 Hello and welcome to lecture number 22, part B. In the last lecture, we were discussing about aspects of compressible flow and what are the different terminologies that one needs to understand in order to analyze and make use of compressible flows. So, there are several aspects of compressible flows which are slightly different from what we have been analyzing so far and in our analysis which we carried during initial part of our course, there was an inherent assumption that the density does not change or the changes with the flow does not really have much kinetic energy and so, changes associated with that kinetic energy was always neglected. But, if you look at flows which involve higher speeds, then kinetic energy term can no longer be neglected and also it is possible that density changes cannot really be neglected. So, how do we take these into account? So, these were through stagnation properties, we have already defined and derived equations for stagnation properties like stagnation enthalpy, stagnation pressure, temperature and so on. And also we have seen that in the absence of any heat or work interactions, stagnation enthalpy does not change which means that for an ideal gas the stagnation temperature does not change across an area of constant of flow through a duct and this is in the absence of any heat or work interactions. But, it is possible that the static pressure or in fact the stagnation pressure can change even if the stagnation enthalpy does not change, stagnation pressure may change because of frictional effects and so, frictional effects can occur even in the absence of any heat or work interactions. So, in a duct flow if there are no heat or work interactions and even if there is and in spite of that if there are frictional losses, then it is possible that stagnation pressure might change, but stagnation temperature enthalpy does not change. So, these were some of the aspects we had discussed in the last lecture. So, what we are going to discuss today is a continuation of some of the aspects we discussed in the last lecture. Let us take a look at what we had discussed in what we are going to discuss in today's lecture. We shall be talking about what are meant by shock waves and expansion. We shall then continue to discuss about different types of shock waves, normal shocks, oblique shocks and Prandtl Mayer expansion waves. So, this is what we will begin our lecture with. We will then continue on to discuss about duct flow with heat transfer and negligible friction. So, these flows are usually classified as Rayleigh flows and we will also look at some aspects of property relations for Rayleigh flows. And towards the end of the lecture, we will be discussing about duct flow with friction, but without heat transfer and these flows are known as Fano flows and so these are some of the topics that we are going to discuss in today's lecture. And what we will begin our lecture is on discussion on shock waves. And if you recall during the later part of the previous lecture, I was discussing about flow through a converging diverging nozzle. And we saw that as you change the back pressure at a certain back pressure, there is sonic flow at the throat and then the flow becomes supersonic and then abruptly it becomes subsonic. I mentioned in the passing during last lecture that this is because of the presence of a shock wave in the divergent section of the nozzle. So, let us look at what we mean by shock waves. So, shock waves are basically certain aspects of a flow in a supersonic flow where there could be abrupt changes in fluid properties. Now, we have already defined what is meant by speed of sound. Now, sound waves are caused by infinitesimally small pressure disturbances and they travel through a medium at the speed of sound. But under certain flow conditions, there could be abrupt changes in fluid properties which can occur through a very thin section and that is known as a shock wave. So, shock wave is a very thin section in a fluid flow across which there could be a sudden change, an abrupt change in fluid properties like pressure, temperature, density and so on and also Mach number. And shock waves are characteristic of supersonic flows. It means that shock waves cannot exist in subsonic flows. So, shock waves occur only in supersonic flows and the reason for this is that in a supersonic flow, the speed at which the fluid moves or the vehicle moves is greater than the speed of sound. And we have seen that sound waves are essentially pressure waves which means that as a vehicle or fluid moves at a speed which is greater than the speed of sound, then the information travel does not occur upstream which means that if there is a vehicle which is moving at a supersonic speed, then the presence of this vehicle is not known to fluid particles which are ahead of the vehicle. And therefore, what happens is that the fluid particles strike the vehicle and since they have to take the shape of the vehicle, they have to flow over the vehicle, this has to occur through the presence of an abrupt change in fluid properties like velocity, temperature and so on. And that occurs through the presence of shock waves. And so, shock waves are very thin sections in a supersonic flow across which there are abrupt changes in all fluid properties. There are some fluid properties which do not change, we will discuss that. And so, since there are abrupt changes taking place through the shock wave, flow through a shock wave is highly irreversible. And therefore, flow through a shock wave is not to be considered as isentropic. So, flow across a shock wave or in the section of the shock wave is non-isentropic. So, you cannot consider that shock wave, flow through shock waves are isentropic. And so, that is one of the aspects of a shock wave. And so, what we will do is what we will first analyze what is meant by a normal shock wave. So, to do that I will take you back to the pressure variation across a convergent divergent nozzle, where we had discussed that at certain back pressure there is a shock wave. So, let us take a look at let us take a relook at what is happening across a convergent divergent nozzle. So, this is what we had discussed in the last lecture that this is a convergent divergent nozzle. And as you change the back pressure that is P B, then the fluid properties across the nozzle changes in particular manner for different values of P B. So, as you reduce P B at a certain pressure there is no change in this the pressure across the nozzle. As you reduce back pressure further then the flow accelerates in the convergent section of the nozzle, it reaches the minimum pressure at the throat. And then it again increases because it is a divergent section divergent section in a subsonic flow acts as a diffuser. So, the pressure rises again and it exits at a certain pressure. Now, if you reduce the back pressure further it becomes it reaches the minimum pressure at the throat. And that is the pressure which is the critical pressure basically and that pressure the flow becomes sonic that is you get a Mach number is equal to 1. Subsequent to that it again rises. If the back pressure is further reduced to let us say a value P D then the flow from the throat accelerates it becomes supersonic. But again at a certain point abruptly it becomes subsonic. So, this abrupt change in the pressure and also the Mach number which you can see here Mach number was one at the throat and then it becomes supersonic. And after this point it again becomes subsonic suddenly. So, this is because of the presence of a shock I had mentioned that there is a shock in the nozzle. So, usual form of a shock which can occur in such flow is a normal shock. So, this is because of the presence of a normal shock that is across a normal shock. There is an abrupt change in the properties which is what has which is what has happened here that across the shock the fluid properties have changed abruptly. And so, this is because of the presence of a normal shock. So, what do we mean by a normal shock? So, a normal shock essentially is a shock across which we will wherein the shock wave and the flow and directions meet at 90 degrees that is the flow direction is normal to the shock wave itself. And so, that is why they are called normal shock waves. And so, we have seen in the case of the convergent divergent nozzle if there is a normal shock wave the flow becomes subsonic across the nozzle. And so, a normal shock wave can occur in supersonic flows. And so, how much the Mach number downstream of the normal shock is depends upon the upstream Mach number and also the basically it just depends upon the upstream Mach number. And the property of a normal shock wave is that downstream of the normal shock wave the fluid becomes subsonic that is downstream which is what we saw in the supersonic nozzle case that after the normal shock wave the flow becomes subsonic and the nozzle now becomes or behaves like a diffuser. So, downstream of the normal shock the flow becomes subsonic. So, let us look at what are the features of a normal shock wave. So, shock waves that occur in a plane which is normal to the direction of the flow are called normal shocks a supersonic flow across a normal shock wave essentially becomes subsonic. Now, since in this case or in the case where there are there is no heat or work interactions we have seen that conservation of energy principle states or requires that the enthalpy remains a constant which means that the stagnation enthalpy at station 1 just before the shock and at station 2 just after the shock becomes the same or is the same. So, h 0 1 is equal to h 0 2. So, if you consider the gas to be an ideal gas with constant specific heats then it follows that from the energy equation that the stagnation temperature before the shock is equal to the stagnation temperature after the shock. So, this is a property of a normal shock that because you do not have any heat or work interactions the stagnation temperature does not or cannot change across the normal shock. But, there are other parameters which change substantially across the shock Mach number was one parameter I mentioned that Mach number downstream of the shock becomes subsonic. So, that is one parameter which changes since Mach number is changing it follows that the velocity also will change. And so, what are the other properties or parameters which will change across the shock which we shall analyze shortly. And we will take a look at what properties across the shock can change and how this can change and how can you correlate the properties downstream of the shock with the properties upstream of the shock. Let us take a look at how we can correlate them. So, here we have flow through a duct it is a generic duct it could be either converging diverging whatever or a constant area. And what is indicated by this blue line here is the shock wave it is a normal shock. And we have taken a very thin control volume which surrounds the shock wave. So, upstream of the shock wave we have a supersonic Mach number and we have properties of the fluid which is velocity of v 1 pressure p 1 static enthalpy h 1 density rho 1 and entropy s 1 downstream of the shock where the Mach number is subsonic we now have velocity v 2 pressure p 2 enthalpy h 2 density rho 2 and the entropy s 2. So, given these properties we will now try to correlate properties upstream and downstream of the shock. So, basically what we will try to do are to apply the governing equations of fluid motion we have primarily 4 in fact 5 governing equations equation for mass energy momentum entropy and the equation of state. So, if you take up 4 of these governing equations let us look at the mass equation or conservation of mass which states that mass flow rate before the shock and after the shock should be the same. So, rho 1 a 1 v 1 should be equal to rho 2 a 2 v 2 conservation of energy states that stagnation enthalpy does not change. So, h 0 1 is equal to h 0 2 conservation of momentum states that area into p 1 minus p 2 is equal to m dot into v 2 minus v 1. Increase of entropy principle states that s 2 minus s 1 is greater than or equal to 0 this is followed this is basically a follow up of the third law of thermodynamics. So, what happens is that if you were to combine 2 of these equations let us say we combine the mass and energy equation and then we plot the combined equations on an h s diagram that is enthalpy entropy diagram then the resultant curve that we get which is basically a combination of the mass and momentum mass and momentum energy equation it is basically known as the Fano line and such flows are basically known as Fano flows we will analyze Fano flows later on in the lecture. Similarly, if we combined mass and momentum equation we get a another equation which when plotted on h s diagram we get a line which is known as the Rayleigh line and such flows are known as the Rayleigh flows. So, if you combine the mass and energy equation we get the Fano line the mass and momentum equation combines we get the Rayleigh line and it follows that these 2 curves when plotted on the same h s diagram will intersected 2 different points and the solution of these 2 points refers to the flow through the shock wave. So, let me illustrate that through an h s diagram. So, on this h s diagram you can see that I have plotted the Fano line equation as well as the Rayleigh line equation the Fano line equation is shown by the red line and the Rayleigh line equation is shown by the blue line these 2 curves meet at 2 points 0.1 and 0.2. So, if you join these 2 lines which is shown by this dotted line that indicates the flow across the shock wave. So, let me explain this h s diagram little more detail let us take a closer look at what is happening across these 2 different curves. So, we have seen that stagnation enthalpy does not change across a shock wave and therefore, it should mean that at 0.1 and 0.2 the stagnation enthalpy should be the same which is what is shown here h 0.1 is equal to h 0.2 and that is joining these 2 points which are intersection of the Fano and the Rayleigh lines and it means that the static enthalpy is obviously can be different. So, static enthalpy at station 1 h 1 plus v 1 square by 2 is basically equal to h 0.1 similarly, at station 2 h 0.2 is equal to h 2 plus v 2 square by 2 and which means that since across a shock wave velocities will be different Mach numbers across a normal shock the Mach number becomes subsonic velocities also change it follows that v 2 will be less than v 1. And therefore, we have h 2 and h 1 which are not equal and h 2 being greater than h 1 and what about entropy? Entropy across the shock increases I mentioned that shock wave is an irreversible process it cannot be considered to be an isentropic process. Therefore, s 2 is greater than s 1. So, we have an increase in entropy here and stagnation pressure now since stagnation enthalpies are same, but static enthalpies are different and there is a loss of pressure stagnation pressure across a nozzle across a shock wave. We now have p 0 1 not equal to p 0 2 p 0 2 is in fact, less than p 0 1 which is why these 2 lines are shown as separate lines the constant pressure lines p 0 1 p 0 2 are different. You can also see that I have indicated 2 different points here. These are the points at which on the fanno as well as Rayleigh line which we will analyze in detail little later. These are the points at which the curve changes its direction and those are the points which correspond to sonic flow that is Mach number is equal to 1 occurs at point A and point B on the fanno and Rayleigh lines respectively. Below these lines we have a supersonic flow and above the lines we have a subsonic flow. So, basically from the H s diagram what we can get is that what we can understand is that the shock wave is something which you can derive from solving the fanno line and the Rayleigh line equations and the point at which these 2 curves intersect basically refers to the flow through the shock wave. And we have also seen that since stagnation enthalpy cannot change because of conservation of energy principle the static enthalpies can be different. Static enthalpy in fact downstream of the shock is higher than the static enthalpy upstream. Stagnation and pressure drops across the shock wave and similarly let us also look at what happens to static pressure and static temperature. We have already seen stagnation temperature does not change and stagnation pressure drops across the shock wave what about static pressure and static temperature. So, if you were to analyze that we have to relate the properties upstream and downstream of the shock wave and let us take a look at how we can relate these 2 upstream and downstream properties. So, if you have to derive expressions before and after the shock wave. So, before the shock wave we have the flow is isentropic before the shock wave. And so we have T 0 1 by T 1 is equal to 1 plus gamma minus 1 by 2 m 1 square where m 1 is the upstream Mach number T 0 1 is stagnation temperature upstream T 1 is static temperature upstream. Similarly, T 0 2 by T 2 is equal to 1 plus gamma minus 1 by 2 m 2 square where m 2 is the downstream Mach number T 0 2 and T 2 are the temperatures downstream of the shock wave. So, these are followed from the isentropic expressions. Now, we know that the stagnation temperatures are equal. So, T 0 1 is equal to T 0 2 if you do that we get an expression in terms of temperature ratios and this we can again further simplify in terms of isentropic relations because P 2 by P 1 can be related to T 2 by T 1. And so we have P 2 by P 1 is equal to m 1 into square root of 1 plus m 1 square gamma minus 1 by 2 divided by m 2 into square root of gamma 1 plus m 2 square gamma minus 1 by 2. So, this is primarily the final line equation for an ideal gas with constant specific heat. So, this is basically looking at the mass and energy equations and solving them we primarily get the final line equation where we can relate the upstream and downstream pressures in terms of in terms of the corresponding Mach numbers. Similarly, we can combine and simplify the mass and momentum equations and what we get is an equation for Rayleigh line. So, if we were to combine the mass momentum equations we basically get the Rayleigh line equation and we correspondingly can relate some of the properties that are upstream and downstream. And so if you look at these two equations and simplify them what we can do is that we can relate the downstream Mach number which is m 2 with the upstream Mach number. So, what we get is m 2 square which is downstream Mach number is equal to m 1 square plus 2 by gamma minus 1 divided by 2 into m 1 square gamma divided by gamma minus 1 minus 1. So, the upstream Mach number and downstream Mach number can be related through the simple equation and what we can see is that they primarily depend upon the ratio of specific heats which for an ideal gas is 1.4 typically. So, we can relate the upstream and downstream Mach numbers through a very simple equation. Similarly, the other parameters like the temperature pressure and so on can actually be related and we can either calculate these and since we can see that they depend specifically on certain properties for an ideal gas where gamma is equal to 1.4 we can actually write down tables for calculating the downstream properties given the upstream properties. And so these properties are actually available in tabulated form and these are known as the shock tables. And if you refer to any book on thermodynamics or in compressible flows towards the end in the appendix you will definitely find these properties which are listed in the form of tables and these are known as the shock tables and usually referred to as either the normal shock tables. In fact, what we will see little later is that when we talk about oblique shocks that oblique shock properties can also be derived from the normal shock tables assuming or simplifying the velocity vectors which are there on an oblique shock and we can calculate the properties across an oblique shock from a normal shock table. So, that is something we will discuss little later. So, basically what we can do is that we can relate the properties that are downstream of a normal shock with that of the upstream properties using simple relations which we have just seen in which primarily depend upon the ratio of specific heats and a few other properties. And so for an ideal gas it is possible that we can get tabulated forms of these normal shock properties and they are basically related to equations of the fano line as well as the Rayleigh line. So, if you look at the previous equation I was talking about where we relate the Mach number which is downstream of the shock with the upstream Mach number it basically represents the intersections of the fano and the Rayleigh lines. If you recall during the discussion on the fano and Rayleigh line I mentioned that there are two points where they intersect and which is basically the shock the flow through the shock and so this equation basically represents those two intersection points. So, to summarize across a normal shock what happens is the upstream Mach number is supersonic, downstream Mach number becomes subsonic and across a normal shock these are the variations of different properties. There is an increase in the static pressure and correspondingly there is a decrease in the stagnation pressure. So, static pressure across a shock increases stagnation pressure across a shock decreases velocity decreases Mach number also decreases across a normal shock and the static temperature increases across a normal shock. Whereas the only property which remains a constant across a normal shock is the stagnation temperature. So, stagnation temperature across a normal shock remains a constant it cannot change because there is no heat or work interaction taking place and entropy across a normal shock increases because the process is highly irreversible entropy increases. So, these are the different variations of the properties across a normal shock and some of these properties will also be these variations will also be valid for an oblique shock which is what we will discuss next. That when there are flow situations when the when the shock need not necessarily be normal to the flow. So, under these circumstances the shock wave can be inclined at a certain angle to the flow and such shock waves are known as oblique shocks and there are several flow situations where we encounter oblique shocks and so flow downstream of the oblique shocks may be subsonic or it may be sonic or it may remain to be supersonic depending upon the Mach number and the turning angle and so on. So, let us look at what we mean by oblique shocks. So, shock waves that are inclined to the flow at an angle are basically known as oblique shocks and why do we have an oblique shock in the first place you already seen why normal shocks occur the same reason applies for an oblique shock as well that in a supersonic flow the presence of obstacles cannot be felt by the flow which is upstream and therefore, the flow has to take an abrupt turn when it hits an obstacle. So, this abrupt turning basically takes place through the presence of shock waves and in the case of obstacles which are like a wedge or a cone in a supersonic flow the turning of the flow takes place through the presence of oblique shocks. So, the angle through which the fluid turns is known as the deflection angle or the turning angle usually denoted by theta and the inclination of the shock or the is basically known as the shock angle or the wave angle that is when a supersonic flow hits an obstacle the angle through which it turns is basically known as the deflection angle of the shock or the turning angle denoted by theta and the angle of the shock wave is basically known as the wave angle or the shock angle. So, we will now look at how we can or the various terminologies associated with the normal shock like deflection angle, shock angle etcetera. How do you find out these angles given a certain geometry in a supersonic flow? So, let us consider a simple example here. So, what we have here is a two dimensional wedge. So, wedge has a half angle of delta and so, we have supersonic flow Mach number greater than 1 m 1 which is using the wedge and as it hits the wedge because the presence of the wedge is not known to the fluid because information is travelling at a speed greater than the speed of sound. So, the presence of this in terms of pressure waves cannot travel upstream and so, what happens is that the fluid knows that the there is an obstacle only after it hits it and so, the fluid has to take an abrupt turn and this abrupt turning of the fluid occurs through the presence of these oblique shocks. So, with the black lines which are shown here are the oblique shocks. So, the presence of this oblique shock causes the downstream flow to be deflected by a certain angle which is known as the deflection angle or the turning angle which is theta which is basically equal to the angle half angle of the wedge itself. So, the flow downstream of the oblique shock takes a direction which is parallel to the wedge itself. So, m 2 will have a direction which is parallel to the this wedge the side of the wedge and the angle at which the oblique shock is inclined is known as the wave angle or the shock angle and that is denoted by beta. So, beta is the angle of inclination of the oblique shock. So, downstream Mach number m 2 will be a value which is different from m 1 which will be certainly less than m 1, but it need not necessarily be a subsonic Mach number and like a normal shock where the downstream Mach number is always less than 1 in an oblique shock the downstream Mach number may continue to remain supersonic, but less than the upstream Mach number or it could become sonic or it could become subsonic and that depends upon the upstream Mach number and these deflection angles. So, like we have discussed already for a normal shock oblique shocks are also possible only in supersonic flows and so flow downstream of the shock could either be subsonic or it could remain supersonic or it could be sonic and this depends upon the upstream Mach number and the turning angle. So, how do we analyze an oblique shock? So, to analyze an oblique shock what we do is that we decompose the velocity vectors upstream and downstream of the shock into normal and tangential components. So, we have seen that the flow approaches an oblique shock at a certain angle or the oblique shock is at a certain angle to the flow both upstream as well as downstream. So, what we do is that we decompose this velocity both the upstream and downstream velocity vectors into their normal and tangential components and then from the normal components we can use the shock tables. The normal shock tables we have already discussed and calculate the properties downstream of the shock from the normal shock tables and then using algebraic manipulation we can find out the properties downstream of the oblique shock. So, let us let us take a look at how we could do this. If you look at this illustration here we have an oblique shock and there is an upstream velocity which is v 1 velocity downstream of the oblique shock is v 2. What we do is we decompose this velocity vector in terms of its normal component and tangential component both for upstream as well as the downstream cases. So, v 1 n is the velocity vector that is normal to the shock upstream v 2 n is the velocity vector downstream of the shock and normal to the shock v 1 t is the tangential component v 2 t is the tangential component of velocity downstream. So, it can be shown that for an oblique shock the tangential component does not change across the shock and so v 1 t will be equal to v 2 t, but v 1 n and v 2 n obviously cannot be the same. And so basically we can now relate the velocity vectors which are upstream and downstream of the shock using the shock angle beta and the deflection angle theta. So, it basically depends upon three parameters the deflection angle theta the shock angle beta and the mach number upstream mach number m and all these three parameters are closely interlinked and we will see how they are interlinked in the form of a chart where which relates mach number deflection angle theta and the shock angle beta. So, the same set of vectors or the same diagram if it is tilted and made normal. So, in the previous case the oblique shock was inclined at a certain angle. Now, if we tilt it and make this normal what we get is the following. So, what we have done is the previous diagram tilted and made normal. So, that the shock we have now the oblique shock is now oriented like this. So, what we see here is that now the flow or the normal component of this velocity which is approaching the shock which is v 1 n hits the shock at 90 degrees and it leaves the shock also at 90 degrees. So, this is very similar to what we have discussed for a normal shock and. So, from normal shock tables we should be able to find out the properties downstream of the shock because now we have one component of the flow which is normal to the shock itself. So, in normal shock sense we have the corresponding mach numbers m 1 n which is greater than 1 and we have already discussed that flow downstream of the normal shock has to be subsonic. So, in terms of the normal components v 2 n when converted to mach number m 2 n will be less than 1, but it is not necessary that m 2 will be less than 1. The normal component upstream and downstream of the shock of an oblique shock will be subsonic, but not necessarily the absolute mach number and. So, if you look at the flow angles we have the shock angle which is beta which is the angle at which the oblique shock is inclined and this is the deflection angle theta and this difference is beta minus theta. So, from these angles we can relate that m 1 n which is the mach number normal component of the upstream mach number is equal to m 1 sin beta that is from this velocity triangle if you see m 1 n is this component it is basically equal to m 1 sin beta. Similarly, m 2 n is for the downstream of the shock m 2 n is equal to m 2 sin beta minus theta. So, that is this angle beta minus theta. So, m 2 n is equal to m 2 sin beta minus theta which is primarily this angle here m 1 n is equal to v 1 n by c 1 and m 2 n is v 2 n by c 2 c 1 is the upstream speed of sound and c 2 is the downstream speed of sound which will be equal to square root of gamma r t 1 for c 1 and for c 2 it will be square root of gamma r t 2. So, what follows is that if we decompose the velocity vectors into normal and tangential components since the tangential component remains unchanged we can use the normal shock tables which we discussed for solving oblique shock in the sense that we can find the downstream Mach number the normal component of the downstream Mach number from the normal shock tables and once we know the beta and theta angles we can now calculate the absolute Mach number which is for downstream of an oblique shock which means all the shock tables and equations which are applicable for a normal shock can be extended for the normal components of the velocity vectors that are upstream and downstream of the shocks and using those relations we can relate the properties that are upstream and downstream of the shock and that is how you could solve an oblique shock problem very similar to that of a normal shock problem and of determine properties which are upstream and downstream of the shock. So, it is possible for us to relate the deflection angle theta to the shock angle beta with the Mach number and if we correlate all the three we can plot the values of theta beta and m for different values of or for a range of beta theta as well as Mach numbers. So, if we do that we have a chart or a graph or a plot for theta beta m where we have the deflection angle theta on the y axis the shock angle beta on the x axis and Mach numbers different contours of constant Mach numbers. So, we have Mach numbers starting from very low Mach numbers sonic Mach numbers all the way up to Mach number tending towards infinity upstream Mach number tending towards infinity. So, you can immediately see that for a particular deflection angle theta and a particular Mach number it is possible that you have two different wave angles. Let us say for example, we take a deflection angle theta of 10 degrees and a Mach number upstream Mach number of Mach number of 2 then it is possible for us to get two different deflection angles one is around 41 degrees around 40 degrees here and another angle which is on the higher side it is more than double of that angle. So, what I have indicated here is that depending upon which angle the wave angle is the shock wave could either be a weak shock or a strong shock correspondingly it is also possible that the Mach number can either be supersonic or it could become subsonic or in the limiting case it could also be sonic. So, you can see that for as at a certain point where Mach number is equal to 1 if you were to join all those points we get a constant sonic Mach number line which is indicated by this dotted line and the different points at which the Mach number becomes one is indicated here. To the left of this line we have supersonic Mach number to the right of this line we have subsonic Mach number. Similarly, you could also have a maximum theta for a particular Mach number below which the shock becomes weak shock or after which the shock is a strong shock. It basically means that there are certain values of theta and beta as well as upstream Mach number for which the downstream Mach number continues to remain supersonic though it would be less than the upstream Mach number it would continue to remain supersonic and there are also cases where it could either be sonic or it could become subsonic depending upon the solution of the oblique shock equation. So, for different values of Mach number what we can we can understand from this chart of theta beta m is that there are basically two possible values of beta for any value of theta which is less than theta max and. So, if you look at the theta is equal to theta max line these are lines on which on the left of which we have weak oblique shocks and on the right of this line we have strong oblique shocks. So, let us take a look at that once again. So, this is the line which joins all the theta max lines that is if you need to have an attached oblique shock for a given Mach number let us consider Mach 2 which is what we were discussing. If you look at an angle which is greater than this point that is around 22 degrees any angle greater than that for this Mach number would lead to a shock which is not attached to the surface. So, you would have a case where the shock is detached from the surface. So, for an attached shock this is the maximum theta which is permissible for this particular Mach number and it also follows that on the left of this line we have weak oblique shocks and on the right of this line we have strong oblique shocks. We could also join all the Mach number equal to 1 points on all these Mach number lines and on the left of this line we have supersonic flow and on the right of this line we have subsonic flow. So, for a given upstream value of upstream Mach number there are basically 2 shock angles and so on the left of the constant Mach number line constant sonic Mach number line we have supersonic flow and on the right hand side we have subsonic flow which means that if you have a deflection angle of let us say 10 degrees and the Mach number is let us say 2 then if you have a weak oblique shock then it means that the Mach number continues to remain supersonic because the solution comes on to the left of this line. And if it is a strong of solution that you have the Mach number can become subsonic downstream of the oblique shock which is. So, in most of the cases we tend to see the weak oblique shock case that is the Mach number continues to remain supersonic, but it is also possible that we can get a subsonic flow downstream of the oblique shock. And for any given value of Mach number and deflection angle beta is equal to beta min or minimum beta represents the weakest possible oblique shock at that Mach number which is basically known as a Mach wave that is for any particular Mach number let us say Mach 2. So, the minimum Mach beta that is possible is about 30 degrees here. And so the Mach number or the shock waves that occur at this particular instance are known as Mach wave that is their very weak shock waves that are present and they are basically known as Mach waves. And so far we have been discussing about shock waves which wherein we have an increase in the static pressure and static temperature and so on downstream of the shock. And there are also flow situations which we will discuss now wherein if let us say the wedge which I had shown for discussing the oblique shock is inclined at a certain angle to the flow. What happens to the flow which is upstream and downstream of the wedge that is on certain corners of the wedge we would have shock waves present because the flow is taking a compression corner. And on the other side of the wedge we may have what are known as expansion waves or expansion fan which are present through which the flow will accelerate. So, in a supersonic flow which is expanding we might encounter very weak waves or sonic waves which are basically known as the expansion waves or expansion fan as we denoted. So, if you look at for example, a two-dimensional wedge which we had taken up for the oblique shock case and so if it was inclined at a certain angle and the flow is likely to expand on one of the corners of the wedge then we see an infinite number of Mach waves which will originate from a particular point on the wedge on the wedge and that is basically known as a Mach wave. And these Mach waves are also often referred to as the Prandtl Mayer expansion waves and we will take a look at what we mean by Prandtl Mayer expansion waves. So, Prandtl Mayer expansion waves basically can denote an infinite number of Mach waves which form when we have a wedge which is at a certain angle of attack or under any other expansion corners. And the Mach number downstream of the expansion fan increases unlike a shock wave where the Mach number decreases and also pressure, temperature and density decrease which is exact opposite of what happens across a shock wave. So, this was the wedge I had shown for an oblique shock. Now, if the wedge was not aligned to the flow and it is at a certain angle then the flow encounters a compression corner on one surface and on the other surface it encounters an expansion corner. So, on the compression corner we continue to have an oblique shock whereas, on the expansion corner because the flow has to now expand it is a supersonic flow and there is an increase in area as you can see here and so it has to be an expansion flow. And so this occurs through the presence of these expansion waves. And the inclination of these expansion waves are usually denoted by the symbol mu. And so there could be infinite number of these expansion waves I have shown only a few of them. And so downstream of these expansion wave the Mach number increases that is m 2 would be greater than m 1. So, there is an increase in Mach number across an expansion wave whereas, there is a decrease in the Mach number across an oblique shock. So, Prandtl Mayer function or Prandtl Mayer expansion waves are inclined at a local Mach number or local Mach angle which is basically denoted by mu. So, mu is the Mach angle for the first expansion wave mu 1 is can be shown to be equal to sin inverse of 1 by m 1. Similarly, mu 2 is equal to sin inverse 1 by m 2. So, mu where mu 2 is the expansion where it is basically the angle for the last expansion wave. And so the turning across the expansion fan turning angle across the expansion fan theta is equal to mu of m 2 minus mu of m 1 where mu of m is known as the Prandtl Mayer function. That is here we denote a function mu which is a function of the Mach number and of course, it is also related to the ratio of specific heats. So, turning angle across an expansion fan can be related to the Prandtl Mayer function at Mach 2 and at Mach 1 where the Prandtl Mayer function can be related to the Mach number in the form of this expression which is basically equal to square root of gamma plus 1 by gamma minus 1 into tan inverse square root of gamma plus 1 by gamma minus 1 multiplied by m square minus 1 minus tan inverse square root of m square minus 1. So, this is this basically denotes the Prandtl Mayer function from the Prandtl Mayer function for upstream and downstream Mach numbers you can calculate the Prandtl Mayer function and the difference between these two functions are basically denotes the turning angle for such a case. So, what we will discuss next are slightly different from what we have discussed of course, it is related in some sense because we are going to talk about Rayleigh and Fano functions and Fano processes. And so, we will first take up a duct flow with heat transfer with negligible friction. So, there is a duct flow case where we consider a duct of constant area and there is heat transfer into or from the system, but there is negligible friction. And this is encountered in several engineering problems like for example, in a combustion chamber we have heat transfer into the combustion chamber, but if we assume friction to be negligible then we can we can approximate this particular process in a simple way that is we basically model combustion as a heat gain process and of course, we neglect chemical composition across the duct. So, this is what we had done for analyzing heat transfer across a combustion chamber. So, the one dimensional analysis or flow through of an ideal gas with constant specific heat through a duct of constant area with heat transfer and negligible friction are known as Rayleigh flow. So, Rayleigh flow is basically heat transfer into an area in a into a duct of constant area of an ideal gas with negligible friction. So, we are going to assume that there is no friction occurring here, there is only heat transfer which causes change in properties across the control volume. So, if you have a gas which has an inlet property which a set of inlet properties P 1, T 1, density rho 1, V 1 and entropy S 1 which are known the exit properties can be calculated from the 5 governing equations of mass, momentum, energy, entropy and equation of state. So, if you were to represent the Rayleigh flow on a T s diagram that is known as a Rayleigh line. So, Rayleigh line represents the locus of all physically attainable downstream states corresponding to an initial state. So, if you define a particular initial state with pressure, temperature, density, velocity and entropy Rayleigh line represents all the properties downstream which are physically attainable which primarily come from solution of all the 5 governing equations. So, if you plot the Rayleigh process on a T s diagram temperature entropy diagram, then we have a very interesting phenomena that is taking place here that is we have this blue line that is shown that is the Rayleigh line. And we can see that as we heat of as we continue to add heat in a supersonic flow which is greater than 1, then as we continue to heat then it approaches Mach number equal to 1. Similarly, in a subsonic flow if we continue to add heat it approaches its Mach number increases and approaches Mach number equal to 1 again and the reverse happens for cooling as well. And there are two distinct points I have shown here one is the point of maximum entropy which in the case of supersonic flow occurs when it reaches it is a limiting Mach number that is Mach number equal to 1. And that is also happening for a subsonic flow where its Mach number increases and finally, reaches a Mach number equal to 1 at point a which is the point of maximum entropy. In the case of subsonic flow we also have a point of maximum temperature T max which means that as you continue to add heat in a subsonic flow it attains a maximum temperature which is given by the Rayleigh line up to Mach max beyond which if you continue to add heat the temperature actually reduces. That means that up to point or between point a and b in a subsonic flow if you add heat you could actually it could actually lead to drop in temperature. So, if you were to summarize this Rayleigh line equation the Mach number at point a is corresponding to sonic Mach number which is Mach 1 point of maximum entropy on the upper arm of the Rayleigh line that is above point a the flow is subsonic and the states on the lower side of point a are supersonic. Heating increases the Mach number for a subsonic flow but it decreases for supersonic flow and both in both the cases Mach number approaches unity during heating that is in both subsonic as well as supersonic flow the Mach number approaches unity during heating. So, what happens in a Rayleigh line process is that if the Mach number is subsonic then if it is subsonic flow then for a given or predefined temperature upstream downstream stagnation temperature increases because you are adding heat. So, stagnation temperature has to increase static temperature may increase or decrease depending upon where you are on the Rayleigh line this is on a subsonic flow. In a supersonic flow case stagnation temperature increases static temperature also increases because the limiting case for that is Mach number equal to 1 there is no change of curve there whereas, in a subsonic flow there is a T max after which the temperature reduces for a certain period. Now, we shall consider another set of flow which is a duct flow with friction but negligible heat transfer the Rayleigh equation or Rayleigh line represented duct flow with heat transfer with negligible friction. Now, an adiabatic flow with friction of an ideal gas with constant specific heat is known as Fano flow. Similarly, Fano line represents the states obtained by solving the mass and energy equation we have seen this earlier when we are talking about normal shocks. So, for an adiabatic flow the entropy must increase in the flow direction because there is friction and so in the case of subsonic flow the Mach number increases due to friction in supersonic flow friction acts to decrease the Mach number in the case of supersonic flow. So, on an H s diagram we can represent a Fano line similar to that of a Rayleigh line that is in a supersonic flow because of friction entropy is increasing there is no heat transfer and so due to friction the Mach number increases and in the limiting case it reaches Mach number equal to 1. Beyond this the Mach number cannot reduce and this state is known as the choking which we have seen for nozzle flows as well. In the case of subsonic flows with friction or due to friction the Mach number would increase and in the limiting case that is at choking point the Mach number reaches 1 beyond which if you try to pass more flow the mass flow would actually it will actually lead to decrease in mass flow. So, in a Fano line it basically states that in supersonic flow due to friction it basically acts to reduce the Mach number to the limiting Mach number of Mach number 1 in subsonic flow it leads to increase in Mach number up to a Mach number of sonic Mach number that is unity. So, the point where Mach number is 1 is known as choking and so it is possible that in a subsonic flow you can accelerate the flow it basically happens because of friction in the case of supersonic flow the flow decelerates limiting case it reaches a Mach number of 1. So, let me summarize what we had discussed in today's lecture. We had discussion on few aspects of compressible flows in which primarily happen in supersonic flows the presence of normal shocks, oblique shocks and Prandtl Mayer expansion waves and subsequently we discussed about two different duct flow cases one was duct flow with heat transfer and negligible friction known as the Rayleigh flow and the second case was a duct flow with friction and negligible heat transfer that was known as the Fano flow. So, in both these cases we have discussed about how the properties of the fluid vary how Mach number changes and what is the limiting case for each of these duct flow problem. So, these were some of the aspects we had discussed during this lecture on compressible flows this was primarily an extension of what we had discussed in the last lecture to begin with on compressible flows and we had more discussion on shock waves and different types of shock waves in today's lecture.