 we will continue the discussion on normal shocks. So, last class actually we are discussing that for a shock if we know the properties at the upstream of the shock we are interested to know the properties at the downstream section. That means for a given flow properties at the upstream of the shock what are the flow properties after the shock waves that means across the shock waves there is a chain in flow properties and we are interested to find that. So, let us continue the discussions in this way that let this is the duct and let this is there is a shock here shock wave and if the upstream section is designated by x with all given properties that max sorry m a x p x rho x t x we are interested in all those properties across the shock. So, last class we derived one very important relation relating to the first the mach number after the shock downstream of the shock and we probably if you recall we get two solutions from a quadratic equation connecting the mach number at the section y after the shock and the before the shock that means m a y and m a x and we found one trivial solution that is m a y is m a x that is equality another one is the which is the possible solutions that m a y square in this way can be written as and it is very simple from that equation two by gamma minus one by two by gamma by gamma minus one m a x square minus one. So, this is the relationship that means if we know a value of m a x that means the mach number at the upstream of the shock then the downstream of the shock we get the mach number which is given by this after that what we was discussing is the non dimensional of just for the moment we come to a different aspect that non dimensionalization of velocity non dimensionalization of velocity in compressible flow. Now, the velocity in compressible flow is non dimensionalized by three reference velocity now here one very pertinent question comes that this v by a the mach number at any section is itself a dimensionless parameter, but this cannot be used as a dimensionless velocity this is because a change in mach number implies both in change in v and a this is because when the flow changes from section to section the flow velocity along with that the velocity of the sound also changes because the state properties change. So, therefore, mach number cannot be used in as a dimensionless or non dimensional flow velocity because the reference value a which is used in the denominator here by its definition also change with the change in v. So, therefore, three reference velocities actually this was discussed in our last class also, but I feel that I was little fast. So, there were some confusions. So, that is why you want to repeat it again. So, there are three reference velocities used in this connection one is the maximum velocity one is the maximum velocity maximum velocity is given by root over two c p into t zero you know relating to stagnation condition in an isentropic flow if we write the energy equation h plus v square by two is equal to h zero stagnation state and any other state given by velocity v and for a perfect gas c p by c p t that h is c p t is equal to c p t zero you know that. So, in general v is equal to root over two c p for any adiabatic flow it is the equation like that. So, the maximum velocity by theory comes when t is equal to zero this hypothetical case. So, this a theoretical velocity can be used as the maximum velocity another one is the this is number one number two is the acoustic velocity a at the stagnation condition this is given by gamma r t zero well another one is the acoustic velocity at the critical condition a star which is given as gamma r t star. Now, usually incompressible flow the velocity is dimensional non dimensionalized or is made dimensionless by use of this velocity as the reference velocity it is the velocity of sound at the critical condition gamma r t dash t star. Now, if we use this v with a star the ratio of the flow velocity to the a star. So, a star has a unique value in a particular flow condition it does not change from section to section it has a unique value that is the value of velocity of sound at the critical condition this is defined as the m a star. So, confusion here was that the last class the asterisk or the star is used for all properties at the critical condition for example, by p star we mean the pressure at the critical condition by t star we mean the pressure temperature at the critical condition by rho star we mean the density at the critical condition similarly a star used here mean means the sound velocity at the critical condition that means it is nothing, but dot over gamma r t star in case of perfect gap, but this critical condition is defined when the velocity is equal to the velocity of the sound flow velocity. That means v star the flow velocity at the critical condition is a star. So, therefore m a at the critical condition is one, but we do not use the symbol m a star to denote the mach number at the critical condition rather m a star there may be a confusion this symbol this is the convention m a star is used to denote the dimensionless velocity v by a star. So, m a star is not the mach number at the critical condition because this cannot be used as m a star because this is a unique and this is the value of this is one at the critical condition. So, we take the critical condition as defined by m a is equal to one when the flow velocity is equal to velocity of sound, but m a star is not the critical mach number at the critical section it is v by a star. Now, from a simple energy equation if I write the energy equation with at any section given by the velocity v and the critical section we can write h star plus v star square by two all right. Now, h can be written again c p t plus v square by two is equal to c p t star plus v star square by two. Now, what is c p c p t c p t here we can write c p t is equal to gamma by gamma minus one r t. So, this is nothing, but a square so a square by gamma minus one. So, if I substitute this here we get a square by gamma minus one plus v square by two is equal to similarly this we will write will be a star square by gamma minus one plus v star square by two. Now, I write this first so v square by two plus a square by gamma minus one. Now, v star is a star that is true at the critical condition. So, this can be written as if I take this common so two gamma minus one gamma plus one into a star square. Now, if I divide by a star square this equation left hand side and right hand side we get m a star square by two v by a star by definition is m a star plus one by gamma minus one a square by two v by a star square is equal to gamma plus one divided by two gamma minus one. Now, a by a star can be replaced like that by definition what is a m a is v by a that means a is v by m a and m a star is that is the dimensionless velocity. Mac number is also a dimensionless quantity containing velocity, but the flow velocity and the sound velocity, but here it is the flow velocity and a reference velocity which is a sound velocity at a particular condition. So, that is the difference here both the quantities change with the flow, but here only the flow velocity change and this is being normalized with a reference quantity at the denominator this is the difference. So, therefore, one can write a by a star a by a star is equal to m a star by m a that means we divide this by this a by a star. So, if I substitute this a by a star in terms of m a star by m a we get an equation this is this we get an equation connecting m a and m a star all right m a and m a star let me write this if we write this we get an equation can you see that. So, we can write an equation m a star square by two plus one by gamma minus one a by a star means m a star square by m a square is m a star plus one by two gamma minus one m a star by m a is a by a star. So, a by a star square is m a star square that means this equation now can be expressed in two fashion one is that m a as a function of m a star very simple one can do or m a star as a function of m a either of these two if it is written you will get most important one is m a star square m a square m a as a function of m a star two by gamma plus one m a star square divided by one minus gamma minus one by gamma plus one m a star square you do not have to remember all this formula you have to know the logic and the steps through which it is they are being deduced similarly one can express m a star in terms of m a as gamma plus one by two m a square divided by one plus gamma minus one by two m a square now out of this two this one is used this is very important that mach number is mach number is expressed in terms of the dimension less velocity m a star now if we use this in the earlier one now we stopped here now if I express m a y in terms of m a y star and m a x in terms of m a x star what we get now before that you see that this is the expression well no this is not the thing where I did it sorry I think I think we should concentrate here otherwise we will be in trouble so m a y these are the two solutions we get we got m a y in terms of m a x now here one thing is sure that if m a x is greater than one we already recognized that the shock takes place when the upstream condition is supersonic that means in a supersonic flow then from this equation one can prove that when simple mathematics can prove this that when m a x is greater than one m a y will be less than one with feasible values of gamma between one point six seven as you know the gamma for compressible fluids that gases varies between one is the absolute minimum and one point six seven is the maximum one for all polyatomic gases so taking any representative value between that it can be proved that when m a x is greater than one m a that means the shock changes a supersonic flow to subsonic flow this will be more clear if we express this mach number at the two section in terms of the m star that means dimension less velocity that means by the use of this equation the equation which we just yes by the use of this equation that means m a y m a that means the m a y or m a x whatever you tell that means the mach number in terms of the corresponding m a star if we do that then we will get the relationship like that m a y star so this is a very simple relationship is equal to one that means here it is obvious that the gamma factor is not there so when m a star m a x star is greater than one that means supersonic m a y star is less than one that means supersonic to subsonic now there are certain routine calculations for the pressure values now we are interested with temperature and pressure that means our basic motto is again to find out all the quantities after the shock quantities are temperature pressure density rho y these are the main flow properties or the state point so this we will calculate how we will calculate if you remember the temperature ratio now it is very simple when we have already derived the relationship between m a y and m a x now if we can write or we recall the relationship between t y and t x in an isentropic flow for a perfect gas it will be one plus gamma minus one by two m a x square plus divided by one plus gamma minus one by two we deduce it with the help of the relationship between the stagnation properties and the local properties that is one plus gamma minus one by two m a square so logic is like that and this was derived starting from the basic energy equation for an adiabatic flow and considering the flow is inviscid and using the perfect gas as the ideal gases as the working fluid well here if we use m a y if we just here sorry here m a y in terms of gamma minus one m a x then we get a relationship like this which is a big one you do not have to remember again I am telling just the logical step you have to know let me write one plus gamma minus one by two m a x square into two gamma by gamma minus one m a x square minus two gamma by gamma minus one m a x square minus one divided by gamma plus one whole square divided by two gamma minus one so objective is that when we know the property at the upstream of the shock by the m a x then we can find out the temperature ratio that means knowing t x and m a x we can find out the t y that means the temperature after the shock that means the temperature ratio t y to t x is expressed in terms of m a x with the same philosophy the pressure ratios are expressed in terms of the mach number as a function of m a x the procedure is like that if we recall the most simple ratios of p y by p x which was derived by the exploitation of the impulse function equality of the impulse function which sorry f y better to write f y is f x which was derived with the use of momentum equation with the use of momentum equation equation of motion then probably if you recall this was gamma m a x square one plus this deductions are very simple so only thing is that you will have to know the logic that this is from this so simply again the substituting m a y in terms of m a x that means the same equation m a y in terms of m a x we can derive the p y to p x p y by p x is equal to two gamma by gamma plus one m a x square minus gamma minus one by gamma plus one now here one interesting thing you can immediately prove you can immediately prove that for any value of gamma greater than one if m a x is greater than one p y by p x is also greater than one. This can be proved immediately that means this quantity is greater than one provided m a x is greater than one and gamma is greater than one this can be proved easily because if you subtract minus one from here then you will see that two gamma by gamma plus one will be thing that means if you subtract minus one it will that means p y by p x minus one is a simple thing that you can do two gamma by gamma plus one into m a x square that means for any value of m a x square greater than one so this is positive that means p y by p x is greater than one so this can be proved this is a very important conclusion that means after the shock the flow reaches subsonic that means the flow is decelerated and at the same time the pressure is increased that means p y is greater than p x. This is the ratio then again this is at the routine affair that rho y by rho x as a function of m x very simple thing so rho y by rho x we have already found out the function that p y by p x as a function of m x is known similarly we have found out t y by t x so if we can express rho y by rho x by these two ratios from the equation of state we can find it we know that p is equal to rho r t so rho y by rho x is p y by p x into t x by t y so t x by t y as a function we know t y by t x similarly this as a function of m a x we know so we can find out rho y by rho x so similar way we can find out v y by v x v y by v x from the continuity is rho x by rho that means rho y by rho x means the reverse v x by v y so this way we can find out the functional relationship m a x let this is some function this is one by function m a x so another function of m a x so this way we can find out the ratio of the properties any properties p t rho v after the shock to that before the shock in terms of the mach number m a x now one important thing is this stagnation pressure p o y and p o x now see in a shock p o y and p o x are not maintained same this is because shock is an irreversible process friction is there so stagnation pressures are not same later so this is the measure of the irreversibility measure of irreversibility irreversibility now before going for a routine evaluation of this you must know this thing that in a flow the stagnation temperature remains constant when the flow is adiabatic because stagnation temperature is the index of the stagnation enthalpy all right because in a perfect gas when there is no heat transfer in adiabatic flow the stagnation enthalpy remains constant total energy enthalpy plus the kinetic energy and in case of an ideal gas the enthalpy can be expressed as c p into t so therefore c p into t plus v square by two is known as the stagnation enthalpy that means c p t zero that is h zero so stagnation temperature is fixed provided there is no energy added or energy taken out but stagnation pressure will not be same stagnation pressure by definition is the pressure which could be reached if the flow is decelerated or comes to the stagnation condition isentropic without friction that means the entire kinetic energy is converted only in the pressure energy understand only in the pressure energy not in the enthalpy through the internal energy that is a very useful concept i think in flow mechanics also you know that is why in energy conservation whether friction is there or not there is no loss total energy remains same but difference is that when friction is there some part is converted in intermolecular energy but in in visit flow or isentropic flow without heat transfer the stagnation pressure remains same that means there is no degradation in the intermolecular energy that means the entire kinetic energy becomes zero when it comes to stagnation test the entire kinetic energy is converted only to the pressure energy and that pressure is known as the stagnation pressure so stagnation pressure will be equal when the flow will be frictionless but for any natural flow when the flow is brought to rest that is the stagnation condition the pressure there is not exactly the stagnation pressure by its definition so therefore you see the ratio of the stagnation pressure in any flow is the measure of the irreversibility that means its departure from one is the measure of the irreversibility let us find out p o y by p x the routine procedure is very simple we can express in terms of their local values and p x by p o x and we know p o y by p y p x by p o x that means the ratio of the stagnation properties to the local properties in case of pressure will be one plus gamma minus one by two m s square gamma by gamma minus one and p y by p x also this ratio also we have deduced in terms of the mach number that means we know this p y by p x that means p y by p x just now i derived that means p y by p x is this so we can replace p y by p x in terms of the m x p o y by p y in terms of the m a y and p x by p o x in terms of the m a x and this m a y is again substituted in terms of m a x so finally we get an expression only in terms of m a x like this this is a big expression again i am telling you do not have to remember this only thing is that you will have to remember the way it is being deduced one plus gamma minus one by two m a x s square to the power gamma by gamma minus one well i think is equal to here then divided by two gamma by gamma plus one m a x s square minus gamma minus one by gamma plus one whole to the power one by gamma minus one so this is the relationship between p o y and p o x and we can see the value of p o y by p is not unity that means it is different from one so stagnation pressure and we can prove that p o y is less than p o x when m a x is greater than that means this is because of the friction when m a x is greater than one that means supersonic flow is change to subsonic flow through a process shock where p o y is less than p o x that means it is the effect of friction that this stagnation pressure is lower than the initial stagnation pressure now after this we are interested to find out what is the change of s entropy let specific entropy small s we are interested in that what is the change of entropy earlier in h s plane we have recognized that for a shock to occur the entropy has to increase according to second law of thermodynamics so what is the value of this so routine calculation starts from the thermodynamic property relation t d s is d h minus v d p for a perfect gas we can write t d s d h is c p d t and v is what is v r by p well v is p v is equal to r t r t by p d p so therefore d s is equal to very simple that you have already learned in your basic thermodynamics r d p by p so now the job is to integrate only d s from x to y this upstream to downstream section so c p d t by t x to y minus integral we can take r outside the characteristic gas constant which is constant so we can take outside the integral now therefore we can write s y minus s x is simply c p l n t y by t x is equal to minus r l n p y by p x now this r can be written with this formula c p you know is equal to gamma by gamma minus 1 into r that means r is equal to gamma minus 1 by gamma alright into c p so if I replace this value of r here and take c p common that is l n then it becomes l n t y by t x and then this coefficient can go as a power here that means p y by p x it is a minus so l n x minus l n y as l n x by y using this formula to the power gamma minus 1 by gamma so I can write this gamma minus 1 by gamma c p c p is coming common so l n t y by t x minus l n p y by p x rest gamma minus 1 by gamma this comes so this is the s y minus s x now this can also be written in terms of this stagnation properties and it is very simple that we can use this stagnation properties this can be straight way written as let me write it again otherwise there will be problem s y minus s x is equal to c p l n t y by t x well divided by p y by p x to the power gamma minus 1 this can be straight way written c p l n t o y by t o x this can be straight way written c p divided by p o y by p o x now you may say sir why are you writing like this as if it appears the ratio of the properties is equal to the ratio of the stagnation properties it is not so but by the relationship it appears so that means if you write t o y by t y that is is equal to 1 plus gamma minus 1 by 2 m a y square similarly if you write t o x by t x is equal to 1 plus gamma minus 1 by 2 m a x square that means the ratio t y by t x is not equal to the ratio t o x by t o y by t o x but they will carry the ratio between these two but at the same time the p o y by p y if you do this calculation it will be obvious immediate m a y square to the power just the reciprocal of it gamma by gamma minus 1 and similarly p o x by p x will be the same quantity with mach number x square to the power gamma by gamma minus 1 so with the help of these four equations if you express t y by t x the ratio of the temperatures in terms of their stagnation temperature and the ratio of the pressures in terms of their stagnation pressures you will see this gamma minus 1 gamma gamma by so these ratio of these two quantities will cancel from the numerator and denominator so that ultimately it is very interesting that the same ratio that means the variables in terms of the local properties can be just changed or substituted in terms of the stagnation properties so that we can express this now here one thing is that t o y is equal to t o x so this is one in case of shock because it is an adiabatic condition there is no energy is added or extracted so therefore the stagnation temperatures are same that is one that means minus c p l n p o y by p o x by p o x to the power gamma minus 1 now if i take this thing here in the coefficient then it can be written and again replaced it by r l n p o y by p o x so one simple expression that s y minus s x by r it is a non dimensional quantity this side l n now since p o y is already we have proved that p o y is less than p o x so therefore this is always greater than zero that entropy is always increased alright now a very routine and very tedious calculations one can make again to have the same conventional things that the entropy change in terms of the mach number that has a function of upstream mach number what he has to do he has to substitute this big expression of p o y by p o x in terms of the mach number somewhere earlier we have deduced it that p o y by p o x in terms of the well in terms of the mach number m a x if we do that we ultimately get the expression s o y that means we get the expression s y minus s x i think you have understood that already we have deduced this so this is very interesting equation in very simplified form but if you want to express in terms of mach number at the upstream section m a then we have to substitute this ratio as a function of m x which we have deduced earlier and final result is that s y minus s x by r is equal to it is a very big one gamma by gamma minus one there is no reason to remember it but you must know this gamma plus one m a x square plus gamma minus one by gamma plus one plus gamma minus one by gamma plus one plus one by gamma minus one l n well two gamma by gamma plus one m a x square minus gamma minus one by gamma plus one now usual convention is to draw now we can show in figure the variation of entropy change s y minus s x by r with mach number m a x now it has been proved already that p o y is less than p o x so s y minus s x is positive greater than well greater than zero i am sorry greater than zero so greater than zero that means entropy increases greater than zero it is not greater than one greater than zero so this can be expressed in terms of m a x now we can plot it so if you take a value of gamma between one to one point six seven let us consider a value any value one point four four for diatomic gases then we can plot this and we will see that well so this is the one mach number let this increases three like that in a supersonic region this is supersonic supersonic and this is subsonic well subsonic so we will see the mach number if i this entropy is going like this so therefore this is an impossible this is the possible shock waves where the entropy increases possible shocks possible shocks and this is the impossible region that means where the s y minus s x are negative that means the possible shocks corresponding to an upstream mach number which is greater than one but for an upstream mach number which is less than one a process will reduce the entropy in an adiabatic flow which is violating the second law of thermodynamics mathematically we can see that there is an asymptotic approach to the minus infinity when the mach number reaches a particular value in the subsonic range this can be found out from this expression here you see that this argument becomes zero implying an infinite value for this in the negative axis then we can find that a particular value of m a x square which becomes equal to gamma minus one by two gamma plus one all right so that value of m a x square makes this argument zero so this argument zero this is the l n so therefore this becomes so a particular value of m a this becomes that this part of the curve is impossible impossible this part of the curve is impossible all right so this is the change in entropy now we just show at the end the come to this to have an idea of stagnation pressure temperature let us draw the phenoline in t s diagram we know that we drew phenoline earlier in h s diagram in a perfect gas h is equal to c p t for a calorically perfect gas c p is constant so h and t are same that means they are just change with a scale factor so the same graph we can draw in t s diagram with the same qualitative picture like this so this is phenoline all right so this is phenoline this is phenoline phenoline so if you recall the rally line which is like this this this is the rally line this is the rally line and the intersection between the phenoline and the rally line is the shock that means this is the direction of the shock that means the shock takes place in such a way that there is a change in entropy so this is the region where mac is greater than one m a that means supersonic this is the region where mac is less than one so this is rally line I must draw that this is rally line I must write r a y l e i g h rally line all right so therefore we see that these are the intersection now here you one thing is very clear that this is the upstream side and this is the downstream side that means this is the x and this is the y we have already discussed the shock wave the upstream and downstream points must match both the rally line conditions and phenoline conditions so this is the direction of the shock so shock takes place in these directions from x to y now therefore we see that in a shock this stagnation temperature is fixed that is t o x because this is adiabatic is equal to t o y that means if I draw an isentropic line from here vertical line so this cuts here the constant pressure lines in t s diagram like this these are the constant pressure line for a gas for example the perfect gas here so this is the p o x but now this state if I draw an isentropic line if I that means if I draw an isentropic line from x this cuts the isentropic temperature corresponding to this state and this point corresponds to a particular constant pressure line which physically signifies this stagnation pressure corresponding to this state that means this stagnation pressure corresponding to the pressure at this state similarly this is valid for any state that state y it has got a pressure p y and it has got a temperature t y and it has got a stagnation temperature t o y if x and y are the points in an adiabatic flow that means between these two points no energy is added or extracted that means the isentropic a stagnation temperature is same that means if I draw the isentropic line and if you just come up to the same stagnation temperature because t o x is t o y so we come up to this point and the constant pressure line passing through this point will indicate this stagnation pressure corresponding to this point since this point is right of this it is obvious geometrically also that means the entropy has increased for this point from this point then a isentropic that a vertical line will cut the same horizontal line to a point where the constant pressure line p o y which is less than p o x that means this is the difference in the stagnation pressure stagnation pressure is decreased where the stagnation temperature remains same so this is precisely the graphical representation of the shock this is the final line flow this is the rally line again I am telling this curve is the final line flow that means the flow without heat transfer but with friction and the steady state condition that is same mass flow and this is the rally line where the flow takes place without friction but in general heat transfer is there and the steady state mass flow condition is maintained that the same mass flow but the shock occurs in such a way that upstream and downstream both satisfies the conditions of no heat transfer conditions of no heat transfer conditions of no friction and also the steady state condition it is very important and there realize the concept that the shock is such that interior details of the shock friction is there shock is a frictional process and natural process which increases the entropy but upstream and downstream of the shock refers to the conditions that no energy is added and ultimately the friction is taken to be negligible in a sense that when you take the momentum equation that across the shock wave the control volume is so thin that we have neglected the momentum neglected the frictional effect so this is an approximation but it is a very good approximation so that equality of impulse functions are valid so therefore the both the upstream and downstream sections corresponds to the intersect corresponds to both the final line and rally line conditions so therefore they correspond to the intersections of rally line and final line points so therefore again coming to this picture you see so this are the two points so this has to be upstream this has to be downstream because of the second law and this figure shows you the corresponding stagnation pressure where does stagnation temperature remains same all right I think okay any question question okay thank you