 We would be dealing largely with one-dimensional flow in this in this module. Later on when we do nozzle flows, we will categorize the nozzle flow as a quasi one-dimensional flow. I will explain what that is when we when we come to that part of the lecture. But for now it is sufficient to say that we will be looking at one-dimensional flows. What do we mean by one-dimensional flow? What we mean is the following if the flow properties change only along the flow direction and the velocity component along the flow direction alone is non-zero. That means there is only one velocity component and that is along the flow direction usually called the stream wise component. So we may write the equations that govern frictionless adiabatic steady one-dimensional compressible flow like this. We are not going to derive this but we will write it. So I assume that you know you would have gone through maybe some courses in fluid mechanics so you should be able to recognize these governing equations. So this equation is called the continuity equation or mass conservation equation. This equation is usually called the momentum equation. So when you write it in this form, you are writing it in the so called non-conservative form and when you write it like this after using the continuity equation, this is called the conservative form of the governing equation. Conservative because everything inside is a perfect differential. This you will recognize as the steady flow energy equation. This of course is second law. So you can see the interconnection between thermodynamics and compressible flow. What is that density now is a function of two variables temperature and pressure and because in the case of ideal gas, it obeys the ideal gas equation of state. So V is a fluid dynamic quantity, all the other quantities are thermodynamic quantities. So that is how compressible flow is linked to thermodynamics which is why flow through nozzles is taught as a part of a course on applied thermodynamics. It can also be taught as a part of a course on gas dynamics but customarily for mechanical engineering students it is taught as a part of applied thermodynamics for this reason only. V is a fluid dynamic quantity and all the other quantities rho, p, h, s they are all thermodynamic properties and the two are very strongly related incompressible flows particularly flow through nozzles. Because flow through nozzles is such a ubiquitous example. Nozzles as well as turbomissionary blade passages is such a ubiquitous application for mechanical engineers, this knowledge of such flows is very, very important. Now because all these equations are perfect differential this, this and this, we may actually integrate them then write them like this rho 1 v 1 equal to rho 2 v 2. So we have a flow field that is station 1, station 2 we are simply integrating this equation between those two stations. Now one interesting aspect about this which is not very clear you know on the face of it is that discontinuous or wave like solutions are also admissible. In other words density need not be continuous from 1 to 2 although they may be located just let us say a very small distance apart. So for example we may have a wave like this so this could be 1, this could be 2 and this could be a flow across which we are seeing a wave. By wave what I mean is something like a sound wave which is pressure disturbance which is propagating in the fluid or if you go to a frame of reference where the wave is stationary and the fluid moves then that is the frame of reference that we are looking at here. Notice that across the wave properties will be discontinuous all the properties for instance density, velocity, enthalpy, entropy all of them will be discontinuous can be discontinuous that is permitted in this I mean by this set of governing equations. So although 1 and 2 are located only a small distance apart the thickness of the wave itself is very very small negligibly small. So 1 and 2 are located only an infinitesimal distance apart but the values are vastly different. So if for example let us say I plot for example density or something like that just for the sake of illustration so density could go like this and then all of a sudden increase and then go like this. So this is rho 1 and this is rho 2 for example pressure could be like this and all of a sudden becomes something like this. So the compressible form of the governing equations alone that we have listed here admit such wave like solutions. These are perfect differentials which is why they admit wave like solutions. That is a very interesting aspect of compressible flows and something that we will discuss in some detail as we go along because it is possible to encounter wave like solutions during this sort of applications that we discussed. For example sound wave we have already mentioned sound wave several times that is a wave like solution which is admitted by this governing equation. Normal shock is another wave like solution which is admitted by this governing equation and normal shocks are routinely encountered in the diverging part of nozzles and also in thermomissionary blade passages and it is very detrimental to performance. So we need to have a good idea of why and why it is detrimental to performance and why it occurs and what possible remedies can be can be adopt to make sure that normal shocks do not appear in these sorts of applications. So the first solution to this set of governing equations that we are going to look at is acoustic wave propagation speed basically propagation of sound wave in a compressible media. So let us say that we have a sound wave which is going out. Let us say this is a source like for example what happens when you throw a stone into a pond. Circular spherical waves start emanating from the location where the stone was dropped they start traveling upward. So initially it will look like this and eventually it may look something like this. So the radius of the spherical wave front eventually becomes quite large. So what we do now is we take a small portion of this wave and try to calculate the speed which is the wave propagate. So basically what we are doing here is to take a small part of this wave front, small compared to its radius so that it essentially appears. So the flow propagates like this and this is a one-dimensional flow field. When you look at this portion of the wave you can see that it is a one-dimensional flow field because the velocity is non-zero only along the flow direction. So in fact we can illustrate the situation like this. Notice that there is no curvature that is shown in the wave front here because the radius of the sphere is much larger than the size that we are looking at here. So we are taking a small portion and then studying it. So you can see that for an observer who is stationary on the ground as this spherical wave front propagate it looks as if the wave front is moving with the speed which is equal to the speed of sound. So this part of the fluid is so the waves propagate outward from the location where the stone is dropped. So the pond is very quiet in this part so this is called the quiescent fluid and there is some up and down motion in this part behind this wave front as the waves are going through and that is the situation we are looking at here. So the fluid behind this wave front may acquire a small velocity as a result of passage of the wave and that is what we have indicated here. The wave itself moves with the speed equal to speed of sound. That is what an observer who is sitting on the show or the banks of the pond that is what he or she will see. The wave front moving out like this water behind the wave front acquiring a small speed may be up and down actually but it is acquiring a small speed but the water ahead of the wave front is quiet. Now let us say that we move to a frame of reference where this observer who was sitting on the show starts running along with the wave front with this small piece of wave front. Then the situation looks like this. So this is a frame of reference where moving frame of reference where the observer is moving with the wave. So in this frame of reference it seems to be it seems that the fluid is approaching the observer with the speed equal to the wave speed and when the observer looks back the flow seems to be receding with the speed which is not equal to the wave speed but something slightly less than that or something slightly different from that less or more we will see. But something slightly different from that not exactly the same slightly different. The emphasis here is on the word slightly different that is the solution we are looking for other solutions are possible. We are right now looking for a solution where it is only slightly different and I am saying slightly because after I drop the stone there is a slight disturbance only to the water. The water is not like boiling and bubbling or anything like that there is a small disturbance you see some waves which are going through. For example, if there are leaves on the water you will see them like bobbing up and down maybe moving slightly downstream also. It is only a slight disturbance. So that is the situation. So V2 is different from V1 less or more we do not know. So this is the solution that we are seeking. So as a result of going through the acoustic wave the flow properties change by an infinitesimal amount and the process is assumed to be isotropic. There is no heat addition here and the changes are sufficiently small that we will assume the process to be isotropic which means that the solution that we are seeking to this set of governing equations is one where S2 is equal to S1. So we may write V2 as being V1 plus d V1. The plus sign here does not should not suggest to you that V2 is more than V1. It only says that V2 is slightly different from V1 and so on. V2 equal to P1 plus d P1 rho 2 equal to rho 1 plus d rho 1. So we substitute this into the original equations. Remember this one was rho 1 V1 equal to rho 2 V2 and this one was P1 plus rho 1 V1 square is equal to P2 plus rho 2 V2 square. So we substitute it into the first two equations and because d V1 d P1 d rho 1 are much much smaller than 1 they are already very small and products of these quantities will be even smaller. So we may neglect those sorts of terms and if you do that then you end up with an expression that looks like this. For example, if I take this I will keep the rho 1 V1. So this quantity here if you look at this product. So that product is rho 1 V1 plus rho 1 d V1 plus V1 d rho 1 plus d rho 1 d V1. So this is a product term so we say that this is negligibly small we throw it out and this rho 1 V1 cancels this rho 1 V1 so we end up with an equation like this and the same thing is done here also. Now if I rearrange combine these two equations I may write like this d P1 over d rho 1 is equal to V1 square which means that V1 equal to A is equal to square root of d P over d rho S equal to constant remember we have already said S is equal to constant and we said that A equal to V1 because we are in the frame of reference where we are moving with the wave which means we are moving with the speed equal to speed of sound. So that means V1 equal to A. So this is the expression for speed of sound in a compressible medium. So this is the wave like solution that the governing equations admit for the propagation of an infinitesimally weak wave across which the change in properties is isentropic. Now in case the fluid is a calorically perfect gas like say air since the process is isentropic from TDS relation we can get this. So d P over d rho we may get to be equal to gamma RT or the speed of sound itself to be equal to square root of gamma RT for a calorically perfect gas only not for steam or refrigerant. Notice that we explicitly did not use the fact that S2 equal to S1 when deriving this but we stipulated that the changes are infinitesimally small. So the stipulation that the change across the change in properties be infinitesimally small across the wave requires the process to be isentropic. If it were not the case then the process will not be isentropic. We will derive that solution next when we relax the requirement that the change should be infinitesimally small or process should be isentropic. Once you relax that there is another wave like solution which is admissible which is the normal shock wave solution across which the entropy increases. So although we did not explicitly use S2 equal to S1 in deriving this the requirement that the change in property be infinitesimally small demands that the process should be isentropic. So what we will do in the next lecture is to now that we have derived the expression for speed of sound we will take a closer look at Mach number and then move on to look at certain other fundamental aspects of one-dimensional flows.