, there is a change in pattern. It may not look like this. There is a change in pattern. That is called biasing. As we keep on increasing this speed, still the flow is subsonic, some information will always propagate. There will be some effect of bias, but it will always propagate upstream also. If you keep on increasing the flow speed, when it becomes sonic, then now this information will just rarely be able to because what happens is that it is trying to go this side and the flow is coming on this side with the same speed. It will stop the information from propagating. When it becomes supersonic, then it will not propagate upstream at all. It will only go in the downstream region. So, then if I draw the schematic of propagation of this information, it will look like this, like a cone and this is how the information is going to propagate. We call point B. This is point A. So, at point A, we have a source of disturbance. This disturbance propagates with a like a spherical wave at a particular speed, which is the speed of sound. Now, the center of the spherical wave is moving downstream with a velocity u. This is the center of the spherical wave. This is moving downstream with a velocity u. So, at time t, it has moved from this location to this location and the distance between these two will be equal to u t because the flow is moving this source away. Now, at any time, if I look from time t equal to 0, it was here. Time t equal to t plus 0 plus delta t, it was here. Slowly, it is moving away and at every time, then depending on the time t, it is moving at certain location and it is going also. So, the influence of this, that is if I put a microphone here, it will not hear it. Only if the microphone is put here, it will hear this. So, the domain of influence is only limited to this cone and then, this cone is given by this angle alpha. So, to alpha, this is the limit of influence. So, the zone of influence is limited to a cone of half angle alpha here. Now, first of all, how do we get this half angle? At time t, the disturbance emitting from here is propagating with the speed of sound a. At time t, this source has reached this point. This distance is u t. The disturbance that emitted from here will reach this point. The distance will be a times t, where a is speed of sound. So, this is a t. So, the radius of this sphere is a t. So, at every location, that is how we get this radius. So, therefore, now this angle alpha, I am not drawn it properly, it will be something like this. So, angle alpha will be sine inverse a t by u t. This is my angle alpha, this is a t by u t. Actually, this will be h, p by h. So, I can cancel this thing and what is u by a? This mark number. So, therefore, this is equal to sine inverse 1 by m. So, or alpha will reach this point alpha equal to tan inverse a t upon square root of u t square minus a t square. This can also be written as tan inverse 1 upon m square minus 1. So, alpha is equal to tan inverse m square minus 1. Remember that I said at the beginning that is applicable only to supersonic flow and that is why it is coming from here. If mark number is less than 1, this becomes imaginary. We cannot define alpha. So, therefore, alpha is defined only if mark number is greater than 1. Even at mark number equal to 1, this is 0. So, it is infinite. So, that is something that is the limiting case. So, alpha is now the domain of influence rather the mark angle. Alpha is called the mark angle and this is the domain of influence for this sound source. So, this is how now the method of characteristics I will come to first. I have defined the mark angle alpha. Now, we will come to the method of characteristics. Let us consider now and this lines by the way are called mark lines. So, the limit of influence is essentially bounded by the mark lines. So, what do we see here? For this point here, what will be the angle of the mark lines given by alpha depends on the incoming flow mark number. So, essentially how much the mark lines will turn depends on the incoming flow mark number. So, that is what the mark line is. So, if I look at these two mark lines, this is the upstream of this mark line. If the flow upstream of a mark line is uniform like in this case, then what we can see here that the mark line is going to be straight line. That is what in this case is, but if this flow is not uniform, then at different locations we have different mark numbers, the mark line will curve. So, another point here is that on this domain of influence everywhere, the flow properties are uniform. So, within the limit of influence, the flow properties are uniform. Here, the flow is uniform coming in. Therefore, the mark line is straight. So, if you have a straight mark line, the flow properties are uniform downstream of the mark line. So, these are the few things that we actually use when we employ a method of characteristics that if we have uniform flow, the mark lines are straight lines and within the limit of influence, the flow properties are uniform. These are the things that we are going to use. Now, let us look at a nozzle and see that how we use this information in the case of a nozzle. So, let me now go to a proper nozzle, proper shaped nozzle. Let me consider a shaped nozzle like this. We are not still going to the full nozzle. I am just showing a part of it. This is our central line. We have a let us say supersonic flow coming here into this nozzle. Now, we have a point here a. From this point, here we have uniform flow. So, from this point, a mark line will emerge. Go like this. Since we are talking about a symmetric nozzle, just opposite side, there is another mark line, another point a dash from where another mark line will emerge and go like this. So, if nothing happens in between here and here, this mark lines will be straight and travel like this. But now, if this was a straight section, this is what is going to happen. But here it is curved. As it is curved, if I look at a supersonic flow over a curvature, what happens? The flow comes here. Because of the curvature, there is an expansion fan about it, it turns and then goes like this. So, it accelerates. So, the mark number here is greater than the mark number here because of this curvature. So, if I look at this small portion here, there is a curvature. Because of this, the flow is accelerated. So, now, if I come to this point, at this point the mark number is more than at this point. So, now, another mark line will emit from here and for that mark number, this is higher. This mark number is higher. So, what happens to alpha? So, this is reducing. So, alpha is increasing because sin alpha is 0, sin 0 is 0, sin 90 is 1. So, alpha is, mark number is increasing, alpha is reducing because sin inverse alpha is reducing. So, alpha is reducing. So, alpha decreases which means something that was at this angle now may be at this angle, slightly this angle. So, if I drop another mark line from here, it will go like this, right, slight decrease in mark angle. Now, if I look at these two, this mark line was straight with certain property, uniform property. This has straight with some other property, but these two mark numbers are different. So, when they intersect, they will bring in a change, right, which will be dependent on the mark number here and the mark number here. So, now, this mark lines will start to curve, right. It is no longer straight line because there are non-uniform properties merging. So, the mark lines will start to change. Let me draw it on the one side only. Similarly, from this side. So, we get a pattern like this. We get a pattern like this, like this. We get a pattern now. As you can see, this looks like a grid pattern. Where the mark lines are now intersecting at different points. So, we got a point D here, a point E here, like that. At different points, this mark lines are intersecting. Now, in between these two mark lines, the flow is still uniform. But now, since this and this mark line are different, these are two different mark lines. So, the uniform flow will neither be this nor be this. It will be a different condition, although it is uniform, but it will be a different condition. And that is how the flow will propagate. That when it goes along, there is going to be change in the mark number. So, now, this mark number change will depend on this curvature and the incoming flow mark number. So, the initial curvature that is provided that will dictate what kind of flow will be emerging. So, let me just summarize. Suppose, we have a uniform flow which enters the diverging portion of the nozzle here. The wall curvature initially, like we have shown here, will establish pressure gradients that are going to turn the stream lines. So, the stream lines are going to be turned. So, because the fluid will be flowing along the wall, like we have shown here, the stream lines are turned like this. Since, we are neglecting friction in this case, the fluid can be assumed to be sliding freely. So, the flow is like in this case in the expansion plan, it is sliding smoothly over the surface. So, at this point, we have a small variation in angle d theta. Let us say, a small variation in angle d theta which causes a small pressure disturbance d p to be produced. And now, this disturbance will propagate along this line. From here to here. And at what angle it will move, depends on the incoming flow mark number. So, it creates a mark line there. Similar disturbance will be propagating from this point. So, this small curvature here, now is our source of disturbance. The flow was coming smoothly here. There is a small curvature that creates the disturbance. So, that is the initial point which creates the initial mark lines. So, these are the two mark lines that are created. Now, we have a continuous curvature at this point, beyond this. So, because of this curvature from every point, there will be mark line that will be coming up. And by the way, if I look at the mark lines here, the mark lines which are emitting from here are moving in this direction. The mark lines which are emitting from here are moving in this direction. The mark lines emitting from the lower duct wall are called left running mark lines. So, these are called left running mark lines. And emitting from the upper wall are called right running mark lines. So, this is the nomenclature that is used. So, we have left running mark lines and right running mark lines. And they are crossing each other in this exact manner as we have drawn here. Now, this designation of left running and right running refer to the direction in which the lines approximate to propagate approximately propagates or rather appears to propagate downstream to an observer looking downstream. So, if I look, let us say if you have a observer sitting here, observer is standing let us say at this point and looking downstream. And there that observer looks at this line. So, this line with respect to that observer is left. This line with respect to that observer is right. That is why these are called left running and these are called right running. So, observer is sitting here and looking downstream. That is why this nomenclature is used. Now, if I look at this region a b c or this region a prime b prime c prime let us say. If I look at this region or that region where if I look at this region there is only one type of mark line here. Of course, it is bounded by the other type. Similarly, on this region there is only one type of mark line bounded by other type here. So, it is only same type of mark line present in both of this. Whereas, we have already discussed that if we have a single type of mark line then the fluid properties are going to be constant just upstream or downstream of each line. So, if I look at this region the fluid properties are going to be constant. Similarly, in this region the fluid properties are constant. Of course, it is going to change from here to here because there are two mark lines that are present, but within a particular region the fluid properties are going to be constant the same type of mark number. So, specifying the properties along any streamline within this. Now, the stream lines are moving like this. Let us say choose this streamline. We specify the property along any streamline will specify that in that entire region. So, specifying the properties along any streamline will be sufficient to describe everything in that region. Now, what is our limiting streamline? The wall. So, if we can specify the property along the wall this is our limiting streamline then the properties all along these are specified within this zone. Of course, within this zone and this zone. So, specifying the wall properties will specify the properties in between everywhere. Now, let us see that what is happening in this region more closely. This region let us say that this point there is a curvature of d theta the flow is turning by amount d theta which is given here. Let us see that how do we get the property variations now when the flow is turning by this amount d theta. So, for that I will draw this diagram a bigger diagram to represent the flow. Let us consider this that we have a velocity coming in this direction u and the flow is turning by a small angle d theta because of that there is a mark line that is emitting. So, upstream mark number here is m this is the upstream mark number m the disturbance propagate at an angle alpha. So, this is my angle alpha this is the mark angle that we have already discussed and we can estimate alpha as tan inverse 1 upon m square minus 1. So, this now any fluid crossing this mark line will have a change in its direction. So, the stream line any stream line crossing the mark line must have a direction change. So, that it continues to be parallel to this wall. So, there is going to be a change in the direction. So, let me consider this this is the stream line it is coming here and this direction gets changed. So, now it moves in a manner parallel to this right. So, this is what is the change direction stream line is going to move and now just to draw an analogy this is like a Prandtl Mayer flow expansion path. So, we know that when the flow supersonic flow changes direction because of the increase in angle it accelerates. So, this velocity is going to increase. So, let us say the velocity now is u plus d u whereas, this flow velocity was u this is the same flow I am just drawing it on this side velocity is u. Then for this I can have a there is a change in direction now of the flow and this angle is alpha this is better. So, this is u this is u plus d u this term is u which is the increase in axial velocity u plus d u and d v velocity is now changing we have a velocity in the v direction as well this angle is d theta. So, if I draw this a triangle just for the increase d u d v this is the overall increase this is the vector diagram only for the change part. So, the axial velocity increases by time on d u there is a d v time. So, the total velocity increases d capital d u. So, there is a if you look at this there is no change in momentum component parallel to the mark line parallel to the mark line it does not change. So, the change in velocity is particularly let me just this diagram is. So, the change in velocity is primarily in the v direction not in the u direction from this diagram we can get d v is u d theta this is my u this is u d theta d v is u d theta and d u from here is equal to this is the change in the x component. Let me write a different notation capital u for the actual velocity otherwise because I am representing the change also with the small u. So, I can write a different notation here. So, d v is equal to capital u d theta and d u is equal to d u which is the change in velocity. Therefore, from there we can get tan alpha is equal to d u by d v and this then is equal to therefore, now from here what do we see tan alpha in place of d u I write this here I write u d theta. So, this is equal to d u by u d theta right and tan alpha I know is equal to d u by u d theta this value. So, tan alpha is equal to 1 upon square root of m square minus 1 right. So, this is equal to 1 upon square root of m square minus 1. So, I can get now an expression for d theta that is the change in angle which is equal to square root of rather let me write this therefore, d u by u equal to d theta upon square root of m square minus 1. So, this is the change in velocity because of this change in angle and as we can see the change in velocity is function of the change in angle as well as the incoming flow mark number. So, based on that we get this expression. Now, from the definition of mark number u square is equal to m square gamma r t right and from isentropic flow assumption this actually is applicable to any adiabatic flow did not be isentropic. We get the isentropic relationship t naught by t given like this I can get t coming here. So, let me just write it here this gives me u square equal to gamma r t naught m square divided by 1 plus gamma minus 1 by 2 m square. So, the velocity now can be written as a function of t naught which is the stagnation temperature mark number that is more important is it can be written in terms of mark number. So, we have represented velocity in terms of the mark number now what we can do is we can differentiate this. So, if you differentiate this differentiating that expression we get d u by u is equal to d m square upon 2 m square 1 plus gamma minus 1 by 2 m square. Let me call this equation 1 what actually what we are trying to do is to change get an expression for change in mark number when there is a curvature what we got is a change in velocity we want to represent it as change in mark number. So, we get this expression from this then now if I combine this this and that expression if I combine then I will get d m square is equal to 2 m square 1 plus gamma minus 1 by 2 m square by m square minus 1 d theta let me call this equation 2. So, let me again see what we have done we have derived this expression here which is relating the velocity change to the angle change and of course, is function of the mark number. The velocity we have expressed in terms of the mark number then we differentiated that and got an expression for d u by u in terms of mark number and here we have a expression for d u by u in terms of theta and mark number when we combine this two we get this expression. So, what is this equation representing the change in mark number as a function of incoming flow mark number and change in angle theta d theta right. So, this is what we wanted to get because we have the curvature we want to know when there is a curvature how much change in mark number we can expect or we can estimate from that from this equation. So, the change in mark number here relate to a change in direction of streamline for isentropic flow because d theta is the change in direction of streamline and we are considering isentropic flow completely here this is something that you have to keep in mind that all this derivation isentropic flow is inherent. So, what we are doing now is completely for isentropic flow it is applicable only to isentropic flow. So, therefore, because this was isentropic this description this is isentropic. So, we are doing it for isentropic flow. So, this is the variation in mark number because of change in streamline which is brought about by change in this wall angle. So, now, once we have this mark number change we can estimate the change in other properties also d t by t will be equal to by just differentiating the isentropic relationships we can get this let me call this equation 3 this is a change in temperature. So, once again differentiating that isentropic relationship we get this and since we are talking about an isentropic flow without any work done there is no work done here an isentropic flow therefore, stagnation temperature remains constant. So, t naught is constant in this case therefore, when we differentiate this t naught goes away goes away similarly stagnation pressure is constant. So, we can take the isentropic relationship for pressure differentiate it we can get the relationship for change in pressure. So, that will be given by change in pressure relationship will be given as d p by p is equal to minus gamma by 2 t m square 1 plus gamma minus 1 by 2 m square let me call this equation 4. Similarly, we can get the change in density d rho by rho is minus d m square upon 2 1 plus gamma minus 1 by 2 m square let me call this equation 5. So, we have 5 equations listed here 1 2 3 4 5 all this equation actually represent the change in flow properties when the angle theta or the wall curvature is changed for a supersonic flow with the mach number m right. We are now getting a change in velocity change in mach number change in temperature pressure and density and this is what we wanted to estimate how the properties change when we have the variation in angle. So, now the wall curvature then is giving us all the required changes that we are looking for. So, the wall curvature now can be replaced here we have in this case what we have done is we have considered a small curvature right. So, using this small curvature which make the calculate this changes right. So, then what we can do is this is our finite curvature we can break it into many small curvatures right. So, for this we know how much is d theta we just get all this then come to this when we come to this point now this mach number has come from that solution and then we have another change in theta. So, another change will occur we can get the conditions here. So, we move downstream we keep on changing the wall curvature and we keep on changing getting the new flow properties because of change in wall curvature till we come to this point at the end. So, the wall curvature can be replaced by finite number of straight line segments right. So, this is a curved wall what we have done is we have removed replaced it by many many straight line segments. And thus, there is a change in the wall the flow properties in the flow field corresponding to each infinite symbol small turn can be calculated now right. So, if you break this curvature into many many small straight line curvatures we can calculate the entire property variation here during this third portion. Now, so what we have done so far once again go back to our original drawing we are now focusing on this portion only we are focusing on this is the straight line C A C B A dash B dash remember what I said is that in this portion either in this region or in this region there is only one type of mach line on this side we have all right learning mach lines here we have all left learning mach lines. So, therefore, the property variation essentially if I get the variation in the wall that is good enough. So, from here to here then knowing the wall curvature here I can get all the property variations and that is what we are doing. So, what we are doing is the from here to here we are breaking this wall into many small segments and calculating the variation here. And since for each of them we have the same property nothing changes. So, we can get the full property variation within this zone. So, this is the first part of our problem our problem actually has multiple parts. First part is when we have only single type of mach lines that we have now explained how to get the property variation one thing for this portion we need to know theta variation right. So, this curvature for this portion must be specified. So, for this portion it is not designed it is initial curvature is specified. So, we get all this from there. So, this theta variation is specified on this wall we get all this. Now, coming back to this now when we come to this zone. So, here do it just like this. So, as I as we have seen here this portion and this portion we have only single type of mach lines. When we come to this portion now we see that the mach lines are crossing left running and right running are crossing right. And not only that if I look at this domain it has come from here and here, but if I look at this one it has come from some other location right. So, therefore, this mach line up to this is ok when it crosses this it has also cross this mach line. So, the properties have changed here. So, at this point there is a different property when it comes here again is a different property whereas, this has seen only one change right. So, there is a continuous change in the properties. So, now we have mach lines crossing each other or the flow crossing each other with different properties and now that makes it little more complex. So, what now we have is intersection of two mach lines first of all it can be a intersection of one left running essentially always a left running and right running mach line will intersect. But the point is that these two mach lines may not have emitted from the same disturbance. So, therefore, the mach number and the flow properties corresponding to these two mach lines may not be same they can be same also. For example, if I look at this point this mach line and this mach line are same. So, when we come to this domain it has the same property, but this and this are there going going across they may not be coming from the same mach line or same emitted from the same property. In that case one mach line has emitted from a flow with mach M 1 other has emitted from a flow with mach M 2 and then the mach angles are going to be different for these two. So, that needs to be accounted for then we the next thing what we are going to see is that when the two mach line intersect how do the properties change and this is one part finally, what is our goal is to make the flow straight or design is not complete. So, next thing what we will do is we will look at how the mach lines intersect what will be the property variation and then so far here the wall itself was enough to give us the conditions. Now, we will design the wall curvature that what kind of the theta should be given here the theta was specified. Now, we will change the theta in such a way that we get finally, a uniform flow at the exit. Let us say this is our nozzle we will get a flow which is uniform like this. This is what we want to do for that first we have to understand how the properties will change when the mach lines intersect and the next is to complete the design. So, I think we have spent almost used up the time today. So, what I will do is I will stop here today in the next class we is first talk about the intersection of mach lines then continue with the design process using method of characteristics and finish this discussion. After that we talk little bit about plug nozzles which are other other type of shape nozzles and then the effect of friction we will talk about the effect of heated heat transfer we will talk about little bit there will be just small descriptions. The main focus in the next class will be completing this method of characteristics particularly when the mach lines cross each other what happens. So, I will stop here now in the next class we will continue from here.