 You may recall that in the previous lecture, I had defined the Mach number m to be the ratio of the speed at that location divided by the speed of sound at the same location. So, v here is actually the speed. So, in case we have for example, a two dimensional or a three dimensional flow, you would take square root of u square plus v square plus w square where u v w are the velocity components in each one of the coordinate direction. But since we are dealing with one dimensional flow now, we can simply take this to be the swim wise component of velocity. And for a calorically perfect gas, you know that the speed of sound, we showed the speed of sound to be equal to square root of gamma RT. Now, Mach number is defined as a ratio. It is a very fundamental quantity in gas dynamics and compressible flow. But it also, I mean a lot of care is required in drawing inferences from values of Mach number. Primarily because it is defined as a ratio, that is ratio of the local speed of the fluid or flow divided by the speed of sound. So, any change in Mach number can be accomplished by either changing the velocity or by changing the temperature, which will have the effect of changing the speed of sound or both. So, when we look at a flow and see that the Mach number changes by so much. It could be due to a change in the speed itself without any other change or change in the speed of sound or a combination of both. Usually it is a combination of both because as the flow expands or as it is diffused in an actual flow field, the thermodynamic properties also change, meaning the speed of sound will change. The fluid dynamic quantity, which is the streamwise velocity also changes. So, usually it is a combination of both. The other complicating factor is that the speed of sound itself is not a constant. It varies from point to point in an actual flow field because the temperature will vary in an actual flow field depending on whether the flow is actually undergoing an expansion process or a diffusion process. So, any directions on the velocity or temperature. So, given a Mach number distribution or change in Mach number, we cannot readily say or readily infer any change in velocity or any change in temperature from this. One has to be careful. Now in gas dynamics, a few reference states are customarily used. Now, reference states allow the governing equations to be simplified and written in a dimensionless form. So, that important parameters can be identified. For instance, many of the expressions that we will derive are usually written in terms of Mach number. Showing that Mach number is the controlling parameter and the equations themselves then can be made dimensionless. So, this also allows tabulated value of quantities in as a function of Mach number. So, and we will use these tables later on for calorically perfect gases. For working substances which are not calorically perfect, the Mach number of course cannot be evaluated very readily because this is not applicable for say steam or R134A or refrigerant. So, Mach number cannot be readily evaluated. But since a vast majority of applications in gas dynamics involve air as the working substance, which we have assumed to be calorically perfect, governing equations being written in terms of being written to include parameters is helpful whether the working substance is calorically perfect or not. The parameters this is just like what we did with for instance the air standard Brayton cycle or Otto cycle or diesel cycle. Although in the real application, the working substances did not execute cyclic process. These cycles were nevertheless immensely useful because they allowed parameters that control the performance to be identified and also how these parameters affect the performance of the cycle, which carries over to the actual engines. So, that was why we studied that in the same manner whether it is calorically perfect or not, writing them by highlighting the parameters in the governing equations is very helpful. Solution procedure can also be made simpler, which I will demonstrate in a minute. But more importantly, important physics in the flow can be brought out by the use of these reference states. Let us see how this comes about. We will discuss only two reference states in this particular lecture. One is the sonic state, the other one is the stagnation state. Now, sonic state as the name suggests, sonic state is a state of the fluid at that point in the flow field where the velocity is equal to the speed of sound. So, we have a flow and at a certain point in the flow field, the velocity becomes equal to the speed of sound. The local velocity becomes equal to the local speed of sound. Such a state is usually denoted using a superscript star. So, the pressure at this location or at this state is denoted P star, temperature is denoted T star and so on. So, the sonic reference state may be thought of as a global reference state, because it usually occurs only at one or two points in the flow field. For example, if you are looking at say flow through a convergent divergent nozzle, then the sonic state is usually encountered at the throat of the nozzle. So, at the throat of the nozzle, the local velocity of the fluid is equal to the speed of sound. And such a flow is said to be choked and we will explain the meaning of this when we discuss flow through nozzles. Now, in an actual flow situation, for example, something like this, the sonic state may not or may not necessarily be accomplished in the, may not necessarily be accomplished in the throat section. So, we could have a legitimate flow through a nozzle where the flow speed at the throat is not equal to the speed of sound. In this case, we can actually imagine an alternative scenario, for example, where we construct say an imaginary convergent nozzle after our actual convergent divergent nozzle and then say that at the throat of this nozzle, the local speed, local fluid speed is equal to the speed of sound. So, what I am saying is the reference sonic state as a reference state is very useful even if it does not occur or it is not realized in the flow field. So, for example, in this particular nozzle flow field, the velocity of the fluid will not be, need not be equal to the speed of sound anywhere in the nozzle. But I can always construct an imaginary nozzle like this and accelerate the fluid to the speed of sound at that location. So, then this becomes our sonic reference state. Now, the other reason why the sonic state is an important reference state is that it demarcates lines of, it demarcates regions of flow which can be accessed from certain directions and regions of flow which cannot be accessed. You may recall that sound travels in, I am sorry, disturbances, weak disturbances travel in compressible fluid with the speed of sound. So, when I have a situation like this where let us say we do get the sonic state here. So, let us say the velocity here is equal to A or the Mach number is equal to 1 at the throat. Now, in this case, once the Mach number has become equal to 1, let us say that I try to influence the flow in this region of the flow field. Let me draw this with a slightly different color. So, let us say that I try to influence the flow in this part of the flow field, the convergent part of the nozzle by connecting this nozzle to a passage which contains a valve. So, let us say this is a valve. So, by opening or closing the valve, let us say I try to control the mass flow rate through the nozzle or the flow in certain parts of the nozzle because any disturbance that I give in the flow by adjusting the valve, when I open or close the valve, I am basically creating a disturbance in the flow field. So, these disturbances travel outwards with the speed of sound as we saw earlier. So, these disturbance, these disturbances begin to propagate upstream also and in case, let us assume that the flow speed is less than the speed of sound in the divergent part of the nozzle. If that is the case, then once these disturbances are traveling with the local speed of sound and when they hit the throat of the nozzle, the flow there is already moving with the speed of sound. So, these disturbances will not be able to propagate beyond the throat which means that this region of the flow field cannot be controlled by having a valve here. If I want to control this region of the flow field, then I must have a valve upstream with which I can control this region of the flow field. So, the sonic state separates regions of flow that are accessible to control from a certain direction to regions that are not accessible. Now, of course, if the flow had to be supersonic in the divergent portion of the nozzle, then these disturbances would not be, would not even be able to get inside the nozzle at all because the flow is already moving with the speed greater than the local speed of sound. So, any disturbance when it meets the flow field will be carried along by the flow and will not be able to propagate further upstream. The next reference state is the stagnation state. Let us say that we have one-dimensional flow and let us say that the state at a location is completely known which means that the pressure, the temperature and the velocity at this point are known. Now, we carry out a thought experiment in which we take the flow from this state or we take the fluid from this state to one with zero velocity by means of an isentropic process. So, we take the fluid at this state and decelerate it to zero velocity by means of an isentropic process. The resulting state is called the stagnation state corresponding to this state. That is very important. The resulting state is known as the stagnation state corresponding to this state. Now, this is very important because for instance, if you take the sonic reference state, notice that the sonic reference state is a global reference state and only one or two sonic states in the entire flow field normally and the sonic state is relevant for the entire flow field. It is not a local reference state, it is a global reference state. Whereas, here we take fluid at a thermodynamic state at a point and decelerate it isentropically to the stagnation state which means that when I do that the fluid will attain let us say certain pressure called the stagnation pressure, certain temperature called the stagnation temperature. Now, it is quite possible that at another point in the flow field, if I repeat the same experiment, I may end up with a different stagnation pressure and stagnation temperature. Even though the final velocity is the same, it is quite possible that I may end up with a different stagnation pressure and stagnation temperature. So, which means that stagnation state is a local reference state. So, we denote properties at the stagnation state by using a subscript 0. So, pressure at the stagnation state is denoted P 0, temperature is denoted T 0, density 0 0 and so on. So, if I take my energy equation in differential form and integrate it from the given state at which the pressure, velocity and temperature are known to the stagnation state 0. So, this is the given state and this is the stagnation state. Notice that the velocity is 0 at the stagnation state, if I do that I end up with this expression is 0 equal to h1 plus v1 square over 2. Now, for a calorically perfect gas since dh equal to cp dt, we may simplify this expression further and write it like this and eventually write it in terms of the Mach number like this. Notice that in writing this we have made use of the factor cp is equal to for a calorically perfect gas, cp is equal to gamma r over gamma minus 1 and the speed of sound a is equal to square root of gamma RT. Now, the stagnation process is isentropic, remember we are going from state 1 to state 0 along an isentropic process. So, s equal to let me just write it slightly differently. So, s equal to constant. So, here p1, t1, v1, p0, p0. Now, we know how t1 and t0 are related from this expression we know that. So, now we can actually get an expression for relating p1 to p0 because the process is an isentropic process. I can write this expression for the for the pressures in connecting the pressure and temperature for an isentropic process of a calorically perfect gas. This is our familiar p raise to gamma, t raise to gamma minus 1 equal to constant. So, I can apply that between state 1 and state 0 and then write it like this. Notice that if we have a subscript 1 here then it is customary to have subscript 1 there also to denote the factor this is the stagnation state corresponding to state 1 and hence for that reason stagnation state is a local reference state. So, we may write p01 over p1 equal to t01 over t1 raise to gamma over gamma minus 1 and if I substitute for t0 over t1 from here then I end up with this expression which is written in terms of the Mach number m. We have dropped the subscript 1 here there is no danger of any ambiguity. Stagnation density can be evaluated by using the ideal gas equation of state. There is a very subtle point regarding the s equal to constant process that we are utilizing here. Notice that we are not actually taking the fluid in the flow field and decelerating it. We are carrying out a thought experiment where we have thermodynamic state like this and we are trying to find out the corresponding stagnation state which means that the energy must be maintained whatever we have in the beginning should be conserved. However, because we are not making the fluid undergo we are not making the actual fluid undergo a deceleration process mass conservation is not applicable. So, we are not taking fluid a certain amount of fluid and then decelerating it through a stream tube. We are only connecting two thermodynamic states here. So, mass conservation is not valid because if mass conservation were to be valid then when I go from the actual state to the stagnation state where the velocity is 0 you will realize that the density will become infinity or density will become infinite. So, you have to be very careful about your understanding of the stagnation process. It is an imaginary deceleration process where we take a certain thermodynamic state and then relate it to another thermodynamic state by means of an isentropic process. So, the stagnation state as I have already mentioned is a local reference state contrary to the sonic state. So, in principle stagnation state can change from one point to the next in the flow. So, as I mentioned earlier I can have a certain value. So, this could be let us say state 1, this could be state 2. So, I can have P01, T01 and so on and this would be different P02 comma T02. So, the reference state can change from point to point. So, in that sense it is a local reference state. However, if for example the stagnation pressure and stagnation temperature remains the same between states 1 and 2 then it becomes a global reference state in this region of the flow it becomes a global reference state because it remains the same. The other important thing subtle thing about the about the stagnation process is that the stagnation process. So, I may have for example flow in a nozzle like this. So, for example I can take a state point here and so the P comma T comma V at this location are all known. I can calculate the corresponding P0 and T0 by means of an isentropic process. So, so what it means is that since it is an isentropic process S0 is equal to S. What is that S0 is equal to S but we are not saying that the flow itself is isentropic. So, entropy may change between these points. So, when I evaluate this stagnation let us call this location 1 I am sorry let us call this location 1 let us call this 2 let us call this 3. So, when I do this S01 is equal to S1 S02 is equal to S2 S03 equal to S3 and S1 need not be equal to S2 need not be equal to S3. So, the flow need not be isentropic only the the stagnation process is an isentropic process that is very important. The other important aspect about stagnation quantity is that they are frame dependent. What do we mean by this when we talked about the wave solution the speed of sound we looked at different frames one where the observer was stationary and the sound wave moved like this. Another one where the observer was moving along with the sound wave. Now, if I measure the static pressure or static temperature irrespective of whether I am in a stationary frame of reference or moving frame of reference the static quantities are all frame independent. So, in fact, we will make the distinction now static quantities or frame independent. What are the static quantities pressure temperature density these are static quantities. So, whatever we were calling pressure is actually the static pressure in gas dynamics whatever we were calling temperature is actually the static temperature density same way because there is a stagnation counterpart to each one of these quantities which is P0 stagnation pressure T0 stagnation temperature rho naught stagnation density. So, these quantities stagnation quantities are frame dependent. So, when I measure any one of these quantities regardless of whether I am stationary or moving I get the same value. In contrast stagnation pressure evaluated in a stationary frame will be different from stagnation pressure evaluated in a moving frame because stagnation quantities involve the velocity. Velocity is frame dependent. So, the wave had a finite speed of A when I am in the stationary frame of reference and the wave moves like this when I am I am moving along with the wave the speed of the wave itself becomes 0. So, the speed changes when I switch frames which is why stagnation quantities are frame dependent. So, let us work out an example involving sound wave. So, consider propagation of sound wave into quiescent air at 300 Kelvin and 100 kPa with reference to the previous figure determine T01 and P01 in the stationary and moving frames of reference. So, stationary frame of reference looks like this 1 2 or like this. So, the sound wave propagates into quiescent air which means the velocity here in this frame of reference is 0 static temperature 300 Kelvin static pressure 100 kPa. So, in the stationary frame of reference V1 is equal to 0. So, the stagnation temperature is equal to the static temperature and the stagnation pressure is equal to the static pressure. Now, in the moving frame of reference so here I am sorry this moves with the speed equal to speed of sound and in the moving frame of reference the wave itself is stationary. So, this is 1 this is 2. So, the flow approaches the wave with the speed equal to the speed of sound and then recedes with a slightly different speed B2. Static pressure and static temperature remain the same. So, this is still 300 Kelvin frame independent. So, this is 100 kPa. Since V1 is equal to A1 Mach number is equal to 1 and if you substitute M equal to 1 into this this expression and this expression we get the stagnation temperature to be 360 and the stagnation pressure to be 189. So, this shows that the same region of fluid can have different stagnation pressures depending on which frame we are in. So, stagnation quantities are frame dependent.