 good morning, welcome you to this session, today we will be discussing normal shocks, now is a very important phenomena, the shock is a very important phenomena in a compressible flow, now earlier we have seen that in case of a convergent divergent flow through a convergent divergent duct, well convergent divergent duct, it is not possible to have the solution for an isentropic flow under all possible conditions of pressures inlet and outlet pressures that we have seen that for a given inlet pressure in case of a convergent divergent duct that is a throat in between, there has to be a design pressure at the outlet to have a solution for the isentropic flow that means to have an undisturbed flow, but what happens if the pressure at the downstream section is kept in between, we have seen that there occurs a sudden discontinuity in the flow field that means there is a sudden change in the pressure and velocities in the flow field within the duct, sometimes outside the duct if the pressure at the outlet in this below that of the design pressure, these are made by the phenomena known as shocks, so usually if we define the phenomena shock, we can define this way that in any fluid flow actually in certain circumstances we will see after that it happens in case of supersonic flow that a sudden changes a very rapid change is in pressure and velocity takes place, because of certain circumstances in the flow and this changes takes place within a very short distances, these are effected within a very short distances in the flow field which is in the order of molecular distances that is order of mean free path few times of mean free path it is so small that means we can simulate it for its analysis by a short discontinuity in the flow field and in our analysis we will be more interested in knowing the flow properties before this discontinuity or before this shock and after this shock without going into the details of what happens within this shock which is in the order of a mean free path whose thickness is in order of mean free path, so without going to the interior details of the shock wave we are interested in the flow properties before and after the shock wave, so well this can be done with a simulation type with a simulation like this we can consider a discontinuity in the flow field that means a section upstream of which the flow properties are steady and downstream of which flow properties are also steady, but there is a change in the flow properties, usually we will find later on that this shock are generated or take place in case of supersonic flow, where after this discontinuity the flow becomes subsonic that means the velocity is reduced that flow is decelerated and subsequently the pressure is increased, it is true that this increase of pressure and decrease of velocity is substantial that means there may be an increase of pressure by five times sometimes five times it becomes that decrease in velocity becomes even three to four times, but still the thickness of the shock waves are very very small, so now let us try to make a formulation in a simplified form for one dimensional shock before that I tell you that there are two types of shocks sometime when the shock font is normal to the direction of the flow that means this discontinuity takes place in a direction perpendicular to the direction of the flow, then the shocks are called normal shocks in cases when the shock font is oblique to the direction of the flow or the discontinuity takes place in an oblique direction with respect to the direction of flow the shocks are called oblique shock just I give you some examples of normal and oblique shock just you see that example in case of a convergent divergent duct you see in case of a convergent divergent duct this is a shock if the design pressure that is the back pressure is not is equal to the design design pressure that means if it is greater than the design pressure as you know corresponding to an inlet stagnation pressure then a shock occurs in the divergent part when the flow is supersonic, so this shock font is normal to the direction of the flow and this region upstream of the shock the flow is supersonic downstream it becomes subsonic and thickness of the shock is extremely small and there is a shock discontinuity in the pressure pressure increases sharply and the velocity decreases this is an example of normal shock normally shock similarly a oblique shock can be seen in case of flow past shock can occur in case of both flow through a duct flow past a body past a surface like that where the flow takes place in this direction let us consider this stream lines like this the flow takes place then a oblique shock may occur like that, so the direction of flow is that the shock font is oblique, so it is oblique shock these are some examples of shocks in case of supersonic flow past a body for example let us consider a wedge type of thing sometimes depending upon the flow situation is supersonic flow when it approaches the body this is supersonic this we will prove that always it the shock takes place from supersonic to subsonic flow this is subsonic this is definitely an oblique shock and this is defined as attached shock attached attached shock which is attached to this body sometimes depending upon the situation the shock font may not be attached this is known as detached shock this is the shock font where this is the flow supersonic this is a wedge shaped structure the flow is supersonic if the supersonic flow faces this type of obstacle then a shock occurs that means in the flow field there occurs a sudden discontinuity sudden jump from supersonic to subsonic flow with an increase in pressure and decrease in velocity so this is subsonic and this type of detached detached shock this type of shock is known as detached shock so these are certain pictures how the shock takes place creating a sharp discontinuity in the flow field now for a mathematical analysis let us consider things in this way that okay let us consider a duct certain part of the duct where the shock takes place and let us consider this is a thin shock wave and let us specify the properties upstream the shock wave by a suffix x let the enthalpy is defined as hx the velocity is vx the density is rho x the pressure is px and all this quantity at the downstream of the shock is given by a suffix y that is the enthalpy velocity density pressure now well if we write the energy equation for a control volume enveloping the shock we can write that hx plus vx square by 2 is equal to hy plus vy square by 2 you know writing the energy equation in this form assumes that with in the control volume there is no heat transfer actually what happens this takes place very rapidly and we can consider this process that means through which the property changes across a very thin shock wave is adiabatic that means if we impose the conditions of this adiabaticness that means no heat transfer condition then we can write this enthalpy plus the velocity head that is the kinetic energy per unit mass basis is equal to hy plus vy square when this equals to the stagnation enthalpy hx or hy which means the stagnation enthalpy corresponding to the situation at upstream x is equal to that corresponding to the situation y because stagnation enthalpy will change only when there will be an energy addition on energy depression energy either energy is added or energy is extracted otherwise this will remain same this is the very important equation this comes this is the energy equation now if I write the continuity equation continuity equation if I write the continuity equation we see that the cross sectional area remains same across the shock because of this thinness so we can write this rho x into vx is equal to rho y into vy since the cross sectional area remains same that is another important conservation equations which comes from the continuity equation now if I write the momentum equation so this is energy this is continuity well now if I write momentum equation or momentum theorem for this control volume then what I can write I can write that net force acting in the direction of flow px minus py into a now see that since the shock wave is very very thin I have told just now that it is in the order of molecular distances that the mean free path we can neglect the frictional force in this small control volume so therefore without friction the only force says acting at the pressure forces and that must be equal to the mass flow rate m dot times v mass flow rate remains the same under steady condition this is the momentum a flux in this direction of flow now mass flow rate mass flow rate can be expressed as rho x vx times the area cross sectional area so accordingly we can write rho a into rho x v sorry sorry rho y vy square minus rho x v x square this area is getting cancelled so we can write px plus rho x v x square is equal to py plus rho y vy square now this term can be written as the impulse now in compressible flow we define a function as impulse function so this is the outcome of momentum now we define an function known as impulse function impulse function in case of compressible flow we define the impulse function f as the sum of pressure plus the product of density in square of velocity that means p plus rho v square this is defined as an impulse function which is also a flow property is a combination of flow properties like this pressure plus rho v square so therefore with this definition we can write the outcome of the momentum equation is that the impulse force impulse function sorry impulse function at the upstream is equal to impulse function of the downstream so these three equations are obtained from the conservation equations conservation of energy conservation of mass that is continuity and the conservation of momentum that is the equation of motion for this control volume along with that along with this we have just see along with this we have the equation of state defining h is we can write in implicitly as a function of s and rho or s as a function of p and rho these are the implicit functional relationship as you know for a one component one single phase flow that thermodynamic properties any of the property can be expressed as any two independent properties that means two properties are independent so therefore h can be expressed as a function of entropy density entropy can be expressed as a function of pressure and density so these are expressed in an implicit form that implicit relationship because the explicit relationship depends upon the type or the nature of the system so for any system we can write the implicit relationship of thermodynamic properties which are nothing but the equations of state these are defined known as the equations of state that means thermodynamic equations of state can be written like that now you see that if I am interested with the help of this energy equation continuity equation momentum equation and the equation of state to find out the locus of points in h s plane which satisfy try to understand which satisfy these equations which satisfy this energy equation continuity equation and the equation of states so physical implication will be made clear after some time now just you follow in a routine manner that if I try to draw the locus or the points having the same stagnation enthalpy satisfying the continuity that means for the same mass flow rate and equation of state not the momentum equation not the momentum equation then we can draw the locus like this how can we draw this first we fix a particular point x that means to do this what we can do so we can choose particular conditions now before that let me tell you my basic intention of doing that you must follow it very carefully and seriously here lies the physical concept the basic motivation is that if I have got certain fixed state here so whether it is possible by shock to attain several states here or not or other way we can put the questions for a given properties here are all state points with different properties are accessible through shock or not the answer to this is no there is an unique state mathematically there is an unique state corresponding to a given state at the upstream which can be achieved through shock but to do this we will have to go through this let us find out the locus of points all points in h s plane which corresponds to these states that means which have the same stagnation enthalpy with this state that means points of constant stagnation enthalpy and satisfying the continuity and the equation of states so a routine process of doing this mathematically that we consider first fix this h x v x rho x p x everything so that a stagnation enthalpy is defined then we choose a v y here any arbitrary value we choose a v y we can find out h y so okay when we find h y v y is if we select v y we find h y now when we know v y from the continuity we can know rho y since we know rho y from a thermodynamic equation of state which can be expressed enthalpy as a function of s and rho we can know s so therefore we can know the value of h and x by doing so with different values of v y arbitrarily choosing v y we can find out the corresponding h y for a given stagnation enthalpy and the corresponding rho y and finally the corresponding h and we can construct a curve like this this curve is known as phano line very important phano line this curve is known as phano line this has got a point here where this curve shows a maximum or minimum as you tell that is the maximum in s so this curve is known as phano line physically this curve implies the locus of the points which mathematically first which satisfy the energy equations and continuity equations and the equation of state now you see this energy equation does not put any restriction to the friction friction may or may not be there where this stagnation enthalpy remains equal that means this energy equation this put constant only on the heat transfer that means the flow is adiabatic that means this is an adiabatic flow which satisfies the steady state condition that is the same mass flow rate across the sections and also the equation of state for the particular system used as the working fluid so therefore this type of curve represents the points having the same mass flow but flowing with adiabatic boundary conditions that means without heat transfer but friction may or may not have this is because we have not put any restriction of fiction so long we have not used this equation that is equality of impulse function which comes from the equation of motion or the momentum theorem or momentum equation where definitely the constant of zero friction has been put has not been taken in deriving this locus so therefore this refers to a flow where the flow takes place without heat transfer that means d q is zero but there may be friction that means we can tell the flows this refers to flows which are frictional adiabatic flow these are not isentropic flows frictional adiabatic flows or irreversible adiabatic flows that means adiabatic flows irreversible adiabatic flows these flows are irreversible with frictions but no heat transfer these flows are known as phano line flows that means the state points in these flows are determined are on this line that means frictional adiabatic flows or a phano line flows that is irreversible adiabatic flows are flows with constant stagnation enthalpy so they are enthalpy and entropy changes along this line well now it can be shown just I will show you that this region of this curve that is the phano line represents the supersonic region m less than 1 supersonic this part represents the subsonic region m less than 1 sorry sorry subsonic region and this is the point let this point is o this is sonic region. So, before showing that let us again see that we can have a an idea that in a supersonic flow you see in a the effect of friction is to reduce that means if I have got a point here now try to understand that if if I got a point at any point here let we consider this is the supersonic point that means at any point here in the supersonic region along this line we have to move in this direction. So, starting from any point that point cannot come that means if the flow takes place if the upstream point is this one. So, downstream point cannot come here it has to go there why can you tell that for any point let example in a supersonic flow the upstream point if we fix at particular point x at upstream point why the downstream point will be always along the curve in the right direction entropy cannot very good very good this is because this is the flow is adiabatic. So, therefore, entropy of the system will increase because the second law of thermodynamic stress that the entropy of an isolated system is always greater than zero entropy change of an isolated system is always greater than zero since the system is an isolated system there is no heat transfer with the surrounding. So, entropy change of the system will be greater than zero similarly if I have a point in this region upstream. So, downstream point will be towards this direction that means the process should take place in such a way that means the flow should take place in such a way that it must increase the entropy of the system since the entropy change of the surrounding is zero because of no heat interaction the entropy change of the system will correspond to the entropy change of the universe. So, therefore, the entropy change of the universe to make entropy change of the universe greater than zero we must make the entropy change of the system greater than zero. So, entropy will always increase. So, flow will take place along the right. So, from this we can also conclude one thing that in a supersonic flow therefore, the effect of friction is that it will decrease the velocity it will move towards the sonic flow that means you can visualize this that in a supersonic flow if you increase the friction usually this is manifested in terms of increasing the length of the duct you will go on decreasing the supersonic flow to the sonic region up to the point two after which a further increase in friction will not change the flow to subsonic region until and unless the inlet condition is altered. Similarly, in a subsonic flow the influence of friction that means this friction can be visualized in terms of the increasing the duct length it will change the for a given mass flow rate to accommodate it will change the subsonic flow towards the sonic region where you see that if you go on increasing the duct length and two have maintained the same mass flow rate you will change the flow more towards sonic region. So, when sonic flow will be reached then for the same mass flow and increase in duct length will not change the flow from sonic to supersonic region until and unless the inlet conditions are changed. So, this is the physical explanation of this fanno line now let us mathematically prove that at o the sonic condition is reached. So, let us prove the condition at o let us consider a infinite small process in any part of the subsonic line ok infinite small process now you see that the energy equation could be written in this way d h plus d of v square by two is zero in case of an adiabatic flow the integrated form of which is h plus v square by two is constant which already we have used. Let me see that which already we have used that h x plus v x square by two h y plus v y square by two that means in differential form it is d h plus d v square by two is equal to zero or d h plus v d v is equal to zero all right. So, this is the energy equation for adiabatic flow the continuity equation tells rho v is equal to constant since the cross sectional area remains constant across the shock. So, this can be written as d rho by rho plus d v by v logarithmic differentiation is zero all right from which we can write d v is equal to minus is equal to sorry minus v into d rho by rho now thermodynamic property relations can be written as t d s is equal to d h minus d p by rho. That means, t d s is d h minus v d p by rho v d p v is the specific volume which I use which we use in case of thermodynamics we use in case of fluid mechanics as one by rho. So, we know this relations we have already discussed earlier t d s is d h minus d p by rho now if I write this equation t d s d h in this form is equal to one minus d p by rho divided by d h now you see therefore, this is the relationship developed from the thermodynamic property relations this is the outcome of the continuity equation and this is the outcome of the energy equations. Now, for final final lines we have used this energy equations for adiabatic flow this is the equations for continuity and this is the thermodynamic property relations now at the point s we see that d s d h is zero d s d h is zero true ok. So, at the point o d s d h is zero that means, this is zero at o. So, therefore, what we get d h is equal to what we get d h is equal to d p by rho. So, now, if I write d v is equal to minus d h is equal to minus d p by rho. So, what is d v d v is minus v d rho by rho. So, if you put that you will get v square is d p by d rho if you put that d v is minus d rho by rho then we get v square is d p by d rho. So, therefore, simply we get very simple expression d p by d rho. Now, since the process is considered adiabatic and if we consider the shock as an isentropic process in case of shock we can write v is equal to del p del rho at constant. So, therefore, in case of shock this is at there the o attains the sonic condition that is the sonic velocity similar way we can write that this part of the curve is supersonic and this part of the curve is subsonic. How is a very simple intuition in that case we will not put d s d h zero here in case of the lower portion of the curve that means, you see this part of the curve d s d h is positive and this part of the curve d s d h is negative very good. So, this part of the curve if you consider that d s d h is positive then d p by rho divided by d h has to be less than one that means d p by rho has to be greater than d h. So, if you d h has to be less. So, if you put like that let us see this thing let us see this thing we have got d s d h is one minus d p by rho by d h all right. So, we can show that in this part d h d s is positive that means, d s d h is also positive that means. So, in this part what will happen. So, this will be less than one that means d p by rho will be less than d h d p by rho will be less than d h otherwise this will not be less than sorry it will be d s by d h is positive that means positive means this will be less than one that means in this region d s by d h is greater than zero in the lower part of the curve which means this has to be less than one. So, d p by d rho is less than d h all right. Then we can use this curve use this equation v d v is equal to minus d h. So, using this we can prove that v d v is minus d h, but here what we will prove that d p by rho. So, v d v. So, let us put d v d v is what let us see here d v is. So, v d v is equal to minus d h. So, d v is minus v d rho by rho. So, minus cancels. So, v square d rho by rho is equal to d h. So, all right. So, v square this is the general expression is d h rho by d rho. So, we write this expression that v square this expression is equal to d h. So, v square is equal to minus d h rho by d rho. In earlier case we had d h is d p by d rho when d s d h is zero at the point o. So, therefore, it becomes d p by rho, but in this case d h is greater than d p by rho because d p by rho is less than d h. So, v square is greater than d p by d rho. That means v is greater than root over d p by d rho. So, putting the additional constant in case of shock that the flow is isentropic without friction that we can write v is greater than del p del rho at s. That means greater than the acoustic speed where earlier we prove that v is equal to acoustic speed. That means, obviously by exploiting this relationship that means it is m greater than 1. That means exploiting the relationship that the slope of the curve h s is positive here, slope of the curve is negative here and the point o it is zero that means d s d h is zero otherwise d h d s is infinite. We can use the equations like this the property relations thermodynamic property relations the continuity equations and the energy equation to finally express v in terms of a d h v square d h rho by d rho and using this expression when d s d h is equal to zero d p by rho is d h then v square is d p by d rho. When d h d h is less than 1 then we can prove that v is greater than this and similarly when d h d h is greater than 1 that means in the upper part of the curve this will be less than this that you can prove when this corresponds to Mach number less than 1. So, therefore this represents a fanno line but you must understand one thing very clearly that this fanno line represents the locus of the points which follow the adiabatic conditions and the steady state continuity equation. That means these are the locus of the points having the same mass flow rates without any heat transfer and following a particular equation of states we will be deriving afterwards these equations for perfect gases. That means this is precisely referring or this precisely referred to frictional adiabatic flow because the frictional that is the zero friction or absence of friction is not a constant put here. So, this is simply the locus of all these state points in a situation of flow there may be friction but no heat transfer no heat transfer and also following the steady state conditions. That means which mathematically satisfies the continuity equation the energy equation h plus v square by 2 is constant that means the energy equation for adiabatic flow and at the same time the thermodynamic equation of state those flows are called fanno line flows. Now, you may ask a question said you started with the shock and immediately you referred to derive the equation locus of points in h s plane which corresponding to certain flow with that where friction may be incurred. Really this has got directly no reference to the shock but ultimately you will find that through this we can explain certain restriction in shock because our final motto is to prove that shock will take place only in supersonic flow and it will decelerate the supersonic flow to subsonic flow not that a supersonic flow will again go to a more supersonic flow the shock is a phenomena where the fluid velocity is decelerated and it jumps from a supersonic to subsonic flow it is just like your hydraulic jump have you read the hydraulic jump that trunk will to rapid flow or rapid flow to hydraulic jump takes place at the rapid flow becomes trunk will flow similar to that is supersonic flow becomes subsonic flow with an increase in pressure and velocity or decrease in velocity. So, for a given supersonic flow there is a unique subsonic state where the flow will reach because of the shock to prove that we have to go through this that means step by step. So, first we deduce the locus or draw the locus of the points in H s plane for a type of flow where friction may be there, but there is no heat transfer and the for the the flow obeys the steady state condition that means the same mass flow rate across each section. So, this is called fanno line flow well next we can consider. So, therefore you understand this is the subsonic and supersonic flow and in this case again I repeat that at any point if we start either in subsonic or in supersonic region to know the downstream point that in which direction the flow occurs or flow takes place we will have to move along this curve definitely for fanno line flows towards the right. So, therefore the influence of friction by making a more depth more duct length of the duct because all the points in this supersonic or subsonic region towards the right represents the length of a duct physically. That means if we increase the length of a duct in a supersonic flow the flow tends to become sonic ultimately when it will reach one o the point o the flow will be choked. So, a further increase in length will not change the flow until and unless the this condition that means this inlet condition with is changed. Similarly, the influence of increasing duct length or friction in a subsonic flow without heat transfer will change the subsonic flow towards the sonic flow the point moves towards the sonic flow we will have to proceed to along the cut to the right said following the friction the second law of thermodynamics that is the increase in entropy for the system without heat transfer will be greater than 0. Now, well the time we will have to see. So, next we see that we consider a flow where the heat transfer is there now if we try to well now if we try to draw the locus of points which satisfy this equation that is the continuity equation well which satisfy now now we take this equation the impulse function that means this momentum equations for 0 friction and the thermodynamic equation of states that means the implicit functional relationship like this depends upon the particular working system and if we try to draw locus in the h s plane the locus will be like this if we draw that in h s plane the locus will be like this. So, there will be a point like this. So, what does this curve represent this curve is known as Rayleigh line r a y l e i g h now again I tell first look mathematically the Rayleigh line is drawn by joining the points or the locus of the points which satisfy the continuity equation the momentum equation where the equality of impulse functions are made and the equation of states that means h is a function of implicit form s rho and p is a function of or s is a sorry s is a function of any way you can write s is a function of p and rho. That means thermodynamic equation of state equation of motion and continuity it refers to a situation of flow where friction is not there because we have satisfied this equation the flow is steady. But this condition we have not satisfied that means in case in this case if we find h o x at any point x and any point y we will see that that may not be equal to this. That means h o x minus h o y we will have some value let us consider these values as d q because we know this difference in stagnation enthalpy is the heat transfer that means h o x minus h o i if we find or rather if we write h o y minus h o x x considering any upstream and y considering any downstream section and let the quantity be denoted as d q which is a physical significance as like this if this is positive then we will consider there is an increase in stagnation enthalpy which represents an amount of heat transfer heat is added to the flow or if this is negative heat is taken away from the flow. That means it represents a flow where friction is absent but there is an heat transfer that means heat is either added d q with a cut i use because to differentiate it from perfect differential or heat is taken away. So, this refers to flow which are reversible or frictionless reversible or frictionless you write frictionless diabetic sorry frictionless diabetic flow. That means flow with heat transfer frictionless or diabetic flows flows with heat transfer but without friction that are known as Rayleigh line flow Rayleigh line flows. That means here the points represent the locus or of the state points in a flow without friction but with heat transfer. So, I think time is up today. So, next class we will be discussing again this Rayleigh line flows. So, this is Rayleigh line flow that means these are the locus which satisfy the equation of motion for an frictionless flow the continuity equation and the thermodynamic equation of state. So, therefore, the locus in h s plane represent the state points in a flow without friction with heat transfer and obeying the steady state conditions that is the same mass flow these flows are known as Rayleigh line flows. They are reversible or frictionless diabetic flow diabetic flow means there is heat transfer either heat is added or heat is taken out. That means the stagnation enthalpy will change. That means all the points in this locus the in this curve the different points represent the different in general the different stagnation enthalpy. Thank you.